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showonlyrefs=truedem theoTheorem[section] coro[theo]Corollary lemma[theo]Lemma prop[theo]Proposition definition defi[theo]Definition remark[theo]Remark notation[theo]Notation exam[theo]Example hypo[theo]Hypothesis equationsection Least energy nodal solutions of Hamiltonian elliptic systems with Neumann boundary conditions Alberto Saldaña[ Institut für Analysis, Karlsruhe Institute for Technology, Englerstraße 2, 76131, Karlsruhe, Germany, [email protected]]& Hugo Tavares[CAMGSD, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal; [email protected]] [Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, Edifício C6, Piso 1, Campo Grande 1749-016Lisboa, Portugal; [email protected]] December 30, 2023 ==========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================We study existence, regularity, and qualitative properties of solutions to the system-Δ u = |v|^q-1 v in Ω, -Δ v = |u|^p-1 u in Ω,∂_ν u=∂_ν v=0 on ∂Ω,with Ω⊂^N bounded; in this setting, all nontrivial solutions are sign changing. Our proofs use a variational formulation in dual spaces, considering sublinear pq< 1 and superlinear pq>1 problems in the subcritical regime. In balls and annuli we show that least energy solutions (l.e.s.) are foliated Schwarz symmetric and, due to a symmetry-breaking phenomenon, l.e.s. are not radial functions; a key element in the proof is a new L^t-norm-preserving transformation, which combines a suitable flipping with a decreasing rearrangement. This combination allows us to treat annular domains, sign-changing functions, and Neumann problems, which are non-standard settings to use rearrangements and symmetrizations. In particular, we show that our transformation diminishes the (dual) energy and, as a consequence, radial l.e.s. are strictly monotone.We also study unique continuation properties and simplicity of zeros.Our theorems also apply to the scalar associated model, where our approach provides new results as well as alternative proofs of known facts. 2010 MSC 35J50 (Primary); 35B05, 35B06, 35B07, 35J47, 35J15Keywords Dual method,subcritical, Hamiltonian elliptic systems, flipping techniques, symmetry breaking,unique continuation.§ INTRODUCTION Let N≥ 1, Ω⊂^N be a smooth bounded domain, and consider the following Hamiltonian elliptic system with Neumann boundary conditions -Δ u = |v|^q-1 v in Ω, -Δ v = |u|^p-1 u in Ω,∂_ν u=∂_ν v=0 on ∂Ω,where ν is the outer normal vector on ∂Ω and we consider p,q>0 in the sublinear (pq<1) or the superlinear (pq>1) cases satisfying a subcritical condition, that is,p,q>0, pq≠ 1, and 1/p+1+1/q+1>N-2/N. In fact, in this setting the more general notion of linearity is pq=1 or, equivalently, 1/(p+1)+1/(q+1)=1 <cit.>. On the other hand, the last inequality in (<ref>) means that the exponents (p,q) arebelow the critical hyperbola (i.e., (p,q) is subcritical) <cit.>, and this condition is trivially satisfied if N=1,2 or if pq<1.Systems with a Hamiltonian structure such as (<ref>) have been extensively studied in the past 25 years and many results are known regarding existence, multiplicity, concentration phenomena, positivity, symmetry, Liouville theorems, etc.We refer to the surveys <cit.> for an overview of the topic and to <cit.> for more recent results.Most of these papers use Dirichlet boundary conditions and, up to our knowledge, the few papers addressing Neumann problems are <cit.>, where existence of positive solutions and concentration phenomena are studied, and <cit.>, which centers on existence of positive radial solutions. However, these papers focus on a different operator of the form Lw=-Δ w+V(x) w, with V positive.In comparison with (<ref>), the shape of solutions changes drastically; for instance, the operator L with Neumann b.c. induces a norm, and this allows the existence of positive solutions, while all nontrivial solutions of (<ref>) are sign-changing. Indeed, if (u,v) is a classical solution of (<ref>), then by the Neumann b.c. and the divergence theorem, ∫_Ω |u|^p-1u=∫_Ω |v|^q-1v=0.Since u≡ 0 if and only if v≡ 0, (<ref>) is only satisfied if (u,v) is trivial or if both components are sign-changing.Condition (<ref>) is called a compatibility condition.As far as we know, our paper is the first to study problem (<ref>).We remark that if p=q>0 and (u,v) is a classical solution of (<ref>), then u≡ v in Ω and (<ref>) reduces to the scalar equation -Δ u = |u|^p-1 in Ω with ∂_ν u=0 on ∂Ω, see Lemma <ref>. Therefore, all our results cover the single equations case.Condition (<ref>)together with Sobolev embeddings and the Rellich–Kondrachov theorem implies thatW^2,p+1/p(Ω)↪ L^q+1(Ω) and W^2,q+1/q(Ω)↪ L^p+1(Ω) compactly.A strong solution of (<ref>) is defined as a pair (u,v)∈ W^2,q+1/q(Ω)× W^2,p+1/p(Ω) satisfying the equations a.e. in Ω, and the boundary conditions in the trace sense.Problem (<ref>) has a variational structure, and (<ref>) are the Euler-Lagrange equations of the energy functional(u,v)↦ I(u,v)= ∫_Ω∇ u·∇ v-|u|^p+1/p+1-|v|^q+1/q+1dx.We define a least energy (nodal) solution as a nontrivial strong solution of (<ref>) achieving the levelc:=inf{ I(u,v): (u,v)≢(0,0), (u,v)is a strong solution of (<ref>)}.In view of (<ref>) and (<ref>), the functional I is well defined at strong solutions. Our main result is concerned with existence, regularity, and qualitative properties of least energy solutions. Let N≥ 1, Ω⊂^N be a smooth bounded domain, and let p and q satisfy (<ref>).The set of least energy solutions is nonempty. If (u,v) is a least energy solution, then (u,v)∈ C^2,ε(Ω)× C^2,ε(Ω) is a classical solution of (<ref>) and the following holds. (i) (Monotonicity in 1D) If N=1 and Ω=(-1,1), then u'v'>0 in Ω; in particular, u and v are both strictly monotone increasing or both strictly monotone decreasing in Ω.(ii) (Symmetry & symmetry breaking) If N≥ 2 and Ω=B_1(0) or Ω=B_1(0)\ B_δ(0) for some δ∈(0,1), then there is e∈∂ B_1(0) such that u and v are foliated Schwarz symmetric with respect to e and the functions u and v are not radially symmetric.(iii) (Unique continuation property) If pq<1, then the zero sets of u and v have zero Lebesgue measure, i.e., |{x∈Ω: u(x)=0}|=|{x∈Ω: v(x)=0}|=0.Results for the scalar equation follow immediately from Theorem <ref>, see Corollary <ref> below.Our approach to show Theorem <ref> is based on a variant of the dual method <cit.>. Later in this introduction we motivate the use of this approach and also the relationship between our results and previously known results for the single-equation problem. To describe the dual framework, we introduce some notation used throughout the whole paper. Let p and q satisfy (<ref>) and, for s>1, letX^s={f∈ L^s(Ω):∫_Ω f =0},α:=p+1/p,β:=q+1/q,and X:=X^α× X^β,endowed with the norm (f,g)_X=f_α+g_β. Let K denote the inverse (Neumann) Laplace operator with zero average, that is, if h∈ X^s(Ω), then u := Kh∈ W^2,s(Ω) is the unique strong solution of -Δ u = h in Ω satisfying ∂_ν u=0 on ∂Ω and ∫_Ω u = 0, see Lemma <ref> below.In this setting, the (dual) energy functional ϕ:X→ is given byϕ(f,g):=∫_Ω|f|^α/α+|g|^β/β-gK f dx, (f,g)∈ X.Since (<ref>) holds for any nontrivial solution, we require a suitable translation of K. For t>0, let K_t:X^t+1/t→ W^2,t+1/t(Ω) be given by K_t h:=Kh +c_t(h) for somec_t(h)∈ such that ∫_Ω |K_t h|^t-1K_t h=0.Then, a critical point (f,g) of ϕ solves the dual system K_q f=|g|^1/q-1g and K_p g= |f|^1/p-1f in Ω, see Lemma <ref>.In the sublinear case (pq<1), the operator ϕ achieves its global minimum in X (see Lemma <ref>); whereas, in the superlinear case (pq>1), ϕ is unbounded from below, but it can be minimized (see Lemma <ref>) in the Nehari-type set𝒩 :={(f,g)∈ X\{(0,0)} :ϕ'(f,g)(γ_1f,γ_2g)=0} with γ_1:=β/α+β,γ_2:=α/α+β.Therefore, we often refer to a minimizer (f,g)∈ X satisfyingϕ(f,g)=inf_Xϕ ifpq<1or ϕ(f,g)=inf_ Nϕ ifpq>1.In particular, if (f,g) satisfies (<ref>), then (u,v):=(K_p g, K_q f) is a least energy solution, that is, ϕ(f,g)=I(u,v)=c, with c as in (<ref>) and solves (<ref>), see Lemmas <ref>, <ref>, and <ref> below. We now describe in more detail the different techniques involved in the proof of Theorem <ref>.The existence is obtainedusing the subcriticality assumption (<ref>) and the compactness of the operator K, while the regularity follows from abootstrap argument (Proposition <ref>).The unique continuation property is shown by extending the techniques from <cit.> to the setting of Hamiltonian elliptic systems and to the dual-method framework (later in this introduction we compare in more detail the results and strategies from <cit.> with ours). Moreover, the fact that least energy solutions are foliated Schwarz symmetric is due to a characterization of sets of functions with this symmetry in terms of invariance under polarizations, see Lemma <ref> below and <cit.> for similar results in other settings.One of our main contributions, from the methodological point of view, is the proof of the strict monotonicity of solutions (for N=1) and of the symmetry-breaking phenomenon (i.e., that least energy solutions are not radial).These results require first a deep understanding of the radial setting: consider Ω⊂^N to be eitherΩ =B_1(0) and fix δ:=0(for any N≥ 1) orΩ =B_1(0)\ B_δ(0) for some δ∈(0,1)(for any N≥ 2).If Y is a set of functions, we use Y_rad to denote the subset of radial functions in Y and we call (u,v):=(K_p g, K_q f) a least energy radial solution if (f,g)satisfies ϕ(f,g)=inf_X_radϕ ifpq<1or ϕ(f,g)=inf_ N_radϕ ifpq>1.Our next result shows that these radial solutions are remarkably rigid. We denote the radial derivative of w by w_r and use a slight abuse of notation setting w(|x|)=w(x).Let p,q satisfy (<ref>) and Ω, δ as in (<ref>). The set of least energy radial solutions is nonempty and, if (u,v) is a least energy radial solution, then (u,v)∈ C^2,ε(Ω)× C^2,ε(Ω) is a classical radial solution of (<ref>) and u_r v_r>0 in (δ,1); in particular, u and v are both strictly monotone increasing or both strictly monotone decreasing in the radial variable.The existence and regularity follows similarly as in Theorem <ref>.The strict monotonicity relies on the following new transformation denoted with a superindex ⋇: let Ω, δ as in (<ref>) andI:L_rad^∞(Ω)→ C_rad(Ω), Ih(x):=∫_{δ<|y|<|x|} h(y) dy =Nω_N∫_δ^|x|h(ρ)ρ^N-1 dρ,: C_rad(Ω)→ L_rad^∞(Ω), h:=(χ_{ Ih>0}-χ_{ Ih≤ 0})h.Then, for h∈ C_rad(Ω), the ⋇-transformation of h is given by h^⋇∈ L^∞_rad(Ω), h^⋇(x) := (𝔉 h)^#(ω_N |x|^N-ω_N δ^N).where # is the one-dimensional decreasing rearrangement (see Subsection <ref>) and ω_N= |B_1| is the volume of the unitary ball in ^N.The function h can be seen as a suitable flipping of h on the set { h≤ 0}, and this step is very important to construct monotone decreasing solutions.In Remark <ref> we motivate further the definition of h^⋇ and in Figure <ref> in Section <ref> we illustrate the construction of h^⋇ in a ball and compare it with the Schwarz symmetrization h^*(x)=h^#(ω_N|x|^N). Observe that annuli, sign-changing functions, and Neumann boundary data are non-standard conditions to work with rearrangements; in fact, many results relying on symmetrizations fail in these settings; for example, the standard Polya-Szegő inequality only holds for nonnegative functions with zero boundary data.Our approach is able to cover these cases mainly because of two reasons: the radiality assumption and the fact that we use Lebesgue spaces within a dual framework, which gives us more flexibility in the construction of our transformation; in this sense, we are transforming the dual variables (f,g) to obtain, together with variational techniques, monotonicity information of solutions (u,v):=(K_p g, K_q f).In general, h^⋇ is not a (level-set) rearrangement of h, since the maximum value of h may vary due to the flipping 𝔉h; however, L^t-norms and (zero) averages are preserved (see Proposition <ref>) and the transformed functions have less energy, as stated in the following result.Let p,q>0, Ω as in (<ref>), and let f,g:Ω→ be continuous and radially symmetric functions with ∫_Ω f = ∫_Ω g = 0. Then (f^⋇,g^⋇)∈ X andϕ(f^⋇,g^⋇)≤ϕ(f,g).Furthermore, if f and g are nontrivial and ϕ(f^⋇,g^⋇)=ϕ(f,g), then f and g are monotone in the radial variable and if (u,v):=(K_p g,K_q f), then u and v are radially symmetric and strictly monotone in the radial variable.The proof exploits the one-dimensionality of the problem and uses elementary rearrangement techniques. Note that, if Ω is a ball, then h^⋇ is actually the Schwarz symmetrization of 𝔉h. The flipping , however, is necessary, since it can be shown that (<ref>) does not hold in general using merely the Schwarz symmetrization (even in the scalar case p=q), see Remark <ref> below. Theorem <ref> is the main tool to show the monotonicity claims in Theorem <ref> and Theorem <ref>. These results are of independent interest and are new even in the single-equation case (<ref>). Furthermore, Theorem <ref> is also the starting point for the proof of the symmetry-breaking phenomenon, which, in the dual setting, reads as follows.If (f,g) satisfies (<ref>), then f and g are not radial.The proof of this claim is done by contradiction: if (f,g) is radially symmetric and (u,v):=(K_p g,K_q f), then (u_x_1,v_x_1) is a strong (Dirichlet) solution of-Δ u_x_1 = q |v|^q-1 v_x_1=:g̅,-Δ v_x_1 = p |u|^p-1 u_x_1=:f̅ in Ωwith u_x_1=v_x_1=0 on ∂Ω.Therefore, from the minimality of (f,g) (approximating ϕ” in a suitable sense) we infer that∫_Ω(e_1) p|u|^p-1u_x_1(u_x_1-K g̅ )+q|v|^q-1 v_x_1(v_x_1-K f̅)dx≥ 0,Ω(e_1):=Ω∩{x_1>0}.Here, Theorem <ref> is very important to control the (possibly singular) terms |u|^p-1 and |v|^p-1, since the strict monotonicity implies that the nodal sets of u and v are merely two inner spheres. In the end, a contradiction is obtained by showing—using the radiality assumption, maximum principles, and Hopf's boundary point Lemma—that the Neumann solution (Kg̅,Kf̅) dominates the Dirichlet solution (u_x_1,v_x_1) in Ω(e_1), and this would imply that the integral in (<ref>) is strictly negative.Observe that the symmetry-breaking statement in Theorem <ref> follows directly from (<ref>).For other symmetry-breaking results for single equations we refer to <cit.> for Dirichlet boundary conditions, to <cit.> for Neumann boundary conditions, and to <cit.> for a (perturbative) symmetry-breaking result for Dirichlet Hamiltonian systems. See also the survey <cit.> and the references therein. We now focus on the particular case p=q, where (<ref>) reduces top>0, p≠ 1, (N-2)p<(N+2).In this situation, we show in Lemma <ref> below that any classical solution (u,v) of (<ref>) satisfies u≡ v, and problem (<ref>) is equivalent to-Δ u= |u|^p-1uin Ω,∂_ν u=0 on ∂Ω,whose solutions are critical points of E(u):=I(u,u)/2=∫_Ω1/2|∇ u|^2 - 1/p|u|^p dxThen, as a particular case of Theorems <ref> and <ref> we have the following.Let N≥ 1, Ω⊂^N be a smooth bounded domain, and let p satisfy (<ref>). The set of least energy solutions of (<ref>) is nonempty and it is contained in C^2,ε(Ω).(i) (Unique continuation) If p<1, then the zero set of every least energy solution has zero Lebesgue measure.(ii) (Monotonicity, symmetry, and symmetry breaking)If N=1 and Ω=(-1,1), then every least energy solution is strictly monotone in Ω.If N≥ 2 andΩ is either a ball or an annulus as in (<ref>), then every least energy solution is foliated Schwarz symmetric and it is not radially symmetric.(iii) (Radial solutions) Let Ω be a ball or an annulus as in (<ref>), then the set of least energy radial solutions is nonempty. If u is a least energy radial solution, then u∈ C^2,ε(Ω) is a classical radial solution of (<ref>) and u is strictly monotone in the radial variable.Up to our knowledge, the monotonicity of least energy radial solutions (part (iii) of Corollary <ref>) is new. Part (ii) is also new in the subcritical superlinear regime as well as the symmetry breaking result for annuli in the sublinear case. In this sense, this corollary complements the results in <cit.>, where unique continuation, symmetry, and symmetry breaking for least energy solutions of (<ref>) in the case 0< p<1 are studied (the case p=0 is also considered in <cit.>, interpreting (<ref>) as -Δ u=sign(u)). Note that the unique continuation property when p>1 is classical, see for instance <cit.>.Next, we comment on the similarities and differences between our approach and that of <cit.>; in particular, we explain the difficulties when passing from the single-equation (<ref>) to system (<ref>) and why the dual method is convenient in this last case.To show existence of solutions (in the sublinear case 0<p<1), the authors in <cit.> minimize the functional (<ref>) on the setN_p={u∈ H^1(Ω): ∫_Ω |u|^p-1u=0},proving that the least energy level is achieved (note that N_p is not a C^1-manifold because p<1). Several difficulties arise when trying to use such a direct method for a Hamiltonian system. For instance, in dimension N≥ 3, the (direct) functional I, given in (<ref>), is not well defined in H:=H^1(Ω)× H^1(Ω) under (<ref>). In fact, even working on a range of (p,q) where I is well defined in H (or using (<ref>) with p,q>1 and a truncation method as in <cit.>), the functional I is strongly indefinite, in the sense that the principal part ∫_Ω∇ u·∇ v does not have a sign, and it is actually positive in an infinite dimensional subspace, and negative in another.To control this difficulty, a direct approach is based, for example, on abstract linking theorems or on special Nehari-type sets. Different alternatives are available in the literature to study Hamiltonian systems variationally (see, e.g., the survey <cit.>), each one with its own advantages and disadvantages. For example, a common strategy is to reduce the system to a single higher-order problem; however, it is not clear how to study Neumann b.c. in this setting and using higher-order Sobolev spaces brings additional complications (for the use of rearrangements, for example). Dual methods, on the other hand, offer a flexible and elegant alternative. This approach entails the challenge of controlling the effects of the nonlocal operator K in the functional ϕ; but it compensates this difficulty with many advantages, for example, the compatibility conditions (<ref>) translate to ∫_Ω f=∫_Ω g=0 in the dual formulation; in particular, this allows in the sublinear case to minimize and differentiate the functional ϕ in the Banach space X (recall that in <cit.> the functional (<ref>) is minimized on (<ref>), which is not a manifold); whereas, in the superlinear case, the dual formulation allows to minimize in a Nehari manifold in a Banach space. The use of a direct or a dual approach has, of course, a strong influence on the methods to study qualitative properties of solutions; this is particularly clear in the proof of the symmetry breaking, described above, where our proof relies on a transformation in dual spaces to obtain monotonicity and then on comparison principles to control the effects of the nonlocal operator K.The proof of the symmetry breaking result in <cit.> (although is also done by contradiction finding a direction along which the energy would decrease) is very different from ours, and it relies on a unique continuation property for minimizers, which is obtained using known uniqueness results <cit.> and nonoscillation criteria <cit.> for sublinear ODEs of type y”+a(t)|y|^p-1y=0. These general theorems are not known for systems of ODEs, and in fact, they may fail in general. Using elementary manipulations in an ODE setting, one can find some extensions of these techniques to systems. Although we do not need these results for any of our proofs, we believe they can be of independent interest; in particular, we use them to show a result on the simplicity of zeros of any radial solution that satisfies a unique continuation property. Let p,q>0, N≥ 2, Ω, δ as in (<ref>), and (u,v)∈ [C^2,ε(Ω)]^2 be a radial classical solution of (<ref>) such that u^-1(0)∩ v^-1(0) has empty interior. Thenu,v∈ W:={ w∈ C^1(Ω) : ∇ w(x) ≠ 0ifx∈Ω satisfies|x|>δ andw(x)=0}, This theorem yields that, if (u,v) is a radial solution of (<ref>) satisfying a (weak) unique continuation property, then the nodal set of u and of v is at most a countable union of spheres and thus |u^-1(0)|=|v^-1(0)|=0. To close this introduction, we mention some open questions about system (<ref>): it is unknown whether or not least energy solutions are (up to rotations) unique; in fact, uniqueness is open even for least energy radial solutions. Moreover, when pq<1 it is unclear if the unique continuation property (i.e., that the nodal sets of solutions have zero Lebesgue measure) holds in general for all solutions, and not just for minimizers.When pq>1, with the exception of the single equation case p=q, the unique continuation propertyis an open question even for least energy solutions. In Remark <ref> below we observe that, in the superlinear case, if a least energy solution (u,v) vanishes on a set of positive measure, then it has a zero of infinite order. For single equations of type -Δ u+V(x)u=0 such a result is available for a large class of potentials (see<cit.> and <cit.>) andhaving a zero of infinite order typically implies that a solution is identically zero (see <cit.>). The paper is organized as follows. In Section <ref> we provide some general auxiliary results in our dual framework; in particular, we show Lemmas <ref> and <ref>, which prove the existence and regularity claims in Theorem <ref>, as well as Lemma <ref> which shows (together with Theorems <ref> and <ref>) Corollary <ref>.Section <ref> is devoted to the study of monotonicity properties of radial minimizers, here the full details of the construction of the ⋇-transformation can be found as well as the proof of Theorems <ref> and <ref>. These results are then used in Section <ref> to prove the symmetry breaking result in Theorem <ref> as well as the foliated Schwarz symmetry claim. Finally, in Section <ref> we show Theorem <ref> as well as the unique continuation statement in Theorem <ref>. §.§ Notation Throughout the paper p and q always denote the exponents in (<ref>). We divide our proofs in two groups:Sublinear:pq <1 , Superlinear and subcritical:pq >1 and 1/p+1+1/q+1>N-2/N.Note that pq<1 readily implies the subcriticality condition, since pq<1 implies 1/(p+1)+1/(q+1)>1>(N-2)/N. If N=1,2 the subcriticality condition is satisfied by any p,q>0. We fixα:=p+1/p,α'=p+1,β:=q+1/q,β'=q+1.Note that α' and β' are the corresponding conjugate exponents of α and β, that is α^-1+(α')^-1=1 and β^-1+(β')^-1=1. Moreover, we define X, X_s, K, and K_s as in the Introduction. For s≥ 1, ·_s and ·_2,s are the standard norms in L^s(Ω) and W^2,s(Ω) respectively and, if s>1 is such that N>2s, then we denote by s^*:=Ns/N-2s>s the critical Sobolev exponent, that is, s^* is the biggest exponent which allows the (continuous) embedding W^2,s(Ω)↪ L^s^*(Ω). If N=1,2 we set s^*=∞.If Y is a vector space, then [Y]^2:=Y× Y. We denote by B_r the ball in ^N of radius r>0 centered at 0.Finally, the nodal set of u:Ω→ is denoted by u^-1(0):={x∈Ω :u(x)=0}.For a measurable set A⊂^N, |A| denotes its Lebesgue measure, and the function χ_A is the characteristic function of A, that is, χ_A(x)=1 if x∈ A and χ_A(x)=0 if x∉A. Finally, we use ω_N to denote the measure of the unitary ball in ^N. § VARIATIONAL FRAMEWORK AND EXISTENCE RESULTS In this section we describe our dual approach. In the following Ω is a smooth bounded domain in ^N, p and q satisfy (<ref>), and let X^s, X, K, K_t, α, and β be as above. §.§ The dual methodThe following lemma recalls some well-known regularity for Neumann problems, see for example <cit.> (see also <cit.>). If s>1, Ω be a smooth bounded domain in ^N, and h∈ L^s(Ω) with ∫_Ω h =0, then there is a unique strong solution u∈ W^2,s(Ω) of-Δ u = h in Ω,∂_ν u=0 on ∂Ω,∫_Ω u = 0in particular, ∫_Ω∇ u∇φ = ∫_Ω h φ for all φ∈ W^1,s'(Ω), s'=s/s-1, and there is C(Ω,s)=C>0 such that u_2,s≤ Ch_s. Recall that if f∈ L^s(Ω) for some s>1, then Kf is the solution of (<ref>) with h=f.Observe that ∫_Ω gK_q f = ∫_Ω gK f = ∫_Ω Kgf = ∫_Ω K_p gf,by integration by parts and because ∫_Ω f=∫_Ω g=0. Our next result shows that ϕ is continuously differentiable.The functional ϕ:X→ as defined in (<ref>) is continuously differentiable in X with ϕ'(f,g)(φ,ψ)=∫_Ω |f|^α-2fφ+|g|^β-2gψ-ψK f-φK g dx for (f,g),(φ,ψ)∈ X.Let ϕ=ψ-T, where Ψ(f,g):=∫_Ω|f|^α/α+|g|^β/βdxandT(f,g):=∫_Ω gK f, (f,g)∈ X.Since α,β>1 it is standard to show that Ψ is continuously differentiable withΨ'(f,g)(φ,ψ)=∫_Ω |f|^α-2fφ+|g|^β-2gψdx,(f,g),(φ,ψ)∈ X.Now we show that T is bilinear and bounded. Indeed, this follows directly from the fact that K(λ h) = λ K h for h∈ L^s(Ω), s∈{α,β}, and the following integration by parts T(f,g)=∫_Ω g K f = ∫_Ω f K g, (f,g)∈ X. Moreover, by Hölder's inequality, Lemma <ref>, and the first embedding in (<ref>), there are C_1,C_2>0 such that|T(f,g)|≤1/2g_βK f_β'≤ C_1g_βK f_W^2,α(Ω)≤ C_2g_βf_α. Thus T is a (continuously differentiable) bilinear bounded mapping and, by (<ref>),T'(f,g)(φ,ψ)=∫_ΩψK f+φK g dx,(f,g),(φ,ψ)∈ X.Therefore, (<ref>) follows from (<ref>) and (<ref>).Before we argue existence of solutions, we establish a one-to-one relationship between critical points of ϕ and strong solutions of (<ref>), see <cit.> for the Dirichlet case. An element (f,g)∈ X is a critical point of ϕ, i.e.,ϕ'(f,g)(φ,ψ)=0 for all (φ,ψ)∈ X,if and only if (u,v):=(K_p g,K_q f)∈ W^2,β(Ω)× W^2,α(Ω) is a strong solution of (<ref>), that is, (u,v) solves (<ref>) a.e. in Ω. Let (f,g)∈ X satisfy (<ref>) and let (u,v):=(K_p g,K_q f). Then, using (<ref>),∫_Ω (|f|^α-2f- u)φ+(|g|^β-2g-v)ψ dx = 0 for all (φ,ψ)∈ X=X^α× X^β.Let w:=|f|^α-2f- u and, for a function ζ∈ L^α(Ω), set ζ:=|Ω|^-1∫_Ωζ∈. Then (ζ-ζ)∈ X^α and0=∫_Ω w(ζ-ζ) = ∫_Ω wζ-ζ∫_Ω w=∫_Ω wζ-w∫_Ωζ = ∫_Ω (w - w)ζ for all ζ∈ L^α(Ω),and therefore w-w=0 a.e. in Ω, which implies that |f|^α-2f=u+w a.e. in Ω and thus f = |u+w|^p-1(u+w) a.e. in Ω.Furthermore, since (f,g)∈ X and u=K_p g, we have that ∫_Ω|u|^p-1 u = 0 = ∫_Ω f = ∫_Ω|u+w|^p-1(u+w), which implies that w=0, and therefore -Δ v = -Δ (K_q f) = f = |u|^p-1u a.e. in Ω. Analogously, -Δ u = |v|^q-1v a.e. in Ω, as claimed.For the converse implication, let (u,v):=(K_p g,K_q f)∈ W^2,β(Ω)× W^2,α(Ω) be a strong solution of (<ref>) for some (f,g)∈ X. Then, necessarily f=|u|^p-1u and g=|v|^q-1v a.e. in Ω, which implies that u=|f|^α-2f and v=|g|^β-2g a.e. in Ω, which implies (<ref>), by (<ref>).The next Proposition states that critical points of ϕ are in fact classical solutions. We refer to <cit.> and <cit.> for analogous results in the superlinear Dirichlet case.Let (f,g)∈ X be a critical point of ϕ, then (u,v):=(K_p g,K_q f)∈ [C^2,ε(Ω)]^2 for some ε>0 and satisfies (<ref>) pointwise. The proof follows closely <cit.>.For t,s≥ 1 denoteW(s,t):=W^2,s(Ω)× W^2,t(Ω) andL(s,t):=L^s(Ω)× L^t(Ω).Let (f,g)∈ X ⊂ L(α,β) be a critical point of ϕ then, by Lemma <ref>, (u,v)∈ W(β,α)↪ L(β^*,α^*) is a strong solution of (<ref>).For n∈_0, let (β_0,α_0):=(β,α), (β_n+1,α_n+1):=( α_n^*/q,β_n^*/p )ifN>2α_n,N>2β_n.Here we are using the notation given in Section <ref> for the critical Sobolev exponent. Note that (|v|^q,|u|^p)∈ L(α^*/q,β^*/p)=L(β_1,α_1) and then Lemma <ref> and (<ref>) imply (u,v)∈ W(β_1,α_1)↪ L(β^*_1,α_1^*). But then (|v|^q,|u|^p)∈ L(β_2,α_2), which gives (u,v)∈ W(β_2,α_2)↪ L(β^*_2,α_2^*). Iterating this procedure we obtain that (u,v)∈ W(β_n,α_n) as long as N>2α_n-1 and N>2β_n-1.We claim that N≤ 2s for somes∈{α_n,β_n : n∈_0},and then the proposition follows from Lemma <ref>, the embedding W^2,s(Ω)↪ C^μ(Ω) for some μ>0, and standard Schauder estimates (see <cit.> and the Remark after the theorem), since t↦ |t|^r-1t for r>0 is Hölder continuous.Indeed, assume by contradiction that N> 2s for all s∈{α_n,β_n : n∈_0} and consider the sequence S_n:=(qβ_n,pα_n). By the subcriticality assumption we have that W(β,α)↪ L(α',β')=L(pα,qβ) and thereforeS_1=(qβ_1,pα_1)=(α^*,β^*)>(qβ,pα)=S_0 and (S_n)_n∈ is, by induction, a (component-wise) increasing (bounded) sequence. Let l_1,l_2>0 be such that S_n→ (l_1,l_2) and note that, by the monotonicity,l_2 > pα = p+1.Moreover(l_1,l_2)=lim_n→∞(α_n^*,β_n^*)=(Nl_2/Np-2l_2,Nl_1/Nq-2l_1), which implies l_2=N(pq-1)/2(q+1).Since the function t↦ h(t):=N(pt-1)/2(t+1) is increasing for t>0 and q<t_0:=2p+N+2/pN-2p-2, by the subcriticality condition (<ref>), we have that l_2= h(q)<h(t_0)=p+1 , which contradicts (<ref>). Then (<ref>) holds and this ends the proof. The next Lemma shows the relationship between the dual and the direct energy functionals when evaluated on solutions.Let (f,g)∈ X be a critical point of ϕ and let (u,v):=(K_p g,K_q f).Then ϕ(f,g)=I(u,v):=∫_Ω∇ u∇ v - |u|^p+1/p+1-|v|^q+1/q+1dxLet f,g,u and v as in the statement. By Proposition <ref>, we have that (u,v) is a solution of (<ref>) and, integrating by parts, ϕ(f,g) =∫_Ω|f|^α/α+|g|^β/β-g K f dx= ∫_Ω p|u|^p+1/p+1+q|v|^q+1/q+1 - ∇ u ∇ v dx.Therefore ϕ(f,g)-I(u,v)= ∫_Ω |u|^p+1+|v|^q+1- 2∇ u ∇ vdx=0, by (<ref>).We finish this Section with a result in the case p=q, in which (<ref>) reduces to a single equation.The proof is the same as in the superlinear Dirichlet case <cit.> and we include it for the reader's convenience. Let p=q>0 and (u,v)∈ C^2(Ω)× C^2(Ω) be a classical solution to (<ref>), then u=v.By testing (<ref>) with u,v and integrating by parts we have that ∫_Ω |∇ u|^2=∫_Ω |v|^p-1vu, ∫_Ω |∇ v|^2 = ∫_Ω |u|^p-1uv,and v_α'^α'=∫_Ω∇ u∇ v =u_α'^α'.Then, by Hölder's inequality,∫_Ω |∇ u|^2+∫_Ω |∇ v|^2 ≤u_α'v_α'^p+v_α'u_α'^p=2∫_Ω∇ u∇ v,which implies ∇ u=∇ v in Ω and then v-u≡ c in Ω for some constant c∈.Therefore, by (<ref>), |u|^p-1u=-Δ u=-Δ (u+c) =|u+c|^p-1(u+c) in B. If u≡ 0 then also v≡ 0. If u≢0 then u changes sign and there is x∈ B with u(x)=0, but then |c|^p-1c=0, and u=v in Ω as claimed. §.§ Existence of least energy solutions: sublinear case Assume (<ref>), that is, p,q>0 and pq<1 and recall the notation (<ref>).The functional ϕ:X→ as defined in (<ref>) is coercive. Let ε:=1/2min{α^-1,β^-1}. Then from (<ref>), Young's inequality, and the fact that β'<α (since pq<1) there is C_2(α,β)=C_2>0 such that |T(f,g)|≤ε(f^α_α+g^β_β)+C_2. This yields that ϕ(f,g)≥ε(f^α_α+g^β_β)-C_2,that is, ϕ is coercive. The functional ϕ achieves a negative minimum in X, that is,min_X ϕ = ϕ(f,g)<0for some (f,g)∈ X.Moreover, ϕ'(f,g)=0 in X, (u,v):=(K_p g, K_q f)∈ [C^2,ε(Ω)]^2 is a classical solution of (<ref>), and (u,v) is a least energy solution, that is, I(u,v)=c with c as in (<ref>). For n∈ let x_n:=(f_n,g_n) be a minimizing sequence in X, which exists by Lemma <ref>.Since X is reflexive there is x:=(f,g)∈ X such that x_n⇀ x weakly in X and (x_n) is bounded in X in virtue of (<ref>). Since the operator K is compact by Lemma <ref> and the compact embedding (<ref>), we have that K(f_n) converges strongly to Kf in L^β'(Ω). Using the lower semicontinuity of norms we have that min_Xϕ=lim inf_n→∞ϕ(f_n,g_n)≥ϕ(f,g), and therefore ϕ achieves its minimum.To see that the minimum is strictly negative, let T and Ψ as in (<ref>) and let φ∈ C^∞_c(Ω)\{0} such that ∫_Ωφ = 0. Then (φ,φ)∈ X\{(0,0)} andT(φ,φ)=∫_Ωφ K φ =∫_Ω (-Δ u) u =∫_Ω |∇ u|^2> 0, where u:=K φ.Thus, since pq<1 is equivalent to αβ>α+β,ϕ(t^β/α+βφ,t^α/α+βφ)= t^αβ/α+βΨ(φ,φ)-tT(φ,φ)<0for t>0 small enough.Then ϕ'(f,g)=0 in X by Lemma <ref>, (u,v):=(K_p g, K_q f)∈ [C^2,ε(Ω)]^2 is a classical solution of (<ref>), by Proposition <ref>, and (u,v) is a least energy solution, by Lemma <ref>.§.§ Existence of least energy solutions: superlinear caseLet p,q∈(0,∞) such that (<ref>) holds.γ_1:=β/α+β,γ_2:=α/α+β, and γ:=αβ/α+β<1,where γ<1 because pq>1. In particular, γ_1α = γ_2β=γ and γ_1+γ_2=1. We define the Nehari-type set𝒩 :={(f,g)∈ X\{(0,0)} :ϕ'(f,g)(γ_1f,γ_2g)=0}={(f,g)∈ X\{(0,0)} :∫_Ωγ_1 |f|^α + γ_2 |g|^βdx=∫_Ω f Kg}. Let (f,g)∈ X\{(0,0)} such that ∫_Ω f K g>0, thent:=t(f,g):=(γ_1 ∫_Ω |f|^α + γ_2 ∫_Ω |g|^β/∫_Ω fKg)^1/1-γis the unique maximum of the function s↦ϕ(s^γ_1f,s^γ_2g) and (t^γ_1f,t^γ_2g)∈. Since ϕ(s^γ_1f,s^γ_2)=s^γ∫_Ω|f|^α/α+|g|^β/βdx-s∫_Ω f K g, γ<1, and ∫_Ω f K g>0, the map s↦ϕ(s^γ_1f,s^γ_2g) has a unique positive critical point, which is a global maximum. The claim now follows by direct computations. There is (f,g)∈ such that ϕ(f,g)=inf_ϕ. Moreover, ϕ'(f,g)=0 in X, (u,v):=(K_p g, K_q f)∈ [C^2,ε(Ω)]^2 is a classical solution of (<ref>), and (u,v) is a least energy solution, that is, I(u,v)=c with c as in (<ref>). 1) The functional ϕ is bounded from below on 𝒩, sinceϕ(f,g)=1-γ/α∫_Ω |f|^γ+1-γ/β∫_Ω |g|^β≥ 0 for(f,g)∈𝒩.Thus inf_𝒩ϕ∈.Take a minimizing sequence (f_n,g_n)∈𝒩. From (<ref>), we have that f_n_α and g_n_β are bounded and thus, up to a subsequence, f_n⇀ f in L^α, g_n⇀ g in L^β, for some (f,g)∈ X. 2) Recalling (<ref>) and using Young's inequality, we haveγ_1 f_n_α^α+γ_2 g_n_β^β=∫_Ω f_nKg_n ≤ Cf_n_αg_n_β≤ C(γ_1 f_n_α^1/γ_1+γ_2g_n_β^1/γ_2)for some C>0 and, since 1/γ_1>α, 1/γ_2>β, there exists δ>0 such that ∫_Ω f_nKg_n ≥δfor every n.3) Combining this with the properties of weak convergence and the compactness of the operator K, we have (f,g)≠ (0,0) and, by definition of 𝒩,0<∫_Ωγ_1 |f|^α+γ_2 |g|^βdx≤∫_Ω f Kg.Therefore, by Lemma <ref>, there exists 0<t≤ 1 such that (t^γ_1 f, t^γ_2g)∈𝒩. Then,inf_ϕ ≤ϕ(t^γ_1 f , t^γ_2 g )=t^γ (1-γ) ∫_Ω|f|^α/α+|g|^β/βdx≤ (1-γ) ∫_Ω|f|^α/α+|g|^β/βdx ≤lim inf_n→∞ϕ(f_n,g_n)=inf_ϕ,thus t=1, and (f,g)∈𝒩 and achieves inf_𝒩ϕ.4) Defining τ(h,k)=ϕ'(h,k)(γ_1h, γ_2 k) for (h,k)∈𝒩, we haveτ'(h,k)(γ_1 h,γ_2 k)=γ_1(γ-1) ∫_Ω |h|^α + γ_2 (γ-1) ∫_Ω |k|^β<0.Therefore 𝒩 is a manifold and, since (f,g) achieves inf_𝒩ϕ, then by Lagrange's multiplier rule there exists λ∈ such that ϕ'(f,g)=λτ'(f,g). By testing this identity with (γ_1 f,γ_2 g), we see that actually λ=0 and (f,g) is a critical point of ϕ in X. Thus (u,v):=(K_p g, K_q f)∈ [C^2,ε(Ω)]^2 is a classical solution of (<ref>) by applying Proposition <ref>. Finally, (u,v) is a least energy solution, by Lemma <ref>. § MONOTONICITY OF RADIAL MINIMIZERSIn this subsection Ω⊂^N is eitherΩ =B_1(0) and fix δ:=0(for any N≥ 1) orΩ =B_1(0)\ B_δ(0) for some δ∈(0,1)(for any N≥ 2).We emphasize that the case B_1(0)\{0} is not considered in any of our results.Our main goal is to show Theorem <ref>.This result is of independent interest, but it is also instrumental in the proof of the symmetry-breaking result in Theorem <ref>.The proof of Theorem <ref> is based on a new transformation using rearrangement techniques and a suitable flipping compatible with our dual method approach.We introduce first some notation and show some preliminary results. §.§ The decreasing rearrangement Let U⊂^N be an open set and h:U→ a measurable function. The decreasing rearrangement of h is given by h^#:[0,|U|]→, h^#(0):=ess sup_U h h^#(s):=inf{t∈ :|{h>t}|<s}, s>0.In particular, h^# is a non-increasing and left-continuous function <cit.>, h^# is a (level-set) rearrangement of h, which yields in particular that L^p-norms and averages are preserved, that is,h_L^p(U)=h^#_L^p(0,|U|) and ∫_U h = ∫_0^|U| h^#,see <cit.>.By <cit.>, we know that if E⊂ U, then∫_E h(x) ≤∫_0^|E|h^#.We now focus on the case where U=I:=[0,l]⊂, for some l>0. If h:I→ is non-increasing in I, then h=h^# a.e. in I, see <cit.>. We use the following decomposition for the integral of a product of two functions.Let φ,ψ:I→ be measurable functions with φ∈ L^1(I), ψ∈ L^∞(I), and a,b∈ with a≤ψ≤ b. Then∫_I φ ψ = a∫_Iφ +∫_a^b ∫_{ψ>t}φ(s) dsdt. A well-known consequence of Lemma <ref> and (<ref>) is the Hardy-Littlewood inequality. [Particular case of Theorem 1.2.2 in <cit.>]Let φ∈ L^1(I) and ψ∈ L^∞(I) be (possibly sign-changing) functions, then∫_I φ ψ≤∫_I φ^# ψ^# Under some additional assumptions, the equality case in (<ref>) can be used to deduce monotonicity properties of the involved functions. Let φ,ψ:I→ with φ∈ L^1(I) and ψ∈ C(I) be a strictly decreasing function. If ∫_I φ ψ = ∫_I φ^# ψ, then φ=φ^# a.e. in I. Let b:=ψ(0) and a:=ψ(l). Then, by Lemma <ref>,a∫_I φ + ∫_a^b ∫_{ψ>t}φ(s) ds dt = ∫_I φ ψ=∫_I φ^# ψ=a∫_I φ^# + ∫_a^b ∫_{ψ>t}φ^#(s) ds dt.Since ψ is strictly decreasing and continuous, {ψ>t}=[0,ψ^-1(t)) for every t∈ [a,b]. Therefore, by (<ref>), ∫_{ψ>t}φ(s) ds≤∫_{ψ>t}φ^#(s) ds for all t∈ (a,b) and thus, by (<ref>),∫_{ψ>t}φ(s) ds = ∫_{ψ>t}φ^#(s) ds for all t∈ (a,b).Furthermore, for every t∈ I we have {ψ>ψ(t)}=[0,t), because ψ is strictly monotone decreasing and continuous. Then, (<ref>) yields that ∫_0^tφ = ∫_0^tφ^# for all t∈ I, and φ=φ^# a.e. in I, by Lebesgue differentiation theorem.§.§ A decreasing transformation in dual spaces In the following we do a slight abuse of notation and use w(|x|)=w(x) for a radial function w. We use L^∞_rad(Ω) and C_rad(Ω) to denote the subspace of radial functions in L^∞(Ω) and C(Ω), respectively.Let Ω, δ as in (<ref>) and letI:L_rad^∞(Ω)→ C_rad(Ω), Ih(x):=∫_{δ≤|y|≤|x|} h(y) dy =Nω_N∫_δ^|x|h(ρ)ρ^N-1 dρ: C_rad(Ω)→ L_rad^∞(Ω), h:=(χ_{ Ih>0}-χ_{ Ih≤ 0})h.Then, for h∈ C_rad(Ω), the ⋇-transformation is given by h^⋇∈ L^∞_rad(Ω), h^⋇(x) := (𝔉 h)^#(ω_N |x|^N-ω_N δ^N).where # is the decreasing rearrangement (see the previous section) and ω_N= |B_1| is the volume of the unitary ball in ^N.The function h can be seen as a suitable flipping of h on the set { h≤ 0}, and this step is very important to construct monotone decreasing solutions, while preserving the (zero) average and the L^p-norm of h, see Proposition <ref> and Remark <ref>.See also Figure <ref> below for some examples. In general, the function h^⋇ is not a (level-set) rearrangement of h, since the maximum value of h may vary due to , but it has the following important properties.Let h:Ω→ be a continuous radial function such that ∫_Ω h = 0. Then, for any t≥ 1, ∫_Ω h^⋇ = 0,∫_Ω |h^⋇|^t = ∫_Ω |h|^t, and( h) = | h|≥ 0in Ω.The second claim in (<ref>) follows by (<ref>), the definition of f^⋇, the fact that | h|=|h| in Ω, and polar coordinates, since∫_Ω |h^⋇|^t=Nω_N∫_δ^1 |(𝔉 h)^#(ω_N ρ^N-ω_N δ^N)|^t ρ^N-1 dρ=∫_I |(𝔉 h)^#|^t=∫_Ω |𝔉h|^t=∫_Ω |h|^t.For the last property in (<ref>), we claim that,(χ_{ h>0}h) = χ_{ h>0} hand(χ_{ h≤ 0}h) = χ_{ h≤ 0} h in [δ,1].Indeed, by continuity and since h(δ)= h(1)=0, the set { h>0} is open in (δ,1)and therefore{ h>0} = ∪_i=1^∞ I_i, where (I_i)_i∈ are disjoint open intervals in (δ,1). Let I_i=(a,b) for some δ≤ a<b ≤ 1, then Nω_N∫_I_i h(s) s^N-1 ds =h(b)- h(a) =0, since a,b∈∂{ h>0}.Let r>δ arbitrary and let ζ:=max{ x :x∈ [δ,r]∩{ h≤ 0} }∈{ h≤ 0}(recall that h(δ)=0 and hence ζ is well defined).If ζ=r then χ_{ h>0}(r)=0 and { h>0}∩ [δ,r] is open in (δ,r), therefore (h χ_{ h>0})(r) =Nω_N∫_δ^r h(s) χ_{ h>0}(s)s^N-1 ds=Nω_N∑_i=1^∞∫_I_i∩(δ,r) h(s)s^N-1 ds=0= h(r) χ_{ h>0}(r).If ζ<r, then χ_{ h>0}(r)=1, h(ζ)=0, and(h χ_{ h>0})(r) =Nω_N∫_δ^r h(s)χ_{ h>0}(s)s^N-1 ds=Nω_N∫_ζ^r h(s)s^N-1 ds= h(r)- h(ζ)= h(r)χ_{ h>0}(r).Thus the first equality in (<ref>) follows, but then, by the linearity of the integral,(h χ_{ h≤ 0})=(h) - (h χ_{ h> 0})=(h)χ_{ h≤ 0},which shows the second equality in (<ref>). The last property in (<ref>) follows from the definition of h, because( h) = (χ_{ Ih>0}h)-(χ_{ Ih≤ 0}h)=(h)χ_{ Ih>0}-(h)χ_{ Ih≤ 0}= | h|≥ 0.Finally, since∫_Ω h^⋇ = ∫_I ( h)^# = ∫_Ω h =( h)(1)=|h(1) |=|∫_Ω h|=0, the firstequality in (<ref>) also holds, and the proof is finished.§.§ Monotonicity resultsIn this subsection we show the energy-decreasing property of the ⋇-transformation. We introduce first a useful notation.Recall that Ω, δ are as in (<ref>) and I:=[0,|Ω|].Forh∈ L_rad^1(Ω) letτ:[δ,1]→ I,τ(r):=ω_N (r^N-δ^N)h:I→,h(s):=h ∘τ^-1(s)= h ((ω_N^-1s+δ^N)^1/N)(recall that we use h(|x|)=h(x)).The function h is the one-dimensional equimeasurable version of h, since, for s∈ I and r=τ^-1(s)∈ [δ,1],∫_0^sh=Nω_N∫_δ^rh(τ(ρ)) ρ^N-1 dρ=Nω_N∫_δ^r h(ρ)ρ^N-1 dρ= h(r).Similarly, observe that |{h>t}|=∫_Ωχ_{h>t}=Nω_N∫_δ^1 χ_{h>t}(r) r^N-1 dr=∫_0^|Ω|χ_{h>t}=|{h>t}| fort∈,and therefore,h^#(s)=(h(τ^-1(s)))^#=h^# (s) for s∈ I. Let f and g as in the statement. By Proposition <ref>, we have that (f^⋇,g^⋇)∈ X and ∫_Ω|f|^α/α+ |g|^β/β = ∫_Ω|f^⋇|^α/α+ |g^⋇|^β/β,so it suffices to show that ∫_Ω g K f ≤∫_Ω g^⋇ K f^⋇.Let (u,v):=(K g,K f) and observe that-(u_r r^N-1)_r/r^N-1=g and-(v_r r^N-1)_r/r^N-1=f in[δ,1]. Then,u_r(r)=-g(r)/Nω_Nr^1-N and v_r(r)=-f(r)/Nω_Nr^1-N for r∈[δ,1].Thus, by Proposition <ref>, ∫_Ω g K f =Nω_N∫_δ^1 -(u_r r^N-1)_r v=Nω_N∫_δ^1 u_r v_r r^N-1 dr=(Nω_N)^-1∫_δ^1f(r) g(r) r^1-N dr≤ (Nω_N)^-1∫_δ^1 | f(r)|| g(r)| r^1-N dr=(Nω_N)^-1∫_δ^1 ( f)(r)( g)(r) r^1-N dr.By (<ref>), ∫_δ^1 ( f)(r) ( g)(r) r^1-N dr=∫_δ^1∫_0^τ(r) f(σ) dσ∫_0^τ(r) g(σ) dσ r^1-N dr=(Nω_N)^-1∫_0^|Ω|∫_0^s f(t) dt∫_0^s g(σ) dσ (ω_N^-1s+δ^N)^2(1/N-1) ds. Let φ(s):= (Nω_N)^-1(ω_N^-1s+δ^N)^2(1/N-1)≥ 0. Since f^# is non-increasingin I and ∫_If^#= ( f)(1)= 0 (by (<ref>) and because averages are preserved by rearrangements), we have that ∫_0^s f^#≥ 0 for s∈ I . Moreover,by Proposition <ref>, ∫_0^s g=( g)(τ^-1(s))≥ 0 for s∈ I.Therefore, t↦∫_t^|Ω|∫_0^s g(σ) dσ φ(s) ds and σ↦∫_σ^|Ω|∫_0^s f^#(t) dt φ(s) dsare non-increasing in I. Then, by (the Hardy-Littlewood inequality) Theorem <ref> and Fubini's theorem, ∫_I∫_0^s f(t) dt∫_0^s g(σ) dσ φ(s) ds =∫_I f(t) ∫_t^|Ω|∫_0^s g(σ) dσ φ(s) ds dt≤∫_I f^#(t) ∫_t^|Ω|∫_0^s g(σ) dσ φ(s) ds dt= ∫_Ig(σ)∫_σ^|Ω|∫_0^s f^#(t) dt φ(s) ds dσ≤∫_Ig^#(σ)∫_σ^|Ω|∫_0^s f^#(t) dt φ(s) ds dσ= ∫_I ∫_0^s g^#(σ) dσ∫_0^s f^#(t) dt φ(s) ds.However, using (<ref>), (<ref>) and reasoning as in (<ref>), we have ∫_I ∫_0^s g^#(σ) dσ∫_0^s f^#(t) dt φ(s) ds =∫_I ∫_0^s ( f)^# (σ) d σ∫_0^s ( g)^#(σ)d σ φ(s)ds=∫_I ∫_0^s f^⋇ (σ) d σ∫_0^sg^⋇(σ) d σ φ(s)ds =∫_δ^1 (f^⋇)(r)(g^⋇)(r) r^1-N dr. Therefore, arguing as in (<ref>), we obtain that ∫_Ω g K f≤ (Nω_N)^-1∫_δ^1 (f^⋇)(r) (g^⋇)(r) r^1-N dr = ∫_Ω g^⋇ K f^⋇,and (<ref>) follows. We now show that if f and g are nontrivial and ϕ(f^⋇,g^⋇)=ϕ(f,g), then f and g are monotone in the radial variable. Observe that, since f is nontrivial, f^# is non-increasing, and ∫_I f^#= ( h)(1)= 0, then ∫_0^s f^#>0 for all s∈ (0,|Ω|). But then the function σ↦∫_σ^|Ω|∫_0^s f^#(t) dt φ(s) ds is a strictly decreasing continuous function in I and, in virtue of Lemma <ref>, equality in (<ref>) may only hold ifg= g^# a.e. in I, which, by (<ref>), implies that g coincides a.e. with a radially monotone function. Then g must also be radially monotone, because | g|=|g| in Ω, g is a nontrivial continuous function, and ∫_Ω g =0.Arguing similarly using (<ref>) we conclude that f is monotone in the radial variable as well.As a consequence, if (u,v):=(K_p g,K_q f), then u and v are strictly monotone in the radial variable by (<ref>) and the fact that either f>0, g>0 or f<0, g<0 in (δ,1). This ends the proof.To motivate the definition of f^⋇, observe that the flipping f is very natural in virtue of (<ref>), but further insight on f can be gained by observing that, by (<ref>), the (Neumann radial) solution of -Δ v =f in Ω satisfies that v_r(r)=-(Nω_N)^-1r^1-N ( f)(r) ≤ 0 in (0,1), i.e. v is decreasing in the radial variable.By (<ref>), this is indeed a very simple way to guarantee that there is at least one minimizer which is monotone in the radial variable; but the flipping f is too simple to extract any further information.To show that all least energy solutions (u,v):=(K_p g,K_q f) are strictly monotone decreasing, we need to combine the flippingwith the decreasing rearrangement # in order to apply the powerful properties of rearrangements (Theorem <ref> and Lemma <ref>).In general, it is not true that ϕ(f^*,g^*)≤ϕ(f,g), where * denotes the Schwarz symmetrization and (f,g) is as in Theorem <ref>. A first observation is that f^* is defined in Ω^* (a ball with the same measure as Ω), and this would be a first complication when considering annular domains. But even if Ω is a unitary ball centered at zero B and we consider the scalar problem (i.e., p=q, see Lemma <ref>), we exhibit next a continuous radial function f:B→ with ∫_B f =0 such that ϕ(f^*,f^*)>ϕ(f,f).Let N=5, ε=1/2, and f(x):= -(ω_N |x|^N+ε)^-1/2+k, where k:=2ω_N^-1((ω_N+ε)^1/2-ε^1/2) is such that ∫_B f=0. In particular, f(s)=-(s+ε)^-1/2+k for s∈ I=[0,ω_N]=[0,8π^2/15]. Observe that f is a strictly increasing function; therefore, by (<ref>), f^*(x)=f^#(ω_N|x|^N)=f^#(ω_N|x|^N)=f(ω_N-ω_N|x|^N).We have that∫_I (∫_0^s f)^2(ω_N^-1s)^2(1/N-1) ds ≈ 5.22726 > 2.24448 ≈∫_I (∫_0^s f^#)^2(ω_N^-1s)^2(1/N-1) ds.These integrals can be computed via hypergeometric functions; for simplicity, here we just give a numerical approximation. Then, since f^#=f^* in I, we may use the techniques and notation from the proof of Theorem <ref> and we find that∫_B fKf=(Nω_N)^-1∫_I (∫_0^s f)^2φ(s) ds>(Nω_N)^-1∫_I (∫_0^s f^*)^2φ(s) ds=∫_B f^*Kf^*,where φ(s):= (Nω_N)^-1(ω_N^-1s)^2(1/N-1) for s∈ I. Therefore ϕ(f^*,f^*)>ϕ(f,f), because f^*_α=f_α by (<ref>).The Neumann b.c. are used at (<ref>) (if Ω is an annulus) and in the integration by parts performed in (<ref>). The Neumann b.c. is also the reason to consider zero-average functions.(Sublinear case) Let (p,q) satisfy (<ref>) and Ω and δ be as in (<ref>).The set of minimizers of ϕ in X_rad is nonempty. If (f,g)∈ X_rad is such that ϕ(f,g)=inf_X_radϕ, then f and g aremonotone in the radial variable and, if (u,v):=(K_p g,K_q f), then u_rv_r>0 in (δ,1); that is, u and v are both strictly monotone increasing or strictly monotone decreasing in the radial variable. Arguing as in Lemma <ref> we have that the set of minimizers of ϕ in X_rad is nonempty and contains only nontrivial functions. By Proposition <ref> we have that all such minimizers (f,g) are continuous up to the boundary ∂Ω and (u,v):=(K_p g,K_q f) solves (<ref>) pointwise, in particular (f,g)=(|u|^p-1u,|v|^q-1v). By Theorem <ref>, u and v are strictly monotone.To show that u_rv_r>0 in (δ,1), assume without loss of generality that u is increasing; then u_r>0 in (δ,1) and, by (<ref>), (<ref>) and (<ref>), v_r=-(Nω_N)^-1 f(r) r^1-N=-(Nω_N)^-1 (|u|^p-1u)(r) r^1-N>0 in (δ,1), thus v is also strictly increasing.This ends the proof. (Superlinear case) Let (p,q) satisfy (<ref>) and Ω and δ be as in (<ref>). The set of minimizers of ϕ in _rad is nonempty. If (f,g)∈ N_rad is such that ϕ(f,g)=inf_ N_radϕ, then f and g aremonotone in the radial variable and, if (u,v):=(K_p g,K_q f), then u_rv_r>0 in (δ,1). Arguing precisely as in Lemma <ref>, there exists at least one minimizer (f,g) of ϕ in _rad. Since the L^s norms are preserved by the ⋇-transformation, and (<ref>) holds, then reasoning as in part 3) of the proof of Lemma <ref> we have that (f^⋇,g^⋇) is also a minimizer of ϕ in 𝒩_rad.The rest of the proof follows as in the proof of Theorem <ref>. Since the proof of Theorem <ref> is based on one-dimensional techniques, one can also apply them to a least energy solution in the case N=1 to obtain monotonicity properties. Let Ω=(-1,1), (f,g) as in (<ref>), and (u,v):=(K_p g,K_q f). Then u_rv_r>0 in (-1,1), that is, u and v are both strictly monotone increasing or strictly monotone decreasing in the radial variable. For N=1 and Ω=(-1,1), let I_1:L^∞(Ω)→ C(Ω) and _1: C(Ω)→ L^∞( Ω) be given byI_1h(x)=∫_-1^xh(ρ) dρ and _1 h:=(χ_{ I_1h>0}-χ_{ I_1h≤ 0})h.Let τ_1:[-1,1]→[0,2], x↦τ(x)=x+1, and for h∈ C(Ω), the ⋇_1-transformation of h is given by h^⋇_1∈ L^∞( Ω), h^⋇_1(x) := (𝔉_1 h)^#(x + 1).The proof now follows by replacing I_1, _1, τ_1, and ⋇_1 instead of I, , τ, and ⋇ in the proofs of Proposition <ref> and Theorems <ref> and <ref> using δ:=-1 and doing some obvious changes (for example, the use of polar coordinates is not needed).§ SYMMETRY AND SYMMETRY BREAKING OF MINIMIZERS§.§ Symmetry breaking in radial domains This subsection is devoted to the following result.Let N≥ 1, Ω as in (<ref>), (<ref>) hold,and (f,g) satisfy (<ref>), then f and g are not radial. The proof of this result is long, and we split it in several lemmas. The argument goes by contradiction and is also based on the results from Section <ref>, i.e., that if f and g are radial, then both functions are either strictly increasing or strictly decreasing. This allows to construct a direction (φ,ψ) along which the second variation of ϕ at (f,g) is strictly negative, contradicting the minimality property of (f,g).In the following, Ω⊂^N is either a ball or an annulus as in (<ref>).We begin with an auxiliary lemma. Let U⊂^N be a domain of class C^2,1 andW(U) :={ w∈ C^1(U) : ∇ w(x) ≠ 0wheneverx∈U satisfiesw(x)=0 },which is an open subset of C^1(U). For s>0 the following holds. (1) There is δ>1 depending only on s such that the map γ: W(U)→ L^δ(U), γ(w):=|w|^s-1 is well-defined and continuous.(2) If w∈ W(U), then (|w|^s-1w)_x_i=s|w|^s-1w_x_i, i=1,…,N, and∫_U |w|^s-1w∂_x_ih = s ∫_U |w|^s-1w_x_i h for allh∈ C^∞_c(U).The proof of Lemma <ref> follows from (the proof of) <cit.>, where only the case s∈(0,1) is considered, but the case s≥ 1 follows easily from standard arguments.The next result state that the second variation of ϕis nonnegative along some directions.We use X_rad to denote the subspace of radially symmetric functions in X. Let pq<1 and (f,g)∈ X_rad be a global minimizer of ϕ in X, (u,v):=(K_p g , K_q f), and let (φ,ψ) ∈ [C^1(Ω)]^2∩ X, with (φ)⊂Ω∖ f^-1(0) and (ψ)⊂Ω∖ g^-1(0). Then the map s↦ϕ(f+sφ,g+sψ) belongs to C^∞() andlim_s→ 0d^2/ds^2ϕ(f+sφ,g+sψ) =∫_Ω1/p|u|^1-pφ^2 + 1/q |v|^1-qψ^2 - φ K ψ - ψ K φdx≥ 0.For s∈, we write ϕ(f+sφ,g+sψ)=Ψ(s)-T(s), withψ(s) =∫_Ω|f+sφ|^α/α +|g+sψ|^β/βdxand T(s) =∫_Ω (f+sφ)K(g+sψ)= ∫_Ω fKg + s ∫_Ωφ Kg + fKψdx + s^2∫_Ωφ K ψ.Then T∈ C^∞() and, by integration by parts,T”(0)=2∫_Ωφ Kψdx=∫_Ωφ Kψ + ψ Kφdx. As for Ψ, since α=p+1/p>1 and β=q+1/q>1, it is standard to show thatΨ'(s)=∫_Ω |f+sφ|^α-2(f+sφ) φ + ∫_Ω |g+sψ|^β-2(g+sψ) ψ.Since u∈ C^2(Ω) and U:=(φ)⊂Ω∖ f^-1(0), then f=|u|^p-1u is bounded away from 0 on U, and |f|^α-2∈ C^1(U). Analogously, |g|^β-2∈ C^1(V) with V:=(ψ).Therefore ψ∈ C^∞() andψ”(0)=lim_s→ 01/s(∫_U(|f+sφ|^α-2(f+sφ) - |f|^α-2f ) φ+∫_V(|g+gφ|^α-2(g+sψ) - |g|^β-2g ) ψ)= ∫_U (α-1)|f|^α-2φ^2dx+∫_V (β-1) |g|^β-2ψ^2,as required. Let _rad be the subset of radial functions in . We say that a function φ:Ω→ is antisymmetric with respect to x_1 if φ(x_1,x')=-φ(-x_1,x') for all (x_1,x')∈Ω.Let pq>1 and (f,g)∈_rad be a minimizer of ϕ in _rad, (u,v):=(K_p g , K_q f).Let (φ,ψ) ∈ [C^1(Ω)]^2∩ X be antisymmetric with respect to x_1 and such that (φ)⊂Ω\ f^-1(0) and (ψ)⊂Ω\ g^-1(0). Then the map s↦ϕ(f+sφ,g+sψ) belongs to C^∞() and (<ref>) holds.By Lemma <ref> the map s↦ϕ(f+sφ,g+sψ) is C^2 at s=0. We now argue as in <cit.>. Assume, by contradiction, that there is (φ,ψ) ∈ [C^1(Ω)]^2∩ X antisymmetric with respect to x_1 satisfying (φ)⊂Ω\ f^-1(0), (ψ)⊂Ω\ g^-1(0), and such thatd^2/ds^2ϕ(f+sφ,g+sψ)<-2κ for s∈(-κ_1,κ_1) and some κ,κ_1>0.Let γ_1 and γ_2 as in (<ref>). Using the fact that (f,g) is radially symmetric and (φ,ψ) is antisymmetric with respect to x_1, we obtain that ϕ'(c^γ_1f,c^γ_2g)(c^γ_1φ,c^γ_2ψ)=0 for c∈[1/2,2].Then, by a Taylor expansion,ϕ(c^γ_1(f+sφ),c^γ_1(g+sψ))≤ϕ(c^γ_1f,c^γ_1g)-2κ s^2+o(s^2) for s∈(-κ_1,κ_1),where o(s^2) is a remainder term uniform in c∈[1/2,2]. For s∈(-κ_1,κ_1) sufficiently small, let t=t(f+sφ,g+sψ)>0 be given by Lemma <ref>; using (<ref>), we may assume that t∈ [1/2,2], then, by Lemma <ref> and using that (f,g)∈,ϕ(t^γ_1(f+sφ),t^γ_2(g+sψ)) =sup_c∈[1/2,2]ϕ(c^γ_1(f+sφ),c^γ_2(g+sψ)) ≤sup_c∈[1/2,2]ϕ(c^γ_1f,c^γ_2g)-κ s^2 = ϕ(f,g)-κ s^2<ϕ(f,g),which yields a contradiction to the minimality of (f,g), because (t^γ_1(f+sφ),t^γ_2(g+sψ))∈, and the claim follows.In the case of least energy radial solutions, Lemma <ref> and Lemma <ref> allow us to conclude the following. Let Ω(e_1):={x∈Ω :x_1>0}. Let (f,g)∈ X_rad satisfy (<ref>) and(u,v):=(K_p g , K_q f).If u is increasing in the radial variable and (f̅,g̅):=( p |u|^p-1 u_x_1 , q |v|^q-1 v_x_1 ), then K f̅≥ 0 and K g̅≥ 0 in Ω(e_1) and∞> ∫_Ω(e_1) p|u|^p-1u_x_1(u_x_1-K g̅ )+q|v|^q-1 v_x_1(v_x_1-K f̅)dx≥ 0,Note that (u,v):=(K_p g , K_q f)∈ [C^2,ε(Ω)]^2 \ {(0,0)} is a radial classical solution of (<ref>), by Proposition <ref>.By Theorem <ref> we know that u_rv_r>0 in (δ,1), with δ=inf_x∈Ω|x|.By assumption u_r≥ 0 in (δ,1), thus u and v are strictly monotone increasing in the radial variable. In particular,f̅:=p |u|^p-1 u_x_1≥ 0 and g̅:=q |v|^q-1 v_x_1≥ 0 in Ω(e_1).Moreover, since u and v are sign-changing, there exist r_1,r_2∈ (δ,1) such that u^-1(0)=f^-1(0)={x∈Ω: |x|=r_1}and v^-1(0)=g^-1(0)={x∈Ω: |x|=r_2},i.e., the nodal sets are two spheres contained in Ω and u,v∈ W(Ω), with W(Ω) as defined in (<ref>). To control these nodal lines we use the following cutoff functions: for ε,r≥ 0, let ρ_ε^r be a smooth radial function in ^N such that0≤ρ^r_ε≤ 1,ρ_ε^r(x)=0 if | |x|-r |<ε, and ρ_ε^r(x)=1 if | |x|-r |>2ε,and let (f̅_ε,g̅_ε):=(f̅ρ^r_1_ε,g̅ρ^r_2_ε) for ε≥ 0; note that (f̅_ε,g̅_ε)∈ [C^1(Ω)]^2 for ε>0 and (f̅_ε,g̅_ε) is antisymmetric in Ω with respect to x_1for all ε≥ 0, because (u,v) is radially symmetric in Ω. Moreover, (K g̅_ε,K f̅_ε)∈[C^2(Ω)∩ C^1(Ω)]^2 for ε>0, (K g̅_0 , K f̅_0)=(K g̅ , K f̅)∈[W^2,t(Ω)]^2for some t>0 (see Lemma <ref>), and (K g̅_ε,K f̅_ε) is also antisymmetric in Ω with respect to x_1 for all ε≥ 0, by uniqueness.In particular, K f̅_ε=K g̅_ε=0 on ∂Ω(e_1)∩{x_1=0} and K g̅_ε≥ 0 in Ω(e_1) andK f̅_ε≥ 0 in Ω(e_1),by the maximum principle and Hopf's boundary point lemma. Since (f̅_ε,g̅_ε)→ (f̅,g̅) in L^t(Ω(e_1)) as ε→ 0, then (Kf̅_ε,Kg̅_ε)→ (Kf̅,Kg̅) in W^2,t(Ω(e_1)) as ε→ 0,and therefore (<ref>) implies (<ref>). Furthermore, since the product of two antisymmetric functions is symmetric, we have that2∫_Ω(e_1) f̅_ε(u_x_1ρ^r_1_ε-K g̅_ε)+g̅_ε(v_x_1ρ^r_2_ε-K f̅_ε)dx=∫_Ωf̅_ε(u_x_1ρ^r_1_ε-K g̅_ε)+g̅_ε(v_x_1ρ^r_2_ε-K f̅_ε)dx≥ 0,by applying Lemma <ref> in the sublinear case (<ref>) or Lemma <ref> in the superlinear case (<ref>) with(φ,ψ):=(f̅_,g̅_) for small ε>0.The claim (<ref>) follows from Lebesgue's dominated convergence theorem once we show the existence of a suitable majorant.Indeed, let (ξ_ε,ζ_ε):=(K g̅_ε,K f̅_ε) for ε≥ 0, then, by (<ref>),f̅_ε(u_x_1ρ_ε^r_1-ξ_ε)→ |u|^p-1u_x_1(u_x_1-ξ_0) a.e. in Ω(e_1) as ε→ 0.Moreover, we have that -Δ (ξ_ε-ξ_0)=g̅_ε-g̅=q|v|^q-1v_x_1(ρ_ε^r_2-1)≤ 0 in Ω(e_1), with ∂_ν(ξ_ε-ξ_0)=0on ∂Ω(e_1)\{x_1=0} and ξ_ε-ξ_0=0 on ∂Ω(e_1)∩{x_1=0}. Thus (testing the equation with (ξ_ε-ξ_0)_+ and integrating by parts), 0<ξ_ε≤ξ_0 in Ω(e_1)for ε small, and|f̅_ε(u_x_1ρ_ε^r_1-ξ_ε)|≤ |u|^p-1u^2_x_1+|u|^p-1u_x_1ξ_ε≤ |u|^p-1u^2_x_1+|u|^p-1u_x_1ξ_0.Note that |u|^p-1u^2_x_1∈ L^1(Ω), because |u|^p-1∈ L^t(Ω)⊂ L^1(Ω) and u∈ C^2(Ω).It remains to show that|u|^p-1u_x_1ξ_0∈ L^1(Ω)Since-Δ(u_x_1-ξ_0)=0 in Ω in the strong sense, interior elliptic regularity (see, e.g., <cit.> ) implies that u_x_1-ξ_0∈ C^∞(Ω); therefore ξ_0∈ C^1(Ω)∩ W^2,t(Ω), because u_x_1∈ C^1(Ω). This directly implies (<ref>) for p≥ 1.If p∈(0,1), then let γ>0 be such thatA:={x∈Ω: r_1-γ<|x|<r_1+γ}⊂Ω.Then∫_Ω(e_1) |u|^p-1 |u_x_1ξ_0|=∫_Ω(e_1)∖ A |u|^p-1|u_x_1ξ_0| + ∫_A |u|^p-1 |u_x_1ξ_0| ≤ (min_Ω(e_1)∖ A |u|)^p-1 u_x_1_L^∞(Ω)∫_Ω(e_1) |ξ_0|+ u_x_1ξ_0_L^∞(A)∫_Ω(e_1)|u|^p-1<∞,and (<ref>) also follows.A majorant for the term g̅_ε(v_x_1ρ^r_2_ε-K f̅_ε) in (<ref>) can be obtained similarly, and this ends the proof.The following lemma shows that an antisymmetric Neumann solution dominates the corresponding Dirichlet solution in a half radial domain Ω(e_1).Let t>1 and h∈ L^t(Ω)\{0} be an antisymmetric function in Ω with respect to x_1 and let w^N:=K h∈ W^2,t(Ω), that is, w^N is the unique strong solution of-Δ w^N=h in Ω,∂_ν w^N=0 on ∂Ω, and ∫_Ω w^N =0.Moreover, let w^D∈ W^2,t(Ω)∩ C^1(Ω) be a strong solution of -Δ w^D=h in Ω with w^D=0 on ∂Ω.If h≥0 and w^N≥ 0 in Ω(e_1) then w^D<w^N in Ω(e_1). By uniqueness of solutions we have that w^D∈ W^2,t(Ω)∩ C^1(Ω) and w^N:=K h∈ W^2,t(Ω) are antisymmetric functions in Ω with respect to x_1 and therefore, since w^N≥ 0 in Ω(e_1) by assumption,w^D-w^N=0 on ∂Ω(e_1)∩{x_1=0} andw^D-w^N≤ 0 on ∂Ω(e_1)∩{x_1>0}.Observe that w^D≢w^N in Ω(e_1), because otherwise w^D=w^N≥ 0 in Ω(e_1) with ∂_ν w^D=w^D=0 on ∂Ω(e_1)∩{x_1>0}, which contradicts Hopf's boundary point lemma.Then, since -Δ(w^D-w^N)=0 in Ω, w^D≢w^N in Ω(e_1), and (<ref>) holds, the maximum principle yields that w^D<w^N in Ω(e_1). Let Ω⊂^N as in (<ref>), (f,g) as in the statement,(u,v):=(K_p g , K_q f)∈[C^2,ε(B)]^2, and (f̅,g̅):=(p |u|^p-1 u_x_1,q |v|^q-1 v_x_1).We argue by contradiction. Assume without loss of generality that f is radial.Using the relations u=|f|^1/pf, v=K_q f, and g=|v|^q-1v, we obtain that also u, v, and g must be radially symmetric. By Lemma <ref>, there is some t>1 such that |u|^p-1,|v|^q-1∈ L^t(Ω) and ∂_x_1(|v|^q-1v)=q|v|^q-1v_x_1∈ L^t(Ω). Therefore, we have that -Δ u = |v|^q-1v∈ W^1,t(Ω) and (interior) elliptic regularity (see for instance <cit.>) yields that u∈ W_ loc^3,t(Ω). Thus, we may interchange derivatives, and (-Δ u)_x_1=-Δ (u_x_1) in Ω. Arguing analogously for -Δ v we conclude that (u_x_1,v_x_1)∈ [W^2,t(Ω)∩ C^1,ε(Ω)]^2 is the unique strong solution of the Dirichlet problem-Δ u_x_1 = q |v|^q-1 v_x_1, -Δ v_x_1 = p |u|^p-1 u_x_1 in Ωwith u_x_1=v_x_1=0 on ∂Ω,where the boundary conditions follow from the fact that u and v are radially symmetric and ∂_ν u=∂_ν v=0 on ∂Ω.By Theorem <ref>, we may assume that u and v are strictly increasing in the radial variable (the other case follows similarly). Then u_x_1 and v_x_1 are nonnegative in Ω(e_1) and,by Lemmas <ref> and <ref>,0 ≤∫_Ω(e_1) p|u|^p-1u_x_1(u_x_1-Kg̅)+q|v|^q-1 v_x_1(v_x_1-Kf̅)dx <0,a contradiction.§.§ Foliated Schwarz symmetry In this section we show that least energy solutions are foliated Schwarz symmetric whenever the domain Ω is a ball or an annulus centered at zero in ^N in dimension N≥ 2. For N=1, see Corollary <ref>.We introduce first some notation.Let ={x∈ℝ^N: |x|=1} be the unit sphere and fix e∈. We consider the halfspace H(e):={x∈ℝ^N: x· e>0} and the half domain Ω(e):={x∈Ω: x· e>0}.The composition of a function w:Ω→ with a reflection with respect to ∂ H(e) is denoted by w_e, that is,w_e: Ω→ is given byw_e(x):=w(x-2(x· e)e).The polarization u^H of u:Ω→ with respect to a hyperplane H=H(e) is given byu^H:=max{u,u_e}in Ω(e), min{u,u_e}in Ω \ Ω(e). Following <cit.>, we say that u∈ C(Ω) is foliated Schwarz symmetric with respect to some unit vector p∈ if u is axially symmetric with respect to the axis ℝ p and nonincreasing in the polar angle θ:= arccos(x/|x|· p)∈ [0,π]. We use the following characterization of foliated Schwarz symmetry given in <cit.>. We remark that these kind of characterizations appeared for the first time in <cit.>, see also <cit.> for a survey on symmetry via reflection methods.There is p∈ such that u,v∈ C(Ω) are foliated Schwarz symmetric with respect to p if and only if for every e∈ eitheru≥ u_e,v≥ v_e in Ω(e) oru≤ u_e, v≤ v_e in Ω(e).The main result of this section is the following. For similar results under Dirichlet boundary conditions, we refer to <cit.> and <cit.>.[Sublinear case]Let Ω be either a ball or an annulus centered at the origin of ^N, N≥ 2. Let (p,q) satisfy (<ref>) and (f,g)∈ X be a global minimizer of ϕ in X and (u,v):=(K_p g,K_q f). There is p∈ such that u and v are foliated Schwarz symmetric with respect to p. Let (f,g) and (u,v) as in the statement and fix a hyperplane H=H(e) for some |e|=1.By Proposition <ref>, (u,v)∈ [C^2,μ(Ω)]^2 for some μ∈(0,1), (u,v) solves (<ref>) pointwise, and (f,g)=(-Δ v,-Δ u)∈ [C^μ(Ω)]^2; thus (f^H,g^H)∈ [L^∞(Ω)]^2 and (u,v):=(K_p (g^H),K_q (f^H))∈ [W^2,N(Ω)∩ C^1(Ω)]^2, by Sobolev embeddings.Let V:=v_e+v-v - v_e, then using the definition of f^H we have that -Δ V =f-f^H-(f^H)_e+f_e=0 in Ω, ∂_ν V=0 on ∂Ω.testing this equation with V and integrating by parts we obtain thatV=k for some k∈. Then v_e+v=v_e+v+k in ΩLet Γ_1:={x∈∂Ω(e) :x· e=0},Γ_2:={x∈∂Ω(e) :x· e>0},w_1:=v-v+k/2, and w_2:=v-v_e+k/2. Since v=v^e and v=v_e on Γ_1 we have thatw_1=w_2=0 on Γ_1, by (<ref>), and ∂_ν w_1=∂_ν w_2=0 on Γ_2. Furthermore,-Δ w_1 =f^H - f≥ 0 in Ω(e) and-Δ w_2 =f^H - f_e≥ 0 in Ω(e),which implies by the maximum principle and Hopf's Lemma that w_1 ≥ 0 and w_2 ≥0 in Ω(e). Therefore, using that v_e=v_e+v-v-k (by (<ref>)) and g^H_e=g_e+g- g^H (by definition of g^H),∫_Ω g K f- g^H K f^H dx =∫_Ω gv - g^H v=∫_Ω(e) gv + g_ev_e - g^H v - (g^H)_e v_edx =∫_Ω(e) gv + g_ev_e - g^H v - (g_e+g- g^H)(v_e+v-v-k)dx=∫_Ω(e) (g_e-g^H)w_1 + (g-g^H)w_2 + k/2(g_e+g)dx ≤ 0,because ∫_Ω g = 0.To show that u and v are foliated Schwarz symmetric with respect to the same vector p∈ we use Lemma <ref> and argue by contradiction. Assume that (<ref>) does not hold. Then, without loss of generality, there are e∈ and the corresponding halfspace H=H(e) such that v≠ v^H in B(e) and either v_e≠ v^H in Ω(e) or u_e≠ u^H in Ω(e). Since t↦ h(t):=|t|^s t is a strictly monotone increasing function infor s>-1, this implies thatf = h(v)≠ h(v)^H = f^H and eitherf_e≠ f^H org_e≠ g^H in Ω(e).As a consequence, w_1>0 and 0≠ g-g^H≤ 0 in B(e) and either w_2>0 in B(e) or 0≠ g_e-g^H≤ 0 in B(e). In any case, (<ref>) implies that ∫_Ω f K g < ∫_Ω f^H K g^H. But then, since L^s norms are preserved under polarizations, we obtain that ϕ(f,g)>ϕ(f^H,g^H), a contradiction to the minimality of (f,g).Therefore (<ref>) holds and the theorem follows from Lemma <ref>. [Superlinear case]Let Ω be either a ball or an annulus centered at the origin of ^N, N≥ 2. Let (p,q) satisfy (<ref>) and (f,g)∈ X be a minimizer of ϕ in 𝒩 and (u,v):=(K_p g,K_q f). There is p∈ such that u and v are foliated Schwarz symmetric with respect to p. Arguing by contradiction as in the proof of Theorem <ref>, we obtain (<ref>). Since (f,g)∈𝒩 and L^s-norms are preserved under polarizations we have, by (<ref>),∫_Ωγ_1 |f^H|^α + γ_2 |g^H|^βdx<∫_Ω f^H Kg^Hand ∫_Ω f^H K g^H >0, then by Lemma <ref> there exists 0<t<1 such that (t^γ_1 f^H,t^γ_2 g^H)∈𝒩. This gives a contradiction, because, by (<ref>),inf_𝒩ϕ ≤ϕ(t^γ_1 f^H, t^γ_2 g^H)= t^γ (1-γ) ∫_Ω|f^H|^α/α+|g^H|^β/βdx < (1-γ) ∫_Ω|f|^α/α+|g|^β/βdx =ϕ(f,g). § FURTHER RESULTS§.§ Unique continuation principle for minimizers In this section, Ω is again a general smooth bounded domain. We prove that if (u,v) is a solution of (<ref>) associated to a minimizer of ϕ, then the nodal sets u^-1(0):={x∈Ω: u(0)=0} and v^-1(0):={x∈Ω: v(x)=0} have zero Lebesgue measure. We do it by extending the results and techniques from <cit.> to the setting of Hamilitonian elliptic systems and to the dual method framework.Recall that if (f,g)∈ X is a critical point of ϕ then (u,v):=(K_pg,K_q f)∈ [C^2,ε(Ω)]^2 is a classical solution of (<ref>). Our main result is the following.Let (f,g)∈ X be a critical point of ϕ in X and (u,v):=(K_p g , K_q f),then u^-1(0)=v^-1(0) a.e.Moreover, if p and q satisfy the sublinear condition(<ref>) and (f,g)∈ X is a global minimizer of ϕ in X, then |u^-1(0)|=|v^-1(0)|=0. To show Theorem <ref> we rely on the following preliminary results.Let γ>0 and f:(0,∞)→ [0,∞) be such that f is bounded in[ε,∞) for every ε>0 and lim_r→ 0r^γ f(r) = 0. Then for every r>0 there is s>0 such that f(s)≥ f(r) and f(t)≤ 2^γ f(s) for all t∈ [s/2,2s]. We now characterize the decay of any solution (u,v) of (<ref>) close to a common zero.Let (u,v)∈ [C^2(Ω)]^2 be a solution of (<ref>) with p,q>0, pq<1. If x_0∈Ω is a point of density one for the set u^-1(0)∩ v^-1(0) then |u(x)|^p+1+|v(x)|^q+1=o(|x-x_0|^γ), where γ=2(p+1)(q+1)/1-pq=2(1-1/α-1/β)^-1.In other words,if lim_r→ 0|u^-1(0)∩ v^-1(0)∩ B_r(x_0)| /|B_r(x_0)|=1then lim_x→ x_0|u(x)|^p+1+|v(x)|^q+1/|x-x_0|^γ=0.Without loss of generality we assume that x_0=0 and we set u≡ v≡ 0 in ^N\Ω. Let f:(0,∞)→ [0,∞) be given byf(r):=r^-γsup_|x|=r(|u(x)|^p+1+|v(x)|^q+1). Step 1. We show first that f∈ L^∞(0,∞). Indeed, assume by contradiction that there is a sequence (r_n)_n∈ such that f(r_n)→∞ as n→∞. Then, by Lemma <ref>, there is (s_n)_n∈ such that f(s_n)≥ f(r_n) and f(t)≤ 2^γf(s_n) for all t∈[s_n/2,2s_n] and n∈.Then f(s_n)→∞ as n→∞ and s_n→ 0, by the definition of f.Working if necessary with a subsequence, we may assume that B_2s_n⊂Ω for all n∈. Set Ω_0:= B_2\ B_1/2 and let u_n,v_n:Ω_0→ be given byu_n(x)=u(s_nx)/f(s_n)^1/p+1s_n^γ/p+1 andv_n(x)=v(s_nx)/f(s_n)^1/q+1s_n^γ/q+1.By (<ref>), and since 2-γ/p+1+qγ/q+1=2-γ/q+1+pγ/p+1=0we have that -Δ u_n = A_n|v_n|^q-1v_n and-Δ v_n= A_n|u_n|^p-1u_nin Ω_0,whereA_n=f(s_n)^-2/γ=f(s_n)^pq-1/(p+1)(q+1)→ 0 asn→∞.Observe next that|u_n(x)|^p+1+|v_n(x)|^q+1 = |x|^γ|u(s_n x)|^p+1+|v(s_n x)|^q+1/(|x|s_n)^γf(s_n)≤ 2^γf(s_n |x|)/f(s_n)≤ 4^γfor all x∈Ω_0 and n∈.By (<ref>), (<ref>), (<ref>), and interior elliptic regularity, there are subsequences u_n→ u^*, v_n→ v^* in C^1_loc(Ω_0). Furthermore, by definition of f and the regularity of u,v, there is (x_n)_n∈⊂𝕊^N-1 such that |u_n(x_n)|^p+1+|v_n(x_n)|^q+1=1 for all n∈. Thus, up to a subsequence, x_n→ x^*∈𝕊^N-1 with |u^*(x^*)|^p+1+|v^*(x^*)|^q+1=1.However, since 0 is a point of density one for u^-1(0)∩ v^-1(0), we have that|{x∈Ω_0:u_n(x)≠ 0}|/|B_2|≤|{x∈ B_2s_n :u(x)≠ 0}|/|B_2s_n|→ 0 asn→∞,which implies that u^*≡ 0 in Ω_0. Analogously, we obtain that v^*≡ 0 in Ω_0. This contradicts (<ref>) and therefore f∈ L^∞(0,∞).Step 2. Now, it suffices to show that lim_r→ 0f(r)=0. We argue again by contradiction: assume there is a sequence r_n→ 0 as n→∞ such that f(r_n)≥ε for all n∈ and for some ε>0. Passing if necessary to a subsequence we have that B_2r_n⊂Ω.Since f is bounded (by Step 1), the rescaled functions u_n, v_n:Ω_0→ given by u_n(x)=u(r_nx)/r_n^γ/p+1 andv_n(x)=v(r_nx)/r_n^γ/q+1are uniformly bounded. Moreover, -Δu_n = |v_n|^q-1v_n and-Δv_n =|u_n|^p-1u_n in Ω_0,by (<ref>), and there is (x_n)_n∈⊂𝕊^N-1 such that |u_n(x_n)|^p+1+|v_n(x_n)|^q+1=f(r_n)≥ε for all n∈. But arguing as before, we obtain subsequences u_n →u, v_n →v in C^1_loc(Ω_0), and x_n→x∈𝕊^N-1 such that |u(x)|+|v(x)|≥ε. But (<ref>) with u_n instead of u_n yields that u≡ 0, and we have analogously that v≡ 0, a contradiction. Therefore lim_r→ 0f(r)=0, and the proof is finished. We now use Proposition <ref> to construct directions along which the energy ϕ decreases. Let (f,g)∈ X be a critical point of ϕ, (u,v):=(K_p g , K_q f), and assume that (<ref>) holds. If |u^-1(0)∩ v^-1(0)|>0, then there is (φ,ψ)∈ X such that ϕ(f,g)>ϕ(f+φ,g+ψ). If |u^-1(0)∩ v^-1(0)|>0 then, by Lebesgue's density theorem (see, for example, <cit.>), there exists a point of density one x_0 of u^-1(0)∩ v^-1(0).Without loss of generality we assume that x_0=0.Since (u,v):=(K_p g , K_q f) is a classical solution of (<ref>) by Proposition <ref>, we obtain by Proposition <ref> that|g|^β =|Δ u |^β = |v|^q+1 =o(|x|^γ) and|f|^α=|Δ v|^α = |u|^p+1 =o(|x|^γ) ,as |x|→ 0, where γ is as in (<ref>).Let ζ∈ C_c^∞(^N)\{0} such that suppζ⊂Ω, ∫_Ωζ =0, and fix t>0 such thatc:=t^αβ/α+β∫_Ω|ζ|^α/α+|ζ|^β/β dx -t ζ_L^2(Ω)^4/∇ζ^2_L^2(Ω)<0(such number t exists, since αβ/(α+β)>1, which is equivalent to pq<1).For r>0 small denote Ω_r:=rΩ⊂Ω and let φ_r, ψ_r:Ω→ be given by φ_r(x):= r^γ/αt^β/α+βζ( x/r ),ψ_r(x):= r^γ/βt^α/α+βζ( x/r )Notice that φ_r≡ψ_r≡ 0 in ^N\Ω_r and note that w:=K (ζ(x/r))∈ C^2(Ω)∩ C^1(Ω) solves classically -Δ w = ζ(x/r) in Ω and ∂_ν w =0 on ∂Ω, therefore∫_Ωζ(x/r) K (ζ(x/r))=∫_Ω (-Δ w)w=∫_Ω |∇ w|^2=∇ w_L^2(Ω)^2.On the other hand, multiplying -Δ w = ζ(x/r) in Ω by ζ(x/r) and using Hölder's inequality,r^Nζ_L^2(Ω)^2 =∫_Ω_r |ζ(x/r)|^2=∫_Ω (-Δ w) ζ(x/r)=∫_Ω∇ w ∇(ζ(x/r)) =r^-1∫_Ω_r∇ w ∇ζ(x/r)≤ r^-1∇ w_L^2(Ω) (∫_Ω_r |∇ζ(x/r)|^2)^1/2 = r^-1∇ w_L^2(Ω) r^N/2∇ζ_L^2(Ω),that is, ∇ w_L^2(Ω)≥ r^1+N/2ζ_L^2(Ω)^2/∇ζ_L^2(Ω), and therefore, by (<ref>),∫_Ωζ(x/r) K (ζ(x/r))=∇ w^2_L^2(Ω)≥ r^2+Nζ_L^2(Ω)^4/∇ζ^2_L^2(Ω).Then, by (<ref>),(<ref>), and observing that γ satisfies γ(1/α+1/β)+2=γ (cf. (<ref>)) we obtainϕ(φ_r,ψ_r)=∫_Ω_r|φ_r|^α/α+|ψ_r|^β/β-ψ_r K φ_r dx=∫_Ω_rr^γt^αβ/α+β|ζ(x/r)|^α/α+r^γt^αβ/α+β|ζ(x/r)|^β/β-r^γ(1/α+1/β)tζ(x/r) K (ζ(x/r)) dx≤ r^γ+N∫_Ωt^αβ/α+β|ζ(y)|^α/α+t^αβ/α+β|ζ(y)|^β/β dy-r^γ(1/α+1/β)r^2+N tζ_L^2(Ω)^4/∇ζ^2_L^2(Ω)= r^γ+Nc<0. We claim that E:=ϕ(f+φ_r,g+ψ_r)-ϕ(f,g)<0. Indeed, by (<ref>),∫_Ω_r|f +φ_r|^α-|f|^α-|φ_r|^αdx ≤∫_Ω_r|o(r^γ/α)+r^γ/αt^β/α+βζ(x/r)|^α-r^γ|t^β/α+βζ(x/r)|^α+o(r^γ) dx ≤r^γ∫_Ω_r|o(1) +t^β/α+βζ(x/r)|^α-|t^β/α+βζ(x/r)|^α +o(1) dx≤r^γ+N∫_Ω|o(1) +t^β/α+βζ(y)|^α-|t^β/α+βζ(y)|^α +o(1) dy=o(r^γ+N)and analogously, ∫_Ω_r|g+ψ_r|^β-|g|^β-|ψ_r|^β dx=o(r^γ+N) as r→ 0.Furthermore, by Proposition <ref>, ∫_Ω_r|ψ_r K f| = ∫_Ω_r|ψ_r v|≤∫_Ω_rr^q γ/q+1|t^α/α+βζ(x/r)| o(r^γ/q+1) =o(r^γ+N) ∫_Ω|t^α/α+βζ|= o(r^γ+N) as r→ 0, and analogously, ∫_Ω_r|φ_r K g| = o(r^γ+N) as r→ 0.Therefore, by (<ref>), (<ref>), (<ref>), (<ref>), and the fact that φ_r≡ 0 in ^N\Ω_r,E =∫_Ω|f+φ_r|^α-|f|^α/α+|g+ψ_r|^β-|g|^β/β-(g+ψ_r)K(f+φ_r)+gK f dx=ϕ(φ_r,ψ_r)+∫_Ω_r|f+φ_r|^α-|f|^α-|φ_r|^α/α+|g+ψ_r|^β-|g|^β-|ψ_r|^β/β-ψ_r K f - φ_r K g dx =ϕ(φ_r,ψ_r)+o(r^γ+N)=r^γ+Nc+o(r^γ+N)<0for r>0 sufficiently small, and the proof is finished. We are now ready to conclude the proof of the main result of this section. Let (f,g)∈ X be a critical point of ϕ in X and (u,v):=(K_p g , K_q f). By <cit.>, Proposition <ref>, and (<ref>), we have that |v|^q=|Δ u|=0 a.e. in u^-1(0) and |u|^p=|Δ v|=0 a.e. in v^-1(0), therefore u^-1(0)=v^-1(0) a.e.Now, assume that (<ref>) holds and (f,g)∈ X is a global minimizer of ϕ in X. Then, by Proposition <ref>, we have that |u^-1(0)∩ v^-1(0)|=0 and thus |u^-1(0)|=|v^-1(0)|=0, since u^-1(0)=v^-1(0) a.e. (Superlinear case) Let (u,v)∈ [C^2(Ω)]^2 be a solution of (<ref>) with p,q>0, pq>1. If |u^-1(0)∩ v^-1(0)|>0, then (u,v) has a zero of infinite order; indeed, if x_0∈Ω is a point of density one for the set u^-1(0)∩ v^-1(0) and γ>0, then |u(x)|^p+1+|v(x)|^q+1=o(|x-x_0|^γ) as x→ x_0.A proof can be obtained by repeating almost word by word the proof of Proposition <ref> with the following changes: in Step 1, the functions u_n,v_n defined in (<ref>) satisfy system (<ref>) with A_n:= s_n^2 (sup_|x|=s_n |u(x)|^p+1+|v(x)|^q+1)^pq-1/(p+1)(q+1) instead of A_n as in (<ref>), observe that A_n→ 0 as n→∞; moreover, in Step 2, the functions u_n,v_n defined in (<ref>) solve (instead of (<ref>)) the system -Δu_n=B_n |v_n|^q-1v_n, -Δv_n=B_n |u_n|^p-1u_n, with B_n=r_n^2+γpq-1/(p+1)(q+1)→ 0 as n→∞.§.§ Simplicity of the zeros for radial solution Lett(r)=r^2-N if N≥ 3 ort(r)=1- log (r) if N=2,b:=lim_s→δt(s)∈(1,∞] (b=∞ if Ω=B_1), J:=[1,b), and φ,ψ:J→ satisfy φ (t(|x|))=u(x) and ψ (t(|x|))=v(x).Then (φ,ψ) is a bounded solution of -φ”=ζ |ψ|^q-1ψ, -ψ”=ζ |φ|^p-1φ inJ with φ'=ψ'=0 on ∂ J, where ζ(t):=(N-2)^-2 t^-2N-1/N-2≥ 0 for N≥ 3 and ζ(t)=e^-2(t-1)≥ 0 for N=2. To prove (<ref>) it suffices to show that|φ|+|φ'|>0 and|ψ|+|ψ'|>0 in J. We argue by contradiction: assume there is x∈ J such that ψ(x)=ψ'(x)=0Then, by (<ref>), there is x_0∈(x,b) such that φ(x_0)=0.Indeed, if |φ|>0 in (x,b) then |ψ”|>0 in (x,b) and, in virtue of (<ref>), ψ must be unbounded if b=∞ or |ψ'(b)|>0 if b<∞, which is impossible by assumption, and therefore (<ref>) follows. Now, it suffices to rule out the next three cases: a)φ(x)=0,b)φ(x)φ'(x)≥ 0, c)φ(x)φ'(x)< 0. Case a)Assume that φ(x)=0. Take the first integral Φ:J→ given by Φ:=φ'ψ'+ζ (|φ|^p+1/p+1+|ψ|^q+1/q+1),which satisfies Φ'=ζ' (|φ|^p+1/p+1+|ψ|^q+1/q+1)≤ 0 inJ,by (<ref>) and the fact that ζ is decreasing. Moreover, since φ(x)=0, (<ref>) implies that Φ(x)=0 and therefore φ'ψ'≤ -ζ(|φ|^p+1/p+1+|ψ|^q+1/q+1)≤ 0in[x,b),by (<ref>). By (<ref>), φ(x_0)=0 for some x_0>x.We claim that φ≡ 0 in [x,x_0]. By (<ref>), this implies that also ψ≡ 0 in [x,x_0], which contradicts the assumption that φ^-1(0)∩ψ^-1(0) has empty interior, ruling out case a).To prove the claim, suppose that φ≢0, and takey^*∈ (x,x_0) such that |φ(y^*)|≠ 0. Without loss of generality, assume that φ(y^*)>0 (the other case is analogous). Then (<ref>) implies φ'(y^*)ψ'(y^*)<0 and (by replacing the point y^* if necessary) we may assume that φ'(y^*)>0. But then, using (<ref>) and a simple continuity argument, we have φ,φ'>0 in [y^*,b), in contradiction with(<ref>). Case b)Let φ(x)>0 and φ'(x)≥ 0 (the case φ(x)<0 and φ'(x)≤ 0 is similar).By adjusting the point x_0 from (<ref>) if necessary, we may assume that φ>0 in (x,x_0), and by system (<ref>) also -ψ”>0 in (x,x_0).By (<ref>), this implies that ψ<0 in (x,x_0) and therefore -φ”<0 in (x,x_0), i.e., φ is convex in (x,x_0), which would contradict φ(x_0)=0. Case c) Finally, we rule out φ(x)φ'(x)<0.Suppose that φ(x)>0 and φ'(x)<0 (the case φ(x)<0 and φ'(x)>0 is similar). Observe that, by(<ref>) we deduce ψ”(x)<0, which, combined with (<ref>), yields ψ<0 in some interval (w_0,x)⊆ (1,x). We claim that actually ψ<0 in (1,x), which yields a contradiction since this would imply φ”>0 in (1,x) and thus φ'(1)<0, contradicting the boundary conditions in (<ref>) . To prove the claim, assume w_0>1 and ψ(w_0)=0.Then, by the mean value theorem, there is w_1∈(w_0,x) such that ψ'(w_1)=0.Since ψ'(x)=0 we may use again the mean value theorem to obtain w_2∈(w_1,x) such that ψ”(w_2)=0. But then φ(w_2)=0 by (<ref>), which is impossible since φ(x)<0, φ'(x)>0, and -φ”=ζ |ψ|^q-1ψ>0 in (w_0,x). Since we have reached a contradiction in all cases, we conclude that (<ref>) cannot happen. The case φ(x)=φ'(x)=0 is completely analogous to (<ref>) and therefore the claim (<ref>) follows.Acknowledgments.A. Saldaña is supported by the Alexander von Humboldt Foundation, Germany.H. 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Bound. Value Probl., pages 2013:194, 11, 2013. | http://arxiv.org/abs/1706.08391v3 | {
"authors": [
"Alberto Saldaña",
"Hugo Tavares"
],
"categories": [
"math.AP",
"35J50 (Primary), 35B05, 35B06, 35B07, 35J47, 35J15"
],
"primary_category": "math.AP",
"published": "20170626141509",
"title": "Least energy nodal solutions of Hamiltonian elliptic systems with Neumann boundary conditions"
} |
Let Γ be a finite, undirected, connected, simple graph.We say that a matching ℳ is a permutable m-matching if ℳ contains m edges and the subgroup of Aut(Γ) that fixes the matching ℳ setwise allows the edges of ℳ to be permuted in any fashion.A matching ℳ is 2-transitive if the setwise stabilizer of ℳ in Aut(Γ) can map any ordered pair of distinct edges of ℳ to any other ordered pair of distinct edges of ℳ.We provide constructions of graphs with a permutable matching; we show that, if Γ is an arc-transitive graph that contains a permutable m-matching for m ≥ 4, then the degree of Γ is at least m; and, when m is sufficiently large, we characterize the locally primitive, arc-transitive graphs of degree m that contain a permutable m-matching.Finally, we classify the graphs that have a 2-transitive perfect matching and also classify graphs that have a permutable perfect matching.A Millimeter Continuum Size–Luminosity Relationship for Protoplanetary Disks Anjali Tripathi1, Sean M. Andrews1, Tilman Birnstiel2, & David J. Wilner1============================================================================= § INTRODUCTION All graphs considered in this paper are finite, undirected, and simple, and are connected unless otherwise stated.A matchingis a set of edges of a graph Γ such that no two are incident with a common vertex.A matchingis a perfect matching of Γ if each vertex of Γ is incident with exactly one edge of . In other words, a matchingis the edge set of a 1-regular subgraph of Γ, andis perfect exactly when the 1-regular subgraph is spanning.Let Γ be a graph, letbe a matching in Γ with m edges, and let G be a subgroup of (Γ).We will say thatis a G-permutable m-matching if the restriction of the action of G to the edge set ofis that of the symmetric group S_m , i.e., if G_^E()≅ S_m.If such a group G and matchingexist, we will say that the graph Γ contains a permutable m-matching.The concept of a permutable matching is due to Zaslavsky, motivated by a question involving signed graphs from <cit.>.A group G of permutations of a set Ω is 2-transitive on Ω if, given two ordered pairs of distinct elements (α, β), (γ, δ) ∈Ω×Ω, there exists g ∈ G such that (α, β)^g := (α^g, β^g) = (γ, δ); in other words, G can map any ordered pair of distinct elements to any other ordered pair of distinct elements.We say that a matchingof a graph Γ is a 2-transitive matching if the setwise stabilizer ofin (Γ) is 2-transitive on the edges of .The purpose of this paper is to study graphs that contain a matchingsuch that the setwise stabilizer ofis multiply transitive on the edges of .This paper is structured as follows.In Section <ref>, we provide background information necessary for the later sections.In Section <ref>, we provide various constructions for graphs with a permutable m-matching, showing that there is actually an abundance of such graphs for any m.Moreover, there are even numerous examples when the graph Γ is required to be G-arc-transitive for G ≤(Γ), that is, when G is transitive on the set A(Γ) of ordered pairs of adjacent vertices.In Section <ref>, we prove the following result, which shows that the degree of a vertex cannot be too small in a graph with a permutable matching, up to a single, known family of exceptions. Let G ≤(Γ).If Γ is a connected G-arc-transitive graph with a G-permutable m-matching, then the degree of the graph Γ is at least m unless m = 3 and Γ is the cycle C_3k, where k ≥ 2. Many of the graphs with permutable matchings constructed in Section <ref> contain a system of imprimitivity, i.e., the full automorphism group of the graph preserves a nontrivial partition of the vertex set (and, in some cases, the stabilizer of a vertex α even preserves a nontrivial partition of the neighbors of α).If a group G of permutations of a set Ω is transitive on Ω but G does not preserve any partition of Ω other than the trivial partitions of Ω into singleton sets and the single set Ω, then G is primitive on Ω.Given a graph Γ and G ≤(Γ), Γ is said to be G-locally primitive if, given any α∈ V(Γ), the stabilizer of α in G is primitive on the neighbors of α.Given the constructions in Section <ref> and Theorem <ref>, it makes sense to consider graphs with degree m that are locally primitive and arc-transitive containing a permutable m-matching.In Section <ref>, we provide a characterization of such graphs.The notation and terminology used in the following theorem are explained in depth in Section <ref>.Let Γ be a connected G-arc-transitive, G-locally primitive graph with degree m ≥ 6 that contains a G-permutable m-matching, and suppose G has a nontrivial normal subgroup N that has more than two orbits on vertices.If the normal quotient graph Γ_N does not contain a permutable m-matching, then Γ_N is a near-polygonal graph and (Γ_N, G/N) is locally-S_m. A group G is said to be quasiprimitive on a set Ω if every nontrivial normal subgroup of G is transitive on Ω, and a group G is said to be biquasiprimitive on a set Ω if Ω has a G-invariant partition Ω = Δ_1 ∪Δ_2 such that the setwise stabilizer G_Δ_i is quasiprimitive on Δ_i for i = 1,2.Using this terminology, Theorem <ref> says that, if there exists a graph Γ that is G-arc-transitive and G-locally primitive with degree m ≥ 6 that contains a G-permutable m-matching, then one can keep taking normal quotients of this graph until reaching either (1) a vertex-quasiprimitive graph with a permutable m-matching, (2) a vertex-biquasiprimitive graph with a permutable m-matching, or (3) a near-polygonal graph such that the stabilizer of a vertex can permute the m neighbors in any way; see Section <ref>.Moreover, graphs in each case exist and are constructed in Section <ref>. We do not know if the theorem holds for m≤5; the restriction on m is a result of the technique.Section <ref> is devoted to the proof of the following theorem, which classifies the graphs with a 2-transitive perfect matching. Joins and matching joins are defined following the statement of the theorem. Let Γ be a connected graph on 2m vertices with a 2-transitive perfect matchingcontaining m edges.Then we have one of the following cases:(1) Γ is join between two graphs that are either complete or edgeless: (a) K_m ∨ K_m ≅ K_2m,(b) K_m ∨K_m,(c) K_m ∨K_m ≅ K_m,m. (2) Γ is a matching join between two graphs that are either complete or edgeless (but not both edgeless): (a) K_m ⊻K_m,(b) K_m ⊻K_m. (3) Let m = p^f, where p is a prime and p^f ≡ 34.Then either: (a) Γ is the incidence graph of the Paley symmetric 2-design over (p^f), i.e., V(Γ) = (p^f) ×{0,1}, and (x,i), (y,j) ∈ V(Γ) are adjacent if and only if i = 0, j = 1, and y -x is a square in (p^f); or(b) Γ is the graph obtained by taking the incidence graph of the Paley symmetric 2-design over (p^f) and replacing the independent sets with copies of K_p^f; that is V(Γ) = (p^f) ×{0,1}, and (x,i), (y,j) ∈ V(Γ) are adjacent if and only if either i = j and x ≠ y or if i = 0, j = 1, and y -x is a square in (p^f). (4) Let m = 5.Then either (a) Γ is the Petersen graph; or(b) Γ = C_5 ∨ C_5.Here, Γ_1 ∨Γ_2 denotes the join of the graphs Γ_1 and Γ_2, in which V(Γ_1 ∨Γ_2) = V(Γ_1) ∪ V(Γ_2) and E(Γ_1 ∨Γ_2) = E(Γ_1) ∪ E(Γ_2) ∪{{α, β} : α∈ V(Γ_1), β∈ V(Γ_2)}. The notation Γ_1 ⊻ _ϕΓ_2 denotes a matching join of Γ_1 and Γ_2.In this case, both Γ_1 and Γ_2 must be graphs with |V(Γ_1)| = |V(Γ_2)| and ϕ: V(Γ_1) → V(Γ_2) is a bijection between the vertex sets. The graph Γ_1 ⊻ _ϕΓ_2 is defined to have vertex set V(Γ_1 ⊻ _ϕΓ_2) = V(Γ_1) ∪ V(Γ_2) and the edge set isE(Γ_1 ⊻ _ϕΓ_2) = E(Γ_1) ∪ E(Γ_2) ∪{{α, α^ϕ} : α∈ V(Γ_1)}.When Γ_1 or Γ_2 is a complete graph or an empty graph, then the resulting graph is unique up to isomorphism regardless of the choice of ϕ, and in this case we simply use the notation Γ_1 ⊻Γ_2.As an example of a matching join, consider two copies of C_5: Γ_1 = {1,2,3,4,5} with x adjacent to y if and only if x - y ≡± 15 and Γ_2 = {6,7,8,9,10} with, again, x adjacent to y if and only if x - y ≡± 15.If we define ϕ to be x^ϕ = x + 5, then the matching join Γ_1 ⊻ _ϕΓ_2 is isomorphic to the 5-prism, whereas if we define ϕ: Γ_1 →Γ_2 by 1^ϕ = 6, 2^ϕ = 9, 3^ϕ = 7, 4^ϕ = 10, and 5^ϕ = 8, then the matching join Γ_1 ⊻ _ϕΓ_2 is isomorphic to the Petersen graph.As a corollary of Theorem <ref> we classify all connected graphs with a permutable perfect matching.Let Γ be a connected graph on 2m vertices with a permutable perfect matching .Then Γ is one of K_2m, K_m ∨K_m, K_m,m, K_m ⊻K_m, K_m ⊻K_m, C_6, or K_6 \{3· K_2}≅ K_2,2,2.In particular, Theorem <ref> classifies the possible induced subgraphs on the vertex set of a 2-transitive matchingof size m in an arbitrary graph: either the induced subgraph is disconnected and is m · K_2 (i.e., m vertex-disjoint edges) or it is connected and is one of the graphs listed in Theorem <ref>. Moreover, the induced subgraph on the vertex set of a permutable m-matchingin an arbitrary graph is either m · K_2 or one of the graphs listed in Corollary <ref>. § BACKGROUNDIn this section we review the terminology and theory that will be used in later sections.Let Γ be a graph.Given a subset X of the vertices of Γ, the induced subgraph of Γ on X is denoted by Γ[X].We denote the fact that the vertices α and β are adjacent by writing α∼β. We denote by Γ the complement of Γ.A walk W is defined to be a sequence of vertices (α_0, α_1, …, α_n) such that α_i ∼α_i+1 for 0 ≤ i ≤ n -1.For α∈ V(Γ), we denote the set of neighbors of α in Γ by Γ(α).The degree of a vertex α is |Γ(α)|, and we say that the graph Γ is regular if every vertex has the same degree.A graph Γ is said to be (v,k,λ, μ)-strongly regular if Γ has v vertices; Γ is regular of degree k; if α, β∈ V(Γ) and α∼β, then |Γ(α) ∩Γ(β)| = λ; and if α≠β∈ V(Γ) and α≁β, then |Γ(α) ∩Γ(β)| = μ.§.§ Permutation groups and graph symmetryLet Ω be a set and G a group of permutations of Ω, that is, let G ≤(Ω).For an element ω∈Ω, the orbit of ω under G is denoted by ω^G.For a subset Δ of Ω, we let G_Δ denote the setwise stabilizer of Δ in G.When Δ = {ω}, a single element of Ω, we write G_ω := G_{ω}.If Δ = {ω_1, ω_2, …, ω_k}, thenG_ω_1ω_2 …ω_k := ⋂_i=1^kG_ω_i,that is, G_ω_1ω_2 …ω_k fixes every ω_i.For instance, G_αΔ denotes G_{α}∩ G_Δ, i.e. the set of elements of G which stabilize both the element α pointwise and the set Δ setwise.If H ≤ G_Δ, then we denote by H^Δ the induced action of H on Δ, i.e., H^Δ is the image of the natural homomorphism from H into (Δ).If G is a group of permutations of Ω_1 and G^' is a group of permutations of Ω_2, then G and G^' are said to be permutation isomorphic if there are both a bijection ψ : Ω_1 →Ω_2 and a group isomorphism ϕ: G → G^' such that, for all g ∈ G and ω∈Ω_1, (ω^g)^ψ = (ω^ψ)^g^ϕ.The group of permutations G is said to be transitive on Ω if, for every α, β∈Ω, there exists g ∈ G such that α^g = β.A group G of permutations of a set Ω is said to be regular on Ω if G is transitive on Ω and G_ω = 1 for all ω∈Ω.Additionally, G is said to be primitive on Ω if G is transitive on Ω and G preserves no nontrivial partition of Ω, that is, G preserves no partition of Ω other than the partition into singleton sets and the partition into the single set Ω.If Π is a nontrivial G-invariant partition of Ω, then Π is called a system of imprimitivity and the elements of Π are called blocks.Finally, a group G is said to be biprimitive on Ω if Ω has a G-invariant partition Ω = Δ_1 ∪Δ_2 such that the setwise stabilizer G_Δ_i is primitive on Δ_i for i = 1,2.The group of permutations G is said to be quasiprimitive on the set Ω if every nontrivial normal subgroup of G is transitive on Ω.If G is primitive on Ω, then G is quasiprimitive on Ω; however, the converse is not true.A group G is said to be biquasiprimitive on Ω if Ω has a G-invariant partition Ω = Δ_1 ∪Δ_2 such that the setwise stabilizer G_Δ_i is quasiprimitive on Δ_i for i = 1,2.Let Γ be a graph with vertex set V(Γ) and edge set E(Γ).An automorphism of a graph Γ is a permutation of the vertices that preserves adjacency.The set of automorphisms of Γ forms a group, which is denoted by (Γ).Note that (Γ) ≤(V(Γ)).Let G ≤(Γ).The graph Γ is G-vertex-transitive if G is transitive on the vertices of Γ, and Γ is G-edge-transitive if G is transitive on edges.Similarly, the graph Γ is G-vertex-quasiprimitive (respectively, G-vertex-biquasiprimitive) if G is quasiprimitive (respectively, biquasiprimitive) on the vertices of Γ.An arc is an ordered pair of vertices (α, β) such that {α, β}∈ E(Γ), and Γ is G-arc-transitive if G is transitive on the set A(Γ) of arcs of Γ.More generally, an s-arc of Γ is an ordered (s+1)-tuple of vertices (α_0, …, α_s) such that {α_i, α_i+1}∈ E(Γ) for 0 ≤ i ≤ s-1 and α_j-1≠α_j+1 for 1 ≤ j ≤ s-1.(Repeated vertices are allowed in the walk defined by the s-arc, but there are no returns in the walk.)The graph Γ is said to be (G,s)-arc-transitive if G is transitive on the set of s-arcs of Γ. Given vertices α, β of Γ, we define the distance between α and β to be the length of a shortest path between α and β (measured in edges), and we denote the distance between α and β by d(α, β).Since we are only considering connected graphs, there will always exist a path between any two vertices α and β, so distance is a well-defined, finite-valued function on pairs of vertices.Given a fixed vertex α, for every natural number i we letG_α^[i] := {g ∈ G_α : β^g = β for all β∈ V(Γ)such thatd(α, β) ≤ i},that is, G_α^[i] is the group that fixes pointwise the set of all vertices at distance at most i from α.In particular, G_α^[1] = {g ∈ G_α : β^g = β for all β∈Γ(α)},and G_α^[1] is often referred to as the kernel of the local action of G since, for the induced action G_α^Γ(α) of the vertex stabilizer G_α on the neighbors of α, we have G_α^Γ(α)≅ G_α/G_α^[1].Finally, for vertices α_1, α_2, …, α_k, we defineG_α_1 …α_k^[1]:= ⋂_i=1^kG_α_i^[1],that is, G_α_1 …α_k^[1] is the pointwise stabilizer of the union of the Γ(α_i).Given a permutation group L, a graph Γ, α∈ V(Γ), and G ≤(Γ) such that Γ is G-vertex-transitive, the pair (Γ, G) is said to be locally-L if G_α^Γ(α) is permutation isomorphic to L.The graph Γ is said to be G-locally primitive if G_α^Γ(α) is primitive on Γ(α). §.§ Normal quotient graphs, voltage graphs, and regular coversLet Γ be a graph with transitive group of automorphisms G, and let N be an intransitive normal subgroup of G.The N-orbits of vertices of Γ form a system of imprimitivity for G, and the normal quotient graph Γ_N with respect to the normal subgroup N is the graph whose vertex set is the N-orbits of vertices, and two N-orbits α^N and β^N are adjacent if and only if there is α^'∈α^N and β^'∈β^N such that α^'∼β^'.The graph Γ is said to be a regular cover of Γ_N if, given any two adjacent vertices α^N and β^N in Γ_N, we have |Γ(α) ∩β^N| = 1.The following lemma is a well-known result, and it shows that local primitivity is a sufficient condition for the original graph to be a regular cover of the normal quotient graph.<cit.>Let Γ be a G-vertex-transitive and G-locally primitive graph, where G ≤(Γ), and let N be a normal subgroup of G with more than two orbits on V(Γ).Then Γ is a regular cover of the quotient graph Γ_N, and the quotient graph Γ_N is G/N-vertex-transitive and G/N-locally primitive. An equivalent definition of a regular cover is as follows.A covering projection p: Γ̃→Γ maps V(Γ̃) onto V(Γ), preserving adjacency, such that for any vertex α̃∈ V(Γ̃), the set of neighbors of α̃ is mapped bijectively onto the set of neighbors of α̃^p.For a vertex α of Γ, the set α^p^-1 of vertices that are mapped onto α by p is called the fiber over the vertex α.An automorphism g ∈(Γ) lifts to g̃∈(Γ̃) if the following diagram commutes: (1) at (0,2)Γ̃; (2) at (2,2) Γ̃; (3) at (0,0) Γ; (4) at (2,0) Γ; [->] (1) to node [anchor=south] g̃ (2); [->] (1) to node [anchor=east] p (3); [->] (2) to node [anchor=west] p (4); [->] (3) to node [anchor=north] g (4);The lift of the trivial group (identity) is known as the group of covering transformations and is denoted (p).The graph Γ̃ is a regular cover of Γ if (p) acts regularly on the set α^p^-1 for all vertices α∈ V(Γ).A voltage assignment on a graph Γ is a map ξ:A(Γ) → H, where H is a group, such that (α, β)^ξ = ((β,α)^ξ)^-1, and a voltage graph is a graph Γ together with a voltage assignment.For ease of notation, the voltage of the arc (α,β) will be denoted ξ_αβ, and ξ_W will denote the total voltage of a walk W, that is, ξ_W is the product (or sum, depending on the group operation) of the voltages of the edges in W.The derived covering graph Γ̃ of a voltage graph has vertex set V(Γ) × H, where two vertices (α, h_1) and (β, h_2) are adjacent iff α is adjacent to β in Γ and h_2 = ξ_αβh_1.The following theorem exhibits the deep connection between regular covers and derived covering graphs:Every regular cover Γ̃ of a graph Γ is a derived cover of a voltage graph (and conversely).In addition, suppose the voltage group is generated by the voltages assigned to the edges of Γ. If the edges of a (fixed but arbitrary) spanning tree of Γ have the identity voltage, then Γ̃ is connected.Fix a spanning tree 𝒯 of a graph Γ. Choose α∈ V(Γ), and assume that the edges of 𝒯 have been assigned the identity voltage. This implies that the voltage assignment ξ induces a natural homomorphism of the fundamental group of Γ based at α (generated by all closed walks in Γ based at α) into the voltage group H. Let g∈(Γ). For each closed walk W based at α, W^g will be a closed walk based at α^g. Moreover, the walk formed by the path in 𝒯 from α to α^g, followed by W^g, followed by the path in 𝒯 from α^g back to α, is a closed walk based at α with the same voltage as W^g. This induces a multivalued function g^ϕ_α:H→ H given by (ξ_W)^g^ϕ_α := ξ_W^g.This is not necessarily well-defined, as two walks W_1 and W_2 may have the same voltage while W_1^g and W_2^g may not. Furthermore, g^ϕ_α may not be defined on all of H. With this in mind, the following lemma gives explicit criteria for an automorphism of a graph to lift. Fix a spanning tree 𝒯 of a graph Γ and α∈ V(Γ). Assume the edges of 𝒯 are assigned the identity voltage and that the voltage group H is generated by the edge voltages of Γ. An automorphism g of Γ lifts to an automorphism g̃ of Γ̃ if and only if g^ϕ_α is a group automorphism. Moreover, if H is abelian, the automorphism g^ϕ_α does not depend on the choice of base vertex α. The following lemma also shows that it is quite possible to get the entire automorphism group of a graph to lift.Let Γ be a graph with edge set E, and let T denote the set of edges of a spanning tree 𝒯 of Γ. Let _p denote the cyclic group of order p, where p is a prime. Let H:= _p^|E| - |T|; H is a _p-vector space. Let X be a basis for H, so |X|=|E| - |T|.Define Γ_p to be the derived regular cover of the voltage graph defined by assigning a distinct element of X to each co-tree edge of Γ.Then Γ_p is well-defined, unique up to graph isomorphism, and (Γ) lifts.Moreover, the induced mapping ϕ: (Γ) →(H) is a group homomorphism. §.§ Near-polygonal graphsFollowing <cit.>, we say that Γ is a near-polygonal graph if there exists a distinguished set of c-cycles 𝒞 such that every 2-path of Γ is contained in a unique cycle in 𝒞.If c is the girth of Γ, then Γ is called a polygonal graph.Furthermore, if our collection 𝒞 of c-cycles is in fact the set of all cycles of length girth(Γ), then Γ is called strict polygonal.Manley Perkel invented the notion of a polygonal graph in <cit.> and that of a near-polygonal graph in <cit.>.(Perkel's original definition of near-polygonal graphs required that the length c of the special cycles be greater than 3. In our definition, we allow c=3.)Polygonal graphs are a natural generalization of the edge- and vertex-set of polygons and some Platonic solids (such as the cube and dodecahedron), and one immediately notes that these are themselves strict polygonal graphs, with the special set of cycles being the polygon itself or the faces of the solid, respectively.The complete graph on n points, K_n, is a strict polygonal graph of girth 3, and the Petersen graph is a polygonal graph of girth 5 that is not a strict polygonal graph <cit.>.Very few examples of polygonal graphs are known; see <cit.>.Near-polygonal graphs have appeared in the past when studying quotient graphs of symmetric graphs <cit.>.We mention here the following result, which gives a sufficient condition for a graph Γ to be near-polygonal: Suppose that Γ is a connected (G,2)-arc-transitive graph, where G ≤(Γ).Let (α, β, γ) be a 2-arc of Γ and define H:= G_αβγ.Then the following are equivalent:(i) there exist both an integer c ≥ 3 and a G-orbit 𝒞 on c-cycles of Γ such that Γ is a near-polygonal graph with set of distinguished cycles 𝒞;(ii) H fixes at least one vertex in Γ(γ) \{β};(iii) there exists g ∈ N_G(H) such that (α, β)^g = (β, γ).§ CONSTRUCTIONS OF GRAPHS WITH A PERMUTABLE MATCHINGIn this section, we provide some constructions of graphs with permutable matchings.We begin with a construction that shows that, for any m ≥ 2, there are graphs that are neither edge- nor even vertex-transitive that contain a permutable m-matching.Let Γ be a graph with a vertex α of degree m and a group of automorphisms G such that G_α^Γ(α)≅ S_m.Define a new graph QΓ to be the graph obtained by subdividing every edge of Γ into a path of length 2. It is not difficult to see that QΓ contains a permutable m-matching.In particular, if Γ = K_1,m, then (QΓ) ≅ S_m and (QΓ) has three orbits on vertices and two orbits on edges; the orbit of edges that do not all share a common endpoint is a permutable m-matching.Obviously, it is possible to construct other such examples; we mention another couple here. Let Γ be a graph with a permutable matching .Let Q_Γ be the graph obtained by subdividing every edge not ininto a path of length 2. Let Γ be a graph with a permutable matching .Let Q^Γ be the graph obtained by subdividing every edge ininto a path of length 3. The graphs produced from these constructions may have less symmetry than the original graphs; for instance, these constructions may take vertex-, edge-, or arc-transitive graphs and produce graphs that are not vertex-, edge-, or arc-transitive.For this reason, we will henceforth restrict ourselves to graphs Γ containing a G-permutable matching that are also G-arc-transitive.Perhaps the most obvious examples of graphs with permutable m-matchings are also examples of vertex-biprimitive graphs with permutable m-matchings.For every m ≥ 2, the complete bipartite graph K_m,m has degree m and there exists G ≤(K_m,m) such that K_m,m is G-vertex-biprimitive and K_m,m contains a G-permutable m-matching. We take G = (K_m,m) ≅ S_mS_2.The group G preserves the partition of the vertices into two sets of size m, and any perfect matching will be a permutable m-matching. Inspired by the example of complete bipartite graphs, the following construction demonstrates that it is quite easy to construct arc-transitive graphs with permutable matchings for any m:Let Γ be an arc-transitive graph with automorphism group H and let m be any fixed natural number.Define Γ(m) as the lexicographical product of Γ with K_m: that is, V(Γ(m)) = { (η, i) : η∈ V(Γ),1 ≤ i ≤ m }, with (η, i) adjacent to (θ, j) if and only if η is adjacent to θ in Γ. If Γ(m) is constructed from an H-arc-transitive graph Γ as in Construction <ref> with H=(Γ), then S_mH ≤(Γ(m)). For G:= S_mH, Γ(m) is G-arc-transitive, and, for any edge {α, β} in Γ, the set := {{(α, i), (β, i)} : 1 ≤ i ≤ m } is a G-permutable m-matching of Γ(m). Another construction which yields infinitely many such graphs from a G-arc-transitive graph Γ with a G-permutable m-matching is the following.Let Γ be a G-arc-transitive graph with a G-permutable m-matching = { (α_i, β_i) : 1 ≤ i ≤ m }. Let E denote the edge set of Γ and let T denote the set of edges of a spanning tree of Γ that contains each of the edges of .Let _p denote the cyclic group of order p, where p is a prime. Let H:= _p^|E| - |T|; H is a _p-vector space. Let X be a basis for H, so |X|=|E| - |T|.Define Γ_p to be the derived regular cover of the voltage graph defined by assigning a distinct element of X to each co-tree edge of Γ. By Lemma <ref>, if Γ is a G-arc-transitive graph with G-permutable m-matching = {{α_i, β_i} : 1 ≤ i ≤ m }, then G lifts to a group G of automorphisms of Γ_p, and it follows that _p := {{(α_i, 1), (β_i, 1) } : 1 ≤ i ≤ m }is itself a G-permutable m-matching of Γ_p.What last these two constructions have in common is that the graphs that are produced are not quasiprimitive on vertices: in each case, the full automorphism group of the graph produced contains an intransitive normal subgroup.Moreover, the groups G chosen above for the graphs arising from Construction <ref> are always locally imprimitive.It makes sense, then, to study the G-arc-transitive graphs that have G-permutable matchings that are G-vertex-quasiprimitive or G-vertex-biquasiprimitive.Indeed, such graphs exist.The odd graph O_n has one vertex for each of the (n-1)-element subsets of a (2n-1)-element set, and vertices are adjacent if and only if the corresponding subsets are disjoint.As the following result shows, there is at least one vertex-quasiprimitive (and, in fact, vertex-primitive) graph with a permutable m-matching for every m ≥ 3.For every m ≥ 3, the odd graph O_m has degree m and there exists G ≤(O_m) such that G ≅ S_2m-1, O_m is G-vertex-primitive, and O_m contains a G-permutable m-matching. We identify the vertices of O_m with subsets of size m-1 of {1, 2, …, 2m-1}. Then S_2m-1 is primitive on the sets of size m-1: the stabilizer of each subset is isomorphic to S_m-1× S_m, a maximal subgroup of S_2m-1 which is core-free (that is, the intersection of all conjugates of the subgroup is trivial; see <cit.>).Hence there is G ≤(O_m) such that G ≅ S_2m-1 and O_m is G-vertex-primitive.For each i such that 1 ≤ i ≤ m, define the sets S_i := {1, … m}\{i} and T_i := {i}∪{m+1, …, 2m-2}.Since vertices of O_m are identified with subsets of {1, …, 2m-1} of size m-1, each S_i and each T_i is a vertex of O_m, and, furthermore,:= {{S_i, T_i} : 1 ≤ i ≤ m}is a matching of size m.Let H:= ({1, …, m}) ≤ G.We note that H ≅ S_m and H stabilizessetwise but allows the edges ofto be permuted as we please.Therefore,is H-permutable, sois G-permutable, as desired.One might expect that if Γ has a group of automorphisms G such that (i) Γ has a G-permutable m-matching, (ii) G has a nontrivial normal subgroup N that is intransitive on vertices, and (iii) Γ does not have an induced subgraph isomorphic to K_m,m (i.e., if Γ does not arise from Construction <ref>), then the normal quotient graph Γ_N should also have a permutable m-matching.However, as the following construction shows, more exotic examples can arise.Let Γ be a (G,2)-arc-transitive, near-polygonal graph of degree m ≥ 3 such that Γ does not contain a G-permutable m-matching and (Γ, G) is locally-S_m.Let E denote the edge set of Γ and let T denote the set of edges of a spanning tree of Γ.Let _p denote the cyclic group of order p, where p is a prime. Let H:= _p^|E| - |T|; H is a _p-vector space. Let X be a basis for H, so |X|=|E| - |T|.Define Γ_p to be the derived regular cover of the voltage graph defined by assigning a distinct element of X to each co-tree edge of Γ. The graph Γ_p created from Construction <ref> contains a permutable m-matching.Let α be a vertex of Γ with Γ(α) = {β_1, …, β_m }.By Lemmas <ref> and <ref>, G lifts to a group of automorphisms G of Γ_p and there is a group homomorphism ϕ: G →(H), where the action is induced on a generating set of all closed walks based at the vertex α.Since Γ is near-polygonal and (G,2)-arc-transitive, each 2-arc (β_i, α, β_j) is contained in a unique cycle C_i,j, and G_α is transitive on these cycles. Define h_i to be the voltage of the walk W_i, where W_i is the concatenation of all cycles C_i,j such that j ≠ i.Note that, since m ≥ 3, the cycles C_i,j are distinct, and X is a basis for H, the h_i are all pairwise distinct.If g ∈ G_α and β_i^g = β_j, then the induced action of g on H sends h_i to h_j.If ξ_i is the voltage of the arc (α, β_i), then the matching {{(α, h_i), (β_i, ξ_i + h_i) } : 1 ≤ i ≤ m } is G-permutable. For each m ≥ 3, there exist infinitely many graphs Γ with a group of automorphisms G such that (i) Γ is (G,2)-arc-transitive,(ii) (Γ,G) is locally-S_m,(iii) Γ contains a G-permutable m-matching, and(iv) G has a nontrivial normal subgroup N that has more than two orbits on V(Γ),yet Γ_N does not contain a G-permutable m-matching.For each m ≥ 3, we can take Γ to be K_m+1, the m-dimensional hypercube Q_m, or the folded m-dimensional hypercube, each of which satisfies the hypotheses of Construction <ref>.To see that the hypercube Q_m has no G-permutable m-matching, we first identify the vertices of Q_m with binary m-tuples and note that (Q_m) ≅ S_2S_m. Consequently, G ≤(Q_m)_α for some vertex α, which without a loss of generality is the all zeros m-tuple.Hence G must preserve distances of vertices from α, and so, in order for G to act like S_m on m distinct m-tuples with the same number of 0's and 1's, there is either exactly one 0 or exactly one 1.Hence, without a loss of generality, the edges in the G-permutable m-matching are all from vertices at distance 1 from α to vertices at distance 2 from α.However, since every 2-path is contained in a unique 4-cycle, the action on the edges in the matching cannot be permutable: once an edge in the matching is fixed, necessarily another neighbor of α is fixed, which fixes another edge in the matching, a contradiction to permutability.The argument for the folded m-dimensional hypercube is analogous. The result now follows from Proposition <ref>, taking Γ = Γ_p, where p ranges over all primes.§ THE LOCAL STRUCTURE OF GRAPHS WITH A PERMUTABLE MATCHINGIn this section, we prove results about the local structure of a G-arc-transitive graph with a G-permutable m-matching, that is, we prove results about the stabilizer of a vertex and the size of the neighborhood of a vertex in such a graph.This first result, which has a similar proof to that of <cit.>, provides information about the edge stabilizer of an arc-transitive graph with a permutable m-matching when m is large enough.Let Γ be a G-arc-transitive graph with a G-permutable m-matching, where m ≥ 6.If {α, β} is an edge of Γ, then there is a subgroup U ≤ G_αβ such that U^Γ(α) has a composition factor isomorphic to A_m-1.Letbe a G-permutable m-matching containing {α, β}, where= { e = e_1 = {α, β}, e_2, …, e_m }.Note that G_^≅ S_m and G_e ^\ e≅ S_m-1.Let K:= {g ∈ G_e: e_i^g = e_i, 1≤ i ≤ m}, the kernel of the action of G_e on .We have KG_e, G_e /K ≅ S_m-1, and hence A_m-1 is a composition factor of G_e (since m ≥ 6, A_m-1 is simple).Now, consider the subgroup G_αβ of G_e.We have G_αβ G_e since it has index at most two, and so ( G_αβ^[1])_ G_αβ G_e.Let P = (γ_0 = α, γ_1 = β, γ_2, …, γ_n) be a path in Γ such that G_αβγ_2…γ_n^[1] = 1.Hence1 = (G_αβγ_2…γ_n^[1])_…(G_αβγ_2^[1])_(G_αβ^[1])_ G_αβ. Since A_m-1 is a composition factor of G_e and G_αβ has index at most two in G_e, A_m-1 must be a composition factor of G_αβ.If A_m-1 is a composition factor of either G_αβ^Γ(α) or G_αβ^Γ(β), then we are done.Otherwise, A_m-1 is a composition factor of (G_α^[1])_∩(G_β^[1])_= (G_αβ^[1])_.Let ℓ be the largest integer such that A_m-1 is a composition factor of (G_αβγ_2 …γ_ℓ^[1])_.This implies that A_m-1 is not a composition factor of (G_αβγ_2 …γ_ℓγ_ℓ+1^[1])_, and so A_m-1 must be a composition factor of (G_αβγ_2 …γ_ℓ^[1])_/ (G_αβγ_2 …γ_ℓγ_ℓ +1^[1])_.Since (G_αβγ_2 …γ_ℓ^[1])_/ (G_αβγ_2 …γ_ℓγ_ℓ +1^[1])_≅( (G_αβγ_2 …γ_ℓ^[1])_)^Γ(γ_ℓ+1)G_γ_ℓγ_ℓ+1^Γ(γ_ℓ+1),A_m-1 is a composition factor of G_γ_ℓγ_ℓ+1^Γ(γ_ℓ+1).Since Γ is G-arc-transitive, G_γ_ℓγ_ℓ +1≅ U ≤ G_αβ, and hence A_m-1 is a composition factor of U^Γ(α) for some U ≤ G_αβ, as desired. A consequence of this result is that the degree of a vertex in an arc-transitive graph with an permutable m-matching is at least m when m ≥ 6; in fact, we can classify the graphs with degree less than m and a permutable m-matching. By Proposition <ref>, when m≥ 6, for an edge {α, β} of Γ there exists U ≤ G_αβ such that U^Γ(α) has a composition factor isomorphic to A_m-1.For m ≥ 5, the smallest faithful permutation representation of A_m has degree m.Since U fixes β∈Γ(α), this implies that |Γ(α)| - 1 ≥ m - 1. When m = 1 and m = 2, the result is clear since Γ is connected.When m = 3, since the graph is connected and arc-transitive, the degree of the graph is at least two.If the degree of Γ is exactly two, then Γ is a cycle, and the result follows by noting that Γ must have at least six vertices and that (Γ), which is a dihedral group, must have order divisible by three. We are left with the cases m = 4 and m = 5.In either case, if the degree of such a graph Γ is 2, then Γ is a cycle, and (Γ) contains no section isomorphic to S_4.If the degree of such a graph Γ is 3, then, by a famous result of Tutte <cit.>, the order of a vertex stabilizer divides 48, and hence the order of an edge stabilizer divides (48 · 2)/3 = 32.If m ≥ 4, G = (Γ), and the permutable matching is , then 3 divides the order of the stabilizer of an edge in G, since the stabilizer of an edge ofcan permute three other edges ofin any way. Thus there is no graph of degree 3 with a permutable 4-matching.Finally, assume Γ is regular of degree 4 and thatis a G-permutable 5-matching for G = (Γ). Let = {e_1, e_2, e_3, e_4, e_5}, where each e_i = {α_i, β_i}.Consider a shortest path P_2 from a vertex of e_1 to a vertex of e_2.Without loss of generality, the path is between α_1 and α_2.Sinceis permutable, there are elements g_i in G_ that fix e_1 and map e_2 to e_i, 3 ≤ i ≤ 5, and so there exist shortest paths from e_1 to e_i, where 2 ≤ i ≤ 5, that are all of the same length. We claim that there must exist a path from α_1 to e_i with the same length as P_2 for each i.If α_1^g_i = α_1, then there is a shortest path from α_1 to α_i, so assume that α_1^g_i = β_1.Consider a fourth edge e_j.Suppose first that α_1^g_j = α_1 (so P_2^g_j is a shortest path from e_1 to e_j starting at α_1). Sinceis G-permutable, there is g ∈ G_ such that e_1^g = e_1, e_2^g = e_2, and e_j^g = e_i.If α_1^g = α_1, then the path P_2^g_jg is a shortest path from e_1 to e_i starting at α_1.On the other hand, if α_1^g = β_1, then, since e_2^g = e_2, P_2^g is a shortest path from e_1 to e_2 starting at β_1, and so there is a shortest path from e_1 to e_2 starting at each of α_1 and β_1.Sinceis permutable, this implies that there exists a shortest path from e_1 to e_i starting at α_1.Suppose next that α_1^g_j = β_1 (so that P_2^g_j is a shortest path from e_1 to e_j starting at β_1).Sinceis G-permutable, there exists x ∈ G_ such that e_1^x = e_1, e_2^x = e_i, and e_j^x = e_j.If α_1^x = α_1, then P_2^x is a shortest path from e_1 to e_i starting at α_1, whereas if α_1^x = β_1, then P_2^g_jx is a shortest path from e_1 to e_j starting at α_1, so there shortest path from e_1 to e_j starting at each of α_1 and β_1.Sinceis permutable, this implies there is a shortest path from e_1 to e_i starting at α_1.Therefore, we can always find a shortest path from e_1 to e_i starting at α_1 for each i, 2 ≤ i ≤ 5. If necessary, we relabel the vertices in e_3, e_4, and e_5 so that there is a shortest path from α_1 to α_i for each i, 2 ≤ i ≤ 5, and we denote these paths by P_i.For each i, letP_i = (α_1 = γ_i,0, γ_i,1, …, γ_i,n = α_i). Since |Γ(α_1) \{β_1}| = 3, at least two P_i go through the same neighbor of α_1, say γ = γ_2,1 = γ_3,1.Consider h ∈ G_ such that h acts onas the permutation (3 4 5).Note that, since the induced action of h onhas order 3, if α_1^h = β_1, then we could choose h^2 instead, so we may assume that α_1^h = α_1 and α_2^h = α_2.Assumefirst that γ^h ≠γ.This implies that Γ(α_1) = {β_1, γ, γ^h, γ^h^2} and that there is a shortest path from α_1 to α_2 through each of γ, γ^h, and γ^h^2.However, this implies, by the permutability of , that there is a shortest path from α_1 to each α_i through each of γ, γ^h, and γ^h^2.Thus we may choose each P_i so that γ_i,1 = γ.On the other hand, if γ^h = γ, then there is a path from α_1 to α_i through γ for each i, namely, P_4' := P_3^h goes from α_1 to α_4 and P_5':= P_3^h^2 goes from α_1 to α_5.Hence, in any case we may assume that γ_i,1 = γ for all i.However, γ has exactly three neighbors that are not α_1.We apply a similar argument for the γ_i,2, γ_i,3, etc., and reach a contradiction: either Γ is disconnected or a vertex has degree greater than 4.Therefore, there is no connected graph of degree 4 with a permutable 5-matching, and the result holds. § LOCALLY PRIMITIVE, ARC-TRANSITIVE GRAPHS WITH DEGREE M AND A PERMUTABLE M-MATCHINGGiven that a graph with a permutable m-matching has degree at least m when m ≥ 4 and given the constructions from Section <ref>, it makes sense to study arc-transitive, locally primitive graphs of degree m that contain a permutable m-matching.The following results show that such graphs Γ with a group of automorphisms G do have a nice structure with respect to nontrivial normal subgroups N of G such that N is intransitive on vertices.Let Γ be a G-arc-transitive graph with degree m ≥ 6 and let (Γ, G) be locally-S_m.Either Γ contains a G-permutable m-matching or Γ is near-polygonal.Let Γ be such a graph, and let α be a vertex with Γ(α) ={β_1, …, β_m}.Since (Γ,G) is locally-S_m, Γ is G-locally primitive; in fact, G_α^Γ(α)≅ S_m and, for each 1 ≤ i ≤ m, G_αβ_i^Γ(α) \{β_i}≅ G_αβ_i^Γ(β_i) \{α}≅ S_m-1.Because G_αβ_1 has nontrivial layer (that is, the group generated by its subnormal quasisimple groups is nontrivial; see <cit.>), then, by <cit.>, G_αβ_1^[1] = 1.Thus [G_α^[1], G_β_i^[1]] ≤ G_α^[1]∩ G_β_i^[1] = G_αβ_i^[1] = 1, i.e., for each i, the elements of G_α^[1] and G_β_i^[1] commute.Since G_α^[1] G_αβ_i, G_α^[1]≅ G_α^[1]/(G_α^[1]∩ G_β_i^[1]) ≅(G_α^[1])^Γ(β_i) \{α} G_αβ_i^Γ(β_i) \{α}.Since the only normal subgroups of S_m-1 when m-1 ≥ 5 are 1, A_m-1, and S_m-1, we conclude that G_α^[1] is isomorphic to one of 1, A_m-1, or S_m-1. §.§ 5.1.1: The case where G_α^[1]≅ S_m-1 Suppose first that G_α^[1]≅ S_m-1.DefineL_i := G_α^[1]G_β_i^[1]≅ G_α^[1]× G_β_i^[1]≅ S_m-1× S_m-1.We note that L_i ≤ G_αβ_i.Since G_αβ_i^[1] = 1, we haveG_αβ_i/G_α^[1]≅ G_αβ_i^Γ(α)≅ S_m-1, and so |L_i| = |G_αβ_i| and hence G_αβ_i = L_i.Moreover, when m -1≥ 5, Z(S_m-1) = 1; hence (G_β_i^[1])_β_j^Γ(β_j) \{α} = 1.In other words, for any i ≠ j we have(G_β_i^[1])_β_j = (G_β_j^[1])_β_i = G_β_i^[1]∩ G_β_j^[1]. For each j ≥ 2, we have G^[1]_β_2 …β_j-1β_j+1…β_m≅ S_2,and so we let G^[1]_β_2 …β_j-1β_j+1…β_m = ⟨ g_1,j⟩.If Γ(α) ∩Γ(β_1) ≠∅, since (Γ, G) is locally-S_m, then Γ≅ K_m+1 and G_α^[1] = 1, a contradiction.Thus we may pick γ_1 ∈Γ(β_1) \{α}, and we define γ_i := γ_1^g_1,i.If H = ⟨ g_1,i : 2 ≤ i ≤ m⟩, H stabilizes = {{β_i, γ_i} : 1 ≤ i ≤ m} setwise, and H^≅ S_m, sois an H-permutable m-matching, as desired. §.§ 5.1.2: The case where G_α^[1]≅ A_m-1 Suppose next that G_α^[1]≅ A_m-1.We defineL_i := G_α^[1]G_β_i^[1]≅ G_α^[1]× G_β_i^[1]as above, only now L_i ≅ A_m-1× A_m-1.SinceG_αβ_i^Γ(α)≅ G_αβ_i/G_α^[1]≅ S_m-1,we have that |G_αβ_i:L_i| = 2.As in the last case,(G_β_i^[1])_β_j = (G_β_j^[1])_β_i = G_β_i β_j^[1]≅ A_m-2.However, in this case,G_β_3 …β_m^[1]≅ A_2 = 1,and soG_αβ_3 …β_m = G_β_3 …β_m≅ G_β_3 …β_m/G_β_3 …β_m^[1]andG_αβ_3 …β_m^{β_1, β_2}≅ G_αβ_3 …β_m/G_α^[1]≅ S_2. Hence we choose g ∈ G_αβ_3 …β_m such that β_1^g = β_2 and β_2^g = β_1.We may also assume that γ_3^g = γ_3 for some γ_3 ∈Γ(β_3) \{α}; otherwise, we replace g by gx, where x ∈ G_α^[1]; indeed, this in fact shows that we may assume that g acts as a transposition on Γ(β_3) \{α}.We also remark that G_αβ_i = ⟨ L_i, g ⟩ for i ≥ 3.DefineH:= ⟨ G_β_i^[1] : 1 ≤ i ≤ m ⟩. It is clear that HG_α, and, since G_α^[1]∩ G_β_i^[1] = 1 and (G_β_i^[1])_β_j=(G_β_j^[1])_β_i=G_β_i^[1]∩ G_β_j^[1] for each i and j, we haveH ≅ H/(G_α^[1]∩ H) ≅ HG_α^[1]/G_α^[1]≲ G_α^Γ(α). Moreover, HG_α, so H is isomorphic to a normal subgroup of G_α^Γ(α).Since H has a nontrivial action on Γ(α), either H ≅ A_m or H ≅ S_m.As in the previous case, Γ(α) ∩Γ(β_1) = ∅.We claim now that H has m-1 orbits of size m onD_2(α) := ⋃_i=1^mΓ(β_i) \{α}.Suppose first that x_i, y_i ∈ G_β_i^[1] and β_k^x_i = β_k^y_i.This means x_iy_i^-1∈(G_β_i^[1])_β_k≤ G_β_k^[1]. Hence, if γ∈Γ(β_k), then γ^x_iy_i^-1 = γ and γ^x_i = γ^y_i.Now suppose x_i ∈ G_β_i^[1], x_j ∈ G_β_j^[1], and β_k^x_i = β_k^x_j = β_ℓ.There exists some r ∈{1, …, m}\{i,j,k,ℓ} and there exist y_i ∈ G_β_i^[1] and y_j ∈ G_β_j^[1] such that β_k^y_i = β_k^x_i = β_k^y_j = β_k^x_j = β_ℓ and β_r^y_i = β_r^y_j = β_r.Thus y_i, y_j ∈ G_β_r^[1], and, if γ∈Γ(β_k), then γ^y_i = γ^y_j from what we just proved above.Thusγ^x_i = γ^y_i = γ^y_j = γ^x_j,and so H has exactly m-1 orbits of size m on D_2(α). Now, define X:= ⟨ H,g ⟩.Since X is a 2-transitive group on Γ(α) that contains a transposition, X^Γ(α)≅ S_m. Now, X ≤ G_α and HG_α, so H is a normal subgroup of X.This implies that the orbits of H onD_2(α) are an X-invariant partition, which we use to find our matching: indeed, suppose γ^H is such an orbit.Then (γ^H)^g = (γ^g)^H.Select the orbit γ_3^H, where γ_3 ∈Γ(β_3) and γ_3^g = γ_3 as above.Define γ_i:=γ_3^H∩Γ(β_i).This implies that γ_3^X = γ_3^H, and hence X stabilizes = {{β_i, γ_i} : 1 ≤ i ≤ m} setwise andis an X-permutable m-matching, as desired.§.§ 5.1.3: The case where G_α^[1]≅ 1 The final case is when G_α^[1] = 1.This implies that G_α≅ S_m and G_αβ_1≅ S_m-1 with a faithful action on each of Γ(α) \{β_1} and Γ(β_1) \{α}.Hence G_αβ_1 β_2 fixes a vertex in Γ(β_1) \{α}.Moreover, since Γ has degree m and (Γ,G) is locally-S_m, Γ is a (G,2)-arc-transitive graph.By Lemma <ref>, Γ is near-polygonal, as desired. We can now prove Theorem <ref>, which essentially characterizes arc-transitive, G-locally primitive graphs of degree m with a permutable m-matching. Suppose that Γ is a G-arc-transitive, G-locally primitive graph with degree m ≥ 6 that contains a G-permutable m-matchingsuch that G contains an intransitive normal subgroup N that has more than two orbits of vertices.Let {α, β} be an edge of , let A be the N-orbit containing α, and let B_1 be the N-orbit containing β.Since G is edge-transitive and the N-orbits of vertices are G-invariant, all edges of Γ are between N-orbits; that is, if {γ, δ}∈ E(Γ), then γ, δ are in different N-orbits.Thus A ≠ B_1.Up to relabeling, there are three possibilities for {γ, δ}, where {γ, δ} is another edge of : (i) Neither γ nor δ is in either A or B_1. (ii) γ∈ A, δ∉B_1. (iii) γ∈ A, δ∈ B_1.§.§ 5.2.1: Neither γ nor δ is in either A or B_1 Since {α, β}, {γ, δ}∈, the four vertices α, β, γ, δ are in distinct N-orbits, and, sinceis a G-permutable m-matching, no two vertices of V() are in the same N-orbit.We may thus view the action of G_ on V() as an action on the N-orbits containing the vertices. This means there will be a G/N-permutable m-matching in the quotient graph Γ_N, and we are done. §.§ 5.2.2: γ∈ A, δ∉B_1 Here,is of the form{{α_1, β_1} = {α, β}, {α_2, β_2}, …, {α_m, β_m}},where α_i ∈ A and β_i ∈ B_i for each i.Sinceis permutable, this implies that B_1, …, B_m are distinct N-orbits.Moreover, if ℬ = {β = β_1, β_2, …, β_m}, since each edge contains a vertex in the N-orbit A andis permutable, the stabilizer of A in G acts as the full symmetric group on ℬ, that is, G_Aℬ^ℬ≅ S_m.Moreover, since Γ is G-locally primitive of degree m, (Γ_N,G/N) is locally-S_m, the neighbors of the vertex A of Γ_N are precisely B_1, …, B_m, and the vertex α has a unique neighbor in each of B_1, …, B_m.By Lemma <ref>, the result follows. §.§ 5.2.3: γ∈ A, δ∈ B_1 Now,is entirely contained within the N-orbits A and B_1, i.e.is of the form {{α_1, β_1} = {α, β}, {α_2, β_2}, …, {α_m, β_m}},where α_i ∈ A and β_i ∈ B_1 for all i. Moreover, since Γ is G-locally primitive, the induced subgraph Γ[V(A) ∪ V(B_1)] is a matching.Since Γ is connected, we may select a path P_2 from α_1 to α_2, sayP_2 = (α_1, γ_1,1, γ_1,2, …, γ_1,k, α_2). For each i, let the vertex γ_1,i lie in the N-orbit C_1,i.For each i, consider the orbit C_1,i^G_, and let C_1,0:= A.If, for any i, |C_1,i^G_| > 1, then, if ℓ is the least such i, |C_1,ℓ^G_| = m (sinceis a G-permutable m-matching) and |C_1,ℓ -1^G_| = 1.This implies further that (Γ_N,G/N) is locally-S_m, and the result follows by Lemma <ref>.Finally, if |C_1,i^G_| = 1 for all i, then Γ cannot be connected without some vertex having more than one neighbor in an N-orbit, a contradiction to the G-local primitivity of Γ.Therefore, in any case either Γ_N contains a G/N-permutable m-matching or Γ_N is a near-polygonal graph with (G/N)_A ≅ S_m. As discussed after the statement of Theorem <ref>, this provides a characterization of graphs with degree m containing a permutable m-matching, in the sense that under these conditions, one can keep taking normal quotients of this graph until reaching either a graph with a permutable m-matching or a near-polygonal graph where the stabilizer of a vertex acts on its m neighbors like S_m. Moreover, Theorem <ref> is a best-possible characterization in the sense that graphs in each case do exist.When combined with Construction <ref>, Theorem <ref> shows that for any m ≥ 6 there exists a connected G-arc-transitive, G-locally-primitive graph with a G-permutable m-matching such that G has an intransitive normal subgroup N with more than two orbits of vertices such that Γ_N is G/N-vertex-quasiprimitive and contains a G/N-permutable m-matching.When combined with Construction <ref>, Proposition <ref> shows that for any m ≥ 6 there exists a connected G-arc-transitive, G-locally-primitive graph with a G-permutable m-matching such that G has an intransitive normal subgroup N that has more than two orbits on vertices such that Γ_N is G/N-vertex-biquasiprimitive and contains a G/N-permutable m-matching.Finally, Corollary <ref> shows that for any m ≥ 6 there exists a connected G-arc-transitive, G-locally-primitive graph with a G-permutable m-matching such that G contains an intransitive normal subgroup N that has more than two orbits on vertices, where Γ_N is near polygonal, (Γ_N, G/N) is locally-S_m, but Γ_N does not contain a permutable m-matching. § A CLASSIFICATION OF GRAPHS WITH A 2-TRANSITIVE PERFECT MATCHINGThis section is devoted to the proof of Theorem <ref>, which classifies the connected graphs that contain a 2-transitive perfect matching of size m. Throughout this section, we will use the following notation.We define Γ to be a graph with a perfect matchingof m edges such that (Γ) is 2-transitive on .This implies that |V(Γ)| = 2m and V(Γ) = V().We writeas follows:={e_i={α_i,β_i}}_i=1^m.We also define M ≤(Γ) to be the subgroup of (Γ) preservingsetwise, i.e., M := (Γ)_.We begin with the following observation, which allows us to subdivide the problem into cases.For any i,j such that 1 ≤ i < j ≤ m, Γ[α_i, β_i, α_j, β_j] ≅Γ[α_1, β_1, α_2, β_2]. This follows immediately from the 2-transitivity of (Γ) on . Let {α_i, β_i}∈.Each other edge ofhas at least one endpoint adjacent to either α_i or β_i. Assume thatcontains more than one edge, and let e_i = {α_i, β_i}∈.Since Γ is connected, either α_i or β_i has another neighbor, say γ.Sinceis a perfect matching, γ is α_j or β_j for some j.Since there is at least one edge from an endpoint of e_i to an endpoint of e_j, the result follows by Lemma <ref>. We now subdivide the problem based on the induced subgraph Γ[α_1,β_1,α_2, β_2]. If Γ has a matchingsuch that (Γ)_ is 2-transitive on the edges of , then the induced subgraph Γ[α_1,β_1,α_2, β_2] will be isomorphic to one of K_4, C_4, K_4 \{e}, P_4, or a triangle with a pendant edge.This follows from Lemmas <ref> and <ref> and exhausting the graphs on four vertices.See Figure <ref> for these induced subgraphs.We consider these cases one by one. If Γ[α_1,β_1,α_2,β_2]≅ K_4, then Γ≅ K_2m. Consider any two vertices γ, δ∈ V(Γ). In ,either γ and δ are matched or not. If they are matched, they are the endpoints of some e_i. If not, one is an endpoint of some e_i and the other is an endpoint of some e_j. But by Lemma <ref>, γ and δ are adjacent in this case as well. Then every pair of vertices is adjacent and Γ≅ K_2m.There is no graph Γ such that Γ[α_1,β_1,α_2,β_2] is a triangle with a pendant edge. Without loss of generality, we let α_1 be the vertex with degree 3 and β_1 be the vertex with degree 1.By the 2-transitivity of (Γ) on , there is a g∈(Γ) such that {α_1,β_1}^g={α_2,β_2} and {α_2,β_2}^g={α_1,β_1}. Without loss of generality, β_2^g=β_1 and α_2^g=α_1. But because α_1∼β_2, α_1^g∼β_2^g, we have α_1^g∼β_1. But α_1^g∈{α_2,β_2}, so we have a contradiction.§.§ The case Γ[α_1, β_1, α_2, β_2] ≅ C_4 In order to characterize the graphs when the induced subgraph Γ[α_1, β_1, α_2, β_2] is isomorphic to C_4, we first need some preliminary results.A vertex γ∈ V(Γ) is contained in a unique edge e_i of , so we define γ^c := V(e_i) \{γ}, i.e., γ^c is the unique vertex adjacent to γ in the matching .Assume Γ[α_1, β_1, α_2, β_2] ≅ C_4.Let x ∈(V(Γ)) be the permutation of the vertices of Γ defined by γ^x = γ^c for all γ∈ V()=V(Γ), that is, α_i^x = β_i and β_i^x = α_i for all i.Then x ∈ Z(M). We first need to show that x ∈ M; that is, we need to show that x ∈(Γ) and x preserves the matchingsetwise.Suppose γ, δ∈ V(Γ) and γ∼δ.If δ = γ^c, then γ^x = δ and δ^x = γ, and so γ^x ∼δ^x.If δ≠γ^c, then, since Γ[α_1, β_1, α_2, β_2] ≅ C_4 and γ∼δ, we have that γ^c ∼δ^c, and hence γ^x ∼δ^x.Since x is a permutation of the vertices of a finite graph Γ mapping edges to edges, x ∈(Γ).Since x fixes each edge e_i, x ∈ M.We will now show that x ∈ Z(M).Let g ∈ M.For any γ∈ V(Γ), we have:γ^gxg^-1= ((γ^g)^x)^g^-1 = ((γ^g)^c)^g^-1 = (γ^c)^gg^-1 = γ^c = γ^x.Therefore, gxg^-1 = x for all g ∈ M, and so x ∈ Z(M). Assume Γ[α_1, β_1, α_2, β_2] ≅ C_4.For γ∈ V(Γ), if e is the edge ofcontaining γ, then M_γ is transitive on \{ e }. Let e ∈ and e = {γ, δ}.Since M is 2-transitive on , M_e is transitive on \{e}.Let e_i, e_j ∈\{e}.Then there exists g ∈ M_e such that e_i^g = e_j.If g ∉M_γ, then gx ∈ M_γ and e_i^gx = e_j, where x is as in Lemma <ref>.The result follows. Assume Γ[α_1, β_1, α_2, β_2] ≅ C_4.Define A_i := {α_i}∪{γ∈ V(Γ) : i≠ j, γα_i} = {γ∈ V(Γ): γ∼β_i}and B_i := {γ∈ V(Γ)| γ∼α_i}.If g ∈(Γ) and e_i^g = e_i, then g preserves the partition of V(Γ) into A_i ∪ B_i.Moreover, if α_i^g = α_i, then A_i^g = A_i and B_i^g = B_i; if α_i^g = β_i, then A_i^g = B_i and B_i^g = A_i.SinceA_i = {γ∈ V(Γ) : γα_i} = {γ∈ V(Γ) : γ∼β_i}and B_i = {γ∈ V(Γ) : γβ_i} = {γ∈ V(Γ) : γ∼α_i},we have that A_i ∪ B_i is a partition of V(Γ).Since automorphisms preserve adjacency and nonadjacency, the result follows.Assume Γ[α_1, β_1, α_2, β_2] ≅ C_4.The vertices of Γ can be partitioned into two sets, A and B, such that |A| = |B| = m, each of A and B contains exactly one endpoint from each edge of , and either Γ[A] ≅Γ[B] ≅ K_m or Γ[A] ≅Γ[B] ≅K_m. By Lemmas <ref> and <ref>, for any vertex γ∈ V(Γ), M_γ has four orbits on vertices: {γ}, {γ^c}, Γ(γ) \{γ^c}, and Γ(γ^c) \{γ}.Without loss of generality, we may let {γ, γ^c} = e_1, Γ(γ^c) \{γ} = {α_i : i ≥ 2}, and Γ(γ) \{γ^c} = {β_i : i ≥ 2}.Moreover, by Lemma <ref>, there exists h ∈ M_α_2 such that α_2^h = α_2 and e_1^h = e_3.Suppose first that γ^h = α_3.Let γ = α_1, and let A := {α_i : 1 ≤ i ≤ m } and B := {β_i : 1 ≤ i ≤ m}.By Lemma <ref>, for each i ≥ 2 there exists g_i ∈ M_α_1 such that α_1^g_i = α_1 and α_2^g_i = α_i.Note thatA_2 = (A_2 ∩ A_1) ∪ (A_2 ∩ B_1),B_2 = (B_2 ∩ A_1) ∪ (B_2 ∩ B_1), where A_i and B_i are defined as in the statement of Lemma <ref>.Since α_2^h = α_2, A_2^h = A_2 and B_2^h = B_2, and so either (i) A^h = A and B^h = B or (ii) h swaps (A_1 ∩ A_2) and (B_1 ∩ A_2) and h swaps (A_1 ∩ B_2) and (B_1 ∩ B_2).However, α_2 ∈ A_1 ∩ A_2, so we have A^h = A and B^h = B.Let α_i, α_j ∈ A, i ≠ j.Since A is invariant under M_α_1 and h, there is α_k ∈ A such that α_k^hg_3^-1g_i = α_j.Since α_1 = γ≁α_k, we have α_j = α_k^hg_3^-1g_i≁α_1^hg_3^-1g_i = α_i.Since i,j were arbitrary, A is a coclique.Since Γ[α_1, β_1, α_2, β_2] ≅ C_4, it immediately follows that B is a coclique as well.Suppose now that γ^h = β_3.Let γ = β_1, and let A := {α_i : 1 ≤ i ≤ m } and B := {β_i : 1 ≤ i ≤ m}.The proof now proceeds as above.By Lemma <ref>, for each i ≥ 2 there exists g_i ∈ M_β_1 such that β_1^g_i = β_1 and β_2^g_i = β_i.Note thatB_2 = (B_2 ∩ A_1) ∪ (B_2 ∩ B_1),A_2 = (A_2 ∩ A_1) ∪ (A_2 ∩ B_1), where A_i and B_i are defined as in the statement of Lemma <ref>.Since β_2^h = β_2, B_2^h = B_2 and A_2^h = A_2, and so either (i) B^h = B and A^h = A or (ii) h swaps (B_1 ∩ A_2) and (A_1 ∩ A_2) and h swaps (B_1 ∩ B_2) and (A_1 ∩ B_2).However, β_2 ∈ B_1 ∩ B_2, so we have B^h = B and A^h = A.Let β_i, β_j ∈ B, i ≠ j.Since B is invariant under M_β_1 and h, there is β_k ∈ B such that β_k^hg_3^-1g_i = β_j.Since β_1 = γ∼β_k, we have β_j = β_k^hg_3^-1g_i∼β_1^hg_3^-1g_i = β_i.Since i,j were arbitrary, B is a clique.Since Γ[α_1, β_1, α_2, β_2] ≅ C_4, it immediately follows that A is a clique as well.If Γ[α_1,β_1,α_2,β_2]≅ C_4, then either Γ=K_m⊻ K_m or Γ=K_m,m.This follows immediately from Lemma <ref> and a consideration of the degree of each vertex in the induced subgraph Γ[α_1,β_1,α_2,β_2]≅ C_4.§.§ The cases Γ[α_1, β_1, α_2, β_2] ≅ P_4 and Γ[α_1, β_1, α_2, β_2] ≅ K_4 \{e} The two remaining cases are actually very closely related.We begin with a helpful lemma.Assume Γ[α_1, β_1, α_2, β_2] ≅ P_4 or Γ[α_1, β_1, α_2, β_2] ≅ K_4 \{e}.Either (1) Γ is regular, or(2) Γ has two orbits of vertices: one orbit is a clique, the other a coclique.We know that G is transitive on , so (Γ) has at most 2 orbits of vertices. If (Γ) is also transitive on V(Γ), then (1) holds. If not, Γ has exactly 2 orbits of vertices, and (Γ) will be 2-transitive on each of these orbits. Thus each orbit is either a clique or a coclique.Both cannot be cliques, because otherwise Γ would be regular. Both cannot be cocliques, because otherwise Γ is not connected. So we are in case (2). This allows us immediately to classify these graphs in the event that they are not regular.If Γ[α_1, β_1, α_2, β_2] ≅ P_4 and Γ is not regular, then Γ≅ K_m ⊻K_m.If we have Γ[α_1, β_1, α_2, β_2] ≅ K_4 \{e} and Γ is not regular, then Γ≅ K_m ∨K_m. This follows immediately from Lemma <ref>. The remaining cases are when Γ is regular.This implies that m is odd.Assume Γ[α_1, β_1, α_2, β_2] ≅ P_4 or Γ[α_1, β_1, α_2, β_2] ≅ K_4 \{e}.If Γ is regular, then m is odd.Moreover, if m = 2k+1, then the degree of each vertex is k+1 if Γ[α_1, β_1, α_2, β_2] ≅ P_4 and the degree of each vertex is 3k+1 if Γ[α_1, β_1, α_2, β_2] ≅ K_4 \{e}. Assume that Γ[α_1, β_1, α_2, β_2] ≅ P_4.For each i ≥ 2, the vertices of e_i contribute 0 to the degree of one endpoint of e_1 and 1 to the other, i.e., each e_i for i ≥ 2 contributes 1 to the sum of the degree of α_1 and the degree of β_1.Since Γ is regular, 2 · |Γ(α_1)| = |Γ(α_1)| + |Γ(β_1)| = 1 + 1 + (m-1). The result follows for Γ[α_1, β_1, α_2, β_2] ≅ P_4. The proof is analogous in the case when we have Γ[α_1, β_1, α_2, β_2] ≅ K_4 \{e}.In fact, in these remaining cases when Γ is regular, Γ must be vertex transitive. Assume Γ[α_1, β_1, α_2, β_2] ≅ P_4 or Γ[α_1, β_1, α_2, β_2] ≅ K_4 \{e}.If Γ is regular, then M is transitive on V(Γ). We know that M is transitive on the edges of , so it suffices to show that there is g_i ∈(Γ) such that α_i^g_i = β_i for each i.Assuming Γ contains more than a single edge, it must contain at least three edges since m is odd.In each case we may choose three edges as follows:2 [scale=0.5] [fill, shape=circle, label=above:α_j] (aj) at (0,5) ; [fill, shape=circle, label=below:β_j] (bj) at (0,0) ; [fill, shape=circle, label=above:α_i] (ai) at (3,5) ; [fill, shape=circle, label=below:β_i] (bi) at (3,0) ; [fill, shape=circle, label=above:α_k] (ak) at (6,5) ; [fill, shape=circle, label=below:β_k] (bk) at (6,0) ;[line width=2pt] (aj)–(bj) node[pos=.5, left] e_j; [line width=2pt] (ai)–(bi) node[pos=.5, right] e_i; [line width=2pt] (ak)–(bk) node[pos=.5, right] e_k; [line width=2pt] (ai)–(bj); [line width=2pt] (ak)–(bi); [scale=0.5] [fill, shape=circle, label=above:α_j] (aj) at (0,5) ; [fill, shape=circle, label=below:β_j] (bj) at (0,0) ; [fill, shape=circle, label=above:α_i] (ai) at (3,5) ; [fill, shape=circle, label=below:β_i] (bi) at (3,0) ; [fill, shape=circle, label=above:α_k] (ak) at (6,5) ; [fill, shape=circle, label=below:β_k] (bk) at (6,0) ;[line width=2pt] (aj)–(bj) node[pos=.5, left] e_j; [line width=2pt] (ai)–(bi) node[pos=.5, right] e_i; [line width=2pt] (ak)–(bk) node[pos=.5, right] e_k; [line width=2pt] (ai)–(bj); [line width=2pt] (ak)–(bi); [line width=2pt] (aj)–(ai); [line width=2pt] (ai)–(ak); [line width=2pt] (bj)–(bi); [line width=2pt] (bi)–(bk);By the 2-transitivity of M on , there is g ∈ M such that e_i^g = e_i and e_j^g = e_k.The g_i that we seek is this g, and the result follows. We now show that there is a bijection between regular graphs in these two cases, i.e. that the two cases correspond.There exists a regular graph Γ_0 on 2m vertices with Γ_0[α_1, β_1, α_2, β_2] ≅ P_4 if and only if there exists a regular graph Γ_1 on 2m vertices with Γ_1[α_1, β_1, α_2, β_2] ≅ K_4 \{e}, and there is a natural bijection between such graphs. Suppose we have such a graph Γ_1.Note that M is the setwise stabilizer ofin (Γ_1), which is transitive on V(Γ_1) but preserves the matching .However, M has (at least) two orbits on the edges of Γ_1: the edges ofand the edges not in .The complement Γ_1 also has M as a group of automorphisms.We define Γ_0 to be the graph with vertex set V(Γ_1) and edge set E(Γ_1) ∪.The group M is still 2-transitive on a perfect matching in this case, but Γ_0[α_1, β_1, α_2, β_2] ≅ P_4.The proof in the other direction is analogous.After considering Lemma <ref>, there are really only three cases left.We may assume that Γ[α_1, β_1, α_2, β_2] ≅ P_4, and one of the following holds: (i) (Γ) is primitive on V(Γ), (ii) Γ is bipartite, or (iii)itself is a system of imprimitivity.(Any other system of imprimitivity is ruled out by the 2-transitivity of M on .) §.§ The case where Γ[α_1, β_1, α_2, β_2] ≅ P_4 and G=(Γ) is primitive on verticesAssume Γ[α_1, β_1, α_2, β_2] ≅ P_4 and G = (Γ) is primitive on vertices.Then Γ is a (G,2)-arc-transitive graph. Since G is primitive on V(Γ), G_α_1 is a maximal subgroup of G.On the other hand, since there is g ∈ M such that α_1^g = β_1, β_1^g = α_1 (see Lemma <ref>), we have M_α_1 < M_e_1 < M ≤ G, and so M_α_1 is not a maximal subgroup of M.Thus M < G.By the 2-transitivity of M on , M has two orbits on E(Γ):and E(Γ) \.Since M < G, there is h ∈ G \ M, i.e., there is an automorphism that does not preserve .This implies that h takes an edge into an edge in E(Γ) \, and so G is transitive on E(Γ).Finally, we note that (i) Γ is G-vertex-transitive, (ii) Γ is G-edge-transitive, (iii) there is an element sending the arc (α_1, β_1) to the arc (β_1, α_1), and (iv) G_α_1β_1 is transitive on Γ(α_1) \{β_1}, which implies that Γ is a (G,2)-arc-transitive graph.Consider the labeling of the vertices as in Figure <ref>.If we define D_i(γ) := {δ∈ V(Γ) : d(γ, δ) = i }, i.e., if D_i(γ) is the set of vertices at distance i from the vertex γ, we can guarantee the distance of all vertices in the graph from α_1 except for the set X; all we know is that X ⊆ D_2(α_1) ∪ D_3(α_1).Assume Γ[α_1, β_1, α_2, β_2] ≅ P_4, G = (Γ) is primitive on vertices, and the subset X is as defined above.Then X ∩ D_2(α_1) ≠∅. Suppose that X ∩ D_2(α_1) = ∅, that is, X = D_3(α_1).By Lemma <ref> and the fact thatis a 2-transitive perfect matching, Γ is distance-transitive with diameter 3.We will show that, if γ, δ∈ X, then γ≁δ.Indeed, suppose γ∈ D_3(α_1) = X.This means that d(β_1, γ) = 2.Since X ∩ D_2(α_1) = ∅, there are no edges from D_1(α_1) to X.Similarly, since Γ is vertex-transitive, there are no edges from D_1(β_1) = Y_1 ∪{α_1} to D_3(β_1) = Y_2 (see Figure <ref>).Hence, if δ∈ D_1(β_1), δ has no neighbors in Y_2.Since Γ is distance-transitive, this means that no vertex in D_2(α_1) has any neighbors in D_2(α_1), i.e., all edges in Γ are from A = {α_1}∪ Y_1 ∪ Y_2 to X ∪ D_1(α_1).However, this means that Γ is bipartite, in contradiction to Γ being vertex-primitive.Therefore, X ∩ D_2(α_1) ≠∅. Assume Γ[α_1, β_1, α_2, β_2] ≅ P_4 and G = (Γ) is primitive on vertices.Then Γ is isomorphic to the Petersen graph. We again assume that vertices are labeled as in Figure <ref>.By the 2-transitivity of M on , M_α_1 is transitive on X, and so X ⊆ D_2(α_1).Hence Γ has diameter 2, and, by Lemma <ref>, Γ is a distance-transitive, diameter 2, triangle-free strongly regular graph.(These are known as rank 3 graphs since, for any vertex α∈ V(Γ), the stabilizer of α is a primitive group of rank 3 on vertices.)We note that Γ is a (4k+2, k+1, 0, μ)-strongly regular graph.By the classic equation relating the parameters (see <cit.>),(k+1)k= [(4k+2)-(k+1)-1]μ,and so μ =(k+1)/3.The eigenvalues of the adjacency matrix for this graph and their multiplicities are known (again, see <cit.>). There are three eigenvalues: k+1, with multiplicity one, and two others. The multiplicities of these other two eigenvalues are1/2((4k+1)±(4k+1)(k+1/3)-(2k+2)/√((k+1/3)^2+8k+1/3)) =1/2((4k+1)±4k^2-k-5/√((k+1)(k+25))) ∈. This implies that(4k^2-k-5)^2/(k+1)(k+25)=16k^2-424k+10585-264000/k+25is a perfect square. The last term allows us, via factoring, to come up with a list of values of k to check, which yieldsk=2ork=24.But, together with μ=k+1/3∈, we rule out k=24, so the only graph in this case is strongly regular with parameters (10,3,0,1) (corresponding to k=2), which is the Petersen graph.It can be verified that the Petersen graph has a 2-transitive perfect matching by direct inspection.For instance, if the vertices of the Petersen graph 𝒫 are represented as subsets of size two of {1,2,3,4,5}, then (𝒫) = S_5 is 2-transitive on the matching = {{{1,2},{3,4}}, {{3,5},{2,4}}, {{1,4},{2,5}}, {{2,3},{1,5}}, {{4,5},{1,3}}}. §.§ The case where Γ[α_1, β_1, α_2, β_2] ≅ P_4 and G=(Γ) is imprimitive on verticesAssume Γ[α_1, β_1, α_2, β_2] ≅ P_4 and that Π = {{α_i, β_i} : 1 ≤ i ≤ m} is a system of imprimitivity on V(Γ).Then m = p^f, where p is a prime and p^f ≡ 34, and Γ is isomorphic to the incidence graph of the Paley symmetric 2-design over (p^f). Suppose Π = {{α_i, β_i} : 1 ≤ i ≤ m} is a system of imprimitivity on V(Γ).This implies that G = M.We remove the edge orbitfrom Γ to create a new graph Γ'; since G = M,is an orbit of the edges of Γ under G, and G still acts 2-transitively on the system of imprimitivity Π.However, each block in Π is now an independent set.The quotient graph Γ'_Π will be the complete graph K_m, and there is exactly one edge between any two blocks in Γ'.By the 2-transitivity of M on Π, Γ' is M-arc-transitive.Hence Γ' is a symmetric spread of the complete graph K_m (see <cit.>).By inspection of <cit.>, the only possibility for Γ is the incidence graph of the Paley symmetric 2-design over (p^f).Moreover, if Γ is such a graph, then V(Γ) = (p^f) ×{0,1}, and (Γ) acts 2-transitively on each copy of (p^f) (simultaneously).Hence the matching {{(x,0), (x,1)} : x ∈(p^f)} is a 2-transitive perfect matching. Our final case is when Γ[α_1, β_1, α_2, β_2] ≅ P_4 and Γ is bipartite.Assume Γ[α_1, β_1, α_2, β_2] ≅ P_4 and that Γ is bipartite.Then m = p^f, where p is a prime and p^f ≡ 34, and Γ is isomorphic to the incidence graph of the Paley symmetric 2-design over (p^f). Assume Γ[α_1, β_1, α_2, β_2] ≅ P_4 and that Γ is bipartite.If G := (Γ) = M, then this case has been resolved by Lemma <ref>.Hence we may assume that M < G.By the 2-transitivity of M on , M has two orbits on E(Γ):and E(Γ) \.Since M < G, there is h ∈ G \ M, i.e., there is an automorphism that does not preserve .This implies that h takes an edge into an edge in E(Γ) \, and so G is transitive on E(Γ).Since (i) G is transitive on the edges of Γ, (ii) G is transitive on the vertices of Γ, (iii) there is an element sending the arc (α_1, β_1) to the arc (β_1, α_1) by Lemma <ref>, and (iv) G_α_1β_1 is transitive on Γ(α) \{β_1}, we have that Γ is a (G,2)-arc-transitive graph.Since Γ is bipartite, using the labeling of Figure <ref>, we have D_2(α_1) = Y_1 ∪ Y_2 and D_3(α_1) = X.Since Γ is (G,2)-arc-transitive and M_α_1 is transitive on X, Γ is a distance-transitive graph of diameter 3.By <cit.>, Γ is the incidence graph of a symmetric 2-design.The points of the design are represented by one of the biparts of Γ.The stabilizer of a point (i.e., of α_1, say) has at most three orbits on points: (i) {α_1}, (ii) the set of all points incident with “block” β_1, and (iii) set of all points not incident with “block” β_1.This means that Γ is the incidence graph of a rank 2 or 3 symmetric 2-design.Such symmetric 2-designs have been classified <cit.>.The only possibilities, other than the Paley symmetric 2-designs, are: the Hadamard design with 11 points where each point is incident with 5 blocks, which gives the same incidence graph as the Paley symmetric 2-design on 11 points; the design with 35 points where each point is incident with exactly 17 blocks, which is ruled out since the only 2-transitive groups on 35 points are A_35 and S_35, which are not involved in the automorphism group of this design (the unique minimal normal subgroup of the automorphism group of this design is isomorphic to A_8); and the design with 15 points where each point is contained in exactly 7 blocks.In this last case, the unique minimal normal subgroup of the automorphism group of the design is isomorphic to A_6.While A_6 has a rank 3 action on 15 points, the stabilizer of a point in this action has orbits of size 1, 6, and 8.However, if the incidence graph of this design had a 2-transitive perfect matching, then the stabilizer of a point would have orbits of size 1, 7, and 7.Therefore, the only such graphs Γ with Γ[α_1, β_1, α_2, β_2] ≅ P_4 and Γ bipartite are isomorphic to incidence graphs of Paley symmetric 2-designs.We are now ready to complete the proof of Theorem <ref>. The result follows from Lemmas <ref>, <ref>, <ref>, <ref>, <ref>, <ref>, <ref>, <ref>, and <ref>.Finally, we prove Corollary <ref>. The result follows from Theorem <ref> and noting which graphs in cases (3) and (4) have an induced symmetric group on the matching.Since the group acting on the matching in each of (3) and (4) has a minimal normal subgroup that is elementary abelian and acts regularly on an odd number of edges, we conclude that the only option in cases (3) and (4) is when m = 3.The result follows. Acknowledgements. The authors wish to thank Thomas Zaslavsky for his numerous editorial suggestions and comments on earlier versions of this paper and the anonymous referees for their helpful reports.plain | http://arxiv.org/abs/1706.08964v3 | {
"authors": [
"Alex Schaefer",
"Eric Swartz"
],
"categories": [
"math.CO",
"05C25, 20B25, 05C22"
],
"primary_category": "math.CO",
"published": "20170627175229",
"title": "Graphs that contain multiply transitive matchings"
} |
Drowning by numbers: topology and physics in fluid dynamics Amaury Mouchet Laboratoire de Mathématiques et de Physique Théorique, UniversitéFrançois Rabelais de Tours, CNRS (UMR 7350),Fédération Denis Poisson, 37200 Tours, France[2cm] Since its very beginnings, topology has forged strong links with physics and the last Nobel prize in physics, awarded in 2016 to Thouless, Haldane and Kosterlitz “for theoretical discoveries of topological phase transitions and topological phases of matter”, confirmed that these connections have been maintained up to contemporary physics.To give some (very) selected illustrations of what is, and still will be, a cross fertilization between topology and physics[A more general review is proposed by <cit.> and a systematic presentation on the topological concepts used by physicists can be found in <cit.>.], hydrodynamics provides a natural domain through the common theme offered by the notion of vortex, relevant both in classical ( 2) and in quantum fluids ( 3). Before getting into the details, I will sketch in1 a general perspective from which this intertwining between topology and physics can be appreciated: the old dichotomy between discreteness and continuity, first dealing with antithetic thesis, eventually appears to be made of two complementary sides of a single coin. § THE ARENA OF THE DISCRETE/CONTINUOUS DIALECTIC One century after Thales of Miletus had proposed that water was the natural principle of all things, the first atomists Leucippus and Democritus advocated for a discrete conception of matter. The existence of an ultimate lower limit of divisibility, materialised by the atoms, may have beena logical answer to the Zeno's paradoxes (; ). In some westernmost banks of the Mediterranean sea, the Pythagorean school was concerned by a line of thought following quite an opposite direction: the discovery of the irrational numbers counterbalanced theconception of a universe exclusively driven by the integer and rational—in the original acception of the word—numbers.For twenty-five centuries, the dialectic between continuity anddiscreteness has never stopped nurturing natural philosophy. At our daily life scales, the ones for which the brains have been shaped by Darwinian evolution[In modern times physics and chemistry were not, by far, the only scientific disciplines to be shaken by violent debates between discrete and continuous schools; in the xixþcentury Lyell's uniformitarianism in geology, by contrast with catastrophism, had an important influence on the young Darwin. By the way, one can notice that the binary opposition between discreteness and continuity provides by itself a meta self-referring epistemological dichotomy, so to speak.], discreteness appears to be an inevitable way for intelligence to model the world[However, neurology shows that numerical cognition is more analogical than numerical: beyond few units, the numbers are encoded and treated by the brain as fuzzy entities <cit.>. ]. Furthermore, operationally speaking, any measurement is reduced, in the last resort, to a reproduciblecounting <cit.>. Etymologically, “discrete”, “critical”, “criterion”, and “discernment” share the same greek root κρίνω (krī́nō, to judge)[The etymology lines of these words can be easily traced back with .]. However, the boundaries of macroscopic objects, considered both in space and time, remain inevitably blurred. For instance, consider one cherry; through absorption and desorption, a perpetual exchange of matter takes place at small scales on the skin of the cherry, and no one can really identify with a precision of one second the time when this cherry has appeared from a blossom or destroyed by natural deterioration[In a contribution to the previous volume of this series <cit.>Emmer+15a I have tried to show how symmetries play a crucial role in the process of abstraction and conceptualisation of a macroscopic object like a cherry.]. This ambiguity was known from antiquity and supply the sorites paradox (what is the minimum number of grains in a heap of sand?)—and the paradox of the ship of Theseus (Plutarch asks if, after decades of restauration, once her last plank has been replaced, the ship remains the same Theseus's ship <cit.>).In the second part of the xixþcentury, experiments allowed to move the debate beyond speculations into the microscopic world. In the same movement, mathematics saw the emergence of a new discipline, topology, where were identified some discrete classifications—first in geometry, then in analysis and algebra—up to continuous invertible transformations (homeomorphisms).The integer numbers upon which the classes of, say, graphs, knots, surfaces, fixed points of a flow, critical points of a real map, are discriminated provide, by essence, a robust quantization; they are topological invariant.To put it in a nutshell, there cannot be “half a hole”. The dimension of a space[In fractal geometry, the Hausdorff dimension of a set, which can be irrationnal, is not preserved by a homeomorphism.], its connectedness (π_0), its homotopy groups (π_1, π_2 and more generally π_n), the signature of the Hessian of a function at a critical point, are examples of such discrete quantities. In the beginning of the xxþcentury, quantum physics refuted so masterfully the Leibniz continuity principle (Nature does not make jumps) that it bears this claim in its very name. The general rule—known by Pythagoreans for music—according to which a stable wave in a bounded domain has its frequencies quantized (that is, function of integer numbers) now applied at a fundamental level to the Schrödinger waves, which described the states of elementary particles, when bounded.The discrete classification of chemical elements successfully proposed in 1869 by Mendeleev and the discrete spectral lines corresponding to the Balmer series, the Paschen series, the Lyman series etc. observed in radiation, could be explained within a unifying scheme offered by quantum theory.Eventhough it appears that each atomic energy level has actually a continuous bandwidth, due to the coupling to the electromagnetic field whose scattering states belong to a continuum (the photon has no mass), it is nevertheless quantum theory that confered to “being an integer” a genuine physical property. So far, neither the quantification of the spin nor the quantification of the electric charge, say, can be seen as an approximation of a continuous model and the analogous of the Mendeleiev table in the Standard Model contains a finite number of species of elementary particles—about twenty, non counting as distinct a particle from its associated antiparticle—characterised by a handful of quantum numbers[The discrete character of some observable properties is all the more strengthened that there exists some superselection rules that make irrelevant any continuous superposition of states differing by some discrete values of this observable. ]. Many attempts have been made for finding a topological origin of these quantum numbers, one of the motivation being that topological invariance is much harder to break than symmetry invariance. In condensed matter, topology offers a protection against the effects of impurities or out-of-control perturbations and therefore participatesto the reproductibility and the fiability of measurements <cit.>. The seminal attempt in this direction is Dirac's model of magnetic monopole <cit.> whose existence would imply the quantization of the electric charge; however, so far, all the quantizations that have been explained find their root in algebraic properties of the symmetry groups used to build a basis of quantum states[Topological properties of these Lie groups, obviously their dimensions but also their compactness, their connectedness and their simple connectivity, do play a role but the algebraic commutation relations of their generators remain the maincharacteristics, which are local ones, that allow to build the irreducible representations defining the one-particle states. ] (in the absence of evidence of elementary magnetic monopoles, the fact that the electric charges appear to be always an integer multiple of one unit remains mysterious).Despite these (temporary?) failures of finding topological rather than algebraic roots for the discrete characteristics of what appears to be elementary particles, the quantum theory of fields offers the possibility of describing some collective effects of those particles whose stability is guaranteed by topological considerations. There exists some configurations of a macroscopic number of degrees of freedom that cannot be created or destroyed by a smooth transformation without passing through an intermediate state having a macroscopic, and therefore redhibitory, energy. Depending on the dimension of the space and of the field describing the model, several such topological defects can be considered (point, lines or surfaces) and have been observed in various condensed states <cit.> including, of course, the quantum fluids where the defects are characterised by quantized numbers that can be interpreted as topological invariants.Vortices, which will be the object of the next two sections,provide typical examples of such topological defects along a line in a 3-dimensional space or localised at one point in a 2-dimensional space for a complex scalar field (or a real bidimensional vector field).Under certain circumstances, these collective effects share many properties with the so-called ordinary particles. Since, theoretically, the distinction between the quasi-particles and particles appears, after all, to be just a matter of convention on the choice of the vacuum and of the particles that are considered to be elementary, one may have the secret hope that at a more fundamental level, having the Standard Model as an effective theory, topology shall have the next, but presumably not the last, word. § CLASSICAL VORTICES …when I first opened my eyes upon the wonders of the whirlpool…Edgar Allan Poe. A Descent into the Maelström (1841).§.§ How vortices participate to the dynamics of the world according to Leonardo and Descartes By strong contrast with the still, rather mineral, backgrounds of his paintings, Leonardo da Vinci's interest for the dynamics of water is manifest in his drawings and writtings all along his life. Vortices in water, in air, and even in blood <cit.>, were a recurrent source offascination for him[<cit.>OMalley69a saw in the exuberance of the terms used by Leonardo and in the profusion of his drawings an attempt to classify the vortices, a line of investigations he kept in mind throughout his life.]. Not only as esthetical motifs (fig. <ref>), not only because of their crucial role for understanding hydraulics and fly, not only because they inspired him fear as a disordered manifestation of flooding or deluge, but also because they provided a central key forhis global conception of the dynamics of the world: l'acqua, vitale omore della terreste macchina, mediante il suo natural calore si move. (water, vital humour of the terrestrial machine, moves by means of its natural heat)[Folio H95r, whose facsimile and transcription can be found on .] <cit.>.More than a century later, most probably without any influence from Leonardo, Descartes put the vortices in the very core of his cosmological model. Rejectingthe atomist concept of a vacuum separating matter <cit.>, he writes[…] putandum est, non tantum Solis & Fixarum, sed totius etiam coeli materiam fluidam esse.([…] we think that not only the matter of the Sun and of the Fixed Stars is fluid but also is the matter of all the sky, trad. am) <cit.>Being aware of the proper rotation of the Sun (it takes 26 days for the sunspots to complete one turn <cit.>) and of the different orbital period of the planets, he pursues further the hydrodynamical analogy[…] putemus totam materiam coeli in qua Planetae versantur, in modum cuiusdam vorticis, in cuius centro est Sol, assidue gyrare, ac eius partes Soli viciniores celerius moveri quam remotiores […]([…] we think that all the matter of the sky, in which the Planets turn, rotates like a vortex with the Sun at its center; that the parts near the Sun move faster than the remote ones […], trad. am) <cit.> Descartes'model was overuled by Newton's theory planetary motion but, somehow, in contemporary astrophysics, vortices are still present—in a complete different way, of course, from Descartes'— and triggered by gravitational field acting through the interstellar vacuum: one may think of protoplanetary accretion disks (turbulence plays a crucial role, in particular in the initial molecular cloud for explaining the scattered births of stars) and, at much larger scales, of galaxies, cosmic whirlpools spinning around a giant black hole. §.§ Accompanying the birth of topology in the xixþcentury His study of the physical properties of organ pipes led Helmholtz to scrutinize the motion of the air near sharp obstacles and the influence of viscosity.The memoir he published in German in 1858 on the subject had a decisive influence on the physicists of the Scottish school including Maxwell, Rankine, Tait and Thomson (who was ennobled in 1892 as Lord Kelvin), all the more that Tait translated it into English in 1867 under the title On the integrals of the hydrodynamical equations, which express vortex-motion <cit.>.Inspired by the parallel between mechanics of continuous media and electromagnetism <cit.>, Helmholtz showed that, given a field of velocities v⃗, its curl, the vorticity field,ω⃗=curl v⃗ is a vector field proportional to the local rotation vector of the fluid. Helmholtz introduced the notion of vortex line (a curve tangent to ω⃗ at each of its points) and vortex filament/tube (a bunch of vortex lines) and proved that during its evolution each vortex line follows the motion of the fluid. The dynamical equation of ω⃗ allowed him to study precisely the dynamics of straight (fig. <ref>) and circular vortex tubes (fig. <ref>). A thin vortex ring whose radius R is much larger than the radius of the cross section of the tube that defines it moves perpendicularly to its plane with the velocity of its center increasing with R[In particular, when two rings moving along the same direction get close,the flow created around the leading ring tends to shrink the following one which, conversely, generates a flow that tends to expendthe ring ahead. Therefore the leading ring slows down while the second one is sped up until it overtakes the former by passing through it, and the role of the rings are exchanged. This tango, predicted and observed by Helmholtz, is described in the end of his 1858 memoir.].Based on the similar mathematical problem arose in electrostatics and magnetostatics, Helmholtz understood that the topology of the irrotational part of the flow was essential to determine globally the velocity potential α: in the set of the points P where ω⃗(P)=0 one can always locally define a scalar field α such thatv⃗=grad αbutIf we consider [a vortex-filament] as always reentrant either within or without the fluid, the space for which [equation (<ref>)] holds is complexly connected, since it remains single if we conceive surfaces of separation through it, each of which is completely bounded by a vortex-filament. In such complexly connected spaces a function [α] which satisfies the above equation can have more than one value ; and it must be so if it represents currents reentering, since the velocity of the fluid outside the vortex-filaments are proportional to the differential coefficients of [α], and therefore the motion of the fluid must correspond to ever increasing values of [α]. If the current returns to itself, we come again to a point where it formely was, and find there a second greater value of [α]. Since this may occur indefinitely, there must be for every point of such a complexly-connected space an infinite number of distinct values of [α] differing by equal quantities like those of tan^-1x/y, which is such a many-valued function […].<cit.>. The topological properties of vortices can also be understood from what is now known as Kelvin's circulation theorem <cit.> which unified Helmholtz results: in an inviscid (no viscosity), barotropic (its density is a function of pressure only) fluid, the flux of the vorticityΓ=∫_𝒮ω⃗·S⃗=∫_∂𝒮v⃗·l⃗through a surface 𝒮 following the motion of the fluid—or equivalently, according to Stokes' theorem, the circulation of the velocity through the boundary ∂𝒮 of 𝒮—is constant. As a consequence, we recover Helmholtz statement that the non simple connectedness of the space filled by the irrotational part of the flow, i.e. the complementary of the vortex tubes, prevents the existence of a continuous globally-defined α and the circulation Γ depends on the homotopy class of the loop 𝒞=∂𝒮. In such an ideal fluid, the vortex lines were therefore topologically stable and Thomson's saw in this stability a key for the description of atomic properties without referring to the corpuscular image inheritated from the atomists of antiquity, which was a too suspicious philosophy for Victorian times <cit.>[Some smoothness into the atom had already been introduced by Rankine in 1851 with his hypothesis of molecular vortices according to which “each atom of matter consists of a nucleus or central point enveloped by an elastic atmosphere, which is retained in its position by attractive forces, and that the elasticity due to heat arises from the centrifugal force of those atmospheres, revolving or oscillating about their nuclei or central points” <cit.>. It is worth noting that Rankine acknowledges the pertinence of William Thomson's comments on the first version of this 1851's proposal.]. Since vortex tubes cannot cross transversaly[But, it seems that neither Helmholtz nor Thomson have considered the possibility of a longitudinal merging of vortex tubes, forming a trousers-like shape <cit.>.] otherwise it is easy to find a 𝒞 that does not satisfy Kelvin's theorem, the knot formed by a closed vortex tube and the intertwinning between several such closed loop remain topologically invariant. The absolute permanence of the rotation, and the unchangeable relation you have proved between it and the portion of the fluid once acquiring such motion in a perfect fluid, shows that if there is a perfect fluid all through space, constituting the substance of all matter, a vortex-ring would be as permanent as the solid hard atoms assumed by Lucretius and his followers (and predecessors) to account for the permanent properties of bodies (as gold, lead, etc.) and the differences of their characters. Thus, if two vortex-rings were once created in a perfect fluid, passing through one another like links of a chain, they never could come into collision, or break one another, they would form an indestructible atom; every variety of combinations might exist. Thomson to Helmholtz, January 22, 1867, quoted by <cit.>.The theory of the vortex atoms offered to Thomson the possibility of making concrete hislong-standing intuition of a continuous conception of the world, as he had confessed it to StokesNow I think hydrodynamics is to be the root of all physical science, and is at present second to none in the beauty of mathematics. Thomson to Stokes, December, 20, 1857,quoted in <cit.>Despite the physical failure of Thomson's ambitious aim <cit.>[As far as classical hydrodynamics is concerned, some progress have been made in the xxþcentury with, for instance, the identification of new integrals of motion constructed from topological invariants like the Calugareanu helicity <cit.>Borisov+08a ; experimentally some not trivial knotted vortices could be produced only recently <cit.>.], the identification of topological invariants on knots, upon which the classification of atoms and molecules would have been based, and the classification of the knots by Tait (see Fig. <ref> for instance) remains a groundbreaking mathematical work, with direct repercussions in contemporary topology.One of the Thomson's greatest hopes, while spectroscopy was gathering more and more precise data, was to explain the origin of the discrete spectral lines with “ […] one or more fundamental periods of vibration, as has a stringed instrument of one or more strings […]” <cit.>. One cannot prevent to find an echo of this motivation in modern string theory where “each particle is identified as a particular vibrational mode of an elementary microscopic string” <cit.>—see also <cit.>.Not without malice, <cit.> was perfectly right to qualify Thomson's dream as a “Victorian theory of everything”. § QUANTUM VORTICES§.§ Topological origin of quantized flux in quantum fluids Unlike what occurs in classical fluids where viscosity eventually make the vortices smoothly vanish, quantum fluids provide a state of matter, much more similar to ideal fluids, where vortices are strongly protected from dissipative processes.Indeed, at low temperature, particles can condensate into a collective quantum state where transport can be dissipationless: this is one of the main characteristics of superconductivity (discovered in solid mercury below 4K by Onnes in 1911), superfluidity (discovered in liquid Helium-4 below 2K by Kapitsa and Allen & Misener in 1938), and Bose-Einstein condensate of atoms (discovered for rubidium below 170 nK by Cornell & Wieman and Ketterle in 1995)[One can find many textbooks at different levels and more or less specialised to one type of quantum fluids. To get an introductory bird's-eye view on quantum fluids and other matters in relation to statistical physics, my personal taste go to <cit.>, <cit.> and the particularly sound, concise, and pedagogical <cit.> (in French). ].There is a second reason, of topological origin,that reinforces the stability of the vortices in quantum fluids: the scalar field α whose gradient is proportional to the current is not a simple mathematical intermediate as in the classical case (see (<ref>)) but acquires the more physical status of being a phase (an angle) that may be measured in interference experiments like in the Aharonov-Bohm effect. As a consequence, on any closed loop 𝒞, the circulation Γ given by (<ref>)has to be an integer multiple of 2π:w[𝒞]1/2π∫_𝒞grad α·l⃗∈ℤ .Since smooth transformations cannot provoque discrete jumps, w is therefore topologically protected.In other words, the flux of curl v⃗—which keeps its physical interpretation of being a vorticity in superfluids as well as in Bose-Einstein condensates of atoms, whereas it represents a magnetic field in superconductors[Compare (<ref>) with the relation B⃗=curl A⃗ between the (gauge) vector potential A⃗ and the magnetic field B⃗.]—is quantized and naturally leads to elementary vortices carrying a unit flux quantum.As a matter of fact, the quantum fluid state is described by a complex field ψ=|ψ|e^iα (the order parameter) and w[𝒞]≠0 denotes a singularity of the order parameter on any surface 𝒮 whose boundary is 𝒞. Vortices constitute a particular case of what is generally called a topological defect whose dimension depends on the dimension of the order parameter and on the dimension of the space.At microscopic scales, very much like in the Rankine model, the vortex is made of a core outside which curl v⃗=0; the vorticity/magnetic lines are trapped inside the core where the density of the superfluid |ψ|^2 tends to zero at its center. Not only, these vortices have been observed in all the three types of superfluids mentioned above but also the triangular lattice they form to minimize the (free) energy due to an effective repulsion between them first predicted by <cit.>, see fig <ref>). When the fluctuations of |ψ| in space and time are negligible, notably at sufficiently low temperatures, the quantum fluid is essentially described by the phase e^iα or equivalently by a bidimensional vector of unit norm oriented at angle α with respect to a given direction (fig. <ref>). §.§ The xy-modelThe latter picture is known as the xy-model, which is also relevant for some classical liquid crystals or for systems of classical spins <cit.>.At macroscopic scales, some collective effects of such model are not very sensitive to the details of the interaction nor to the geometry of the elementary cell in the case of a lattice but depend crucially on the dimension d of the space of positions (the number of components of r⃗).Typically, the energy of the system increases when some differences in the orientation α appears; more precisely the energy density contain a term proportional to (grad α)^2. It is not affected by a homogenous rotation of all the spins,α(r⃗) ↦α(r⃗)+α_0 ,where the angle α_0 does not depend on r⃗.The absolute minimum of the total energy is obtained when all the vectors are aligned, which is the configuration at temperature T=0 K. When T>0, the equilibrium corresponds to more disordered configurations but, for d=3[Surprisingly, as far as the computations are concerned, the integer nature of d becomes secondary and one can formally consider d as continuous. The condition for an order/disorder phase transition at T_critical>0 to exist is d>2. ], some non-zero average value of α can be maintained up to a critical temperature T_critical beyond which the average value of α is zero (fig. <ref>).At d=2, on the contrary, the correlations between fluctuations never decrease sufficiently rapidly at large distancesand the average value of α is zero as soon as T>0. However one can still identify,at somefinite temperature T_critical>0,a qualitative change of behaviour in the correlation lengths, from a power-law decay at large distances to an exponential decay and this phase transition has observable repercussions, notably in superfluids helium films <cit.>. The theoretical description of what appeared to be a new kind of phase transition, now known as topological phase transitions, was proposed by <cit.> who showed that vortices were a cornerstone of the scheme. As soon as their first papers, Kosterlitz and Thouless, talked about “topological order” because they were perfectly aware that this type of phase transition, unlike all the phase transitions known at the time of their publication, relies on topology rather than on symmetry (breaking). As we have seen above on eq. (<ref>), each vortex (now a topological defect of one dimension) is characterised by an integer, called the topological index of the vortex which can be reinterpreted using the concepts introduced by Poincaré in a series of papers that can be considered as the foundations of topology as a fully autonomous research discipline <cit.>.Any direction far away a topological defect of dimension f in a space of dimension d is represented by an element of the rotation group in n=d-f-1 dimensions, in other words such a defect can completely enclosed by a n-dimensional sphere S_n. In d=3 dimensions a wall (a surface of dimension f=2) cannot be enclosed (n=0), a vortex-line (f=1) can be enclosed by a circle (n=1), a point (f=0) can be enclosed by a n=2-sphere.In d=2 dimensions a wall (a line of dimension f=1) cannot be enclosed (n=0) and a point can be enclosed by a circle (n=1). To each direction one can associate the value of the order parameter and therefore to each defect one gets a map from S_n to 𝒫 where 𝒫 denotes the space to which the order parameter belongs. In the examples above 𝒫 is just the set S_1 of the angles α but much more different situations may be encountered.For n=1, any loop 𝒞 around a given point maps on a closed path 𝒞' in 𝒫=S_1 and the topological index w of the point is just the winding number of 𝒞' (figs. <ref> and <ref>).More generally, the topological invariants are given by the group π_n of 𝒫 (for n=0 it provides the connectedness, for n=1 it provides the first homotopy group that is the simple connectedness, etc.).A continuous transformation of the configuration cannot modify w at any point and physically it would require a macroscopic amount of energy to change w. On the other hand, one configuration having one defect can be deformed continuously at low cost of energy into any other configuration having a defect with the same w. In particular, the transformation (<ref>) does not cost any energy at all. One cannot therefore expect to isolated elementary vortex (w=1) or isolated elementary antivortex (w=-1) to be spontaneously created from a perfect ordered state. Nevertheless, a pair of vortex-antivortex is affordable when T>0 (fig. <ref>). The continuous creation (or annihilation) of such a pair can be understood by considering the appearance of a fold on a drapery (back to Leonardo again?).One may intuitively see that this is a generic process, stable with respect to smooth transformations, that describes the creation or the annihilation of a pair of maximal-minimal points on a smooth function (fig. <ref>) or, equivalently, the creation or annihilation of intersection points when two curves that cross transversaly are smoothly locally deformed[Topology is fully at work here and the study of the stability of the critical points of smooth mappings is the object of catastrophe theory whose greatest achievement is to have classified the generic possible scenarios; the simplest one being precisely the fold catastrophe, depicted in figure <ref> <cit.>.].The topological phase transition describes precisely how the creation of an increasing number of vortex-antivortex pairs as the temperature increases eventually lead from a topological order to a state where complete disorder reigns. § CONCLUDING REMARK To come back to issues mentioned in the last paragraph of 1, in quantum theory, the fundamental elementary particles stem from algebraic symmetry considerations. However, we have some clues (topological defects, solitons, instantons, monopoles, etc.)that topology may offer a complementary ground. The parallel between creation/annihilation of particle-antiparticle pairs and creation/annihilation of vortex-antivortex pairs may be more than a simple analogy. Thomson/Kelvin's intuition may take an unexpected but relevant form, after all.Acknowledgement: I am particularly grateful to Pascal Brioist(Centre d'Étude Supérieures de la Renaissance de l'Université de Tours) for his expert advices on Leonardo studies, to Boris Behncke (invg-Osservatorio Etneo)) for letting me use his photo of the vapour ring created by the Etna (see fig <ref>) and to Michele Emmer who triggered the subject of this essay for the conference Matematica e Cultura 2017, Imagine Math 6, at Venice.James99a 49 urlstyle[Abrikosov(1957)]Abrikosov57a A. A. Abrikosov. The magnetic properties of superconducting alloys. J. Phys. Chem. 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"authors": [
"Amaury Mouchet"
],
"categories": [
"physics.hist-ph",
"cond-mat.quant-gas",
"math.HO"
],
"primary_category": "physics.hist-ph",
"published": "20170626132410",
"title": "Drowning by numbers: topology and physics in fluid dynamics"
} |
mnras 000–000 0000 On the SFR-M_* main sequence archetypal star-formation history and analytical models. L. Ciesla1,D. Elbaz1, and J.Fensch1.Received; accepted ===================================================================================== Diffuse, extended radio emission in galaxy clusters, commonly referred to as radio halos, indicate the presence of high energy cosmic ray (CR) electrons and cluster-wide magnetic fields. We can predict from theory the expected surface brightness of a radio halo, given magnetic field and CR density profiles. Previous studies have shown that the nature of CR transport can radically effect the expected radio halo emission from clusters <cit.>. Reasonable levels of magnetohydrodynamic (MHD) wave damping can lead to significant CR streaming speeds. But a careful treatment of MHD waves in a high β plasma, as expected in cluster environments, reveals damping rates may be enhanced by a factor of β^1/2. This leads to faster CR streaming and lower surface brightnesses than without this effect. In this work we re-examine the simplified, 1D Coma cluster simulations (with radial magnetic fields) of <cit.> and discuss observable consequences of this high β damping. Future work is required to study this effect in more realistic simulations. § INTRODUCTION Galaxy clusters are the largest gravitationally bound objects in the universe, and are host to processes described by every area of physics. For some clusters, this includes particle acceleration to very high energies, generating cosmic rays (CRs). Cosmic ray electrons (CRe) at these energies will emit synchrotron emission in the presence of magnetic fields which is observable as radio emission.Galaxy clusters can exhibit radio emission of a few morphological types. We focus here on radio haloes, which refers to extended, diffuse radio emission. Studies have shown that where radio haloes are detected, the total radio luminosity of the halo correlates well with total X-ray luminosity of the host cluster. But many clusters do not have radio haloes at all. Thus there is a peculiar bimodality in the presence of radio haloes in galaxy clusters that must be explained (<cit.>, <cit.>, <cit.>).Related to this problem is the source of the CRe which produce the emission. CRe at the required energies (∼1-10 GeV) have short cooling times (100 Myr) for typical cluster magnetic field strengths (∼3 μG) and densities (∼ 10^-3 cm^-3). The presence of a radio halo in a given cluster therefore indicates relatively recent injection of CRe. The `hadronic model' claims that this is provided by hadronic collisions of CR protons (CRp) with ambient thermal nuclei that result in pions, which then decay into secondary CRe. A rival model, `turbulent reacceleration', hypothesizes that low-energy CRe undergo Fermi-II reacceleration by gas motions during mergers <cit.>. However, the abundance and distribution of CRp is still an important ingredient in this model, as secondary CRe produced during hadronic collisions provide seeds for reacceleration <cit.>.[Acceleration from the thermal pool is precluded by strong Coulomb losses <cit.>.] An excellent summary of the state of the field can be found in <cit.>.An issue with the hadronic model is that high energy CRp have much longer (∼10-100 Gyr) cooling times. If a cluster undergoes some merger or structure formation process that may be responsible for accelerating CRs, then we may naively expect the resulting radio halo to last for as long as or longer than the age of the universe. If this were the case, we would struggle to explain clusters that don't exhibit radio halo emission, since every cluster is expected to accelerate CRp at some point in its history through such a mechanism. A lack of hadronic emission implies either that the assumed acceleration efficiencies are too high or that the CRp are quickly diluted by transport effects. We investigate this second possibility. Note that CRp are frequently invoked in turbulent reacceleration models as well, to generate secondary seed electrons for reacceleration. Observations are best fit by a flat CRp profile, which can be explained by CR streaming <cit.>.Building on the work of <cit.>, <cit.> attempted to resolve this issue by suggesting that, for favorable magnetic field orientations, CRp can travel at high speeds away from cluster centers, quickly reducing the CRp density and thus the expected radio luminosity on appropriate time scales. Bulk CR transport is limited by the streaming instability, which amplifies hydromagnetic waves traveling in the same direction as the mean cosmic ray velocity if the cosmic rays are anisotropic in the frame of the wave <cit.>. However, if other plasma processes damp the wave then a larger streaming speed is required to excite the instability, resulting in faster cosmic ray transport. In <cit.> it was shown that under galaxy cluster plasma conditions the strongest damping mechanism is the turbulent damping process proposed by <cit.>. Galaxy cluster plasmas are characterized by large ratios of thermal to magnetic pressure, a parameter usually denoted as β. In this work we show that, in high β environments, wave damping is actually stronger than the rate quoted in <cit.>. Therefore, CRs can travel even faster than previously thought and radio halos turn off faster than we previously estimated for the same density and magnetic field profile[As we discuss in <ref>, we use a very simple 1D halo model with purely radial magnetic fields for comparison with <cit.>. More sophisticated simulations are needed in future work to examine the effects of this damping in real clusters.]. While this may explain the bimodality of radio haloes in the hadronic model, it cannot reconcile other issues with the hadronic model. Among others, gamma-ray non-detections in the Coma cluster put strict limits on the amount of CRp present (<cit.>). As such, this work should not be seen as an attempt to strengthen the hadronic model of radio haloes so much as a more accurate theory of CR transport in clusters in general, which is still an important ingredient in the reacceleration model (<cit.>).Also, although the time evolution of radio halos is the main focus of the paper, we note that faster streaming may also help to explain the puzzling lack of diffuse γ-rays from galaxy cluster cores (<cit.>). It may also be germane to understanding radio mini-halo luminosities <cit.>. While the hadronic model is problematic for understanding giant radio halos, it remains the chief contender for understanding kinematically quiescent radio mini-halos <cit.>, where the CRp could be sourced by a central AGN. Note that modulo loss processes, the contribution of the AGN jet to CRe could be non-negligible as well.In <ref> we review the theory of MHD wave damping in collisionless, high β plasmas and show how it modifies previously estimated turbulent damping rates. An important part of this calculation is an estimate of the effective collisionality of the plasma under the assumption that microturbulence is present due to pressure anisotropies <cit.>; we argue that the level of microturbulence is low enough that Alfvén waves at the characteristic wavelengths excited by cosmic ray steaming instabilities are nevertheless collisionlessly damped. We also discuss the possible effects of the magnetic field topology, which is not taken into account in our simulations. In <ref> we describe the simulation setup, which is similar to the setup we used previously for the Coma cluster <cit.>. Sections <ref> and <ref> gives the results and conclusion, respectively.§ MHD WAVES IN TURBULENT, HIGH BETA PLASMA§.§ Wave dampingAs mentioned in <ref>, the streaming speed as a function of energy is determined by balancing the growth rate of the streaming instability <cit.> Γ_cr() = ω_cpn_CR(>γ_R)/n_i(v_D(γ_R)/v_A-1) against the mechanism(s) that damp the wave. In (<ref>), ω_cp is the proton gyrofrequency (for proton cosmic rays), γ_R≡ω_cp/c is the minimum Lorentz factor of a resonant cosmic ray, and v_D is the energy dependent streaming speed.Equation (<ref>) is computed assuming a perfectly straight and uniform background magnetic field _0. Under this assumption, the fastest growing waves propagate parallel to _0, with waves of any given wavenumberbeing driven primarily by cosmic rays with gyroradii r_L∼^-1.If _0 has curvature or perpendicular structure, strict parallel propagation is impossible: there is a minimum angle θ_min which depends on the background field structure. <cit.> evaluated θ_min for Alfvén waves excited by cosmic ray streaming in a background field upon which an anisotropic MHD turbulent cascade is imposed, and found it to be tanθ_min = (λ_∥/λ_⊥) ∼(r_L/L_MHD)^1/4∼(r_Lϵ/v_A^3)^1/4, where L_MHD is the length scale at which the characteristic bulk speed is the Alfvén speed v_A, and the turbulent energy cascade rate is ϵ≡ v_A^3/L_MHD. The perpendicular lengthscale λ_⊥ and turbulent velocity v_λ_⊥ corresponding to θ_min are λ_⊥∼ r_L^3/4 L_MHD^1/4∼ r_L^3/4(v_A^3/ϵ)^1/4, andv_λ_⊥∼ v_A(λ_⊥/L_MHD)^1/3∼ v_A(r_L/L_MHD)^1/4. According to the <cit.> scenario, MHD waves are progressively sheared by the turbulent cascade, eventually transferring their power to the dissipation scale. The corresponding “turbulent damping rate" is therefore on the order of the eddy turnover rate at the perpendicular scale Γ_damp∼v_λ_⊥/λ_⊥∼ϵ^1/3/λ_⊥^2/3∼(ϵ/r_Lv_A)^1/2, where in the last equality we have used eqns. (<ref>) and (<ref>). This was the damping rate used in the ZEUS simulations in <cit.>. The <cit.> calculations have recently been generalized by <cit.> to a variety of scenarios.A key assumption in deriving eqn. (<ref>) is that the dissipation scale is much shorter than the wavelength of the wave. If this is not the case, the dissipation rate can exceed the shearing rate. In collisionless, high β plasmas, MHD waves of even small obliquity are subject to strong ion Landau damping <cit.>, a process whereby ions with parallel velocity v_∥ which satisfies the resonance condition v_∥=ω/ absorb energy from the wave due to acceleration by its parallel electric field [This process is not to be confused with nonlinear Landau damping, in which the thermal ions resonate with the low frequency beat wave generated by the interaction of two higher frequency waves <cit.>. NLLD was first invoked for galaxy cluster plasmas by <cit.> and was argued to be subdominant in turbulent galaxy cluster plasmas in <cit.>.]. We briefly review the Foote & Kulsrud calculation here. They defined β = P_g/P_B = ρ k_BT/μ m_p/B^2/8π = v_i^2/v_A^2, and framed their analysis in terms of the rescaled wave frequency ω and wave number k:ν≡ω/k_0v_A,l ≡kcosθ/k_0,k_0 ≡ω_civ_A/v_i^2 ,where ω_ci is the thermal ion cyclotron frequency. They also introduce a parameter α=π^1/2β^1/2l^2tan^2θ.<cit.> show that in the small l limit, which applies to most of the waves amplified by streaming cosmic rays, the dispersion relation can be approximated asymptotically byν = l - iα/4l±1/4l(l^6-α^2+4iα^3/l^2)^1/2Expanding eqn. (<ref>) for small α, the damping rate is given byΓ_damp = -k_0v_AIm(ν) ≈ k_0v_Aα/4lUsing (<ref>) for θ and restoring dimensional units, we arrive at the resultΓ_damp≈√(π)/4k_0v_Aβ^1/2ltan^2θ≈√(π)/4β^1/2r_L^-1v_A(r_Lϵ/v_A^3)^1/2 ⇒Γ_damp≈√(π)/4β^1/2(ϵ/r_Lv_A)^1/2,where we have used lk_0 = kcosθ = r_L^-1.Comparison with (<ref>) shows that with this treatment the wave damping rate is enhanced by a factor of β^1/2. In the cluster environments simulated in <cit.>, where β can be of the order of 100 in the cluster centers, this factor can lead to significantly higher CR streaming speeds than those predicted by (<ref>).The factor of β^1/2 can be heuristically understood as follows. The Landau damping rate is given by the rate at which resonant ions absorb energy from oblique waves; Γ_ damp^ Landau∼ k v_i tan^2θ. The turbulent damping rate is given by the rate at which a pair of interacting Alfvén waves cascade, Γ_ damp^ turb∼ k v_ A tan^2θ. The geometrical factor of tan^2θ is the same for these pairwise interactions. Thus, the Landau damping rate is larger by a factor ∼ v_i/v_ A∼β^1/2. §.§ Collisionality The analysis by <cit.> assumes a collisionless plasma. We must verify that the plasmas in galaxy cluster environments we will be simulating are sufficiently collisionless. In particular, we must determine whether microinstabilities driven by pressure anisotropy can render the thermal ions essentially collisional. The relevant comparison to make here is the ion mean free path to the typical wavelength of the Alfvén waves in question.The long mean free paths,relatively weak magnetic fields, and pervasive large scale turbulence in galaxy cluster plasmas make them attractive candidates for shear driven pressure anisotropy <cit.> through distortion of the magnetic field and preservation of the particles' adiabatic invariants. We follow the notation of <cit.> in the following discussion.Shear in a fluid with velocity field 𝐮 and magnetic field direction 𝐛 will drive pressure anisotropy while collisions will oppose it. Balancing the two yields an equilibrium anisotropy given by <cit.> Δ_i≡P_⊥,i-P_∥,i/P_i= 2.9/ν_ii(𝐛𝐛:∇𝐮-1/3∇·𝐮).In the above, the i subscript indicates we are referring to ions and ν_ii indicates the frequency of Coulomb collisions. These collisions serve to isotropize the pressure. A complication arises if collisions are so infrequent that the pressure anisotropy exceeds the threshold for microinstabilities:Δ_i>1/β_i,mirror instability Δ_i<-2/β_i,firehose instabilityIf Coulomb collisions alone are not sufficient to prevent these instabilities, we may expect two possible responses of the fluid. In one scenario, the shear S≡𝐛𝐛:∇𝐮-1/3∇·𝐮 will adjust itself to reduce the pressure anisotropy until marginal stability is achieved. In this case the collision frequency does not change - it is simply the Coulomb collision frequency and the mean free path is the Coulomb mean free path. This is the working hypothesis in <cit.>.In the other scenario, the shear S remains unchanged and instead the resulting magnetic fluctuations driven by the instabilities will themselves scatter the ions. This increases the effective total scattering frequency ν_i, which now includes these magnetic scatterings in addition to Coulomb scattering, until marginal stability is achieved:Δ_i=2.9/ν_iS=2ξ/β_i→ν_i=1.45β_i/ξS,where ξ is -1 for the firehose instability, and 1/2 for the mirror instability. In this case, the total collision frequency is enhanced, and so the mean free path is shorter than the Coulomb mean free path.The above marginal stability criterion therefore provides a lower limit on the ion scattering frequency ν_i, and thus an upper limit on the ion mean free path λ_i=v_i/ν_i. This limit depends on the shear forcing S. Let us approximate this forcing as S∼ U/L for some characteristic speed U and length scale L. If we assume a turbulent cascade with Kolmogorov scaling, U^3/L=const., we can relate any U and L to the outer scales U_0 and L_0. The marginal stability criterion then becomesν_i∼1.45β_i/ξU/L=1.45β_i/ξU_0/L_0(L_0/L)^2/3The highest collision frequency will therefore be dictated by the smallest scale of turbulence, the dissipation scale L_d. This scale is determined by balancing the cascade rate U_d/L_d with the viscous dissipation rate, which itself depends on the collision frequency, and is of order v_i^2/(ν_iL_d^2). We obtainU_d/L_d=U_0/L_0(L_0/L_d)^2/3∼v_i^2/ν_iL_d^2 ⇒(L_d/L_0)^4/3∼v_i^2/ν_iU_0L_0 L_d/L_0=(v_i^2/ν_iU_0L_0)^3/4Combining (<ref>) and (<ref>) we arrive at an expression for the ion collision rate at marginal stabilityν_i∼1.45β_i/ξU_0/L_0(ν_iU_0L_0/v_i^2)^1/2 ⇒ν_i∼(1.45β_i/ξ)^2U_0^3/L_0v_i^2indicating an ion mean free path ofλ_i∼ 0.48ξ^2L_0/β_i^2(v_i/U_0)^3∼ 0.48L_0/β_i^2ℳ_0^3where we have defined the Mach number at the driving scale ℳ_0≡ U_0/v_i.We can plug this solution for ν_i back in to (<ref>) to find the dissipation length:L_d∼ L_0(v_i^2/β_i^2U_0^4/v_i^2)^3/4∼ L_0ℳ_0^-3β_i^-3/2Note that the turbulent velocity at this scale isU_d=U_0(L_d/L_0)^1/3∼ U_0ℳ_0^-1β_i^-1/2=v_A.That is, under the assumption of marginal stability, the characteristic turbulent velocity at the dissipation scale is about the same as the Alfvén speed. This means the dissipation scale L_d is about the same as the length scale L_MHD defined in <cit.>'s streaming simulations as a measure of the wave damping rate, where it is assumed to be of order 100 kpc.Now we can make an estimate of the mean free path as a function of L_MHD:λ_i∼ L_0β_i^-2ℳ_0^-3∼ L_dβ_i^-1/2∼ 14 kpc(L_MHD/100 kpc)(β_i/50)^-1/2This is much longer than the several AU gyroradii of the ∼100 GeV CR protons that generate the secondary e^± which produce the observed radio emission. The assumption of collisionless wave damping is therefore sound if we assume marginal stability to MHD microinstabilities.We can also compare the above result to the Coulomb mean free path λ_ii. If λ_ii is significantly shorter, then Coulomb collisions alone are enough to keep the pressure anisotropy (<ref>) low enough to avoid MHD microinstabilities. We haveλ_ii=v_it_i=√(kT/m_p)√(m_p)(kT)^3/2/4√(π)lnΛ e^4 n_i≈ 520 cmT_7^2/n_i,-3where we approximate lnΛ≈ 30 and we adopt the subscript notation Q_x=Q/10^x in cgs units.In the central regions of our simulated Coma cluster (see <ref>), T_7=9.5 and n_i,-3=3.4, so λ_ii≈ 4.4 kpc. This is less than the mean free path derived from marginal stability above, implying Coulomb collisions sufficiently isotropize the pressure to avoid instabilities, at least in the cluster center. But this is still well above the AU scale gyroradii, implying we are safely in the collisionless damping regime.The fact that L_ d∼ L_ MHD when the plasma is marginally stable to microinstabilities raises potentially serious issues. It could imply that there is no MHD inertial range, since the Reynolds number at the Alfvén scale is of order unity, and parallel trans-Alfvénic motions simply dissipate. Indeed, recent analytic and numerical work finds an upper limit on shear Alfvén fluctuations of δ B_⊥ /B_0∼β^-1/2 <cit.>, above which the perturbation is rapidly quenched by the firehose instability. While CR-streaming driven turbulence lies below this limit, the background turbulence does not, and it is unclear whether equation (<ref>) (which relies on canonical Goldreich-Sridhar theory) still applies. Such issues lie well beyond the scope of this paper, but they raise important caveats to keep in mind.§.§ Magnetic Topology and Other Unmodeled FactorsAs will be explained in <ref>, our numerical simulations are 1D spherically symmetric. This is primarily for two reasons - first, this simplifies the physics for ease of computation. Second, we want an apples-to-apples comparison with the simulations from <cit.>, which also employed these simplifications for the first reason. We discuss here the possible implications of these simplifications.Real galaxy clusters are, of course, not spherically symmetric. The Coma cluster, which we use as our characteristic cluster, has noticeable azimuthal dependence in the radio surface brightness, as reported by <cit.>. We compare their 1.4 GHz observations of different quadrants of the Coma cluster using the Green Bank Telescope (provided by Larry Rudnick, personal communication) with the azimuthally averaged observations of <cit.> as well as the initial surface brightness in our model cluster in figure <ref>. There is clearly non-symmetric structure in the radio signal. The comparison between the two sets of data is further complicated by subtraction of the foreground signal from our own Galaxy, which is handled differently in each paper. As far as they relate to the construction of our initial CR profile, these differences in the data are of little consequence - streaming times in our simulations are not sensitive to small changes in the initial CR profile.However, the departure from spherical symmetry has other, more significant implications for the evolution of out model CRs. The most important of these is the structure of the cluster's magnetic field. Our 1D simulations necessarily have perfectly radial magnetic fields, which is a best-case scenario for streaming. In the limit of small cross-field diffusion, CRs can only stream along magnetic field lines. They can thus leave the cluster most quickly if the field lines are radial.But in a real cluster the magnetic field may be significantly tangled. Indeed, the very basis of our streaming model is that the presence of MHD turbulence provides a wave damping mechanism. This same turbulence will also tangle the magnetic field on some length scale, increasing the escape time of CRs in our streaming model in some potentially complicated way.Fortunately we can make reasonable estimates of how the escape time is affected by such tangling. Suppose we want to know how long it takes for a CR to travel a distance D away from the center of our cluster. In the 1D symmetric case with perfectly radial magnetic field lines, this time is justt_st∼ D/v_st,where we simplify to a case where the streaming speed v_st is roughly constant in space and time. If instead, the field is tangled on some length scale L, we can treat the transport of CRs as a random walk with steps of length L, travel at speed v_st. CR transport is then effectively diffusive with diffusion coefficientκ∼ Lv_st.The time for CRs to diffuse out to a distance D is thent_diff∼D^2/κ∼D/LD/v_st∼D/Lt_st.In other words, tangling of the field on scale L increases the escape time of CRs by roughly D/L.Is this a large factor? We can approximate the tangling length scale consistently by finding the scale where the kinetic energy density in turbulence equals the magnetic energy density. Below this scale, turbulent motions are too weak to bend the field lines. This occurs when1/2ρ v_t^2 = B^2/8π = 1/2ρ v_A^2Using our definition of L_MHD as the length scale where the turbulent speed v_t is equal to the Alfvén speed v_A (see <ref>), we see that the tangling length scale described above is just of order L_MHD. Our fiducial value is 100 kpc, and our simulated cluster has radius 1 Mpc. We may then expect the effective escape speed to exceed the streaming times in our simulation by about a factor of 10.There are also some observational constraints on the magnetic field structure in galaxy clusters which rely on Faraday rotation measure (RM) observations of radio sources in or behind the clusters. The theoretical background for this technique is described in detail by many authors (see <cit.> and <cit.> for just two examples). In this framework, the structure of the magnetic field is described by a simple power law in Fourier space, |B_k|^2∝ k^-n, between two length scales Λ_min and Λ_max.<cit.> use RM observations in this way to constrain the magnetic field of the Coma cluster in particular. They claim that the field which best fits the RM data is tangled on scales ranging from Λ_min∼ 2 kpc to Λ_max∼ 34 kpc, with a steep Kolmogorov-like spectrum of n=11/3. This suggests that most of the power is on large scales, i.e. most of the “steps” in the field line random walk are of length Λ_max∼ 34 kpc. This is reasonably close to our above estimate of L_MHD. We note, however, that in general the magnetic field structure of galaxy clusters is not well constrained by observations. So while it is hard to say with any certainty, we may reasonably expect our simplified model to underestimate streaming times by a factor of 10 - 30.In the context of our question about radio halo turnoff times, this is not a cripplingly large factor. It extends the turnoff time from hundreds of Myr to a few Gyr, still in the range of reasonable turnoff times for the hadronic model. More importantly for the context of this work, this factor is independent of the high beta effects considered here. If our older simulations from <cit.> are off by a factor of 10, then the new simulations presented here are also off by this same factor, and our main point is unchanged. Of course there are non-linear effects which complicate this - as the CR density drops the streaming speed increases. If escape speeds are initially slower by a factor of 10, the overall shutoff time may be changed by an entirely different factor. Determining the effects of this non-linearity would require non-1D simulations which are beyond the scope of this work. However, we are encouraged that in the apples-to-apples comparison we make here, the high-beta effect causes an increase in the initial streaming speeds by a factor of β^1/2∼ 8 in the central regions of the cluster. So differences brought about by these high beta effects should at least be comparable to the effects of field tangling.Still, it is difficult to assess the effects of field tangling and non-linear evolution other than the above speculation. If we have overestimated the field tangling scale by a factor of about 10, which is possible considering the spectrum in <cit.> extends down to 2 kpc, then our escape speed becomes of the order of the age of the universe even without accounting for the non-linearity of the evolution. Future observational constraints on magnetic field structure in galaxy clusters may therefore prove critical to the question of long range CR streaming.There is another point to be made about the magnetic field topology which may be important. We have talked about the effects of a tangled field, but what if the field lines never leave the cluster? Consider a single field line which extends out to some radius R in the cluster before folding back on itself and returning towards the center. Then along this field line, streaming will even out the CR density out to R, but CRs will be unable to leak out past R. If all the field lines worked this way, no CRs would be able to escape, although they could spread out evenly within radius R. However, if some percentage of field lines leave the cluster, then CRs will always be able to leak out along these lines. The halo dropoff time would then be primarily determined by two factors - the percentage of field lines which exit the cluster, and the rate of cross field diffusion which allows CRs on trapped lines to migrate to neighboring lines which escape.§ SIMULATION SETUP We reproduce here the spherically symmetric ZEUS hydrodynamic simulations of the Coma cluster from <cit.>, but with the damping rate enhanced by the factor of β^1/2 from the analysis in <ref>. As mentioned in <ref>, spherical symmetry is clearly a drastic simplification (L. Rudnick; personal communication) but we assume it here to keep the problem tractable and to focus on the difference between the present treatment and <cit.>. As in <cit.>, we model the Coma cluster with density and temperature profiles given byn_e/10^-3cm^-3=3.4[1+(r/294 kpc)^2]^-1.125 T=8.25 keV[1+(r/460 kpc)^2]^-0.32These profiles are taken from <cit.>, with the density inferred from X-ray observations (<cit.>).The magnetic field is assumed to scale with gas density asB=B_0(n_e(r)/n_e(0))^α_B,with B_0=5 μG and α_B=0.3, as suggested by constraints from <cit.> (<cit.> also considered α = 0.5 which is more in line with <cit.>; the choice of α = 0.3 here is to properly compare with the results from <cit.>. It is conservative in the sense that it implies a lower β and therefore a weaker Landau damping effect). The above density, temperature, and magnetic field profiles are fixed for these simulations - only the CR distribution is evolved in time.The initial CR distribution is of the form used in <cit.>, motivated by cosmological hydrodynamic simulations of galaxy clusters where cosmic rays are accelerated via diffusive shock acceleration:f_p(r,p_p)=C(r)∑_iΔ_i p_p^-α_i Δ=(0.767,0.143,0.0975)α=(2.55,2.3,2.15).Here and in the equations to follow, the momenta p are expressed in units of mc for the appropriate m. Namely, for actual momentum P we will work in terms of p_p = P / m_pc and p_e = P / m_ec. In this framework we have to be careful how we define our distribution functions f_p and f_e. To be unambiguous, let us define them as such:dn_p(r, p_p) = f_p(r, p_p)dp_p dn_e(r, p_e) = f_e(r, p_e)dp_ewhere dn_p(r, p_p) is the differential number density of CRp at radius r and unitless momentum p_p, and similarly for the electrons.We choose the normalization C to be of the form [Although this resembles the normalization in <cit.>, please note the difference between our C(r) (which have dimensions of number density) used here and the C̃(r)=C(r)ρ(r)/m_p (unitless) used by <cit.>. We utilize this formula as a convenience, but our CR density profile is actually much flatter than in their simulation.]C(r)= (C_ vir-C_ center)/1 + ( r/r_ trans)^-β_ C + C_ center.In <cit.>, the parameters C_vir, C_center, r_trans, and β_C are determined from scaling relations. We instead choose values that roughly reproduce the observed synchrotron radiation. For Coma, these are C_ center=6 × 10^-11cm^-3, C_ vir = 5.2 × 10^-11cm^-3, r_ trans=55 kpc, β_ C=1.09. With the above initial distribution we evolve the CRp distribution function forward in time according tof_pt+(𝐮+𝐯_𝐀)·∇ f_p= 1/3pf_pp∇·(𝐮+𝐯_𝐀) +1/p^3∇·(Γ_dampB^2𝐧/4π^3m_pΩ_0v_A𝐧·∇ f_p/|𝐧·∇ f_p|)as in <cit.>. In the above, 𝐮 and 𝐯_𝐀 are the gas and Alfvén velocities, Ω_0 is the non-relativistic gyrofrequency, and 𝐧 is a unit vector which points along the magnetic field. The last term represents diffusion with respect to the Alfvén wave frame due to wave damping, and is directly affected by the discussion in <ref>. More details of this equation and its numerical evolution can be found in <cit.>. We set 𝐮≡ 0 in the simulations discussed here, but include it in eqn. (<ref>) for completeness.As the CRp distribution function evolves, we derive from it at every time step a steady-state secondary CRe distribution function according tof_e(r, p_e)=1/|ṗ_̇ė|∫_p_e^∞dp_e' s_e(r, p_e'),which balances the source function from the CRp hadronic collisionss_e(r, p_e)=4/316m_e/m_pcn_N(r)σ_ppf_p(r,p_p=16m_e/m_pp_e)with the losses from synchrotron emission and inverse Compton (IC) scatteringṗ_̇ė(r, p_e)=4/3σ_Tp_e^2/m_ec(ε_B(r)+ε_cmb).In (<ref>), n_N is the number density of target nucleons and we assume we are far above the pion production threshold.This steady state model includes two implicit assumptions. First, the energy losses of the secondary CRe are dominated by synchrotron and IC losses. Namely this means we assume the CRe don't stream on time scales faster than their loss times ( 100 Myr), a condition we will have to check a posteriori. This assumption will turn out to be well satisfied. Second, we assume that the source function s_e is not changing significantly on these same time scales. This will turn out not to be very well satisfied, as we discuss below in <ref>.Once we have the secondary CRe distribution function at each time step, we can then determine the synchrotron emissivity at frequency ν. From <cit.>,j_ν(r)=0.333√(3)/2πe^3B(r)/m_ec^2∫_0^∞dp_ef_e(r,p_e)F(ν/ν_c).In the above, ν_c=3eBp_e^2/4π m_ec and the function F is an integral of a modified Bessel function, F(x)=x∫_x^∞ K_5/3(x')dx'. With the emissivity in hand, we determine surface brightness and total luminosity from simple spatial integrals[We reiterate that the hadronic model for giant radio haloes is disfavored by gamma-ray non-detections. We include a radio surface brightness prediction for direct comparison with <cit.>. The dimming of the model radio halo can also be thought of as a simple proxy for the evolution of CRp.].§ RESULTS The total 1.4 GHz luminosity of our simulated Coma cluster as a function of time is shown in Figure <ref>. Three CR transport models are compared: in the first, there is no wave damping and CRs stream at the Alfvén speed. In the second, there is wave damping according to (<ref>). In the third, there is wave damping with the high-β correction factor, (<ref>). We see that the increased factor of β^1/2 in the damping rate allows CRs to stream out even faster than in our original simulation, causing the radio luminosity to also drop on faster time scales. The streaming speeds for the relevant 100 GeV CRp are shown in figure <ref>. This figure shows the streaming speed at a fixed radius of 300 kpc as a function of time. Initially the streaming speeds are larger with the high-β correction in comparison to without, and this enhancement grows with time due to the non-linear evolution of the streaming speeds.We may wonder if these enhanced streaming speeds interfere with any of our assumptions, namely our assumption that CRe secondary losses are dominated by synchrotron and inverse Compton losses. CRe will stream at the same speeds as CRp of the same energy, and if the CRe stream out on time scales comparable to their loss times, our steady state treatment is incorrect. Looking at Figure <ref> we may suspect that this is the case, as the ∼ 10^4 km/s streaming speeds imply streaming times of ∼100 Myr across our 1-2 Mpc cluster.However, for the electrons, the relevant energies for 1.4 GHz emission are 1-10 GeV. As shown in Figure <ref>, CRp at these energies stream out much more slowly, even in our high-β model[Note that most CR energy resides at ∼GeV energies, and that even with the high β correction these CRs remain more or less locked to the wave frame. Thus, CR heating in cluster cores <cit.> remains viable.]. We can expect the streaming times of our CRe secondaries to be ∼1-2 Gyr, much longer than their loss times. Note that since the streaming instability growth rate is proportional to n_ CR (equation (<ref>)), and n_ CR,p≫ n_ CR,e, the CRe actually scatter off the wave field created by the much more abundant CRp. If they manage to achieve spatial separation, the CRe could in principle stream much faster.Still, the streaming speeds of the 100 GeV CRp which source these electrons are high, implying a short crossing time. Our steady state model assumes that the CRp distribution changes on time scales much longer than the CRe loss times, such that the production of secondaries `quickly' reaches an equilibrium before the CRp density changes too much. This assumption is no longer satisfied, so a more complete treatment of secondary CRe production and transport may be necessary. However, we have used somewhat artificial initial conditions which do not reflect the cosmological build up of CRp. As long as the injection time is longer than the streaming time (likely to be true for CRs sourced by structure formation (giant radio halos), less so for CRs sourced by AGN activity (radio mini-halos)), CRs will have a flat distribution, which slowly increases in normalization.§ CONCLUSIONDespite significant advances in observations and modeling, the origin and transport of cosmic rays in galaxy clusters remains incompletely understood. As long as this is the case, radiative diagnostics based on cosmic rays will be provisional, and a full assessment of the role of cosmic rays in the dynamics and energy balance of clusters will elude us.In this paper, we have added a new ingredient to the propagation problem: the role of ion Landau damping in suppressing the growth of Alfvén waves excited by cosmic ray streaming.Although waves which propagate exactly along the background magnetic field _0 are undamped, inhomogeneity in _0 precludes perfectly parallel propagation. At the large β≡ 8π P_g/B^2 values characteristic of galaxy clusters, oblique waves are subject to strong Landau damping due to their parallel electric fields.When the propagation angle is calculated assuming _0 is a uniform field with a superimposed anisotropic MHD turbulent cascade, the resulting damping rate (<ref>) is of the same form as the damping rate estimated from the turbulent shearing rate (<ref>), which was taken as the dominant damping mechanism in earlier work <cit.>, but exceeds the turbulent damping rate by a factor of β^1/2. It is worth noting that the assumption of an anisotropic turbulent cascade is key to this result: if the scale of variation of _0 were taken to be a global scale, the resulting damping rate would be very slow <cit.>.Since Landau damping is a collisionless process, it is important to check whether the effective ion mean free path is much longer than the wavelengths of the Alfven waves. Estimating the mean free path λ_ii to Coulomb scattering is straightforward and the result (<ref>) shows that the waves are indeed collisionless with respect to Coulomb interactions. However, we also estimated for the first time the mean free path λ_i due to scattering from microscale instabilities driven by pressure anisotropy (<ref>), and foundthat it too is much larger than the wavelengths of the Alfvén waves in question, fullyvalidating the role of Landau damping in suppressing the Alfvén waves that scatter cosmic rays in turbulent galaxy cluster plasmas. Note that there are potentiallyimportant caveats about the nature of MHD turbulence in a high β plasma <cit.>, which are beyond the scope of this paper.We then reconsidered the evolution of an idealized version of the radio halo of the Coma cluster following <cit.>, but including Landau damping. In this model, the halo is produced by secondary electrons and positrons created in hadronic interactions between cosmic ray protons and thermal cluster gas. As expected, transport of the cosmic ray protons is much faster, quickly depleting them and turning off the halo on even faster timescales than predicted in the original model. This strengthens the case for rapid evolution of radio halos, at least in this idealized (1D, radial magnetic field) model cluster, unless the cosmic ray primary proton source is continuously replenished. This setup is the most optimistic case there is for evolution via streaming, an important caveat of this work. Future simulations with higher dimensionality and more complicated field topologies are necessary to study this effect in real clusters. The problem of cosmic ray electron transport on galaxy clusters was first raised by <cit.>, who pointed out that the radiative loss time of cosmic ray electrons is much shorter than their transport time at the Alfvén speed from the core of the cluster. If the primary CRe could stream out to large distances in less than one cooling time, they could supply the radio emission in the outskirts without being reaccelerated. However, even in our ideal scenario this does not seem to be the case. Although we have not undertaken a detailed comparison of Jaffe's model with ours, the transport speeds of 5 GeV electrons shown in Figure <ref> are well below the ∼ 2000 km/s transport speed that Jaffe estimated was necessary to form the halo with primary electrons. The rapid transport speed of cosmic ray protons due to Landau damping of Alfvén waves could help to explain the stringent upper limits on diffuse γ-ray emission from galaxy cluster cores reported by <cit.>. It also has important consequences for hadronic production of CRe `seeds' for turbulent reacceleration - rapid streaming produces a flat CRp profile, giving a seed population which better fits observational constraints <cit.>. It is also relevant to the tension between observed radio halo luminosities and models of CR heating of cluster cores <cit.>, since high energy CRs responsible for the former stream at much higher velocities than the lower energy CRs relevant for the latter. These issues are beyond the scope of the current paper, but a topic for future work.Acknowledgements:We are happy to acknowledge discussions with Matt Kunz, Eliot Quataert, and Larry Rudnick. JW and EGZ acknowledge support by NSF Grant AST-1616037, the WARF Foundation, and the Vilas Trust. SPO acknowledges support from NASA grant NNX15AK81G. | http://arxiv.org/abs/1706.08525v2 | {
"authors": [
"Joshua Wiener",
"Ellen G. Zweibel",
"S. Peng Oh"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170626180001",
"title": "High $β$ Effects on Cosmic Ray Streaming in Galaxy Clusters"
} |
Satellite conjunction analysis is the assessment of collision risk during a close encounter between a satellite and another object in orbit. A counterintuitive phenomenon has emerged in the conjunction analysis literature, namely, probability dilution, in which lower quality data paradoxically appear to reduce the risk of collision. We show that probability dilution is a symptom of a fundamental deficiency in probabilistic representations of statistical inference, in which there are propositions that will consistently be assigned a high degree of belief, regardless of whether or not they are true. We call this deficiency false confidence. In satellite conjunction analysis, it results in a severe and persistent underestimate of collision risk exposure. We introduce the Martin–Liu validity criterion as a benchmark by which to identify statistical methods that are free from false confidence. Such inferences will necessarily be non-probabilistic. In satellite conjunction analysis, we show that uncertainty ellipsoids satisfy the validity criterion. Performing collision avoidance maneuvers based on ellipsoid overlap will ensure that collision risk is capped at the user-specified level. Further, this investigation into satellite conjunction analysis provides a template for recognizing and resolving false confidence issues as they occur in other problems of statistical inference. belief functionsepistemic uncertaintystatistical inferencecollision risksatellite conjunction analysisprobability dilution § INTRODUCTIONA satellite conjunction is an event in which two satellites or a satellite and a piece of debris are estimated to pass near each other. The goal of satellite conjunction analysis is to determine whether the risk of collision is high enough to necessitate a collision avoidance maneuver. Avoiding collision is not just important to the individual satellite operator. Everyone in the space industry has a stake in minimizing the number of in-orbit collisions. Each collision adds to the debris already in orbit, and in aggregate, this debris has the potential to make low Earth orbit unnavigable <cit.>.So, while some of the issues involved in conjunction analysis may initially seem arcane, correctly resolving those issues is of existential importance to the space industry. §.§ A Modern ProblemOver the past 15 years, aerospace researchers have recognized a counterintuitive phenomenon in satellite conjunction analysis, known as probability dilution <cit.>. That is, as uncertainty in the satellite trajectories increases, the epistemic probability of collision eventually decreases. Since trajectory uncertainty is driven by errors in the tracking data, the seemingly absurd implication of probability dilution is that lower quality data reduce the risk of collision. Several researchers have attempted to synthesize an ad hoc fix to probability dilution <cit.>. Most of them have defined an alternative collision risk metric that is increasing, or at least non-decreasing, as a function of trajectory uncertainty.Alternatively, some satellite operators have adopted a high sensitivity to small collision probabilities <cit.>. However, if probability dilution is indicative of a fundamental problem with collision probabilities, it is not immediately obvious that being sensitive to small collision probabilities will resolve that problem.§.§ A Centuries-Old ArgumentSatellite trajectory estimation is fundamentally a problem of statistical inference, and the confusion over probability dilution cuts to the heart of the long-standing Bayesian–frequentist debate in statistics; for a review, see <cit.>. Satellite orbits are inferred using radar data, optical data, GPS data, etc. which are subject to random measurement errors, i.e., noise. The resulting probability distributions used in conjunction analysis represent epistemic uncertainty, rather than aleatory variability. That is to say, it is the trajectory estimates that are subject to random variation, not the satellite trajectories themselves. In the Bayesian view, that is a distinction without a difference; it is considered natural and correct to assess the probability of an event of interest, such as a possible collision between two satellites, based on an epistemic probability distribution <cit.>. In the frequentist view, however, it is considered an anathema to try to compute the probability of a non-random event <cit.>. This prohibition is expressed in some corners of the uncertainty quantification community as the requirement that epistemic uncertainty be represented using non-probabilistic mathematics <cit.>. Despite these objections, at present, satellite navigators appear to be adopting the Bayesian position. §.§ DefinitionsAn aleatory uncertainty is an uncertainty due to random (i.e., aleatory) variation of the quantity or event being analyzed. An epistemic uncertainty is uncertainty due to imperfect knowledge about the quantity or event being analyzed. One of the many ways in which epistemic uncertainty can arise in a fixed quantity is if the data used to measure or infer it are subject to random errors.A belief function is a function used to represent the amount of positive support a proposition has accrued from the evidence. A plausibility function merely represents the lack of evidence against a proposition. Belief and plausibility are complementary functions; that is, Pls( A ) = 1 - Bel(A ), for every proposition, A, and its complement (i.e., negation) A. As a rule, the plausibility accorded to a proposition is always greater than or equal to the belief accorded to it. An additive belief function is a belief function that satisfies the Kolmogorov axioms and in which, as a consequence, Bel( A ) = Pls( A ) for all propositions <cit.>. An epistemic probability is an additive belief function used to represent epistemic uncertainty. Confidence distributions,[We refer to confidence distributions in the sense of <cit.>, as opposed to <cit.> and <cit.>, which eschew epistemic probabilities.] fiducial distributions, and most Bayesian posterior distributions[One could argue that a posterior probability derived using an aleatory prior, as in <cit.>, is itself an aleatory probability.] are epistemic probability distributions. An aleatory probability is a probability used to represent aleatory uncertainty. It is equal to the fraction of trials in which a specified event occurs—or would occur—over an infinite number of equivalent random trials.The mathematics of aleatory and epistemic probability functions are identical; the distinction is a question of subject matter. §.§ Our ContributionIn this paper, we pursue basic questions raised by probability dilution. Can satellite operators take epistemic probability at face value as a risk metric? If not, why not? What alternatives are safe for use in conjunction analysis? How can we know that those alternatives are safe? We address these questions, in part, by placing conjunction analysis in its context as a problem of statistical inference. Consequently, what we discover about probability dilution has broad implications for statisticians as well as satellite operators. In Section <ref>, we provide a thorough exposition of probability dilution. We show that the counterintuitiveness of probability dilution is entirely due to the fact that, in conjunction analysis, probability of collision is an epistemic probability, rather than an aleatory probability. Moreover, we show that probability dilution is linked to a more severe operational defect that we call false confidence. We then evaluate the efficacy of pursuing small epistemic probabilities as a way to mitigate the problems posed by false confidence.In Section <ref>, we show that false confidence is not limited to conjunction analysis. All epistemic probability distributions are subject to false confidence. That is, in any additive belief function used to represent epistemic uncertainty, there are false propositions that have a high probability of being assigned a high belief value. The false confidence generated by epistemic probability of collision in satellite conjunction analysis is a manifestation of this more general deficiency.In Section <ref>, we demonstrate a framework for identifying statistical methods that are free from false confidence. We build this approach on the “validity criterion” introduced in the Martin–Liu Theory of Inferential Models <cit.>. For a statistical inference to be valid in the Martin–Liu sense, only a true proposition may have a high probability of being assigned a high belief value. An inference satisfying this criterion will not suffer from the false confidence phenomenon described in Section <ref>. Note that the reader does not need to be familiar with Martin–Liu theory to understand the work presented in this paper. Only the validity criterion is used, and its formal definition is provided in Section <ref>.§ EPISTEMIC PROBABILITY OF COLLISIONEpistemic probability of collision is coming into more common use as a risk metric in satellite conjunction analysis <cit.>. In fact, the most-cited literature on satellite conjunction analysis simply takes it for granted that epistemic probability of collision is the correct way to represent collision risk, e.g., <cit.>. However, the counterintuitiveness of probability dilution calls this practice into question, especially considering the unsettled status of epistemic probability in the statistics and uncertainty quantification communities. This part of the paper explores issues surrounding the use of epistemic probability of collision as a risk metric for satellite conjunction analysis. Section <ref> reviews the standard computation of collision probability. Section <ref> describes how probability dilution arises from those mathematics. Section <ref> describes a more severe operational defect in epistemic collision probabilities: false confidence. Section <ref> explores the use of small thresholds for epistemic probability as a way to counteract false confidence.§.§ Computation of Collision ProbabilityAs mentioned in Section <ref>, satellite conjunction analysis is rooted in a problem of statistical inference. The inferred parameter of interest is the vector of positions and velocities of the two satellites at time of closest approach. That is,θ = ( u_1 , v_1 , w_1 , u̇_1 , v̇_1 , ẇ_1 , u_2 , v_2 , w_2 , u̇_2 , v̇_2 , ẇ_2 )^⊤ where u_1,v_1,w_1 are the position of one satellite; u̇_1 , v̇_1 , ẇ_1 are its velocity; u_2 , v_2 , w_2 are the position of the other satellite; u̇_2 , v̇_2 , ẇ_2 are its velocity.The true value of θ is unknown; it is estimated using tracking data, x, via an inferential algorithm called a “filter” that delivers an estimate, θ̂( x ), along with its uncertainty, expressed as a 12 × 12 covariance matrix, C_Θ. In this paper, uncertainty in θ is attributed to random errors in the tracking data. In reality, uncertainty also arises from errors in the dynamics model used in the filter. Uncertainty in the trajectory estimates is usually assumed to have a multivariate normal distribution <cit.>. The epistemic probability density for θ is therefore taken to bef_Θ;x(θ) = {( 2π)^ 12 ( C_Θ ) }^-1/2exp{ -1/2( θ̂( x ) - θ)^⊤ C_Θ^-1( θ̂( x ) - θ) } where ( C_Θ ) is the determinant of C_Θ. This epistemic probability distribution for θ can be rationalized in two ways. One way is to treat filter estimation as an implementation of Bayesian inference <cit.>. The other way is to describe this representation as a fiducial distribution for θ, in the sense of <cit.>, or a confidence distribution for θ, in the sense of <cit.>, on the argument that a random resampled θ̂( X ) would have a multivariate normal distribution with mean θ and covariance matrix C_Θ, where X represents a random realization of the tracking data, drawn from the same distribution as x, the tracking data actually obtained. The opening of Section <ref> briefly describes the conditions under which the normality assumption will and will not hold. Assuming normality holds, the standard epistemic probability of collision calculation proceeds in three steps. First, uncertainty in θ is propagated to uncertainty in the relative offset between the two satellites at closest approach. That is, Δ u = u_2 - u_1 Δ v = v_2 - v_1 Δ w = w_2 - w_1. Since the linear transform of a multivariate normal variable is itself a multivariate normal, the triple ( Δ u , Δ v , Δ w ) has a multivariate normal distribution with the following 3×3 covariance matrix: C_Δ = A C_Θ A^⊤whereA = [ -100000 +100000;0 -100000 +10000;00 -100000 +1000 ] or more compactly C_Δ = C_Θ;1:3,1:3 + C_Θ;7:9,7:9 - 2 C_Θ;1:3,7:9,where C_Θ;m_1:m_2,n_1:n_2 is the sub-matrix of all entries in rows i and columns j such that m_1 ≤ i ≤ m_2 and n_1 ≤ j ≤ n_2. That is, C_Δ is the sum of the covariance matrices for the positions of the two satellites minus the covariance between the positions of the two satellites.In the second step, uncertainty along the axis parallel to the relative velocity vector is integrated out. That is, instead of computing probability of collision as the probability of the true value of the displacement falling within the set of ( Δ u , Δ v , Δ w ) values indicative of collision, probability of collision is computed as the probability accorded to an extruded ellipse containing that three-dimensional set. The theoretical rationale behind this move is that it captures uncertainty in the timing of closest approach <cit.>. The primary practical effect of this step is to reduce a three-dimensional integration problem to a two-dimensional integration problem. More nuanced ways of treating uncertainty in the timing of closest approach are available <cit.>, but the standard two-dimensional treatment is sufficient for our analysis. This transformation is executed in a few sub-steps. It starts with the unit vector in the relative velocity direction, i _Δ V, defined asΔ V = ( u̇_2 , v̇_2 , ẇ_2 ) - ( u̇_1 , v̇_1 , ẇ_1 ) and i_Δ V = Δ V / Δ V .Next, another direction in the plane perpendicular to i_Δ V can be defined arbitrarily. For example it could be taken as in the same direction as the cross-product of i_Δ V and ( Δ u , Δ v , Δ w ), assuming that they are not aligned. Denote this arbitrary direction vector as i_u'. The final direction vector is simply the cross-product, i_v' = i_Δ V× i_u'. Once this system has been established, define the rotated displacement vector as [ u'; v'; w' ] = M [ Δ u; Δ v; Δ w ]whereM = [i_u'^⊤;i_v'^⊤; i_Δ V^⊤ ]is the rotation matrix. The covariance matrix for the rotated ( u' , v' , w' ) vector is C_Δ' = M C_ΔM^⊤.Since ( u' , v' , w' ) has a multivariate normal distribution, uncertainty in the Δ V direction can be integrated out by simply dropping w'. That completes the extrusion step, reducing the problem to the two-dimensional ( u' , v' ) space, with estimate ( û' , v̂' ) and covariance matrix C_Δ';1:2,1:2. The third and final step in the standard calculation of collision probability is to define the probability of collision as the epistemic probability of having u' and v' such that u'^2 + v'^2 ≤ R^2, where R is the sum of the characteristic radii of the two satellites in the conjunction event. In other words, the satellite shapes are approximated as spherical objects. Collision occurs if and only if, at closest approach, the distance between the two satellites is less than their combined size.This probability integral can be re-expressed in a standardized probability space, by performing an eigendecomposition of the displacement covariance matrix <cit.>, which yields [ u'; v' ] = [û'; v̂' ] - E_S [ S_1 0; 0 S_2 ][ ξ_1; ξ_2 ] where S^2_1 and S^2_2 are the eigenvalues of C_Δ';1:2,1:2; E_S is a 2 × 2 matrix whose columns are the normalized eigenvectors of C_Δ';1:2,1:2; and finally, ξ_1 and ξ_2 are independent unit normal variables, with mean zero and variance one. Note that, since E_S is the eigendecomposition of a symmetric matrix, E^⊤_S E_S returns the identity matrix. Defining u” and v” as the E^⊤_S rotation of u' and v', the magnitude of ( u” , v”) is equal to the magnitude of ( u' , v' ). That is, [ u”; v” ] = E^⊤_S [ u'; v' ]u”^2 + v”^2 = u'^2 + v'^2, where u” = û” - S_1 ξ_1 u” = v̂” - S_2 ξ_2.Thus, the following equivalences hold for the integration condition:u'^2 + v'^2 ≤ R^2u”^2 + v”^2 ≤ R^2 ( S_1 ξ_1 - û”)^2 + ( S_2 ξ_2 - v̂”)^2 ≤ R^2.What has been accomplished is that now, instead of integrating a circle in a ( u' , v' ) space that will vary from problem to problem, we are integrating an ellipse in a standardized unit normal ( ξ_1 , ξ_2 ) space. The centroid of that ellipse is located at ( û” / S_1 , v̂” / S_2 ), and its semi-major and semi-minor axes are R/S_1 and R/S_2. Following the logic of <cit.>, the epistemic probability of collision is computed using a contour integral transformation, as follows: Bel( ℂ) = ∬_( S_1 ξ_1 - û”)^2 + ( S_2 ξ_2 - v̂”)^2 ≤ R^2 1/2πexp[ -1/2( ξ^2_1 + ξ^2_2 ) ] dξ_1 dξ_2= ∫_0^2π( 1-e^-r_ψ^2/2/2 π r_ψ^2) ( R^2/S_1 S_2 + û” R/S_1 S_2cosψ + v̂” R/S_1 S_2sinψ)dψ wherer_ψ^2 = ( û”/S_1 + R/S_1cosψ)^2 + ( v̂”/S_2 + R/S_2sinψ)^2. This integral can be approximated to negligible error using the trapezoidal rule, provided that at least 10 ×max( S_1 / S_2 , S_2 / S_1 ) evenly spaced quadrature points are used <cit.>.Because our goal is to understand and explore how the epistemic probability of collision changes for different levels of uncertainty, we pursue the special case where S = S_1 = S_2. Defining the estimated displacement as D^2 = û”^2 + v̂”^2 = û'^2 + v̂'^2, this simplification reduces the epistemic probability of collision to a function of two ratios: estimated relative displacement, D / R, and relative uncertainty, S / R.§.§ Probability DilutionFigure <ref> illustrates epistemic probability of collision as a function of S / R for several values of D / R. These curves follow a common pattern. So long as D/R > 1, for small uncertainties, the probability of collision is small. That much makes intuitive sense. If the satellites are estimated to miss each other, and the uncertainty in those trajectory estimates is small, then it is natural that the analyst have a high confidence that the satellites are not going to collide. Similarly, it makes sense that as uncertainty grows, the risk of collision also grows. However, something odd happens in the curves in Figure <ref>. Epistemic probability of collision eventually hits a maximum, and past that maximum, as relative uncertainty rises, the epistemic probability of collision decreases. This decrease is called probability dilution, and it has an odd implication. Since the uncertainty in the estimates of the trajectories reflects the limits of data quality, probability dilution seems to imply that lowering data quality makes satellites safer. That implication is counterintuitive in the extreme <cit.>. As a rule, lowering the data quality makes any engineering system less safe, and to claim that ignorance somehow reduces collision risk seems foolish on its face. That being said, the mathematics of probability dilution are ironclad and relatively simple. Recall that, when expressed in unit normal space, the integration region determining the probability of collision is an ellipse whose centroid is at ( û” / S_1 , v̂” / S_2 ) and whose semi-major and semi-minor axes are R/S_1 and R/S_2. Figure <ref> illustrates how the integration region changes as a function of S, when S = S_1 = S_2. Combined satellite size, R, and estimated distance at closest approach, D, are held fixed with D/R = 5. For four different values of S/R, the integration region is illustrated, along with the corresponding probability of collision. Increasing the level of uncertainty has two effects on the integration region: It shrinks and draws closer to the origin.The probability density of a unit normal space is highest at the origin, decreasing exponentially as a function of the square of distance from the origin. If D / R > 1, the positive effect due to getting closer to the origin, where there is more probability mass, initially outweighs the negative effect due to shrinking the integration region. Near the origin, though, the probability density curve flattens out. So, after a certain point, there is not much more probability mass to be gained by moving closer to the origin, and the negative effect of shrinking the integration region overtakes the positive effect due to shifting the location of the integration region. Past that point, probability of collision decreases as uncertainty increases. So, in total, probability dilution is due to a simple and straightforward shrinkage of the integration region relative to displacement uncertainty.Even if the normality and shape assumptions were relaxed, the integration region would still undergo the same shrinkage phenomenon that is illustrated in Figure <ref>. Each satellite represents a bounded set of points. The displacement values indicative of collision are therefore also a bounded set of points. If one were to relax the assumptions outlined in Section <ref>, instead of an ellipse, one would have a different shrinking shape in a standardized three-dimensional probability space. Probability of collision would still have the same qualitative behavior as a function of relative uncertainty. Mathematically speaking, the key factor underpinning probability dilution is that the integration region (i.e., failure domain) is bounded.There is no way of reframing satellite conjunction analysis so that the failure domain is not bounded. Therefore, if we take the view that epistemic probability is valid, then it would seem that probability dilution is fundamental to conjunction analysis. Why, then, do aerospace researchers <cit.> find this supposed mathematical inevitability so counterintuitive?Interestingly, were the uncertainty in the satellite trajectories aleatory, rather than epistemic, probability dilution would actually make sense. For example, suppose two satellites were known, with certainty, to be on a collision course, and the satellite operator could only impart an impulse of random magnitude in a random direction, say, via a poorly controlled maneuvering thruster on a tumbling satellite. If the mean of this distribution were the null vector, due to a lack of control over thrust direction, then the higher the variance of the added impulse, the bigger the resulting perturbation and hence the smaller the resulting probability of collision. In this example, the mean of the resulting trajectory distribution would still have the satellites on a collision trajectory, but if one applies a big enough impulse to one of the satellites in a random direction, that poorly controlled collision avoidance maneuver still has a high probability of success. In that context, higher variance in a satellite's trajectory really does reduce the risk of collision. However, in this hypothetical example, it is a variance in the trajectory itself that makes the satellite safer, not a variance in the estimate of that trajectory. It should go without saying that, given two satellites on a sure collision trajectory, simply recomputing the trajectories with lower quality data does not make them safer. There is, therefore, an operational distinction to be made between genuine aleatory variability and epistemic uncertainty. To summarize, the problem with probability dilution is not the mathematics. The mathematics are incontrovertible, and they even make sense in the right context. Probability dilution is only counterintuitive when orbit “variance” reflects the limits of data quality, rather than genuine aleatory variability in the orbits themselves. So, if probability dilution is inappropriate, that inappropriateness must be rooted in a mismatch between the mathematics of probability theory and the epistemic uncertainty to which they are applied in conjunction analysis. §.§ False ConfidenceThe final clue indicating the inappropriateness of probability dilution is the fact that, for a fixed S/R ratio, there is a maximum computable epistemic probability of collision. Whether or not the two satellites are on a collision course, no matter what the data indicate, the analyst will have a minimum confidence that the two satellites will not collide. That minimum confidence is determined purely by the data quality.For all S/R, the maximum collision probability is obtained when D/R = 0; that is, when the best estimate of the two satellite paths indicates that they are on a collision course. Therefore, the curve for D/R = 0 in Figure <ref> yields the maximum computable probability of collision as a function of S/R. For example, if the uncertainty in the distance between two satellites at closest approach is ten times the combined size of the two satellites, the analyst will always compute at least a 99.5% confidence that the satellites are safe, even if, in reality, they are not. A relative uncertainty of ten may sound high, but it may help the reader to keep in mind the magnitude of the trajectories being estimated, relative to the size of the objects on those trajectories. Satellite trajectories are measured in thousands of kilometers; satellites are measured in meters. Trajectory uncertainties in conjunction analysis are usually measured in hundreds of meters <cit.>. For conjunction analysis done more than a week in advance, those uncertainties can grow to kilometers <cit.>. As a consequence, S/R ratios greater than ten are the rule, not the exception. Under those circumstances, even if two satellites are on a collision course, because of probability dilution, the epistemic probability of collision is guaranteed to be small. The biggest problem with probability dilution is not its counterintuitiveness; instead, it is the false confidence that it imbues. Due to probability dilution, satellite navigators using epistemic probability of collision as their risk metric will always compute a high confidence that their satellites are safe, regardless of whether or not they really are. The only rare exceptions are conjunctions in which both satellites are extremely well-tracked and at least one of them is relatively large (for an example, see <cit.>). From an operational perspective, if taken at face value, epistemic probability of collision is a misleading risk metric. §.§ Pursuing Small Epistemic Probabilities Some satellite navigators treat relatively small epistemic probabilities of collision as though they reflect a high level of risk. So, it would be inaccurate to characterize the entire satellite operations community as taking epistemic probabilities at face value. For example, the CARA group based at NASA Goddard monitors all conjunctions with a computed probability of collision greater than 10^-7, and conjunctions with a computed probability of collision greater than 4.4 × 10^-4 are treated as high risk. These thresholds are calibrated in an effort to achieve desired frequentist error rates, in essence treating epistemic probability of collision as a test statistic, using historical data from past conjunctions as a reference population <cit.>.Unfortunately, even the idea of compensating for false confidence by setting a low threshold for epistemic probability of collision is problematic, because relative data quality modulates the relationship between epistemic probability of collision and the real aleatory probability of failing to detect an impending collision. To model this relationship, we consider the aleatory variability of predicted distance, D, between two satellites at closest approach, if the data used to produce that prediction were redrawn from the same distribution as the actual data obtained. Assuming that variability in the estimated direction of relative velocity is small enough to be neglected, the marginalization and eigendecomposition steps can proceed as in Section <ref>. This yields D^2 = Û”^2 + V̂”^2 =( u” + S_1 ξ_1 )^2 + ( v” + S_2 ξ_2 )^2where ( u” , v”) is the true unknown two-dimensional relative displacement between the two satellites; Û” and V̂” are the random resampled analog to û” and v̂” from Section <ref>; S_1 and S_2 are the principal standard deviations of the marginalized two-dimensional covariance matrix, as in Section <ref>; and ξ_1 and ξ_2 are independent unit normal random variables. In the case where S_1 = S_2 = S, this simplifies to( D/S)^2 = ( u”/S + ξ_1 )^2 + ( v”/S + ξ_2 )^2.This can be further reduced to (D / S)^2 = χ^2_2;( D_T/S )^2, where χ^2_j,δ is a non-central chi-squared random variable with j degrees of freedom and non-centrality parameter δ, and D_T^2 = u”^2 + v”^2 is the unknown true distance between the two satellites at closest approach. In Section <ref>, we showed how epistemic probability of collision can be computed as a function of D/S and S/R. Given a fixed S/R, epistemic probability of collision is a decreasing function of D/S. So, given a fixed detection threshold for epistemic probability, the aleatory probability of detecting an impending collision is equal to the probability of obtaining an observed D/S that is less than or equal to the critical D/S that yields an epistemic probability of collision equal to the threshold value. Rather than try to compute this relationship directly, it is easier to simply generate D/S values from its non-central chi-squared distribution and then compute the corresponding epistemic probabilities of collision. This yields epistemic probability threshold as a function of desired detection rate, which can be reversed to give the aleatory probability of failing to detect an impending collision as a function of a satellite operator's chosen detection threshold for epistemic probability of collision. Figure <ref> illustrates this relationship for several values of S/R in two scenarios, an impending head-on collision with D_T = 0 and an impending glancing collision with D_T = R. The two dotted vertical lines mark the lower and upper thresholds for epistemic probability of collision reported in <cit.>. Conjunctions with an epistemic probability of collision less than 10^-7 are treated as lowest risk and ignored. Conjunctions with an epistemic probability of collision greater than 4.4 × 10^-4 are treated as high risk events. Using these thresholds, if S/R = 10, an impending head-on collision has a 91.2% chance of correctly being identified as a high risk conjunction. Similarly, a glancing collision with S/R = 10 has a 91.1% chance of being correctly identified as high risk. So, broadly speaking, one could say that the CARA threshold has reasonably good performance for S/R = 10. Unfortunately, that performance degrades quickly as uncertainty increases. At S/R = 20, the aleatory probability of correctly identifying an impending collision as a high risk conjunction drops to 64.8%. For S/R > 33.74, the aleatory probability of detecting an impending collision using the 4.4 × 10^-4 threshold drops to zero. For example, in a conjunction between two objects with a combined radius of five meters, a satellite navigator using the CARA upper risk threshold will have zero chance of detecting an impending collision unless or until their predicted displacement uncertainty drops below 170 meters.In summary, trying to compensate for false confidence by pursuing small epistemic probabilities of collision appears to be an unproductive strategy. Even small epistemic probabilities end up distorted, and they still result in false confidence under conditions that satellite navigators regularly encounter. Furthermore, the relationship between data quality and false confidence is more complicated than we have depicted in this idealized analysis. It depends not only on S/R but also on S_1/S_2, and in reality, S_1 and S_2 are never equal. Every conjunction analysis has a unique combination of covariance matrices. So, if an analyst wants to hold the risk of failed detection at a consistent level, it is necessary to define a different epistemic probability threshold for each and every conjunction analysis. In other words, epistemic probability of collision could, theoretically, be used as a test statistic,but it cannot be taken directly as a risk metric, even if satellite navigators are sensitive to small epistemic probabilities of collision. There is no single threshold for epistemic probability that will detect impending collisions with a consistent degree of statistical reliability.§ THE FALSE CONFIDENCE THEOREMWhat causes probability dilution and false confidence? As explored in Section <ref>, it is not an error in the mathematics. Rather, it appears to be a mismatch between the mathematics of probability theory and the subject matter to which those mathematics are applied in satellite conjunction analysis. As mentioned in Section <ref>, questions over the appropriateness of epistemic probability have never been settled. As shown in Section <ref>, those questions take on an undeniably practical dimension in satellite conjunction analysis. This part of the paper introduces the false confidence theorem. Section <ref> suggests that probability dilution and false confidence are rooted in the axioms of probability theory. Section <ref> posits a formal definition for false confidence. Section <ref> proves that all epistemic probability distributions assign copious amounts of false confidence. Section <ref> places this theorem in the context of the Bayesian–frequentist debate. §.§ Motivating InsightGiven highly uncertain data, it makes intuitive sense that one would not be certain of collision. An analyst should be reluctant to assign a high degree of belief to any proposition, based on low-quality data. The problem is that, within the confines of probability theory, general non-commitment of belief is not an option. One of the axiomatic properties of probability functions is that they are additive. So, any belief not assigned to “collision” is automatically assigned to “not-collision,” i.e., Bel( ℂ) = 1 - Bel( ℂ).While a low assignment of belief to “collision” might seem natural when uncertainty is high, the consequently high assignment of belief to “not-collision” seems unnatural. The implication here is profound. If additivity is the root cause of false confidence, it means that false confidence is inherent to epistemic probability distributions, due to their mathematical structure. In Section <ref>, we show that this is exactly the case.§.§ Definition of False ConfidenceStrictly speaking, one could characterize any confidence or belief or epistemic probability assigned to a false proposition as “false confidence.” However, the whole point of statistical inference is that one does not know, at the outset, the exact true value of the parameter being inferred. So, it is to be expected that, in the course of any given inference, some false propositions will be assigned some amount of belief. The problem described in Sections <ref>-<ref> is more severe: There is a fixed proposition of practical interest that is guaranteed or nearly guaranteed to be assigned a large epistemic probability, regardless of whether or not it is true. This suggests a formal definition. To state this definition precisely, we need some additional notation.First, let Pro_X;θ denote the aleatory distribution of the data X ∈^n depending on the fixed but unknown true value of the parameter θ∈Ω_θ⊆^m being inferred.Consider a set-function Bel_Θ;x(·) defined on the power set of Ω_θ, depending on data x, such that Bel_Θ;x(A) represents the data analyst's epistemic degree of belief in the truthfulness of the proposition A ⊆Ω_θ about the unknown parameter θ.Then the map x ↦Bel_Θ;x is an inferential method. We say that the inference about some proposition of interest, A, suffers from severe false confidence if, for some unacceptably high belief assignment 1-α where α∈ (0,1) and unacceptably high probability of assignment p ∈ (0,1), there is some putative value of θ∈Ω_θ such that A ∌θandPro_X;θ( { x : Bel_Θ;x( A ) ≥ 1 - α} ) ≥ p. Under this notation, Figure <ref> is a straightforward plot of p as a function of α, where A is the assertion that the two satellites will not collide. As relative uncertainty grows, the level of false confidence seen in satellite conjunction analysis grows arbitrarily severe. Next, we show that, when considered over the full range of measurable propositions that the analyst might assess, most epistemic probability distributions used in practice suffer from arbitrarily severe false confidence.§.§ TheoremMost statistical inference in science and engineering involves one or more real-valued continuous parameter(s). So the epistemic probability distributions (e.g., posteriors) commonly used in practice are, for all values of the observable, continuous distributions over the parameter being inferred. That is, belief assigned by the additive belief function Bel_Θ;x can be represented via an epistemic probability density function, say, f_Θ;x(θ), with respect to Lebesgue measure λ on Ω_θ, depending on the observation, x. This is the case for most Bayesian posteriors, confidence distributions <cit.>, and fiducial distributions <cit.> used in practice. The satellite collision problem described in Section <ref> is one such example.The false confidence theorem below states that all such epistemic probability distributions suffer from arbitrarily severe false confidence. Consider an additive belief function Bel_Θ;x characterized by an epistemic probability density function f_Θ;x on Ω_θ such that sup_ϑ f_Θ;x(ϑ) < ∞ for Pro_X;θ-almost all x, for all θ.Then for any θ∈Ω_θ, any α∈ (0,1), and any p ∈ (0,1), there exists a set A ⊂Ω_θ such that (<ref>) holds. For fixed x, set f̅_x = sup_ϑ f_Θ;x(ϑ).Then for any bounded set B ⊂Ω_θ, Bel_Θ;x(B) = ∫_B f_Θ;x( ϑ) dϑ≤f̅_xλ(B).As a function of X ∼Pro_X;θ, for fixed θ, f̅_X has a distribution; let η=η_θ,p∈ (0,∞) satisfy Pro_X;θ({x: f̅_x ≤η}) ≥ p.Choose a neighborhood B ∋θ with measure λ(B) = α / η, so that λ(B)η = α.Define A as the complement of B. For any belief function satisfying the Kolmogorov axioms, we haveBel_Θ;x(A) = 1-Bel_Θ;x(B).By definition, A ∌θ, and by (<ref>) and (<ref>), we have f̅_x ≤ηBel_Θ;x(B) ≤λ(B)ηBel_Θ;x(B) ≤αBel_Θ;x(A) ≥ 1-α.By definition of η, the left-most event occurs with Pro_X;θ-probability at least p and, therefore, the right-most event occurs with Pro_X;θ-probability at least p, proving the claim.Theorem <ref> is an existence result; so, our proof proceeds by constructing the simplest possible example. This is achieved by defining a neighborhood around the true parameter value that is so small that its complement—which, by definition, represents a false proposition—is all but guaranteed to be assigned a high belief value, simply by virtue of its size. In practice, no one would intentionally seek out such a proposition, but that is beside the point. Every real-world risk analysis problem involves a proposition of interest that is determined by the structure of the problem itself; e.g., “Will these two satellites collide?”. Just as the practitioner will not seek out propositions strongly affected by false confidence, neither do practitioners have the option of avoiding such propositions when they arise. What the false confidence theorem shows is that, in most practical inference problems, there is no theoretical limit on how severely false confidence will manifest itself in an epistemic probability distribution, or more precisely, there is no such limit that holds for all measurable propositions. Such a limit can only be found for a specific proposition of interest through an interrogation of the belief assignments that will be made to it over repeated draws of the data. That is the type of analysis pursued in Section <ref>, which reveals a severe and pernicious practical manifestation of false confidence.Additionally, the proof of the false confidence theorem has a direct connection to the practical manifestation of false confidence seen in satellite conjunction analysis. They are both driven by the fact that the size of a measurable set puts an upper bound on the belief that can be assigned to it, which due to additivity, puts a lower bound on the belief that will necessarily be assigned to its complement. The quantitative relationship between epistemic probability and the relative size of the proposition to which it is assigned is perhaps best illustrated by Figure <ref>. This relationship holds whether or not the proposition of interest (e.g., collision) includes the true parameter value, but in the proof of the false confidence theorem, it is necessary to show that it does indeed hold, even if θ∈ B. That being said, it is important not to over-interpret this commonality.Practical manifestations of severe false confidence are not limited to problems in which the proposition of interest corresponds to an obviously small set. This is explored in recent and on-going work referenced in Section <ref>. Even in the present work, consider the fact that—while the proposition that the two satellites will collide is Lebesgue-measurable in displacement space—in the original state space, collision would correspond to an infinite hypercylinder, which has an unbounded Lebesgue measure. The false confidence theorem puts a lower bound on how widely severe false confidence will manifest itself in practice, but it does not in any way put an upper bound on the pervasiveness of this phenomenon. At the end of the day, any risk analyst who plans to represent the results of a statistical inference using an additive belief function must determine whether their proposition of interest (e.g., “Will these two satellites collide?”) is one of those affected by false confidence. If so, the next question is, “How severe?”, because there is no general limit to how bad it can get. That is the message of the false confidence theorem.§.§ Broader ImplicationsFalse confidence is the inevitable result of treating epistemic uncertainty as though it were aleatory variability. Any probability distribution assigns high probability values to large sets. This is appropriate when quantifying aleatory variability, because any realization of a random variable has a high probability of falling in any given set that is large relative to its distribution. Statistical inference is different; a parameter with a fixed value is being inferred from random data. Any proposition about the value of that parameter is either true or false. To paraphrase Nancy Reid and David Cox,[The exact quote reads, “[E]ven if an empirical frequency-based view of probability is not used directly as a basis for inference; it is unacceptable if a procedure yielding regions of high probability in the sense of representing uncertain knowledge would, if used repeatedly, give systematically misleading conclusions.” <cit.>] it is a bad inference that treats a false proposition as though it were true, by consistently assigning it high belief values. That is the defect we see in satellite conjunction analysis, and the false confidence theorem establishes that this defect is universal.This finding opens a new front in the debate between Bayesian and frequentist schools of thought in statistics. Traditional disputes over epistemic probability have focused on seemingly philosophical issues, such as the ontological inappropriateness of epistemic probability distributions <cit.>, the unjustified use of prior probabilities <cit.>, and the hypothetical logical consistency of personal belief functions in highly abstract decision-making scenarios <cit.>. Despite these disagreements, the statistics community has long enjoyed a truce sustained by results like the Bernstein–von Mises theorem <cit.>, which indicate that Bayesian and frequentist inferences usually converge with moderate amounts of data.The false confidence theorem undermines that truce, by establishing that the mathematical form in which an inference is expressed can have practical consequences. This finding echoes past criticisms of epistemic probability leveled by advocates of Dempster–Shafer theory, but those past criticisms focus on the structural inability of probability theory to accurately represent incomplete prior knowledge, e.g., <cit.>. The false confidence theorem is much broader in its implications. It applies to all epistemic probability distributions, even those derived from inferences to which the Bernstein–von Mises theorem would also seem to apply.Simply put, it is not always sensible, nor even harmless, to try to compute the probability of a non-random event. In satellite conjunction analysis, we have a clear real-world example in which the deleterious effects of false confidence are too large and too important to be overlooked. In other applications, there will be propositions similarly affected by false confidence. The question that one must resolve on a case-by-case basis is whether the affected propositions are of practical interest. For now, we focus on identifying an approach to satellite conjunction analysis that is structurally free from false confidence. § THE MARTIN–LIU VALIDITY CRITERIONThe false confidence theorem establishes that, to be structurally free from false confidence, it is a necessary condition that a statistical inference be represented non-additively. However, while necessary, non-additivity is not sufficient. It is trivially easy to imagine a non-additive belief function that is just slightly non-additive and therefore suffers from a false confidence problem as bad or almost as bad as that depicted in Section <ref>. What we need is a criterion, by which to constrain statistical inference, that is sufficient for eliminating false confidence or, more precisely, managing the severity with which false confidence manifests itself. The Martin–Liu validity criterion states that the aleatory probability with which a false proposition will be assigned some amount of belief must be no more than one minus that level of belief <cit.>. Stated mathematically,Pro_X;θ( { x : Bel_Θ;x( A ) ≥ 1 - α}) ≤α , ∀ α∈[ 0 , 1 ] , A ⊂Ω_θ , s.t.A ∌θ. Since the Martin–Liu validity criterion explicitly limits the rate at which belief is assigned to false propositions, it is almost a tautology that a statistical approach satisfying this criterion will not suffer from the severe false confidence phenomenon described in Sections <ref>–<ref>.This part of the paper implements the Martin–Liu validity criterion as a way to identify statistical methods that are free from false confidence. Section <ref> proves the Martin–Liu validity of confidence intervals and confidence regions. Section <ref> shows that Kσ uncertainty ellipsoids can often be interpreted as confidence regions. This provides a statistically reliable tool for managing collision risk in satellite conjunction analysis. Section <ref> provides confidence ellipses on the two-dimensional displacement between the two satellites at closest approach. §.§ Confidence RegionsFrequentist confidence intervals and confidence regions have traditionally been rationalized via coverage probability. That is, a ( 1 - α) × 100% confidence interval or confidence region is accepted as valid because, over an infinite number of repeated independent draws of the data, at least ( 1 - α) × 100% of the resulting confidence intervals or confidence regions would cover the true parameter value. However, if evaluated as tools for assigning belief to propositions, i.e., sets, confidence intervals and confidence regions also satisfy the Martin–Liu validity criterion.Let Γ_α( x ) be the ( 1 - α) confidence region for θ given the realized data x. The natural assignment of belief and plausibility made by this confidence region are Bel_Θ;x( A ) =1 - α , A ⊃Γ_α( x ) 0 , A ⊅Γ_α( x ) andPls_Θ;x( A ) =1 , A ∩Γ_α( x ) ≠∅ α , A ∩Γ_α( x ) = ∅.That is to say, a confidence region represents the simple assertion that we are 1-α confident that the true value of θ is somewhere inside Γ_α( x ). Any sets containing Γ_α( x ) inherit that confidence; all other sets accrue no positive confidence. It is a coarse way to represent a statistical inference, perhaps more so than is necessary, but it is also safe in the sense that the Martin–Liu validity criterion holds. Indeed, by the coverage probability definition of a confidence region, the event {Γ_α( x ) ∋θ} has Pro_X;θ-probability at least 1-α. So, for any set A ∌θ, the event {Γ_α(x) ⊂ A} has Pro_X;θ-probability at most α. Therefore, for any false proposition, i.e., any set A such that A ∌θ, the probability that said proposition will be assigned a confidence of at least 1-α is less than or equal to α, hence (<ref>) holds. When interpreted this way, a confidence region represents a consonant confidence structure, i.e., a possibility distribution with coverage probability properties <cit.>. A recent paper already proves that all consonant confidence structures satisfy the Martin–Liu validity criterion <cit.>. The reason that we have included the special case proof for confidence regions is that most readers are likely to be unfamiliar with possibility distributions and confidence structures. In contrast, simple confidence intervals and confidence regions are well-established, well-disseminated tools of statistical inference. As proven above, they do not suffer from false confidence.[It may be helpful to remind the reader that, as pointed out in Section <ref>, confidence distributions in the sense of <cit.> do suffer from false confidence, despite also being rationalized in terms of coverage probability. Consonance and coverage probability together are adequate to ensure Martin–Liu validity, but coverage probability alone is not.] §.§ Uncertainty Ellipsoids in Satellite Conjunction AnalysisUncertainty ellipsoids are already well-established as a screening tool in conjunction analysis <cit.>. We propose that they can be used as a principal risk management tool. K σ ellipsoids, also known as covariance ellipsoids, can often be treated as approximate confidence regions. The case for this interpretation starts by treating a satellite's true state, θ, as a fixed unknown with θ̂( x ) as an estimate subject to random errors. If a sufficiently large amount of data is used, the error between a satellite's filter-derived state estimate and its true state can often be treated as approximately normal, even if the observation errors are drawn from non-normal distributions <cit.> and even if the filter used is sampling-based <cit.>. That is, in conjunction analysis, one can often assumeθ̂( X ) - θ∼ n( 0 , C_Θ)where n( 0 , C_Θ) is a multivariate normal distribution with zero mean and covariance matrix C_Θ. This assumption may break down if a conjunction analysis is done too far in advance <cit.> or if too few data are used to estimate the satellite state or if there is an unaccounted for bias error in the filter algorithm or data. For simplicity, we persist in the normality and unbiasedness assumptions. Relaxing these assumptions is left as a topic for future work. This representation of aleatory variability in θ̂( X ) - θ can be eigendecomposed as θ̂( X ) - θ = E_C_ΘΛ_C_Θ^1/2ξwhere X is a (hypothetical) random realization drawn from the same distribution as the data used to estimate the orbits, E_C_Θ is a matrix whose columns are the eigenvectors of C_Θ, Λ_C_Θ is a diagonal matrix whose entries are the corresponding eigenvalues of C_Θ, and ξ is a unit normal vector of the same dimensionality as θ. This can easily be rearranged into a pivot, as Λ_C_Θ^-1/2 E_C_Θ^⊤( θ̂( X ) - θ) = ξ. A pivot is a function of the parameter being inferred and the random data used to infer it whose output is a random variable with a known fixed distribution <cit.>; in this case, ξ is a unit normal vector. Pivots are special because they can be used to derive confidence regions. In particular, we are interested Kσ uncertainty ellipsoids. A K σ uncertainty ellipsoid on θ is generated by projecting a sphere in ξ-space centered at the origin. Let K > 0 be some constant. A confidence region of the form Γ_α( x ) = {ϑ : ξ = Λ_C_Θ^-1/2 E_C_Θ^⊤( θ̂( x ) - ϑ) , ∑_i=1^(θ)ξ_i^2 ≤ K^2 } will map to θ as a (θ)-dimensional ellipsoid. The confidence, ( 1 - α ), associated with this uncertainty ellipsoid can be computed as 1 - α = Pro_X;θ( { x : Γ_α( x ) ∋θ}) = Pro_ξ( {ξ : ∑^(θ)_i=1ξ^2_i ≤ K^2 }) = F_χ^2;(θ)( K^2 ) where F_χ^2;j is the central χ_j^2 distribution function.Software is commercially available that is capable of computing three-dimensional Kσ uncertainty ellipsoids representing uncertainty in the predicted position of a satellite and then determining whether or not the resulting ellipsoids for two satellites intersect <cit.>. If two ellipsoids overlap in a conjunction analysis, a collision avoidance maneuver may be needed. If they do not, the two satellites are considered safe, to within the confidence level associated with the uncertainty ellipsoids.There are a couple of caveats to the approach as currently implemented in the field. First, as traditionally conceptualized, ellipsoid overlap approaches fail to account for the physical size of the two satellites <cit.>. To correct for this, effective ellipsoid overlap should be defined as occurring when the minimum distance between the two uncertainty ellipsoids is less than the combined radius of the two satellites. So long as the position uncertainties are much larger than the satellites' combined size, the distinction is negligible. However, satellite size should not be ignored in problems with relatively small position uncertainties. The other caveat is that, because the two satellites are treated separately, the joint confidence associated with having captured both satellite positions in their respective uncertainty ellipsoids is usually slightly lower than the confidence individually assigned to each ellipsoid. If each ellipsoid has a 1-α confidence attached to it, assuming independence between the data used to infer the two satellite trajectories, the confidence attached to simultaneous coverage is 1 - α' = Pro_X_1,X_2;θ_1,θ_2( { x_1 , x_2 : Γ_α,1( x_1 , A ) ∋θ_1 , Γ_α,2( x_2 , A ) ∋θ_2 }) = ( 1 - α)^2. In reality, the errors in the data used to estimate the two satellite positions may not be entirely independent, but the specter of non-independence is easily resolved. Using Fréchet bounds <cit.>, even under totally unknown dependence between the data, 1 - α' ≥max( 0 , 1 - 2α) . Note that if α is small, ( 1 - α)^2 = 1 - 2α + α^2 ≈ 1 - 2α. So, using 1-2α to represent the joint confidence associated with the two uncertainty ellipses is both conservative and extremely close to the independence case. Thanks to the result about confidence regions presented in Section <ref>, so long as one performs a maneuver whenever the two uncertainty ellipsoids intersect, the rate at which collisions occur over a large number of conjunctions—i.e., the operational aleatory probability of collision—will be capped at α' = 2α. For example, suppose one were to use a pair of 4σ ellipsoids to represent uncertainty in the positions of the two conjuncting satellites. This represents a 99.9% confidence region around each satellite position. Moreover, one can say with 99.8% confidence that both satellite positions are contained within their respective ellipsoids. And, more importantly, if a satellite operator performs a collision avoidance maneuver whenever those 4σ ellipsoids overlap, the operational probability of collision will be capped at 0.227%.All of that being said, ellipsoid overlap does not mean that two satellites are definitely going to collide. Rather, it means that collision is still plausible in light of the data. Performing a collision avoidance maneuver is one way of driving down the plausibility of collision, but it is not the only way. For conjunctions predicted several days in advance, prudent navigators will opt to gather more data, which will allow them to shrink the uncertainty ellipsoids before finally deciding whether or not to make a collision avoidance maneuver. So long as they do not wait too long, this is usually the most reasonable course of action.§.§ Uncertainty Ellipses on DisplacementAn alternative approach based on confidence regions is to derive an uncertainty ellipse on the two-dimensional displacement introduced in Section <ref>. Assuming that variability in the direction of the relative velocity vector is small enough to be ignored, the (random) estimate of displacement at closest approach, ( Û' , V̂' ), has a bivariate normal distribution whose mean is equal to the unknown true displacement, ( u' , v' ). That is,[Û'; V̂' ] = [ u'; v' ] + E_S [ S_1 0; 0 S_2 ][ ξ_1; ξ_2 ] where ξ_1 and ξ_2 are independent unit normal variates and E_S, S_1, and S_2 are the eigendecomposition of the two-dimensional displacement covariance matrix, C_Δ;1:2,1:2, defined in Section <ref>. A K σ confidence ellipse can be constructed on ( u' , v' ) by projecting a circle, ξ_1^2 + ξ_2^2 ≤ K^2, onto ( u' , v' ). The confidence associated with this ellipse is computed from a chi-squared distribution with two degrees of freedom, as outlined in Section <ref>. Recall that collision is associated with the condition u'^2 + v'^2 ≤ R^2, where R is the combined “radius” of the two satellites. So long as the confidence ellipse for ( u' , v' ) does not intersect this circle, then one can say with ( 1 - α) confidence that collision will not occur.The advantage of this approach is that it will yield a lower false alarm rate than the ellispoid overlap approach described in Section <ref>. The disadvantage is that it involves two assumptions that the ellipsoid overlap approach does not, namely that variability in the direction of the estimated relative velocity vector is negligible and that the covariance between the position estimate errors for the two satellites is known. This second assumption is often taken for granted, but it is non-trivial. Additionally, supporting software for the two-dimensional confidence ellipse on displacement is not yet widely available to satellite navigators. § CONCLUSIONSThe work presented in this paper has been done from a fundamentally frequentist point of view, in which θ (e.g., the satellite states) is treated as having a fixed but unknown value and the data, x, (e.g., orbital tracking data) used to infer θ are modeled as having been generated by a random process (i.e., a process subject to aleatory variability). Someone fully committed to a subjectivist view of uncertainty <cit.> might contest this framing on philosophical grounds. Nevertheless, what we have established, via the false confidence phenomenon, is that the practical distinction between the Bayesian approach to inference and the frequentist approach to inference is not so small as conventional wisdom in the statistics community currently holds. Even when the data are such that results like the Bernstein-von Mises theorem ought to apply, the mathematical form in which an inference is expressed can have large practical consequences that are easily detectable via a frequentist evaluation of the reliability with which belief assignments are made to a proposition of interest (e.g., “Will these two satellites collide?”).Our rationale for framing conjunction analysis in frequentist terms is that, as established in the opening of Section <ref>, everyone in the space industry has an interest in limiting the number of in-orbit collisions. Thus, our goal has been to help satellite operators identify tools adequate for limiting the literal frequency with which collisions involving operational satellites occur. Framing the problem in frequentist terms enables us to do that, whereas framing the problem in Bayesian terms would not. The only circumstance in which a Bayesian analysis could directly enable satellite operators to control the frequency with which collisions occur is if it were based on an aleatory prior <cit.> on the satellite states, and this prior would need to reflect the underlying risk of collision due to orbital crowding. As currently practiced, no such aleatory prior is used in conjunction analysis, nor, in our estimation, is one likely to become available in the coming years. An estimate of the aggregate collision risk per unit time seems feasible, but how to parse that into priors on the satellite states is non-obvious. Such an operation may not be well-posed. So, for someone interested in limiting the literal frequency with which collisions occur, it is necessary to treat the satellite states in each conjunction as a fixed, albeit uncertain, reality and then to assess how reliably a proposed risk metric performs. That is the analysis pursued in Section <ref>, and under that analysis, epistemic probability of collision does not appear to be a viable risk metric. There are other engineers and applied scientists tasked with other risk analysis problems for which they, like us, will have practical reasons to take the frequentist view of uncertainty. For those practitioners, the false confidence phenomenon revealed in our work constitutes a serious practical issue. In most practical inference problems, there are uncountably many propositions to which an epistemic probability distribution will consistently accord a high belief value, regardless of whether or not those propositions are true. Any practitioner who intends to represent the results of a statistical inference using an epistemic probability distribution must at least determine whether their proposition of interest is one of those strongly affected by the false confidence phenomenon. If it is, then the practitioner may, like us, wish to pursue an alternative approach. Fortunately, the Martin–Liu validity criterion provides a formal benchmark by which to identify statistical methods that are inherently free from severe false confidence. For example, in Section <ref>, we proved that frequentist confidence intervals and confidence regions satisfy the Martin–Liu validity criterion, thus identifying a widely-disseminated and well-established statistical tool that is free from severe false confidence. Technically, this is a special case of a theorem already proved in <cit.>, but our special case proof is offered in the hope that it will be more accessible to a wider readership. In satellite conjunction analysis, this means that Kσ uncertainty ellipsoids and uncertainty ellipses, which can often be rationalized as confidence regions, are statistically reliable in the same way that epistemic probability of collision is not.§ FUTURE AND ON-GOING WORKMuch work remains to be done exploring the range of real-world uncertainty quantification problems that are practically impacted by false confidence. Risk analysis problems with bounded failure domains are strong candidates for severe false confidence. Recent work demonstrates that false confidence can also arise as the result of non-linear uncertainty propagation, even if a marginalization-specific reference posterior is used <cit.>.As mentioned in Section <ref>, though statistically reliable, confidence intervals and confidence regions are also coarse. More nuanced possibilistic solutions to problems of statistical inference can be derived using consonant confidence structures or Martin–Liu inferential models. In satellite conjunction analysis, the next natural step is to evaluate the statistical performance of the alternative collision risk metrics mentioned in Section <ref> to see which, if any, satisfy the Martin–Liu validity criterion. Finally, a recent conference paper <cit.> asserts that the ellipse overlap approach will motivate too many collision avoidance maneuvers. The authors of <cit.> argue for epistemic probability of collision as a risk metric on the grounds that it is reasonable to treat conjunctions with diluted collision probabilities as akin to the large number of unrecognized conjunctions that occur daily with objects that are too small to be tracked at all, for which no mitigation actions are possible. They view both situations as part of the background risk that is simply accepted as part of operating a satellite. This argument may initially seem attractive, but we see it as promoting an ill-advised tolerance to blind risk. Regardless of the burden incurred by acting on it, the approach we advocate enables satellite operators to quantify and potentially manage their collision risk exposure in the face of high relative uncertainties (i.e., high S/R ratios); epistemic probability of collision does not. We recognize the natural desire to balance the goal of preventing collisions against the goal of keeping maneuvers at a reasonable level, and we further recognize that it may not be possible to achieve an acceptable balance between these two goals using present tracking resources. Nevertheless, committing to a risk metric that cannot reliably distinguish ambiguity from confirmed safety is not, in our view, a durable resolution to this impasse. § COMPETING INTEREST Alexandria Validation Consulting, LLC has developed a proprietary algorithm for computing collision risk in satellite conjunction analysis, based in part on insights derived from the research presented in this paper. To protect the financial interests of M.S.B. and his co-investors, this algorithm is currently being withheld from publication.§ ACKNOWLEDGMENTSThanks go to the anonymous reviewers for the critical-yet-helpful feedback; to Salvatore Alfano for providing M.S.B. with an introduction to the literature on satellite conjunction analysis and probability dilution; and toWilliam Oberkampf for his insights and periodic reminders of the systemic threat posed by in-orbit collisions. Special—though, anonymous—thanks go to the satellite navigator who initially brought satellite conjunction analysis to M.S.B.'s attention. | http://arxiv.org/abs/1706.08565v5 | {
"authors": [
"Michael S. Balch",
"Ryan Martin",
"Scott Ferson"
],
"categories": [
"math.ST",
"stat.TH"
],
"primary_category": "math.ST",
"published": "20170626190921",
"title": "Satellite conjunction analysis and the false confidence theorem"
} |
Auto-Encoder Guided GAN for Chinese Calligraphy Synthesis Pengyuan Lyu1, Xiang Bai1, Cong Yao2, Zhen Zhu1, Tengteng Huang1, Wenyu Liu1 1Huazhong University of Science and Technology, Wuhan, Hubei, China2Megvii Technology Inc., Beijing, China{lvpyuan, yaocong2010}@gmail.com; {xbai, jessemel, tengtenghuang, liuwy}@hust.edu.cn December 30, 2023 =================================================================================================================================================================================================================================================================================== In this paper, we investigate the Chinese calligraphy synthesis problem: synthesizing Chinese calligraphy images with specified style from standard font(eg. Hei font) images (Fig. 1(a)). Recent works mostly follow the stroke extraction and assemble pipeline which is complex in the process and limited by the effect of stroke extraction. We treat the calligraphy synthesis problem as an image-to-image translation problem and propose a deep neural network based model which can generate calligraphy images from standard font images directly. Besides, we also construct a large scale benchmark that contains various styles for Chinese calligraphy synthesis. We evaluate our method as well as some baseline methods on the proposed dataset, and the experimental results demonstrate the effectiveness of our proposed model.§ INTRODUCTION Chinese calligraphy is a very unique visual art and an important manifestation of Chinese ancient culture which is popular with many people in the world. Writing a pleasing calligraphy work is so difficult that it always takes the writer many years to learn from the famous calligraphers' facsimiles. Is there a way to synthesize calligraphy with specified style expediently? We will explore an effective and efficient approach for calligraphy synthesis in this paper.Automatic calligraphy synthesis is a very challenging problem due to the following reasons: 1) Various Chinese calligraphy styles. A Chinese character usually has thousands of calligraphy styles which vary from the shapes of component and the styles of strokes; 2) Deformations between the standard font image and calligraphy image. The standard font image and calligraphy image for the same character are only similar in relative layout of radicals of character but different in the layout and style of strokes.Recently, there are some attempts <cit.> to synthesize calligraphy automatically, which first extract strokes from some known calligraphy characters and then some strokes are selected and assembled into a new calligraphy character. The above mentioned methods are largely dependent on the effect of strokes extraction. However, the stroke extraction technology does not always work well when the Chinese character is too complex or the character is written in a cursive style (Fig. 1(b)) where the strokes are hard to separate and have to be extracted artificially <cit.>.Considering there are some shortcomings in stroke assemble based methods, we treat the calligraphy generation as an image-to-image translation problem and propose a new method which can generate calligraphy with a specified style from a standard Chinese font (i.e. Hei Font) directly without extracting strokes of characters. Over the past few years, many network architectures have been proposed and applied to different image-to-image tasks. However, those networks are all designed to handle the pixel-to-pixel problems, such as semantic segmentation, and poor performance is achieved when there are deformations between the input and target images (Fig. 1(c)).To overcome these problems, we propose a deep neural network based model which consists of two subnets. The first one is an encoder-decoder network acting as image transfer, which encodes an input standard font image to a feature representation and then decodes the feature representation to a calligraphy image with specified style. The encoder-decoder network with similar architecture has been used in <cit.> and show that the feature representation is likely to compress the image content. This network architecture is sufficient to reconstruct an image. But considering that in our task the input images and output images only have the same relative layout among radicals but are different in the layout and style of strokes, it is hard for an encoder-decoder network to yield vivid calligraphy images. So besides the transfer who captures the layout of input standard font image, we also use another encoder-decoder network acting as autoencoder which inputs and reconstructs calligraphy images to guide the transfer to learn the detailed stroke information from autoencoder's low level features. Finally, we train the two subnets together with reconstruct loss and adversarial loss to make the output look real.In summary, the contributions of this paper are two aspects: Firstly, we propose a neural network based method which can end-to-end synthesize calligraphy images with specified style from standard Chinese font images. Compared to some baseline methods, our approach achieves the best results with more realistic details. Secondly, we establish a large-scale dataset for Chinese calligraphy synthesis collected from the Internet. The dataset composes of 4 calligraphy styles and each style contains about 7000 calligraphy images. § RELATED WORK §.§.§ Chinese Calligraphy Synthesis In the past few years, many works on Chinese calligraphy synthesis have been proposed. In <cit.>, Xu et al. propose a method based on shape analysis technique and hierarchical parameterization to automatically generate novel artistically appealing Chinese calligraphy artwork from existing calligraphic artwork samples for the same character. Xu et al. <cit.> propose a method to parameterize stroke shapes and character topology, and successfully transfer font Style Kai into a specific users’ handwriting style by choosing the most appropriate character topology and stroke shapes for a character. Different from the above mentioned methods which follow the stroke extraction and stroke assembly pipeline, we input a standard font image to our model and output a calligraphy image directly.§.§.§ Image-to-Image Translation Image-to-image translation is an extensive concept which covers many tasks such as edge/contour extraction <cit.>, semantic segmentation <cit.>, artistic style transfer <cit.>, image colorization <cit.> et al. in computer vision field. However, in those tasks, image-to-image translation problems are often formulated as pixel-to-pixel translation problem, where the input images and target images have the same underlying structure and without any deformations. In this paper, we focus on another scenario in image-to-image translation where there are deformations between the input and target images. To be specific, in our calligraphy synthesis task, the input standard font images and target calligraphy images only have the similar relative layout among radicals of the same characters but are different in the position and style of strokes.§.§.§ Generative Adversarial Networks Generative Adversarial Networks is proposed by <cit.> which has attracted great interest from the computer vision and machine learning community and has a rapid development <cit.>. GAN is not only used in unsupervised learning such as generate an image from random noise vector but also used with some image-to-image translation tasks <cit.> to make the output look real. Like <cit.>, we train our image transfer using an adversarial loss as well as the reconstruction loss between the output images and target images to generate desirable results. To learn the deformation between the input images and target images,we also reconstruct the low level features of our transfer supervised by the low level feature from an autoencoder.§ PROPOSED METHOD In this section, we describe in detail the proposed method.As shown in Fig. <ref>, our module consists of two encoder-decoder networks which have similar network structure and can be trained together in an end-to-end way. We refer to the two subnets as Supervise Network and Transfer Network respectively, as Transfer Network is used to transfer a standard font image to a calligraphy image with specified style, and Supervise Network can provide supervision information for Transfer Network in training stage. Details of the two subnets are discussed below.§.§ Supervise Network The supervise network is an autoencoder network. The encoder consists of a series of Convolution-BatchNorm-LeakyReLU <cit.> blocks which takes a calligraphy image as input and produces a C×1×1 latent feature representation of that image, where C is the dimension of the latent feature. The decoder is stacked by a series of Deconvolution-BatchNorm-ReLU <cit.> blocks, which takes the latent feature representation from encoder and outputs an image which is similar to the input image. The architecture of the supervise network is a simple CNN based encoder-decoder network but has skip connections between each layer i and layer n-i as <cit.>, where n is the total number of layers of supervise network. The skip connections are essential for the supervise network to output images with photo-realistic details. We have experimented and verified that the simple encoder-decoder network can only output images with the rough layout but almost lost all stroke information, but our supervise network can generate correct strokes as input images. We argue that the feature maps of the bottleneck layer in the simple encoder-decoder lost fine details of input images but the spatial structure is kept, and that skip connections can provide the decoder with detailed information.§.§ Transfer Network The transfer network is also a CNN based encoder-decoder network which inputs a standard font image and generates a calligraphy-like image. The encoder and decoder are similar as the supervise network which is composed by a series of Convolution-BatchNorm-LeakyReLU and Deconvolution-BatchNorm-ReLU blocks respectively, but there is a little difference in skip connections. Chinese characters have diverse and complicated layouts and are hard to transform to calligraphy image from standard font image even the two images have the same layout. Instead of concatenating the feature outputted by layer i and layer n-i directly, we use a residual block <cit.> to connect layer i and layer n-i and sum the feature yielded by the residual block and layer n-i to enhance the capacity to learn the minute difference between the spatial architecture of standard font and specified calligraphy images.The standard font image and the corresponding calligraphy image always have the same character component structure but vary greatly in the layout and style of strokes. The high level features in encoder carry layout information of the input standard font images, but it is not enough to generate calligraphy images with clear strokes and specified style when the model is only supervised by the target calligraphy image. So we use the above supervise network to guide the transfer network. Let S={s_1,s_2,...,s_k} and T={t_1,t_2,...,t_k} denote the low level feature representations yielded by supervise network and transfer network's decoder respectively. We use s_j to supervise t_j in order to guide the decoder of transfer network to learn the feature representations which carry the layout and style of strokes, layer by layer.Generative Adversarial Network (GAN) is recently proposed by <cit.> and has been widely used in image generation tasks <cit.>. Adversarial loss has the effect of learning the same distribution of the ground truth distribution, which can make the output images look real. <cit.> have shown that an image transfer with an adversarial loss can output much sharper results than the one only with L1 loss. We can adjust our transfer network to a GAN framework easily with an additional discriminative model. We treat transfer network as generator G and use a deep network as discriminative model D following <cit.>. In our work, the generator G is optimized to output images which have the same distribution as truth calligraphy images by generating images that are difficult for the discriminator D to differentiate from real images. Meanwhile, D is conditioned on the input standard font images and optimized to distinguish real images and fake images generated by G.§.§ End-to-End Joint Training We train the two subnets jointly in an end-to-end way. Given a pair of training sample (x,y) which is composed of a standard font image x and a calligraphy image y for the same character.For the supervise network, we take calligraphy image y as input and the objective is to reconstruct y. We use L1 loss as our reconstruction loss rather than L2 loss as L1 tends to yield sharper and cleaner image. Let A(.) be the supervise network, the objective of the supervise network can be expressed as: ℒ_supervise=𝔼_y p_data(y)[||y-A(y)||_1]For the transfer network, we input standard font image x and take calligraphy font image y as ground truth. We also reconstruct the low level feature S from the supervise network. We define the reconstruction loss function as: ℒ_reconstruct-1=𝔼_t_1 p_data(t_1)[||t_1-s_1||_1]...ℒ_reconstruct-(k)=𝔼_t_(k) p_data(t_(k))[||t_(k)-s_(k)||_1]ℒ_reconstruct=λ_1×ℒ_reconstruct-1+...+λ_k×ℒ_reconstruct-(k)Besides, we define our adversarial loss as: ℒ_adversarial=𝔼_y∼ pdata(y)[log D(x,y)]+𝔼_x∼ pdata(x)[log(1-D(x,G(x)))]So, our final objective is: G^*=GminDmaxℒ_adversarial+λ_sℒ_supervise+λ_rℒ_reconstruct §.§ Implementation details In this paper, all images are scaled to 256 × 256 and converted to binary images before being fed into the model. In addition, we employ data augmentation to artificially enlarge the dataset for the purpose of reducing overfitting. We flip the image horizontally with probability of 0.5. The encoder of supervise network and transfer network both have 8 stacked Convolution-BatchNorm-LeakyReLU blocks, which yield 1 × 1 latent feature representations of the input calligraphy images and standard font images respectively. The decoder of supervise network and transfer network both have 7 stacked Deconvolution-BatchNorm-ReLU blocks and followed by a Deconvolution layer which will generate 256 × 256 binary images. All Convolution and Deconvolution layers in the above mentioned part have 4×4 kernel size and 2×2 stride. The residual block in transfer net consists of Convolution, Batch normalization and ReLU layers as <cit.> and only exists between the layers whose feature map size are 2×2 and 4×4 of encoder and decoder. The architecture of D is adapted from <cit.>. 7 stacked Convolution-BatchNorm-ReLU blocks are used and followed by a convolution layer and output the probability of the input images like real. The method was implemented in Torch <cit.>. In our experiment, we supervise the decoder of transfer net from the layer with feature map size 16×16 to 128×128 and set λ_1...λ_k to 1, and set λ_s and λ_r to 100. We choose initial learning rate of 0.002 and train the proposed model end-to-end with Adam <cit.> optimization method. This model was trained with batch size set to 16 until the output tends to be stable in training phase. When testing, only the transfer network is used to generate calligraphy images. We also use a median filter to denoise the output image as a post-process method to make the results cleaner.§ EXPERIMENTS In this section, we propose a new benchmark for Chinese calligraphy generation and evaluate our algorithm on the proposed dataset. Besides, we also compare our method with other neural network based image translation methods to prove the effectiveness of our approach. §.§ Dataset As far as we know, there are no existing public datasets for Chinese calligraphy images generation with specified style. Therefore we propose a new dataset for calligraphy images automatic generation collected from the Internet. This dataset contains 4 subsets which are written by 4 famous calligraphers in ancient China, namely Mi Fu, Zhao Zhiqian, Liu Gongquan and Shen Yinmo in different style. Some samples from 4 subsets are shown in Fig .<ref>. What we can see is that the styles of 4 subsets vary from one to another and cover a few categories, such as running script, official script and regular script. As shown, running script shows enormous shape transformation. Official script exhibits wide and flat shapes. Its characters are usually horizontally long and vertically short. Regular script is more clear and neat which is mostly similar to printed fonts. Each subset in our proposed benchmark contains about 7000 images and is split into two set: training set and validation set. We randomly select 6000 images as training set and the rest images are treated as validation set for each style and ensure that the training set and validation set have no overlap in characters. For convenience, we call this dataset Chinese Calligraphy Synthesis-4(CCS-4).§.§ Baseline Methods §.§.§ RewriteRewrite <cit.> is a neural style transfer for Chinese font which is effective to transfer a typographic font to another stylized typographic font. It is a simple top-down Convolution network with big convolution kernel size and 1×1 stride. Each Convolution layer in Rewrite is followed by Batch Normalization layer and a ReLu layer. The architecture of Rewrite is stacked by some above mentioned convolution blocks and end up with a 2×2 MaxPooling layer then followed by a Dropout layer and a Sigmoid layer. The network is minimized by L1 loss and total variation loss.§.§.§ Encoder-Decoder Network Encoder-Decoder network is an effective image-to-image translation model and has been widely used and studied for many image translation tasks, such as semantic segmentation <cit.>, edge extraction <cit.>, image colorization <cit.>, image restoration <cit.> and image style transfer <cit.> etc. We use the architecture proposed by <cit.> as a baseline and train the model with L1 loss and adversarial loss.§.§.§ UNet UNet is proposed by <cit.> and is an extension of the simple encoder-decoder method. In <cit.>, skip connections are used to connect encoder and decoder, based on the fact that in an image translation task, the input and output differ in surface appearance, but both are renderings of the same underlying structure. Besides, the network is also optimized with L1 loss and adversarial loss. §.§ Evaluation and Discussions We evaluate our proposed method as well as other baselines on the CCS-4 dataset. We use the font style Hei as character’s standard font for each method, as Hei font has the least style structures, even thickness and reduced curves. So using Hei font as character's standard font can avoid the similarity between input font style and output calligraphy style, which may increase the difficulty of calligraphy generation but can evaluate the robustness and effectiveness of the evaluated methods. §.§.§ Qualitative Results We train a model for every above mentioned baseline methods as well as our method on the four subsets individually. We show some samples generated by the baseline methods and our proposed method in Fig. <ref>. Our method achieves the best results on all the subsets. Specifically, Rewrite and Encoder-Decoder method achieve the worst results. The images generated by Rewrite and Encoder-Decoder only have the right spatial structure of Chinese character component but the layout and style of strokes are far from satisfactory. As Fig. <ref> shows, In Mi Fu and Zhao Zhiqian subsets, almost all strokes can not match the ground truth strokes.The UNet method achieves a better result than the Encoder-decoder result. Some results are close to ground truth both in global style and local stroke style but have a small part of wrong strokes on the Zhao Zhiqian, Liu Gong quan and Shen Yinmo subsets. However, the results on Mi Fu subset is a little unsatisfactory. We argue that the layout of strokes in Zhao Zhi qian, Liu Gongquan and Shen Yinmo are very similar to the strokes of the standard font, which is much easier to transform for the translation invariance of CNN. But there are significant differences between the standard font and Mi Fu calligraphy which may be too hard for UNet to learn.The best results are obtained by our method. Even evaluated on Mi Fu subset, our model can still generate images with the similar style of the global image and the local stroke. In some scenes, such as the input character has complex stroke structure, our method still can handle well.§.§.§ Effect of The Supervise Network In practice, low level features are hard to learn well in ordinary autoencoder models, so we add Supervise Network in our model as a reference to guide the transfer network to learn detail layout and style of strokes. In Fig. <ref>, we compare our model with the one without Supervise network while other parts maintain the same design. In the aspect of perceptual quality of the generated font images, our model beats the one without Supervise network which holds a general structure of the character while having some wrong fine details.§.§.§ Effect of The Adversarial Loss From the results shown in the Fig. <ref>, we can see that the introduction of GAN Loss helps to improve the quality of the generated image. It is obvious that there are more valid vivid details of the characters are added and the generated image tends to be more sharp with much less blur. Take the first column as example, we can see that the generated image is similar with the ground image in shape layout but loses some details. However, after adding GAN Loss, the image generated is more sharp and detailed. We can draw the conclusion that GAN Loss helps the generator mitigate the blur and add the details which cannot be captured only with the L1 Loss.§.§.§ Analysis of The Standard Font Our approach achieves desirable results when we use Hei Font as the standard font. Here, we use a different font as our standard font to explore the affect of the standard font. As shown in Fig. <ref>, we use Kai font as the standard font, and our model can still output photo-realistic calligraphy images which shows the robustness of our method.§ CONCLUSION In this paper, we propose a model consisting of two subnets: transfer network and supervise network to synthesize Chinese calligraphy. The transfer network can transfer a standard font image to a calligraphy image with specified style and the supervise network can supervise the transfer network to learn detailed stroke information. The two subnets can be trained together. Compared with recent Chinese calligraphy generation works, our approach can generate Chinese calligraphy images from standard font images directly. Besides, we also establish a benchmark for Chinese calligraphy generation and evaluate our method as well as other baseline approaches on the proposed dataset. our method achieves the best results. In the future, we will investigate methods that can handle large deformation between the input target images and expand our method to more general problems.plain | http://arxiv.org/abs/1706.08789v1 | {
"authors": [
"Pengyuan Lyu",
"Xiang Bai",
"Cong Yao",
"Zhen Zhu",
"Tengteng Huang",
"Wenyu Liu"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170627113531",
"title": "Auto-Encoder Guided GAN for Chinese Calligraphy Synthesis"
} |
E-mail: [email protected] Department of Computer Science, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, Aichi, 466-8555, Japan RIKEN Center for Advanced Intelligence Project 15F, 1-4-1, Nihonbashi, Chuo-ku, Tokyo, 103-0027, Japan RIKEN Center for Advanced Intelligence Project 15F, 1-4-1, Nihonbashi, Chuo-ku, Tokyo, 103-0027, Japan Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, JapanThis study investigated typical performance of approximation algorithms known as belief propagation, greedy algorithm, and linear-programming relaxation for maximum coverage problems on sparse biregular random graphs. After using the cavity method for a corresponding hard-core lattice–gas model,results show that two distinct thresholds of replica-symmetry and its breaking existin the typical performance threshold of belief propagation.In the low-density region, the superiority of three algorithmsin terms of a typical performance threshold is obtained by some theoretical analyses.Although the greedy algorithm and linear-programming relaxation have thesame approximation ratio in worst-case performance, their typical performance thresholds are mutually different, indicating the importance of typical performance. Results of numerical simulations validate the theoretical analyses and imply further mutual relations of approximation algorithms75.10.Nr, 89.70.Eg, 89.20.Ff, 02.60.PnTypical Approximation Performance for Maximum Coverage Problem Koji Hukushima December 30, 2023 ==============================================================§ INTRODUCTIONApproximation algorithms, which are important for hard optimization problems, have attracted researchers' interest because, for NP-hard optimization problems, oneencounters difficulty when solving optimal solutions exactly in polynomial time of the problem size. Since the P versus NP problem arose, development and performance analyses of approximation algorithms have persisted as a central issue of computer science and operations research.Two performance evaluations of approximation algorithms exist: worst-case performance and typical (or average-case) performance. As described in this paper, we specifically examine the latter mainly using statistical–mechanical methods, although the former has been investigated mainly in the literature of theoretical computer science <cit.>. Worst-case performance is defined by the pair of an optimization problem and its approximation algorithm. An approximation ratio is then defined by the maximal ratio of an optimal value to an approximation value over all instances if the problem is a maximization problem (and vice versa, otherwise). It is important to provide strict performance guarantee of the algorithm,although the worst-case instance is sometimes pathological. However, the typical performance is defined as the average performance of approximation algorithmfor a given optimization problem over randomized instances. It is sometimes useful for practical use. It sheds light on properties of optimization problems andapproximation algorithms in a perspective that is different from worst-case analysis. An interesting point of the typical property of optimization problemsis found in a close relation to the spin-glass theory in statistical physics. Extensive studies of various computational problems have revealed that the concept of replica symmetry (RS) and its breaking (RSB) in spin-glass theory reflects average computational complexity <cit.> andstructure of the solution space <cit.> of the problems.Typical performance of approximation algorithms often exhibits a phase transition on typical goodnessor accuracy of approximation, which is called threshold phenomena in the literature of theoretical computer science <cit.>. Some approximation algorithms for minimum vertex covers (min-VC) are good examples of threshold phenomena. The problem is defined on a graph. The randomized problem ischaracterized using a random graph ensemble with c being the average degree. From a statistical–mechanical perspective, the problem on Erdös-Rényi random graphshas the RS-RSB threshold at c=e=2.71… <cit.>. Moreover, three approximation algorithms have been investigated: belief propagation (BP)(or message passing) <cit.>, greedy leaf-removal algorithm <cit.>, andlinear-programming (LP) relaxation <cit.>. Approximation algorithms other than BP naively have no direct connection to the spin-glass theory. Nevertheless, for Erdös-Rényi random graphs, these algorithms have the same performance threshold as the RS-RSB threshold. However, subsequent studies of general random graphs <cit.> indicate that their thresholds are not equivalent for some random graphs because of their graph structure. Using statistical–mechanical techniques, typical performance has been studiedfor variants of leaf removal <cit.>and relaxation technique such as LP relaxation <cit.> and semidefinite-programming relaxation <cit.>. These studies have revealed not only typical approximate performance itself but also a suggestive connectionto typical properties of optimization problems, random graph structure, and the spin-glass theory.As described in this paper, we examine the unweighted maximum coverage (max-COV) problem defined in the next section. Although the max-COV belongs to the class of NP-hard,it has several practical applications such as pan and scan problems <cit.>,multi-topic blog watch <cit.>, and text summarization <cit.>. Our main purpose is to examine the typical performance of approximation algorithms for the problem and to compare their performance thresholds. We specifically examine three approximation algorithms: belief propagation, greedy algorithm, and LP relaxation. Some statistical–mechanical methods applied to both analytical and numerical analyses together with mathematical rigorous discussions clarify typical performance of those algorithmsand a suggestive mutual relation among approximation algorithms.This paper is organized as follows.In the next section, we define details of the max-COV and its approximation algorithms. As a random graph ensemble, biregular random graphs are also defined. In Section <ref>, a hard-core lattice–gas model for the problem is introduced. Its BP equations are obtainedbased on Bethe–Peierls approximation. In Section <ref>, we present a study of the typical performance of BP using the RS cavity method. Calculation of the spin-glass susceptibility provides the threshold below which BP typically approximate max-COV with high accuracy. In Section <ref>, the greedy algorithm is analyzed based on a mean-field rate equation of its deletion process. Using the obtained solution, the typical performance threshold of the greedy algorithm is evaluated. In Section <ref>, LP-relaxed approximate values are evaluated rigorously. The theorem is proved using the weak duality theorem. In Section <ref>, we describe some numerical results whichsupport the validity of theoretical analyses presented in the previous sections. We also execute some additional simulations to consider the typical performance of a modified greedy algorithm and randomized rounding of LP relaxation. These results provide suggestive relation between approximation algorithms.The last section is devoted to a summary and discussion of the results. § MAX-COVER PROBLEM AND APPROXIMATION ALGORITHMS §.§ Maximum coverage problemThe max-COV is defined as follows:Let S be a set of M elements and 𝒮={S_1,⋯,S_N} be a collection of subsets of S. For a given positive integer K(≤ N), the problem is to choose at most K subsets to maximize the total number of elementsin the union of chosen subsets. An example of an instance with M=4 and N=3 is shown at the left-hand side of Fig. <ref>. Given that K=2, one can select subsets S_1 and S_3 such that all elements are included in the union of the subsets. Converting an instance to a bipartite graph, one obtains an integer-programming (IP) representation of the max-COV. Let G=(V_1,V_2,E) be a bipartite graph where each edge in E connects to a vertex in V_1 and a vertex in V_2. Vertices in V_1 and V_2 are labeled respectively by i∈{1,…,N} and a∈{1,…,M}. V_1 corresponds to a collection of subsets 𝒮, whereas V_2 represents S. Each edge (i,a) is set if subset S_i includes element a. For Fig. <ref>, the left example is converted to a bipartite graph in the right-hand side. We then introduce binary variables {x_i} and {y_a}, respectively, to V_1 and V_2. Also, x_i is set to one if vertex i (or subset S_i) is selected and zero otherwise. Similarly, y_a is set to one if vertex a is connected to a selected vertex (or element a belongs to the union of chosen subsets) and zero otherwise. The problem is therefore represented by the following IP problem:[ ∑_a=1^My_a,;y_a≤∑_i; (i,a)∈ Ex_i∀ a∈ V_2,; ∑_i=1^N x_i≤ K,; x_i∈{0,1} ∀ i∈ V_1,y_a∈{0,1} ∀ a∈ V_2. ]The inequality in the second constraint can be replaced with an equality sign because the problem is a maximization problem.As described in this paper, typical-case property of the max-COV is analyzed by randomizing its instance. Then, we introduce ρ_x =K/N and assume that ρ_x∈[0,1] is constant. For random bipartite graphs, we specifically examine (L,R)-biregular random graphs where degrees of each vertex in V_1 and V_2 respectively denote L and R. Using this simple random graph ensemble, our statistical–mechanical analyses reveal interesting properties of the problem.To take the large graph limit, the number of vertices in V_2 is rescaled as M=α N with a constant factor α. We assume that a random graph is sparse, i.e., L and R are constant with respect to N. For the randomized max-COV, we define an average optimal cover ratio over random graphs with cardinality N+M and a given ρ_x byρ_y(ρ_x;N)=1/M∑_a=1^My_a^opt(G,ρ_x),where {y_a^opt(G,ρ_x)} represents optimal solutions in V_2 and (⋯) representsan average over random bipartite graphs with size N+M. Its limiting value to N→∞ is denoted by ρ_y. Wesimply call it the average optimal cover ratio.§.§ Approximation algorithmsWe introduce three approximation methods for max-COVs. The first algorithm is belief propagation (BP). The recursive equations called BP equations are derived from Bethe–Peierls approximationfor a spin system corresponding to a given optimization problem. For systems on trees, graphs with no cycles, the Bethe–Peierls approximation and BP are exact. In general, however, there exist cycles in a graph yielding correlations between variables or spins, which results in inexact estimation of solutions (or configurations). BP on graphs with some cycles is regarded as an approximation method. It is called loopy BP <cit.>.The second one is a simple greedy algorithm.At each step, this algorithm has the following procedure: (i) choose one vertex named i with the maximum degree in V_1, (ii) delete vertices neighboring to vertex i from V_2, and (iii) update V_1 to V_1\ i and return to (i) if |V_1|>N-K. This simple algorithm gives an approximation ratio of 1-1/e <cit.>. The problem cannot be approximated within this ratio unless P=NP <cit.>.The last is linear programming (LP) relaxation. An integer programming problem including the max-COV is relaxed to LP problems by replacing integral constraints with real constraints. The LP-relaxed max-COV is given as[ ∑_a=1^My_a,;y_a≤∑_i; (i,a)∈ Ex_i∀ a∈ V_2,; ∑_i=1^N x_i≤ K,;x_i∈ [0,1] ∀ i∈ V_1,y_a∈[0,1] ∀ a∈ V_2. ]Actually, LP approximation value gives the upper bound of the problem. Because LP problems are solvable exactly in polynomial time, LP relaxation is a widely used approximation technique. The approximate solution obtained by LP relaxation usually involves non-integers. One must round those numbers appropriately to obtain an approximate integral solution for the IP problem. Here, we consider randomized rounding <cit.>. Using the obtained approximation solution {x_i^LP}, one selects vertex i∈ V_1 to set I⊂ V_1with probability x_i^LP/K up to K vertices. Then, the rounded solution {x_i^LPr} is set to x_i^LPr=1 for i∈ Iand x_i^LPr=0 otherwise.The rounded approximation value is readily calculated from {x_i^LPr}. In terms of the worst-case performance, LP relaxation and its randomized rounding in expectation has the same approximation ratio as the greedy algorithm. As described in this paper, we study the typical performance of these approximation algorithms. It is evaluated by approximate values averaged over randomized max-COVs defined in the last subsection. Similar to the average optimal cover ratio, the average cover ratio is defined as the average ratio of the approximate valueto the cardinality M of vertex set V_2. It is regarded as exhibiting good typical performance if the average cover ratio obtained by an approximation algorithm is equal to the average optimal cover ratio in the large-N limit. The main aim of this paper is to evaluate the typical performance threshold of ρ_x below which an approximate algorithm exhibits good typical performance with L and R fixed. Evaluation of the approximation algorithms is accomplished by comparing their typical performance thresholds. § BP EQUATIONS FOR MAX-COVAs explained in this section, BP equations for max-COV are derived based onthe statistical–mechanical model for the problem.We set particles on vertices in V_1 and V_2, which respectively occupy vertex i and a if x_i=1 and y_a=1. The hard-core lattice-gas model for max-COVs on bipartite graph G is then naively given as the following partition function.Ξ_0(μ;G)=∑_x∈{0,1}^N∑_y∈{0,1}^Mexp(μ∑_a=1^My_a)H(K-∑_i=1^Nx_i) ∏_a=1^MH(∑_i∈∂ ax_i-y_a ),Therein, H(x)=1 (x≥ 0), 0 (x<0) is the Heaviside step function and ∂ a={i∈ V_1| (i,a)∈ E} stands for a set of neighbors of vertex a∈ V_2.In the partition function, μ represents a chemical potential for particles on V_2. One can construct BP equations for this model. However, it is inconvenient for practical use because of the constraint ∑_i=1^Nx_i≤ K.Therefore, we use the following alternative partition function of Ξ(μ;G)=∑_x∈{0,1}^N∑_y∈{0,1}^Mexp(μ'∑_i=1^Nx_i+μ∑_a=1^My_a) ∏_a=1^MH(∑_i∈∂ ax_i-y_a ),where μ'=μ'(μ,ρ_x;G) is a chemical potential for particles on V_1. The ratio of μ' with respect to μ is defined asκ = -μ'/μ.This parameter is regarded as a Lagrange multiplier for the constraint on the number of selected vertices. The appropriate value of κ must satisfy the condition given byρ_x(μ,κ)=ρ_x,ρ_x(μ,κ)≡⟨1/N∑_i=1^N x_i ⟩_μ,where ⟨…⟩_μ is a grand-canonical average with a given μ. To consider ground states, we take the large-μ limit with parameter κ fixed.First, we construct BP equations for Eq. (<ref>). Using the Bethe–Peierls approximation, the single spin probability P_i(x) which x_i takes x isP_i(x)≃ Z_i^-1e^-μκ x∏_a∈∂ iP_a→ i(x),,where ∂ i={a∈ V_2| (i,a)∈ E} is a set of neighbors of vertex i∈ V_1. Z_∗ is a normalization factor hereinafter. P_a→ i(x) represents the marginal probability of x_∂ a\ i and y_a under the condition x_i=x. Similarly, single spin probability P_a(y) that y_a takes y readsP_a(y)≃ Z_a^-1e^μ y∑_x_∂ aH(∑_i∈∂ ax_i-y )∏_i∈∂ aP_i→ a(x),where P_i→ a(x) is the probability of x_i taking x on the cavity graph G\ a. These probabilities satisfy the following recursive relations under the same approximation. P_i→ a(x) ≃ Z_i→ a^-1e^-μκ x∏_b∈∂ i\ aP_a→ i(x),P_a→ i(x) ≃ Z_a→ i^-1∑_ye^μ y∑_x_∂ a\ iH(x+∑_j∈∂ a\ ix_j-y ) ∏_j∈∂ a\ iP_j→ a(x).To take the large-μ limit later, it is convenient to introduce cavity fields {h_ia} and {ĥ_ai}defined respectively by P_i→ a(x)∝exp(μ h_iax) and P_a→ i(y)∝exp(μĥ_aiy). Then, BP equations for cavity fields are given ash_ia = -κ+∑_b∈∂ i\ aĥ_bi, ĥ_ai = -1/μln[1-1/1+e^-μ∏_j∈∂ a\ i1/1+e^μ h_ja].By rescaling the single spin probability as P_i(x)∝exp(μξ_ix)and P_a(y)∝exp(μξ̂_ay) using local fields{ξ_i} and {ξ̂_a},Eqs. (<ref>) and (<ref>) then readξ_i = -κ+∑_a∈∂ iĥ_ai, ξ̂_a = -1/μln[1-1/1+e^-μ∏_i∈∂ a1/1+e^μ h_ia].One can estimate the single spin probabilityby solving BP equations (<ref>) and (<ref>) as the loopybelief propagation.§ REPLICA-SYMMETRIC SOLUTIONIn this section, typical performance of BP is studied using the RS cavity method based on the simplest RS ansatz. The RS ansatz assumes that cavity fields {h} and {ĥ} are independent random variablesrespectively following probability distributions P(h) and P̂(ĥ). For biregular random graphs, it is apparent that these distributions have no variance because of the absence of fluctuation of degree in V_1 and V_2. We therefore introduce the cavity fields h and ĥ on (L,R)-biregular random graphs. Using BP equations (<ref>) and (<ref>), they satisfy the following RS cavity equations in the large-N limit as h = -κ+(L-1)ĥ, ĥ = -1/μln[1-1/(1+e^-μ)(1+e^μ h)^R-1].Similarly, the local fields on (L,R)-biregular random graphs are set to ξ and ξ̂, which satisfyξ = -κ+Lĥ, ξ̂ = -1/μln[1-1/(1+e^-μ)(1+e^μ h)^R].According to Eq. (<ref>), for a given ρ_x, the local field ξ is determined as e^μξ/1+e^μξ=ρ_x.Then, using Eqs. (<ref>) and (<ref>), the appropriate parameter κ is represented asκ=Lĥ-1/μln r,where r=ρ_x/(1-ρ_x).By substituting Eqs. (<ref>) and (<ref>) to Eq. (<ref>)and by introducing x=e^-μĥ, one obtains the self-consistent equation as (1+r x)^R-1(1-x)=1/1+e^-μ.The order of this solution changes depending on the value of ρ_x. The solution x vanishes as μ becomes large if ρ_x<1/R holds. Using Taylor expansion of the left-hand side of Eq. (<ref>), it is estimated asx= e^-μ/1-(R-1)r+O(e^-2μ).Using Eq. (<ref>), an average density of particles on V_2 readslim_M→∞⟨1/M∑_i=1^M y_a ⟩_μ = 1-(1+rx)^-R/1-(1+rx)^-R+e^-μ= Rρ_x+O(e^-μ)Taking the large-μ limit, the average cover ratio for ρ_x<1/R isobtained as ρ_y^RS= Rρ_x.This relation indicates that each vertex in V_2 is connected to at most one chosen vertex in V_1. However, if ρ_x is larger than 1/R (R≥ 2), then the solution x remains constant for sufficiently large μ.In this case, a simple solution ρ_y^RS=1 is obtained by Eq. (<ref>)because the average cover ratio in Eq. (<ref>) touches to one at ρ_x=1/R.Before closing this section, we discuss the stability of the RS solutionsusing spin-glass susceptibility defined asχ_ SG(μ)=1/N+M∑_i∈ V_1∑_a∈ V_2 (⟨ x_iy_a ⟩_μ-⟨ x_i ⟩_μ⟨ y_a ⟩_μ)^2.Another representation of χ_ SG using cavity fields is known based on the fluctuation–dissipation theorem <cit.>. As shown in <cit.>, in the case of biregular random graphs, it readsχ_ SG(μ)≃∑_d=0^∞λ^d,λ=𝔼[∑_j∈∂ a\ i;b∈∂ j\ a(∂ĥ_ai/∂ĥ_bj)^2],where 𝔼[⋯] represents an average over vertices in V_1 and random graphs. The cavity fields are mutually correlated in general.Actually, BP cannot converge any more <cit.> if the susceptibility diverges. In this sense, the divergence of χ_ SG not only means the instability of the RS solution,but also poor typical performance of BP as an approximation algorithm in the static sense. In the low-density region where ρ_x<1/R holds, it is evaluated asλ = (L-1)(R-1)[r(1-x)/1+rx]^2, →(L-1)(R-1)r^2 (μ→∞).The RS-RSB threshold in the large-μ limit therefore readsρ_x^RS=1/1+√((L-1)(R-1)).Otherwise, the other threshold ρ_x^RS'(>1/R) satisfies(L-1)(R-1)[ρ_x^RS'(1-x^∗)/(1-ρ_x^RS')(1+x^∗)]^2=1,where x^∗ is a solution of(1+ ρ_x^RS'/1-ρ_x^RS'x^∗)^R-1(1-x^∗)=1.In Fig. <ref>, RS-RSB thresholds ρ_x^RS and ρ_x^RS' on (3R,R)-regular random graphs are shown as a function of R. Except for the case R=1, the RS-RSB thresholds separate RS and RSB regions. The RSB region remains for a finite ρ_x while they converge to zero in the large-R limit.§ TYPICAL ANALYSIS OF GREEDY ALGORITHMIn this section, typical behavior of the greedy algorithm on (L,R)-biregular random graphsis investigated in the low-density region of ρ_x<1/R. The corresponding RS solution indicates that one can choose vertices in V_1 without overlapping their neighbors. It is therefore sufficient to analyze the fraction of vertices in V_1 with maximum degreeduring the deletion process. Here we analyze a rate equation that is used frequently for analyses of similar greedy algorithms <cit.>. Let V(T) be the expected number of vertices with maximum degree inV_1 at T-th step of the algorithm. By the definition of the algorithm in sec. <ref>, we find thatNV(T+1) =NV(T)-1-L(R-1)NV(T)/N-RT+O(N^-1),where the assumption L=O(1) is used. By introducing v(t)=V(T)/N and t=T/N,we obtain the following differential equation in the large graph limit. dv(t)/dt=-1-L(R-1)/1-Rtv(t).Under the initial condition v(0)=1, the solution readsv(t)=(L-1)(R-1)(1-Rt)^L(R-1)/R-(1-Rt)/LR-L-R.Let ρ_x^g be a threshold below which vertices with the maximum degree are left at the end of the algorithm. Considering that t represents a fraction of chosen vertices in V_1, we find v(ρ_x^g)=0, that is,ρ_x^g=1/R{1-[(L-1)(R-1)]^-R/LR-L-R}.If ρ_x<ρ_x^g, then no chosen vertex in V_1 has overlapped neighbors in V_2 resulting in good typical performance of the algorithm, i.e., ρ_y^g=Rρ_x. However, the algorithm typically mistakes selections of vertices in V_1 and underestimates the average cover ratio. Results show that ρ_x^g≤ρ_x^RSwhere the equality holds for R=1, indicating that BP is better than the greedy algorithm in terms of the typical performance threshold. § ANALYSIS OF LP RELAXATIONAn LP-relaxed value of the max-COV on any biregular graph is evaluated exactly using the LP duality <cit.> as follows. An LP-relaxed value of the max-COV on any (L,R)-biregular graph is LK if K≤ 1/R holds. x_i=K/N (1≤ i≤ N), y_a=RK/N (1≤ a≤ M) is a feasible solution of LP-relaxed max-COV (<ref>). The value of the cost function is then RKM/N=LK.The Lagrangian function of Eq. (<ref>) is written asL(x,y,p,q)= ∑_a=1^My_a+p(K-∑_i=1^Nx_i)+∑_a=1^Mq_a(∑_i∈∂ ax_i-y_a),where p∈ℝ and q∈ℝ^M. Using the weak duality theorem, one finds the following inequalities.max_x,ymin_p,qL(x,y,p,q) ≤min_p,qmax_x,yL(x,y,p,q)≤max_x,yL(x,y,p,q).Consequently, max_x,yL(x,y,p,q) is an upper bound of the LP-relaxed value. Because Eq. (<ref>) is represented byL(x,y,p,q)= ∑_i=1^N(-p+∑_a∈∂ iq_a)x_i+∑_a=1^M(1-q_a)y_a+pK,where a solution x_i=y_a=1 (1≤ i≤ N,1≤ a≤ M) realizes the maximum of the function with p,q satisfying -p+∑_a∈∂ iq_a≥ 0 and 1-q_a≥ 0 (1≤ a≤ M). Assuming that q_a=q holds for 1≤ a≤ M, one finds thatmax_x,yL(x,y,p,q)=-pN+LNq+M-Mq+pK.On the right-hand side, setting (p,q)=(M/N,K/N) results in LK, which indicates that the upper bound of the LP-relaxed max-COV on (L,R)-biregular graph is LK. Because the approximate solution denoted above achieves this bound, the LP-relaxed value is equivalent to the bound. This theorem claims that ρ_y=Rρ_x (ρ_x≤ 1/R), suggesting that LP relaxation typically findsgood approximate values in the RS regime. The LP-relaxed value is equivalent to ρ_y^RS in the high-density regime where ρ_x>1/R.§ NUMERICAL RESULTSThis section presents description of some numerical results for validation of the theoretical analyses. Here, we set (L,R)=(9,3) as an example. Then, the RS-RSB threshold and the typical performance threshold of the greedy algorithm are evaluated as ρ_x^RS=0.2 and ρ_x^g=(1-2^-4/5)/3=0.1418…. Biregular random graphs are generated based on implementation of the configuration model <cit.>. At least 400 random graphs are used to take a random graph average.§.§ Average cover ratioTo examine the validity of theoretical analyses on the typicalperformance of approximation algorithms, the average cover ratio ρ_y is evaluatedusing several methods. We employ a BP-guided decimation (BPD) algorithm <cit.> as a variant of loopy BPto obtain a feasible solution satisfying all the constraints.The algorithm fixes a value of a variable based on a solution of BP equations (<ref>) until all variables are fixed to either zero or one.When a variable is fixed, BP equations are updated by applying the fixed value.If the number of variables in V_1 fixed to one reaches to K, the remainders of x are immediately fixed to zero to satisfy the constraint.An approximate value of the algorithm is thus a function of the parameter κ, which depends on inputs, i.e., μ, K and a graph.Here, the BP equations with μ=20 are solved iteratively up to 150 steps, which enables the algorithm to fix a variable practicallyeven if the iterations cannot reach to the RS fixed point.The parameter κ is tuned to maximize an approximate value while the detail will be reported elsewhere.In addition, the RS estimation of average cover ratio ρ_y^RS in theRS regime ρ_x<ρ_x^RS is used for comparison to BPDthough its typical performance is possibly evaluated directly as in <cit.>.The theoretical result presented in section <ref> is used for LP relaxation, which is valid for arbitrary biregular graphs. Average approximate values of the greedy algorithm areestimated numerically with N=10^3. We also evaluate the average optimal value using the replica exchange Monte Carlo (EMC) method <cit.>.Results show that single-spin flip updates of the model (<ref>) takes a long relaxation time for equilibration even using the exchange MC, indicating the existence of deep valleys of the free energy.Consequently, it is necessary to accelerate equilibration for a system with sufficiently small chemical potential μ.To avoid such slow relaxation, we consider an alternative lattice–gas model on bipartite graph G represented asΞ_1 (μ;G)=∑_x∈{0,1}^Nexp(μ∑_a=1^Mθ(∑_i∈∂ ax_i)) δ(K,∑_i=1^Nx_i),where θ(x) takes one if x> 0 and zero otherwise, and δ(x,y) is Kronecker's delta. In this model, variables y are eliminated because y_a=θ(∑_i∈∂ ax_i) holdsin the large-μ limit. The ground states of the alternative model therefore correspond to the optimal solutions of the max-COV on the same graph. Moreover, single-spin flip updates in the model are substantially equivalent to multi-spin updatesin the original model (<ref>), which makes the relaxation time to equilibrium states markedly short.Because optimal values are invariant by replacing inequality constraint ∑_i=1^N x_i≤ K to equality one, the equality constraint is adopted and Kawasaki dynamics for a density-conserved system <cit.> is applied. The results are presented in Fig. <ref>.Numerical results are compatible to the RS estimation nearly up to the RS-RSB threshold ρ_x^RS=0.2 for the BPD and EMC, and up to the typical performance threshold ρ_x^g predicted analytically for the greedy algorithm. As shown in the inset of Fig. <ref>,the average cover ratio by the greedy algorithm deviates from theaverage optimal ratio above ρ_x^g, while other estimates split above ρ_x^RS,indicating that the Bethe–Peierls approximation and LP relaxation are no longer appropriate.In fact, as shown in Fig. <ref>, the loopy BP for N=400 fails its convergence above ρ_x≃ 0.195, which may imply the existence of dynamical one-step RSB phase. It is expected that BPD finds approximate solutions with good accuracyif a corresponding loopy BP converges to a fixed point. Otherwise, decimation in BPD is performed based on a wrong estimation by the Bethe–Peierls approximation. We emphasize, however, that BPD finds relatively good approximate solutions even in the RSB phase as shown in Fig. <ref>.From these observations, we confirm that statistical–mechanical analysessuccessfully predict typical performance thresholds by approximation algorithms.§.§ Greedy algorithm and its variantHere we examine the greedy algorithm more closely. To validate the correctness of our analysis, we specifically examine a fraction of selected vertices without the maximum degree. The value r_g is evaluated byr_g=1-ρ_x^g/ρ_x,where ρ_x^g is given as Eq. (<ref>). The algorithm finds an optimal solution of the problem if r_g=0. As depicted in Fig. <ref>, the analytical estimations of r_g averaged over biregular random graphsagree well with the numerical results. The fraction arises at ρ_x^g corresponding to the typical performance threshold. Additionally, we modify the greedy algorithm to reduce the gap of its typical performance to that of BP. As suggested above, the reason lies in the point that the greedy algorithm often chooses wrong vertices leading toshortage of vertices with maximum degree. To avoid the situation, one must select a vertex to preserve as many vertices to a maximum degree as possible, meaning that the optimal selection requires an exhaustive search.We therefore propose a modified algorithm in consideration ofthe influence of the selection on other vertices in V_1, the simplest improved algorithm toward the optimal selection. Let ∂^2 i={j∈ V_1\ i| j∈∂ a (∀ a∈∂ i)}be a set of the second neighbors of vertex i∈ V_1. We also define the subset of vertices with the maximum degree in V_1 by W_1. The modified greedy algorithm is given as follows: at each step, (i) choose vertex named i∈ W_1 so that |∂^2i∩ W_1| is minimized, (ii) delete vertices neighboring to vertex i from V_2, and (iii) update W_1 and V_1 and return to (i) if |V_1|>N-K. In Fig. <ref>, the fraction r_g of the modifiedalgorithm is also shown with that of the original greedy algorithm.As shown in Fig. <ref>, the typical performance threshold of the proposed algorithm isimproved, although it is still below the RS-RSB threshold. This fact suggests that the typical performance of greedy algorithms depends of approximate choices and BP selects vertices better thanthose greedy selections. §.§ Randomized rounding of LP relaxationIn Sec. <ref>, we examine the typical performance of LP relaxation in terms of its approximate value. In this subsection, randomized rounding, a practical means of constructing a feasible integer solution from LP relaxation, is applied to LP-relaxed solutions. It is noteworthy that the typical performance of randomized rounding probably depends on the selection of an LP solver and its setting. Here, we use IBM ILOG CPLEX with a default setting. The approximate value of the rounded solution is compared to that of the RS solution in the RS phase, which is LK. One considers that the rounding finds an optimal integer solution of the problem if two estimates mutually coincide. We define a success ratio p_r by the fraction of random graphs on which the rounding finds an optimal solution. Fig. <ref> presents the average approximation ratio ρ_y^r and the success ratio p_r as a function of ρ_x, which strongly suggests that the randomized rounding exhibits a phase transition in terms of its typical performance. The threshold ρ_x^r is less than 0.1, which is muchlower than ρ_x^g, whereas the LP-relaxed value is regarded as nearly optimal in the RS region ρ_x<ρ_x^RS=0.2. We therefore conclude that the typical performance of the randomized rounding of LP-relaxed solution is inferior to that of the greedy algorithm. § SUMMARY AND DISCUSSIONAs described in this paper, we investigate the typical performance of approximation algorithms called belief propagation, greedy algorithm, and linear-programming relaxation for maximum coverage problem on sparse biregular random graphs. The typical performance of BP is studied by application of the RS cavity method to a correspondent hard-core lattice–gas model. Results show that, in the large-μ limit, there exist two distinct RS-RSB thresholds regarded as typical performance thresholds of BP. In addition, the greedy algorithm performance and LP relaxation were studied especially in the low-density region. Results show that the typical performance threshold of the greedy algorithm is lower than that of BP and that LP-relaxed values are always equivalent to the RS solutions leading to the threshold equivalent to that of BP. Those analytical results were validated by executing some numerical simulations. Results of additional numerical studies suggest that BP typically works better than the modified greedy algorithm andthat randomized rounding of LP-relaxed solutions has a lower threshold than the greedy algorithm.To assess the typical performance of BP as an approximation algorithm, we concentrated on statistical–mechanical analysis of max-COV up to the RS level. Further analyses based on the one step RSB will be necessary to reveal statistical–mechanical properties of the problem and typical performance analysis of another algorithm called survey propagation. Another possible avenue of future work is the extension of our analyses to other random bipartite graphs. As with min-VCs, it is an attractive question whether the magnitude relation of typical performance thresholds changes depending on the random graph ensembles.Our results provide not only respective typical performance of approximation algorithms but also their suggestive mutual relations. Specifically examining LP relaxation, it is worth emphasizing that the LP relaxation finds good approximate values compared to optimal values butthe typical performance of its randomized rounding has the smallest threshold among approximation algorithms studied here. To fill the gap of thresholds, it is important to examine modifications of LP relaxation such as the cutting-plane approach <cit.>. As for the greedy algorithms and their modification, numerical results suggest that evaluation of the influence of their deletion processaffects the marked improvement of the typical performance threshold. The fact that BP is better than the greedy algorithm and its modification indicates that BP incorporates the influence more efficiently. These suggestions are expected to be of great help to understand properties and relations of approximation algorithms in terms of typical performance. In addition, our analyses of the greedy algorithm and randomized rounding of LP relaxation illustrate that typical case and worst case evaluations capture different notions of approximate performance in optimization problems.This fact indicates the importance of the typical-case analysis of approximation algorithms. We hope that the arguments and results presented herein stimulate further studies and that the typical performance analyses of approximation algorithmswill attract the interest of researchers in many diverse fields.ST warmly thanks T. Takaguchi for stimulating this study. The use of IBM ILOG CPLEX has been supported by the IBM Academic Initiative. This research was supported by JSPS KAKENHI Grant Nos. 25120010 (KH), 16K16011 (TM), and 15J09001 (ST). | http://arxiv.org/abs/1706.08721v2 | {
"authors": [
"Satoshi Takabe",
"Takanori Maehara",
"Koji Hukushima"
],
"categories": [
"cond-mat.dis-nn",
"cond-mat.stat-mech",
"cs.DS"
],
"primary_category": "cond-mat.dis-nn",
"published": "20170627083213",
"title": "Typical Approximation Performance for Maximum Coverage Problem"
} |
[pages=1-last]camera_ready.pdf | http://arxiv.org/abs/1706.08574v1 | {
"authors": [
"Zibo Meng",
"Xiaochuan Fan",
"Xin Chen",
"Min Chen",
"Yan Tong"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170626194233",
"title": "Detecting Small Signs from Large Images"
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Center for Artificial Low Dimensional Electronic Systems, Institute for Basic Science (IBS), 77 Cheongam-Ro, Pohang 790-784, Korea Department of Physics, Pohang University of Science and Technology (POSTECH), Pohang 790-784, KoreaDepartment of Physics, Pohang University of Science and Technology (POSTECH), Pohang 790-784, KoreaCenter for Artificial Low Dimensional Electronic Systems, Institute for Basic Science (IBS), 77 Cheongam-Ro, Pohang 790-784, Korea Department of Physics, Pohang University of Science and Technology (POSTECH), Pohang 790-784, KoreaCenter for Artificial Low Dimensional Electronic Systems, Institute for Basic Science (IBS), 77 Cheongam-Ro, Pohang 790-784, KoreaLaboratory for Pohang Emergent Materials, Pohang University of Science and Technology (POSTECH), Pohang 790-784, Korea Rutgers Center for Emergent Materials and Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USALaboratory for Pohang Emergent Materials, Pohang University of Science and Technology (POSTECH), Pohang 790-784, Korea Rutgers Center for Emergent Materials and Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, [email protected] Center for Artificial Low Dimensional Electronic Systems, Institute for Basic Science (IBS), 77 Cheongam-Ro, Pohang 790-784, Korea Department of Physics, Pohang University of Science and Technology (POSTECH), Pohang 790-784, Korea Correlated electronic states at domain walls of a Mott-charge-density-wave insulator 1T-TaS_2 Han Woong Yeom December 30, 2023 =============================================================================================Domain walls in interacting electronic systems can have distinct localized states, which often govern physical properties and may lead to unprecedented functionalities and novel devices. However, electronic states within domain walls themselves have not been clearly identified and understood for strongly correlated electron systems. Here, we resolve the electronic states localized on domain walls in a Mott-charge-density-wave(CDW) insulator 1T-TaS_2 using scanning tunneling spectroscopy. We establish that the domain wall state decomposes into two nonconducting states located at the center of domain walls and edges of domains. Theoretical calculations reveal their atomistic origin as the local reconstruction of domain walls under the strong influence of electron correlation. Our results introduce a concept for the domain wall electronic property, the walls own internal degrees of freedom, which is potentially related to the controllability of domain wall electronic properties.Conductive domain walls and their functionalities <cit.> have been reported in multiferroic insulators <cit.>, magnetic insulators <cit.>, Mott insulators <cit.>, and layered CDW materials <cit.>. CDW with periodic modulations of charge/lattice at low temperature and their domain walls are widely observed in metallic layered transition metal dichalcogenides <cit.>. Among them, 1T-TaS_2 shows a series of CDW transitions coupled to substantial electron-electron interaction. Especially, a metal-insulator transition occurs below 190 K through a transition from a nearly commensurate to a commensurate CDW state <cit.>. The insulating ground state is realized by electron correlation on narrow electron bands formed by the commensurate CDW lattice <cit.>. Recent experimental results have demonstrated that the correlated insulating ground state can be transformed into various quasi-metallic and superconducting CDW states by external and internal control parameters, such as pressure <cit.>, optical (electrical) excitation <cit.>, and chemical doping <cit.>. These conductivity switchings are intrinsically very fast, which may make possible novel ultrafast devices based on correlated electrons.In the metallic excited states, CDW domain walls are common objects and considered as the origin of the metallic property. They have been suggested as highly conducting channels themselves <cit.> and/or to screen electron correlation within Mott-CDW domains <cit.>. Moreover, the emerging superconductivity out of the Mott-CDW insulating phase has been considered being directly related to conducting domain walls <cit.>. However, despite such long discussion, the electronic states of domain walls have not yet been clarified spectroscopically. Here, we show that the domain walls in 1T-TaS_2 have two well-confined and non-metallic in-gap states above Fermi level (E_ F) using scanning tunneling microscopy and spectroscopy (STM and STS). They are located on the domain wall center and edges of neighboring domains, respectively. The theoretical calculations strongly suggest the substantial correlation effect in forming spatially decomposed and non-metallic domain wall states. ResultsDomain walls in the insulating CDW state. Figure <ref>a shows a typical STM image of the commensurate Mott-CDW state in 1T-TaS_2 at 4.3 K. The unit layer of 1T-TaS_2 consists of a Ta layer sandwiched by two S layers with each Ta atom coordinated octahedrally by S atoms. Unpaired 5d electrons of Ta form a half-filled metallic band which is unstable against the CDW formation driven by strong electron-phonon coupling. Ta atoms undergo a reconstruction into a so called David-star unit cell which is composed of 13 Ta atoms; the 12 outer atoms pair up and shift toward the central one <cit.>, where a unpaired 5d electron is left over <cit.>. In the commensurate CDW phase, David-star unit cells (green lines in Fig. 1a) exhibit a regular triangular lattice with a period of 12.1Å and its STM image is dominated by protrusions representing unpaired electrons of central Ta atoms <cit.>. The on-site Coulomb repulsion drives these unpaired electrons into a Mott insulating state, which form otherwise a narrow metallic band. In this way, the insulating state is driven by a cooperative interplay between the CDW and the electron-electron interaction <cit.>. In spite of the clear commensurability and the long range order of the undoped low temperature phase, there exist one-dimensional intrinsic defects, domain walls, across which the phase of CDW is abruptly shifted (see Fig. <ref>b and Supplementary Figure 1). Since there can be 13 atoms within a CDW unit cell,12 anti-phase domain wall configurations are possible as indexed in Fig. <ref>a <cit.>. The configuration of a domain wall can be easily identified by the phase shift (black arrows in Fig. <ref>a, b) of CDW protrusions of neighboring domains (see Supplementary Figure 2). Only few cases among 12 configurations are observed including the most popular ones shown in Fig. <ref>, probably because of the energetics and/or kinetics of the domain wall formation <cit.>. The present work covers two most popular and straight domain walls [indexed as the 2^ nd (upper)and the 4^ th (lower) one following the atomic indices of Fig. <ref>a] which were found to be stable within the experimental time scale. We mainly focus on the2^ nd domain wall which is the majority species <cit.>.Localized in-gap states of the domain wall and edges. Electronic states localized on domain walls are revealed by acquiring STS spectra around domain walls with a high spatial resolution. The previous STS work showed only enhanced spectral weight around the band gap region without observing any distinct electronic states <cit.>. In the present work, for example, STS spectra were measured at each pixel of the STM image containing two domain walls of different but most popular types (Fig. <ref>b).Within the Mott gap region of -0.1 ∼ +0.2 eV around E_ F, one can observe two pronounced localized electronic states.A strong spectral feature appears along the center of domain walls in the empty state at +0.15 eV (two green curves in Figs. <ref>c and <ref>g) and another one also in the empty state at +0.08 eV (a orange curve in Figs. <ref>c and <ref>f) but slightly away from the center of domain walls. The latter is localized along the edge of neighboring domains and a similar but weak feature also appears in the filled state at -0.05 eV (Fig. <ref>d). Note that we do not observe any distinct spectral weight at E_F which is essential for domain-wall-originated metallicity or superconductivity (Fig. <ref>e). The present result is consistent with our recent work for nearly commensurate domain wall networks <cit.>. The localized in-gap state above E_ F were also observed on zigzag domain walls (see Supplementary Figure 3).The dI/dV maps at the energies of the in-gap states show charge modulations along domain walls and edges. Their periodicities are the same as that of the David-star reconstruction. These domain wall and edge states exhibit specific phase relations determined by the phase difference between neighboring domains across the domain wall (see Supplementary Figure 2 and 4). This implies that the domain wall reconstruction is strongly influenced by the periodic potential imposed by the CDW reconstruction in neighboring domains <cit.>.Gapped nature of domain walls and their localized electronic states become more clear in a closer look of point-by-point STS spectra. Figures <ref>a and <ref>b show well resolved STM images of the 2^ nd and 4^ th domain walls, which are the same types as those of Fig. <ref>b. Within domain walls, David-star CDW units cannot be formed completely and the incomplete David-stars reconstruct into pairs of small and large protrusions along domain walls. This will be discussed in more detail below. A series of dI/dV spectra across domain walls are shown in Figs. <ref>d and <ref>g. Away from domain walls, the dI/dV curves reproduce the energy gap of the Mott-CDW state; two Hubbard states construct a Mott gap of 0.44±0.02 eV. The subband splittings (single-headed black arrows in Fig. <ref>d, <ref>g) are footprints of the formation of CDW and the David-star reconstruction <cit.>. Upon approaching the domain wall, CDW protrusions and STS spectra do not show a noticeable spatial variation except for a tiny band bending until they reach the last CDW unit cells of domains, which we call domain edges. As discussed above, at domain edges in both sides of the domain wall, an in-gap state emerges at +0.07 eV. Then, at the domain wall center, the Hubbard states are replaced by a strong spectral feature of +0.15 eV. Note that the domain wall itself is not metallic at all as also shown in the spatial maps at E_ F (Fig. <ref>d). While the edges of domains have higher spectral weight near E_ F, the band gap on the domain edge can be unambiguously defined by the edge of the lower Hubbard band and the new spectral feature above E_ F. Their peak-to-peak splitting are 120 meV and 50 meV on the 2^ nd and 4^ th domain wall, respectively (red double headed arrows in the Fig. <ref>e and <ref>h). Due to thermal and instrumental broadenings, these spectral features leave decaying intensities towardE_ F, which obscure the band gap. However, our higher resolution STS measurements (see Methods) clearly show the zero conductance region at E_ F (see Fig. <ref>e, <ref>h, and Supplementary Figure 5), which makes the existence of the band gap unambiguous. Correlation dependent domain wall and edge reconstructions. In order to elucidate the origin of the domain wall and edge states, first principles calculations were performed.We used a domain supercell in a monolayer 1T-TaS_2 in which a single domain is composed of five columns of David-stars. Each domain is shifted to construct domain wall of the 2^ nd type between neighboring domains. The equilibrium structure after a full relaxation of atomic positions is shown in Fig. <ref>a. Here we use the usual density functional theory framework without putting any extra electron correlation, which substantially underestimates the Mott-CDW band gap at E_ F (Fig. <ref>e) <cit.>. The two overlapped David-star columns in the domain wall region simply split into two symmetric rows of incomplete David-stars with 12 Ta atoms on each. This produces a unique electronic state on the domain wall at the energy of conduction band minimum (arrows in Fig. <ref>e). This domain wall structure and its electronic state deviate qualitatively from what were observed. We then include the extra electron correlation effect with a finite U value within the GGA+U scheme. The U value is tuned up to 2.5 eV in order to properly reproduce the experimentally observed Mott gap of the CDW domain (Fig. <ref>f). With the enhanced electron correlation, the domain wall reconstruction drastically changes;it becomes much narrower with 11 Ta atoms, which split longitudinally into centered hexagons and linear tetramers (see Supplementary Figure 6). The atomic lattice of the domain wall is resolved in our high resolution STM topography while small lattice distortions due to the reconstruction within are hardly detected <cit.> (see Supplementary Figure 7). However, strong charge modulations relevant with the atomic reconstruction are shown clearly in our STM and STS results. The hexagon on the domain wall center exhibits an electronic state at a slightly lower energy than the upper Hubbard band (a green arrow in Fig. 3f).The other states at a lower energy closer to E_ F (a blue and orange arrow in Fig. 3f) emerges at the edges of hexagons and extends to neighboring David-stars units, that is, to domain edges (Fig. 3f). The 4^ th domain wall has also the spatially decomposed in-gap states and a smaller gap feature are reproduced in our calculations U=2.5 eV (see Supplementary Figure 4). This result strongly suggests the crucial role of electron correlation in the formation of the domain wall electronic state.We note that there are still discrepancies between the calculation and the measurement, especially in the spatial distribution of edge states (see Supplementary Figure 4). The maxima of calculated edge states (Fig. 3d) are located closer to the narrow domain wall than those observed while the tendency to spread toward edges of domains is consistent. We suspect that this discrepancy may be related to the interlayer coupling discussed in the previous studies <cit.>, which is not included at all in our calculations. Discussion The electronic state(s) within the band gap observed here is in line with localized states suggested in sharp phase kinks of order parameters in charge or spin ordered insulators <cit.>. Our STM and STS results disclose that the domain wall is not a metallic channel developed by the suppression of the CDW order, which have been assumed by many previous studies <cit.>.On the other hand, a few other studies suggested that free carriers from domain wall electronic states screen the electron correlation in neighboring CDW domains <cit.>. This idea was critically tested in the present work with a clearly negative answer. The domain wall junctions shown in Fig. <ref>b is a good testbed for the screening effect since the CDW domain in between the two domain walls would have an increasingly larger effect of the screening, if any, when approaching closer to the junction point. However, our STS data do not show any substantial change of electronic states from that of the Mott-CDW gap structure as approaching the junction (Fig. <ref>c), except for the domain wall and edge states (the yellow spectrum in Fig. <ref>c). This result clearly rules out the screening scenario to explain the metallicity of the textured CDW states. Of course, the metallic state can be driven by developing a random disorder potential in Mott insulators and decreasing the CDW lateral or vertical ordering <cit.>. However, the single isolated and straight domain wall as discussed here is not related to such a global disorder to melt the correlation gap <cit.>. In summary, we discover distinct electronic states of domain walls of a symmetry-broken correlated insulator for the first time. The domain walls of the Mott-CDW phase of 1T-TaS_2 have well localized non-metallic in-gap states along the center of domain walls and the edges of neighboring domains. Not only the lack of free carriers but also the lack of any substantial screening or doping effects by domain walls are disclosed unambiguously. These results request to rewrite most of the current scenarios on the metallic and superconducting phases emerging from this correlated insulator. The origin of the split non-metallic domain wall states are clearly understood as due to the reconstruction within domain walls composed of multiple atoms and electrons under the substantial influence of electron correlation. That is, the internal structural and electronic degrees of freedom within domain walls are indicated to be very important. The internal structural degree of freedom of domain walls were recently noticed for one <cit.> and two dimensional <cit.> systems, but to the best of our knowledge, the substantial electron correlation effect within a domain wall has not been observed before.These internal degrees of freedom might be exploited to provide controllability of domain wall electronic properties for the functionalization of complex materials with domain walls. phvMethodsphvExperiments. The STM and STS measurements have been carried out with a commercial STM (SPECS) at 4.3 K. Pt-Ir wires were used for STM tips. All STM images were acquired with the constant current mode with bias voltage (V_s) applied to the sample. The standard lock-in technique with voltage modulation V_m=10 mV and frequency f=1 kHz has been adapted for acquiring dI/dV spectra. The high resolution spectra were obtained with V_m=2 mV. A single crystal of 1T-TaS_2 were prepared by iodine vapor transport method, as described in Ref. 18. The samples were cleaved at room temperature and quickly transferred to the pre-cooled STM head.phvTheoretical Calculations. Our calculations were carried out using density functional theory (DFT) with the projector augmented wave method, Perdew-Burke-Ernzerhof (PBE) exchange and correlation functional and GGA+U as implemented in the Vienna ab initio simulation package (VASP). To avoid interactions between the supercell, about 10 Å vacuum is inserted in vertical direction. The plane wave cut-off energy is set to 260.0 eV and the Monkhorst-Pack k-point mesh is 1×2×1. All the atoms are relaxed until forces on the atoms are less than 0.02 eV/Å.phvAcknowledgmentsThis work was supported by the Institute for Basic Science (Grant No. IBS-R014-D1). LW and SWC are partially supported by the Max Planck POSTECH/KOREA Research Initiative Program (Grant No. 2011-0031558) through NRF of Korea funded by MEST. SWC is also supported by the Gordon and Betty Moore Foundations EPiQS Initiative through Grant GBMF4413 to the Rutgers Center for Emergent Materials.phvAuthor contributionsDC and HWY conceived the research idea and plan. DC and JL performed the STM / STS measurements. LW and SWC grew the single crystals. SHL and GG conducted the DFT calculations. DC and HWY prepared the manuscript with the comments of all other authors. 999 phvReferencescatalan2012domain Catalan, G., Seidel, J., Ramesh, R., & Scott, J. F. Domain wall nanoelectronics. Rev. Mod. Phys. 84, 119-156 (2012).seidel2009conduction Seidel, J., et al. Conduction at domain walls in oxide multiferroics. Nat. Mater. 8, 229-234 (2009).meier2012anisotropic Meier, D., et al. Anisotropic conductance at improper ferroelectric domain walls. Nat. Mater. 11, 284-288 (2012).oh2015experimental Oh, Y. S., et al. 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B, 81, 245122 (2010).cheon2015chiral Cheon, S., Kim, T. H., Lee, S. H. & Yeom, H. W.Chiral solitons in a coupled double Peierls chain. Science 350, 182-185 (2015).lahoud2014emergence Lahoud, E. et al. Emergence of a novel pseudogap metallic state in a disordered 2D Mott insulator. Phys. Rev. Lett. 112, 206402 (2014).chiesa2014emergence Chiesa, S., Chakraborty, P. B., Pickett, W. E., & Scalettar, R. T. Disorder-induced stabilization of the pseudogap in strongly correlated systems. Phys. Rev. Lett. 101, 086401 (2008). | http://arxiv.org/abs/1706.08607v1 | {
"authors": [
"Doohee Cho",
"Gyeongcheol Gye",
"Jinwon Lee",
"Sung-Hoon Lee",
"Lihai Wang",
"Sang-Wook Cheong",
"Han Woong Yeom"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170626213125",
"title": "Correlated electronic states at domain walls of a Mott-charge-density-wave insulator 1T-TaS2"
} |
National Tsing Hua University, Department of Physics and Institute of Astronomy, No. 101 Sect. 2 Kuang-Fu Road,30013, Hsinchu, Taiwan National Tsing Hua University, Department of Physics and Institute of Astronomy, No. 101 Sect. 2 Kuang-Fu Road,30013, Hsinchu, Taiwan Astrophysics, Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK The extent of the accretion disk in the low/hard state of stellar mass black hole X-ray binaries remains an open question. There are some evidence suggesting that the inner accretion disk is truncated and replaced by a hot flow, while the detection of relativistic broadened iron emission lines seems to require an accretion disk extending fully to the innermost stable circular orbit. We present comprehensive spectral and timing analyses of six Nuclear Spectroscopic Telescope Array () andobservations of 339 taken during outburst decay during the autumn of 2015. Using a spectral model consisting of a thermal accretion disk, Comptonized emission, and a relativistic reflection component we obtain a decreasing photon index, consistent with an X-ray binary during outburst decay. Although, we observe a discrepancy in the inner radius of the accretion disk and that of the reflector, which can be addressed to the different underlying assumptions in each model, both model components indicate a truncated accretion disk that resiles with decreasing luminosity. The evolution of the characteristic frequency in Fourier power spectra and their missing energy dependence support the interpretation of a truncated and evolving disk in the hard state. Thedataset allowed us to study, for the first time, the evolution of the covariance spectra and ratio during outburst decay.The covariance ratio increases and steeps during outburst decay, consistent with increased disk instabilities. § INTRODUCTION339 can be regarded as the archetypical low-mass black hole X-ray binary. Since its discovery in 1973 by the OSO-7 satellite <cit.>, it showed frequent outbursts of varying strength, and there is a possible ∼200 days long-term variability <cit.>. Both, the mass of the black hole in and the distance to 339, are rather uncertain, estimated at M≃10 M_⊙ and d≃8 kpc by <cit.>. Based onobservations, <cit.> used the spin and inclination they determined from modelling the reflection component as input for the continuum fitting to obtain a mass of 9.0^+1.6_-1.2 M_⊙ and a distance of 8.4±0.9 kpc just from the X-ray spectrum.During most of its outbursts 339 evolves from the so called low/hard state (LHS) through the hard and soft intermediate states (HIMS/SIMS) to the high/soft state (HSS), where it can remain for several weeks, before it returns at lower luminosity into the LHS, passing again through the SIMS and HIMS. Here, we follow the classification of <cit.>; see however <cit.> for an alternative classification. The different states through which the black hole transient evolves can be identified in the hardness intensity diagram <cit.>, as the hard and soft states have distinct spectral properties. In the LHS the X-ray spectrum is dominated by Comptonized emission, which can be described by a power law with a photon index of Γ≤ 1.7 and cut-off energies of ∼ 50 – 100 keV. In the HSS the X-ray spectrum is dominated by thermal emission of the accretion disk. Furthermore, hard and soft states show distinct variability properties. The power density spectra (PDS) of the LHS show strong (up to 40 – 50 per cent) band limited noise (BLN) and quasi-periodic oscillations (QPOs), while PDS of the HSS show weak (few per cent) power law noise. While there is strong observational evidence that for a large fraction of the HSS the accretion disk extends all the way down to the innermost stable circular orbit <cit.>, the accretion geometry in the LHS has been a subject of intense debate in recent years. In the quiescent state, the disk is found to recede <cit.>. This suggests that a truncated disk is required in early stages of an outburst, and that the LHS is dominated by emission of an optically thin, advection-dominated accretion flow <cit.> surrounded by an accretion disk truncated far away from the ISCO <cit.>.During the outburst, the inner accretion disk must then evolve and extend itself closer to the black hole.Based on observations of soft emission in the LHS, which is often attributed to disk emission, the inner disk radius can be determined by modelling this component. Alternatively, the disk radius can be inferred from modelling the reflection component, as the disk acts as a reflector. 339 is a key source in the inner disk truncation debate. Some studies of 339 at luminosities ≥1% L_Edd suggested an inner disk radius consistent with a disk extending down to the ISCO <cit.>. However, <cit.> using the same data as <cit.> and <cit.>, found a large truncation radius. The discrepancy was attributedto the severe pile-up in theMOS data, which broadens the iron line. A highly truncated disk has been also found by <cit.>, who calculated the inner radius both from modelling the disk emission and that of the reflection component. Interestingly, the radii calculated by the two methods do not agree with each other. A study of the 2013 outburst, in which 339 always remained in the LHS, found an inner disk radius of R_in∼20R_g from both disk and reflection spectral components <cit.>. Based on a systematic study of the iron line region, that tracked the evolution of the inneraccretion disk in the LHS, <cit.> found an accretion disk that extended closer to the black hole at higher luminosities, but was consistent with being truncated throughout the entire LHS. <cit.> reanalysed seven archivalobservations of 339 in the hard state at outburst rise, fitting the data with a modified<cit.> plus<cit.> model (they split the normalisation of thecomponent into two factors, one depending on the mass and distance, and one depending on the `true' inner disk radius and inclination, where the latter one can then be linked to the corresponding parameters of themodel). They found that the inner disk radius of the accretion disk could not be set equal to that of the reflector and obtained an evolution of the inner disk radius consistent with the truncation disk model. The discrepancy between the two radii implied that the soft spectral component was not a standard blackbody disk. Studies of 339 at luminosities below 1% L_Edd found a truncated accretion disk <cit.>.Reflection modelling and disk continuum models also allow us to estimate the spin and inclination. Although there is a discrepancy between the results obtained with these two methods, which is likely due to different underlaying assumptions in the models <cit.>, a Schwarzschild ( non-spinning) black hole can be ruled out with high significance <cit.>. Most studies prefer a spin of a>0.9 <cit.>. Only <cit.> argued, based on continuum modelling and assuming an inclination >45, that the spin should be less than 0.9. The inclination of 339 is only weakly constrained by dynamic measurements. It must be less than 60, as the system is non-eclipsing <cit.>, and a plausible lower limit of i>45 is given from the mass function <cit.>. Measurements of the inner disk inclination from the broad iron line prefer rather low values: i=12^+4_-2 <cit.>, i=19^∘±1^∘ <cit.>, i=18.^∘2^+0.3_-0.5 <cit.>, while using relativistic reflection models give values ≥ 30: 36^∘±4^∘ <cit.>, 48^∘±1^∘ <cit.>, 31to 59<cit.>, 30^∘±1^∘ <cit.>. A combined spectroscopy and timing analysis fitting both the lag and spectral data obtained i<30<cit.>.By fitting mean X-ray spectra, one studies the time-averaged spectral shape of a source, learning nothing about how the individual spectral components vary with respect to each other in time. One technique to construct `variability spectra' is to obtain a PDS for each individual energy channel and to integrate the PDS over a given frequency range to measure the variance in that channel. That way one can construct an rms spectrum, and examine the components which vary over that frequency range <cit.>. <cit.> developed a technique called covariance spectra, which is similar to rms spectra and allows to disentangle the contribution of spectral components to variations on different time-scales. The covariance is derived between the channel of interest and a broader reference band, which makes it more robust regarding low signal-to-noise data. Covariance spectra and ratios derived fromdata taken during the LHS at the beginning of an outburst revealed an increased disk blackbody variability with respect to the Comptonized emission below 1 keV at time scales longer than 1 s <cit.>.On shorter timescales variability of the Comptonized component drives the disk variability, consistent with propagating models modified by disk heating at short timescales.In this paper, we present a comprehensive study of the spectral and temporal variability properties of sixandobservations of 339 taken during decay of its 2014/15 outburst.§ OBSERVATION AND DATA ANALYSIS§.§monitoringThe 2014/15 outburst of 339 was detected by /BAT <cit.> and followed-up with thesatellite <cit.>. We analysed all /XRT <cit.> observations taken in windowed timing mode between 2014 October 31st and 2015 October 3rd, using the online data analysis tools provided by the Leicesterdata centre[http://www.swift.ac.uk/user_objects/], including single pixel events only <cit.>. §.§<cit.> followed the decay of the outburst with six observations. The first five observations are taken with a spacing of 5 days, while the sixth observation is taken 13 days after the fifth. Details on individual observations are given in Table <ref>. We solely employed EPIC/pn data in our study, as they provide higher time resolution and are less affected by pile-up than data of the MOS cameras. We filtered and extracted the EPIC/pn event files using standard SAS (version 14.0.0) tools, paying particular attention to extract the list of photons not randomised in time. We analysed the data with the SAS taskto check if the data are affected by pile-up and ended up using a source region that gives us an observed pattern distribution that follows the theoretical prediction quite nicely. We included single and double events (PATTERN≤4) in our study. For our timing studies we selected the longest interval of continuous exposure available in each observation. We produced PDS in the 1 – 2and 2 – 10 keV band. We subtracted the contribution due to Poissonian noise <cit.>, normalised the PDS according to <cit.> and converted to square fractional rms <cit.>. To extract energy spectra we used the total exposures. We extracted energy spectra and corresponding background spectra, redistribution matrices, and ancillary response files for all observations. §.§<cit.> observations taken simultaneously to theobservations are available. Details on individual observations are given in Table <ref>. We analyseddata using the NuSTARDAS (version 1.4) toolsand , with CALDB 20160325. We extracted source photons from a circular region with a radius of 30” located at the known position of 339. A background region of the same shape and size located close to the source on the same detector and free of source photons was used. To investigate short term variability we derived cospectra in the 3 – 30 keV band using MaLTPyNT <cit.>. The energy range between 3 and 30 keV comprises about 97 per cent of the source photons detected within the 3 – 78 keV band. The cospectrum is the cross PDS derived from data of the two completely independent focal planes and represents a good proxy of the white-noise-subtracted PDS <cit.>. § RESULTS §.§ Hardness intensity diagram and light curveBased on the data available from the /XRT monitoring of the outburst, we determined the source count rates in the total (0.8 – 10 keV), soft (0.8 – 3 keV), and hard (3 – 10 keV) energy band, and derived a hardness ratio by dividing the count rate observed in the hard band by the one obtained in the soft band. The HID and the long term light curve is shown in Fig. <ref>and Fig. <ref>, respectively. Due to sun constraints there is a gap of 81 days between the first (2014 Oct. 31) and second (2015 Jan. 21) /XRT observation, where 339 was already on its way to the soft state. The source entered into the HSS and stayed there at least until 2015 July 27. In the next available observation, taken after a gap of 31days, 339 is already in the intermediate states on its way back to the LHS. In the light curve the times when the / observations took place are indicated. §.§ Power density spectraFor thedata, we derived PDS in the 1 – 2 and 2 – 10 keV range, for thedata we derived cospectra in the 3 – 30 keV range (Fig. <ref>). Fordata, we used time bins of three times the frame time, which allowed us to sample frequencies up to ∼28-29 Hz, and stretches of 16384 bins. Fordata we used time bins of 2^-8 s and stretches of 512 s. In general the PDS show band limited noise (BLN) components, fitted with zero-centered Lorentzians. Parameters can be found in Table <ref>. For thedata the soft and broadband PDS can be fitted with two BLN components. The same is true for the first twoobservations, while the remaining fourobservations require three BLN components. The additional BLN component shows up at a rather high characteristic frequency (∼8.7 Hz) that decreases with ongoing outburst decay. An overall decrease of the characteristic frequency also shows up in the other two BLN components of thedata. In thePDS the decreasing trend in the characteristic frequency is less obvious (Fig. <ref>). A QPO at a characteristic frequency of ν_char=√(ν_0^2+Δ^2)∼0.08 - 0.09Hz is present in obs. 6. Parameters, including peak frequency (ν_0) and half width at half maximum (Δ), are given in Table <ref>. Its significance decreases with higher energy (3.7σ in the 1 – 2 keV band, 2.8σ in the 2 – 10 keV band, and 2.4σ in the 3 – 30 keV band) and its Q factor is high (Q=ν_0/2Δ20). The feature is a type-C QPO <cit.>. No harmonics are detected, which is most likely related to the fact that the observation is taken late during outburst decay when the source is already rather faint. In thedata of obs. 1 a peaked noise componentat ν_char∼0.21Hz is present (parameters are given in Table <ref>). Its significance in the 1 – 2 keV band (2.8σ) is higher than in the 2 – 10 keV band (2.3σ), while the Q factor in the soft band (4.0) is smaller than in the 2 – 10 keV band (5.3).§.§ Covariance spectraUsingdata, which cover energies below 3 keV, we derived covariance spectra on short and long timescales and obtained covariance ratios dividing the long timescale covariance spectrum by the short timescale one <cit.>. We used the energy range between 1 and 4 keV as reference band, taking care to exclude energies from the reference band that are in the channel of interest. Taking a look at the PDS, we found that the break frequency where the top-flat part of the PDS goes into the decaying part evolves between different observations (see Fig. <ref>). We can use the following timescales for all observations to compare variability in the decaying part of the PDS to variability in the flat part: 0.05 s time bins measured in segments of 3.5 s for shorter time scales and 12.5 s time bins measured in segments of 625 s for longer time scales. The obtained covariance ratios are shown in Fig. <ref>, where different symbols indicate different observations. We find that the (energy-averaged) covariance ratio increases during outburst decay. Furthermore, we find in all observations at energies below ∼ 2 keV an increase of the covariance ratios towards lower energies. This behaviour has also been observed for observations of 339 taken during outburst rise and has been interpreted as sign of additional disk variability on longer timescales <cit.>. Comparing the long-to-short covariance ratios of different observations with each other reveals that the increase of the covariance ratio steepens with time. While for the first four observations the covariance ratios above 2 keV remain rather flat, we find an increase towards higher energies in the last two observations.In addition to study covariance ratios by comparing spectra on long and short timescales for each observation, we can derive ratios by comparing variability spectra on short (or long) timescales between different observations (Fig. <ref>). The ratios obtained by dividing the short and long timescale covariance spectra of all observations by those of the first observation, show an increase towards higher energies. This indicates that 339 hardens during outburst decay. For the short timescale ratios, we find that the ratio decreases during outburst decay, which is to be expected as the source gets fainter during outburst decay. On the long timescale 339 first gets brighter before it reaches again in obs. 4 a brightness comparable to that of the first observation, and then gets fainter in the last two observations. Deriving ratios relative to obs. 2, we see that on short timescales the ratios (of the later observations) are rather flat, indicating that not much spectral evolution takes place on short timescales after obs. 2, while the spectral hardening continues on the longer timescales.§.§ Energy spectraWe fit averaged energy spectra, using simultaneous /EPICpn anddata, within isis <cit.> in the 0.8 – 78 keV range, wheredata covered the 0.8 – 10 keV range anddata the 4 – 78 keV range. We excludeddata between 3 and 4 keV from our analysis, as the spectral residuals show a clear mismatch of NuSTAR in this energy range. This behaviour has been observed previously and is attributed to pile-up and cross-calibration differences <cit.>, which are taken care of in newer versions of NuSTARDAS and CALDB. We grouped thedata to a minimum signal to noise ratio of three and thedata to a minimum signal to noise ratio of five. For the fourobservations taken in timing mode, we ignored energies between 2.0 and 2.4 keV, as this energy range is affected by a residual feature related to small shifts in energy gain at the Si-K and Au-M edges of the instrumental response <cit.>. To fit the energy spectra we used themodel <cit.> together with a disk blackbody component <cit.>. Themodel allows us to fit the Comptonized emission including relativistic reflection. We included an absorption component <cit.>, using the abundances of <cit.> and the cross sections given in <cit.>. We also added a floating cross-normalization parameter, which was fixed to one forFPMA, to take uncertainties in the cross-calibration between the different telescopes into account.We constrained the foreground absorption to =5.55±0.0921, fixed the inclination at 30 and the spin at 0.95 <cit.>. We used an emissivity index of 3 <cit.>. <cit.> showed that allowing for a variable emissivity index all other parameters did not change significantly.Allowing for a free ionization parameter and iron abundance in themodel, we find that these two parameters show large variability between different observations. The ionization parameter in obs. 2, 3 and 4 is close to zero, indicating a neutral disk, while the disk seems to be highly ionised in obs. 1 and 5, and slightly ionised in obs. 6. The iron abundance decreases from ∼6 in the first observation down to ∼2 in obs. 3 and then increases to the maximum allowed value of 10 in the last observation. This behaviour, especially the one of the iron abundance, seems to be highly unphysical. <cit.> addressed a problem with a too high iron abundance by allowing the photon-indices of the continuum and the input to the reflector to be different. Including an additional<cit.> component, we find an unphysical high iron abundance of 10 in all observations, and an ionization parameter of 3 – 4 in all observations but the last one, where it is close to zero. <cit.>, who reanalysed seven archivalobservations of 339 in the hard state at outburst rise with a model similar to the one used in our study, also found that the iron abundance pegged at the maximum allowed value when thenormalisation is a free parameter. Since adding ancomponent does not help to get a more physical iron abundance, we fixed A_Fe at the best fit value of 1.58 obtained by <cit.>. As we still find a very low ionization parameter in obs. 2, 3, and 4 and a high ionization parameter in obs. 1, 5, and 6, we decide to fit all observations either with a low or a high ionization parameter.The energy spectra of the first and last observation are shown in Fig. <ref>. The obtained spectral parameters are given in table <ref> and how they evolve during outburst decay is shown in Fig. <ref>. We find that the inner disk radius obtained from themodel increases, while the disk temperature, photon index andnormalization decrease monotonically along outburst decay. We derive the inner disk radius from the normalisation of themodel, assuming a distance to 339 of 8.4±0.9 kpc and an inclination of 30±1<cit.>, and including a correction factor of 1.18, to account for spectral hardening (assuming a hardening factor of 1.7) and for the fact that the disk temperature does not peak at the inner radius <cit.>. The reflection fraction, which is always below unity, also shows a decreasing trend. Small reflection fractions below 0.5 have also been measured for the LHS at outburst rise and can indicate either a truncated accretion disk or an outflowing corona <cit.>. Assuming a neutral reflection disk the radius of the reflector obtained from themodel is 20±5 R_g and then increases to more than 75R_g in the last two observations. In the case of a highly ionised disk the radius of the reflector is highly variable (5.5<R_refl<100 R_g). For the last two observations the radius of the reflector of a highly ionised disk is consistent within errors with the radii for a neutral reflection disk.Assuming the lamppost geometry (), and allowing for a free iron abundance, ionisation parameter, reflection disk radius and lamppost height, we obtain an iron abundance around 5, which increases to the maximum allowed value of 10 in the last two observations, an ionisation parameter indicating a highly ionized disk, a reflection disk radius below ∼3 R_g, apart from the last observation where it is not constrained, and a highly variable lamppost height between ∼ 4 and 25 R_g, apart from the last observation where it is not constrained. Assuming an iron abundance of 1.58 and a highly ionised disk, we obtain an increasing reflection disk radius, apart from obs. 4, and a variable lamppost height, which is not constrained in the last two observations. The spectral parameters are given in table <ref>. Fixing the reflection disk radius at the ISCO, does not help to reduce the variability in the lamppost height.§ DISCUSSION In this study we made use of simultaneousanddata obtained during the decay of the 2014/15 outburst of 339, to investigate the evolution of the energy spectra and variability during outburst decay in the 0.8 to 78 keV range. The HID derived from the available /XRT monitoring data clearly shows that the / observations of 339 followed the evolution of the sourcein the decaying branch of the outburst. The coexisting of two different power spectral shapes in the soft and hard band during the hard-to-soft state transition has been reported perviously <cit.>. In this study, the shape of the power density spectra is the same in the 1 – 2 and 2 – 10 keV band, as the PDS of both bands can be fitted by two BLN components. Thus, there is no sign of energy dependence of the power spectral shape during the soft-to-hard state transition. In <cit.>, we found that for observations of 339 and other black hole X-ray binaries taken in the LHS during outburst rise, there is at least one BLN component which has a smaller characteristic frequency in the 1 – 2 keV band compared to its characteristic frequency in the 2 – 10 keV band. In this study, we find this difference in characteristic frequency only for obs. 4 and 5, while for all other observations the characteristic frequencies in these two bands agree within errors. According to the picture given in <cit.>, characteristic frequencies, which are consistent between both bands, imply that the disk ends far away from the black hole and that the photons in both bands experience a similar number of scatterings. The decrease of the break frequency in the PDS along outburst decay, indicates an increasing inner disk radius <cit.>. Investigation of the combined / energy spectra revealed a decreasing photon index, which is in agreement with the hardening of 339 during outburst decay. The evolution of the accretion disk parameters (decreasing disk temperature, increasing inner disk radius) is in favour of the truncation disk model <cit.>. Regarding the inner disk radius obtained from the reflection component we find that the radius stays around 20±5 R_g in the first four observation and then increases dramatically to over 75 R_g, assuming a neutral reflection disk. Given the large inner disk radius, which indicates a truncated accretion disk, a neutral disk is to be expected. If we allow for a free ionisation parameter, we find a highly ionised disk in the first observation, and a neutral disk in obs. 2, 3, and 4. Assuming a highly ionised disk in the first observation results in a smaller inner disk radius (and also in changes in the other spectral parameters) in this observation. The evolution of the spectral parameters is still consistent and in agreement with the truncation disk model.The discrepancy between the inner disk radius of the accretion disk and that of the reflector can be related to the assumptions made in each model. The exact value of the inner disk radius derived from the disk blackbody normalization depends on the value of the hardening factor used (here 1.7). The uncertainty in the value of the hardening factor should not affect the evolution of the inner disk radius, as long as the hardening factor does not evolve during outburst decay. Themodel neglects the zero-torque inner boundary condition at the ISCO <cit.> and the effects of strong disk irradiation by the hard X-rays <cit.>. The later effect should only be important in the first (few) observations, as both model components indicate a disk truncated far away from the ISCO in the last two observations. The value of the inner radius obtained from the reflection component depends on the ionisation state and surface density structure of the disk and on disk inclination <cit.>. Furthermore, there is a degeneracy between the spin and the inner accretion disk, as for current available data, decreasing the spin or increasing the inner edge of the disk can be considered to be almost equivalent. In addition, currently available reflection models describe the illumination of an otherwise cold slab of gas. They do not take into account the hotter surface layers of the disk in the case of a black hole X-ray binary, which will have a significant effect upon the reflection spectrum <cit.>. Effects of the spatial extent of the corona, of coronal elevation and of mild relativistic outflow on the reflection spectrum, are also not included in currently available reflection models. In contrast to spectral studies of 339 during outburst rise based onandobservations <cit.>, our spectra do not show indications of a broad iron line, and we do not require two different photon indices for the Comptonisation and reflection component.Thedataset allowed us to study for the first time the evolution of covariance spectra and ratios during outburst decay. Up-to-now studies of these properties mainly focus on outburst rise <cit.>. Our study reveals that while the source hardens, the energy-averaged covariance ratio increases. This increase shows that the variability on long time scales (low frequencies) contributes more to the overall variability compared to the variability at short time scales (high frequencies) while 339 evolves along outburst decay. This behaviour also shows up in the PDS where the power in the top-flat part of the PDS at frequencies below 0.04 Hz clearly increases while the power in the decaying part of the PDS at frequencies above ∼0.3 Hz shows much less evolution, while the source hardens. Furthermore, we find that the increase in covariance ratio towards lower energies steepens, while 339 hardens. The observed steepening of the increase implies that additional variability on long time scales becomes more and more important at soft energies while the source gets harder. In the scenario, in which the additional variability on long time scales and soft energies is thought to be due to intrinsic instabilities in the accretion disk <cit.>, which can be invoked by damped mass accretion rate variations or oscillations in the disk truncation radius <cit.>, the observed steepening of the increase indicates an increase in the disk instabilities when 339 hardens. The observed evolution suggests that the stable disk of the soft state develops instabilities which get stronger when the source hardens. We thank the referee for thoughtful comments that helped to improve the clarity of our paper. This project is supported by the Ministry of Science and Technology of the Republic of China (Taiwan) through grants 104-281-M-007-060, 105-2112-M-007-033-MY2 and 105-2811-M-007-065. Based on observations obtained with , an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. This research has made use of data obtained through the High Energy Astrophysics Science Archive Research Center Online Service, provided by the NASA/Goddard Space Flight Center. Facilities: , , .apj | http://arxiv.org/abs/1706.08980v1 | {
"authors": [
"H. Stiele",
"A. K. H. Kong"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170627180015",
"title": "NuSTAR and XMM-Newton observations of the 2015 outburst decay of GX 339-4"
} |
theoremTheorem | http://arxiv.org/abs/1706.08448v1 | {
"authors": [
"M. N. Chernodub",
"Shinya Gongyo"
],
"categories": [
"hep-th",
"cond-mat.other",
"cond-mat.quant-gas",
"hep-ph"
],
"primary_category": "hep-th",
"published": "20170626155421",
"title": "Edge states and thermodynamics of rotating relativistic fermions under magnetic field"
} |
firstpage–lastpage 2017 A Millimeter Continuum Size–Luminosity Relationship for Protoplanetary Disks Anjali Tripathi1, Sean M. Andrews1, Tilman Birnstiel2, & David J. Wilner1=============================================================================The mass of the central black hole in a galaxy that hosted a tidal disruption event (TDE) is an important parameter in understanding its energetics and dynamics. We present the first homogeneously measured black hole masses of a complete sample of 12 optically/UV selected TDE host galaxies (down to g_host ≤ 22 mag and z = 0.37) in the Northern sky. The mass estimates are based on velocity dispersion measurements, performed on late time optical spectroscopic observations. We find black hole masses in the range 3 × 10^5 M_⊙ ≤ M_ BH ≤ 2 × 10^7 M_⊙. The TDE host galaxy sample is dominated by low mass black holes (∼ 10^6 M_⊙), as expected from theoretical predictions. The blackbody peak luminosity of TDEs with M_ BH ≤ 10^7.1 M_⊙ is consistent with the Eddington limit of the SMBH, whereas the two TDEs with M_ BH ≥ 10^7.1 M_⊙ have peak luminosities below their SMBH Eddington luminosity, in line with the theoretical expectation that the fallback rate for M_ BH ≥ 10^7.1 M_⊙ is sub-Eddington. In addition, our observations suggest that TDEs around lower mass black holes evolve faster. These findings corroborate the standard TDE picture in 10^6 M_⊙ black holes. Our results imply an increased tension between observational and theoretical TDE rates. By comparing the blackbody emission radius with theoretical predictions, we conclude that the optical/UV emission is produced in a region consistent with the stream self-intersection radius of shallow encounters, ruling out a compact accretion disk as the direct origin of the blackbody radiation at peak brightness. galaxies: bulges – galaxies: nuclei – galaxies: fundamental parameters – accretion, accretion disks – galaxies: kinematics and dynamics§ INTRODUCTION It is currently accepted that supermassive black holes (SMBH) reside in the centers of most, if not all, massive galaxies (e.g.). If there is gas close to the hole, its accretion has directly observable signatures and we designate the center an active galactic nucleus (AGN). However, if there is no gas near the SMBH, indirect methods must be used to infer its presence. Occasionally a reservoir of gas may wander near the black hole in the form of a star. If the tidal forces due to the SMBH are larger than the self-gravity of the star, the SMBH will tear it apart, and about half of the star will be accreted by the central black hole <cit.>. This so-called tidal disruption of a star is accompanied by a luminous flare at X-ray, UV or optical wavelengths, announcing the presence of an otherwise dormant SMBH to the Universe. In the last two decades, about two dozen tidal disruption events (TDEs) have been discovered in various wavelength regimes such as X-rays <cit.>, UV <cit.> and optical light <cit.>. From an observational point of view, there seem to be two broad classes of TDEs: those where X-ray (or even higher energy) emission was detected, and those where optical emission was detected. It should be noted that not all optical TDEs were followed up at X-ray wavelengths, which may partially explain this apparent dichotomy. Two exceptions are already known, including ASASSN–15oi <cit.> and ASASSN–14li, which was detected not only at optical <cit.> and X-ray <cit.> wavelengths but was also observed to produce radio emission <cit.>. In the classical picture of TDEs, the electromagnetic radiation is produced when the bound debris circularizes and falls back to the SMBH <cit.>. An accretion disk forms at a radius of about 2 R_ p, where R_ p is the pericenter radius of the orbit of the disrupted star. The disk forms rapidly and efficiently circularizes due to stream-stream collisions induced by relativistic precession. While this scenario is able to explain the properties of TDEs producing X-rays, the temperatures and luminosities of optical TDEs are an order of magnitude lower than theoretical predictions <cit.>. Several scenarios have been proposed to explain the optical emission mechanism of TDEs, including thermal reprocessing of accretion power by material far from the hole <cit.>, shock emission produced by the self-intersecting debris stream <cit.> or outflows <cit.>. More recently, magnetic stresses have also been considered as the source of both X-ray and optical emission <cit.>. A theoretical framework that can explain the dynamics and energetics of both X-ray and optical emission from TDEs has yet to converge towards a unified theory.Observational studies of TDEs are critical to provide meaningful constraints on key ingredients for theoretical models, such as the dynamical efficiency of stream circularization, the primary TDE power source, and the dominant emission mechanisms. Because of the two-body nature of a TDE, constraining the mass of the black hole component helps to disentangle other aspects of the events, including the dynamics and energetics. For instance, the tidal radius of the disrupted star, the energetics of the accretion phase, the post-disruption dynamics and the expected electromagnetic (and gravitational wave) emission all depend on the black hole mass. Constraining the black hole mass can also provide direct constraints on the accretion efficiency or the amount of mass accreted during a TDE. Currently the mass of the black hole is usually inferred from modelling rather than used as an input parameter because no accurate, systematic measurements are available. Constraining the mass of a black hole in the center of a galaxy has a rich history (see for a review). The discovery of correlations between the bulge luminosity and mass (the M – L relation, e.g.,) or bulge velocity dispersion and mass (the M – σ relation, e.g. or) indicate that there is a tight connection between the evolution and formation of the SMBH and the stellar bulge <cit.>.By exploiting these correlations, it is possible to measure black hole masses even when it is not possible to spatially resolve the sphere of influence of the SMBH (at z ≥ 0.01) and derive the mass from the dynamics of stars or gas that is directly influenced by the black hole. At higher redshifts, using these scaling relations has the advantage of being less data intensive than direct methods such as reverberation mapping. They have therefore made SMBH mass measurements a relatively easy task (compared to direct methods) at redshifts in excess of z ∼ 0.01.A robust method for extracting the velocity dispersion from galaxy spectra is to compare the width and equivalent width of stellar absorption lines with stellar template libraries in pixel space (e.g.,,). Working in pixel space makes masking bad pixels more easy, while it also facilitates the simultaneous modelling of gas and stellar kinematics with other observational effects such as contamination due to emission-line gas <cit.>.In this work we present the first systematic effort to measure the black hole masses of a sample of 12 optically/UV selected TDE host galaxies. In Section <ref>, we describe the sample selection and observations used to perform the measurements. Section <ref> explains the methodology we followed; we present the results and discuss their implications in Section <ref>. Finally, we summarize in Section <ref>.§ OBSERVATIONS AND DATA REDUCTION§.§ Host galaxy sampleWe have obtained spectroscopic observations (Table <ref>) of galaxies hosting optically/UV selected nuclear transients with a blackbody temperature in excess of 10^4 K (which we will refer to as TDEs) located in the Northern sky (declination ≥ 0^∘). Our sample is complete down to a limiting (host galaxy) magnitude of g_ host = 22 mag; the hosts span a range in redshift from 0.016 to 0.37. These transients were discovered by a variety of surveys (see Table <ref> for references to the discovery papers), including the Sloan Digital Sky Survey (SDSS), the All Sky Automated Survey for Supernova (ASAS–SN), the (intermediate) Palomar Transient Factory (PTF), the Panoramic Survey Telescope and Rapid Response System (PS1) and the Galaxy Evolution Explorer (GALEX). Our sample comprises 12 sources out of a total of 13 optically/UV discovered TDEs in the Northern sky[http://TDE.space]. PS1–11af is the remaining source at g_host = 23 and z = 0.405 <cit.>. There is one TDE in our sample for which a discovery article has not yet been published in the literature: iPTF–15af. This TDE was discovered in the galaxy SDSS J084828.13+220333.4 ( ).The observations were performed with the William Herschel Telescope (WHT, Section <ref>) on La Palma, Spain, the Very Large Telescope (VLT, Section <ref>) at Cerro Paranal, Chile and the Keck–II telescope on Mauna Kea, Hawaii. §.§ WHT/ISIS We obtained late time spectra of some TDE host galaxies using the Intermediate dispersion Spectrograph and Imaging System (ISIS,) mounted at the Cassegrain focus of the 4.2m William Herschel Telescope (WHT) located on La Palma, Spain. We used the R600B and R600R gratings in the blue and red arm respectively, with central wavelengths optimized for covering wavelength regions containing host galaxy absorption lines. There is a gap in the coverage between the blue and red arms due to the use of a dichroic at 5300 Å. The wavelength coverage of this setup is 1000 Å around the central wavelength of each arm. A summary of the observations is presented in Table <ref>. We first perform the standard reduction steps such as a bias level subtraction, a flat field correction and a wavelength calibration using iraf. Cosmic rays are removed using the lacos package in iraf <cit.>. The typical root-mean-square deviation (rms) of the applied wavelength solution is ≤ 0.1 Å, which corresponds to at most 0.5 pixels. The absolute wavelength calibration is evaluated by measuring the position of a Hg i sky line at λ4358.33, and when necessary the spectra are shifted to match the same wavelength scale. This ensures that combining multiple spectra of the same source does not introduce an artificial broadening of the absorption lines. The spectra are rebinned to a linear dispersion on a logarithmic wavelength scale. We perform an optimal extraction <cit.>, which weights each pixel along the spatial profile by the inverse variance of the number of detected photons (i.e. pixels containing less signal get down-weighted) to achieve the highest possible signal-to-noise ratio (SNR) for the extracted spectrum. The variance spectra are also calculated and will be used for Monte Carlo simulations (Section <ref>).We measure the instrumental broadening of the different observational setups using arc lamp observations taken together with the science spectra to measure σ_ instr. The resolution of the observations is slit-limited for all spectra. Our observations provide an instrumental resolution FWHM of 1.75 Å in the blue arm for a 11 slit width (or better, if the slit width was smaller), which corresponds to 55 km s^-1 at 3900 Å (Table <ref>). We present the resulting spectra in Figure <ref> (top panel).§.§ VLT/X-shooter For iPTF–16fnl, we have obtained a late time spectrum (∼ 193 days after peak brightness) in which the TDE does not contribute a significant fraction to the total galaxy light on 2016 November 25 (Onori et al. in prep.) with X-shooter <cit.>, mounted on UT2 (Kueyen) of the Very Large Telescope (VLT) at Cerro Paranal, Chile. The 1800 s observation (OB ID: 1617353) was performed using an 08 slit. The spectral resolution provided by this setup is R = 6200, which yields an instrumental broadening equivalent to σ = 20 km s^-1 at 3900 Å. We use the ESO Phase 3 pipeline[http://www.eso.org/observing/dfo/quality/XSHOOTER/pipeline] reduced spectrum of the UVB arm for our analysis, which has an absolute wavelength calibration accurate to 0.3 Å. §.§ Keck/ESI We took medium resolution spectra with the Echelette Spectrograph and Imager (ESI;), mounted at the Cassegrain focus of the Keck–II telescope on Mauna Kea, Hawaii. The instrument provides a wavelength coverage ranging from 3900 – 10000 Å in multiple echelle orders. The observations were performed using a 05 slit, providing a near-constant resolving power of R = 8000. The FWHM resolution is 38 km s^-1, which translates to an instrumental resolution of σ_ instr = 16 km s^-1.The data were reduced using the MAuna Kea Echelle Extraction () software package, which was developed and optimised for the reduction of ESI data. The pipeline performs standard spectroscopic data reduction routines including bias subtraction, flatfielding and spectrum extraction. The standard star Feige 34 was used to compute the trace of the science objects. The position of each echelle order is traced, optimally extracted and wavelength calibrated independently, after which the different orders are rebinned to a linear dispersion on a logarithmic wavelength scale with a dispersion of 11.5 km s^-1 per pixel. The orders are combined using the combine command to produce a 1D spectrum. The wavelength calibration is performed in iraf using two arc lamp (CuAr and HgNe+Xe) exposures. §.§ Further data processingAfter obtaining the 1D spectra from our WHT, VLT and Keck observations, further processing steps are required before we can measure the velocity dispersion. The spectra are normalized by fitting 3^ rd order cubic splines to the continuum in molly. We mask all prominent absorption and emission lines during this process to identify the continuum. We average the spectra, weighting by the mean SNR (variance) of each individual exposure. We extract spectra from two different spatial regions of the host galaxy for each exposure (see Section <ref>). One extraction includes the whole galaxy along the slit, to increase the SNR of the resulting spectrum. The second extraction region is centered on the peak of the light profile, and has an aperture radius equal to the seeing of the exposure. This extraction aims at isolating as much as possible the bulge region of the galaxy, to provide an estimate of the central velocity dispersion rather than the luminosity-weighted velocity dispersion obtained from the entire galaxy. We measure the seeing using point sources present on the slit; if not available, we use measurements of a local seeing monitor (the Robotic Differential Image Motion Monitor, available for the WHT data) as an estimate. In case no measurements are available, we use an aperture equal to the slit width, effectively mimicking a square fiber with sides equal to the slit width.§ VELOCITY DISPERSION MEASUREMENTS We use the penalized pixel fitting (ppxf) method <cit.> to measure the line of sight velocity dispersion function (LOSVD), typically denoted as f(v), of the galaxies in our sample. Briefly, the method consists of convolving a set of template spectra with an initial guess for f(v), which is then compared to the observed host galaxy spectrum. The LOSVD is parametrized by a series of Gauss-Hermite polynomials in the form:f(v) = 1/σ√(2π) exp(1/2(v-V)^2/σ^2) [ 1 + ∑_m=3^M h_m H_m(v-V/σ) ]where V is the mean velocity along the line of sight, σ is the velocity dispersion, H_m are Hermite polynomials and h_m their coefficients. The Hermite polynomials are defined asH_i = 1/√(i!) e^x^2(-1/√(2)∂/∂ x) e^-x^2 where we include terms up to H_4. The terms H_3 and H_4 parametrize the asymmetric and symmetric deviations from a Gaussian line profile, respectively.The best-fitting template is found by χ^2 minimization, using the set of templates convolved with f(v) for the variables [V, σ, h_3, h_4]. The ppxf method was specifically designed to extract accurate kinematical information in the case of low SNR spectra. We refer the reader to <cit.> and <cit.> for more details.§.§ Template libraryWe note that the red part of the WHT spectra does not contain well defined, deep and unblended absorption lines suitable for a robust measurement of the velocity dispersion. At bluer wavelengths, the Ca ii H+K absorption lines at λλ3934,3968 in combination with many smaller absorption lines provide the best means to determine the velocity dispersion. The H Balmer absorption lines are known to be strongly affected by pressure broadening due to collisional or ionizational excitation, and we exclude them from the measurement process. We therefore only use the blue part of the WHT spectra, starting at 3900 Å. We fit the full spectral range, as the use of many absorption lines present in the spectrum will improve the measurement of the velocity dispersion. We mask the H Balmer lines, and in addition emission lines of O iii at λλ4959,5007, the diffuse interstellar band at λ5780 and the interstellar Na i D absorption lines at λλ5890,5895. Based on the highest resolution spectrum and the wavelength coverage of the observations, we choose template spectra from the ELODIE v3.1 database <cit.>. This spectral library contains 1554 templates at R = 10000 at 5500 Å, which implies a velocity dispersion resolution of σ = 17at 3900 Å. By using a large set of templates we minimize the effects of mismatches between the observed galaxy spectra and the templates used to derive the line broadening. The best-fitting parameters are obtained by χ^2 minimization. Because the higher order terms (h_3 and h_4) can only be robustly constrained in the case of high SNR data, the method includes a bias factor which penalizes these terms in the best-fitting solution to 0 in case the SNR is low. We follow the procedure outlined in <cit.> to determine the appropriate value for the penalty in the fitting procedure for each galaxy. During the measurement process (in ppxf) for the Keck spectra, we take into account that the template FWHM resolution (in Å) is independent of wavelength (0.54 Å), but the ESI spectral resolution (in Å) varies with wavelength. We only use the wavelength range where σ_ template ≤ σ_ ESI, starting at 4300 Å and ending at 6800 Å, where the template spectral coverage stops.§.§ Luminosity-weighted LOSVD and central LOSVD In contrast with the IFU/fiber observations that are typically used to measure the kinematics of galaxies (for example in the SDSS Baryonic Oscillations Spectroscopic Survey, ), we measure the LOSVD using long-slit observations. For spectroscopic observations obtained using a fiber instrument with a ∼ few arcsec diameter, one expects an evolution of the measured velocity dispersion with the ability to spatially resolve the bulge of the galaxy, i.e. with redshift (e.g.). For increasing distances, the velocity dispersion is influenced by stars at larger physical radii, and thus depends on the velocity dispersion profile of the galaxy. We use longslit observations, and the measurements including the entire galaxy in the extraction region are effectively luminosity-weighted velocity dispersions. It was shown by <cit.> that such measurements reflect the central velocity dispersion to good degree (to within 5 per cent, see their figure 1) as long as the slit width is smaller than or comparable to the effective light radius of the host galaxy.It should be noted that the sample used by <cit.> consists of galaxies at much lower redshifts and with higher masses. Therefore the bulge region in these nearby, massive elliptical galaxies is more dominant in a long-slit observation than we expect them to be for our sample, which consists of galaxies at higher redshifts and smaller bulge masses, as theory predicts these smaller SMBHs to produce higher rates of TDEs <cit.>. The underlying principle still holds, but the luminosity-weighted LOSVD measurements of our sample must be interpreted with care: its relation to the central velocity dispersion depends on the relative dominance of the bulge region over the rest of the galaxy. For this reason, we provide central velocity dispersion measurements based on the careful extractions outlined in Section <ref>, which aim at isolating the velocity dispersion in the central part of the galaxy.§.§ Robust velocity dispersionsTo robustly estimate the velocity dispersion and its uncertainty induced by the measurements, we perform 1000 Monte Carlo simulations. We resample the original spectrum by drawing flux values from a Gaussian distribution within the errors as obtained from the optimal extraction for each pixel. This ensures that the data quality of each simulation (i.e. the average SNR) remains the same and does not influence our measurements. We fit the resulting distribution of velocity dispersion values with a Gaussian function, and adopt the mean and standard deviation as the best-fitting value for σ and its uncertainty. § RESULTS AND DISCUSSION As an illustration, we show the result of the template fitting procedure in Figure <ref> using the WHT spectrum of TDE1. Overlaid in red is the best-fitting template spectrum broadened to 126 km s^-1. The residuals are shown in green, while blue regions are excluded in the fitting process. The velocity dispersion is well defined and the fit describes the data well, leaving little structure in the residuals. In Figure <ref> we show the distribution of measured σ values and the Gaussian fit used to determine the mean and standard deviation. To obtain black hole masses, we assume that the M – σ relation holds for all the velocity dispersions we measure, and convert the measurements to masses using the relation from <cit.>:M_ BH/10^8 M_⊙ = 1.66 ×(σ/200 km s^-1)^4.86To estimate the uncertainties in the black hole mass, we add the uncertainties of the velocity dispersion measurements linearly with the 0.34 dex systematic uncertainty introduced by using the M – σ relation <cit.>. The uncertainty is dominated by the scatter in the M – σ relation except for D23H–1. In Table <ref> we present the results of the velocity dispersion measurements for our sample. We also include the redshift, host galaxy magnitude and half-light radius, as well as literature values of velocity dispersion measurements for comparison purposes. §.§ Comparison to independent measurements For several sources in our sample, velocity dispersion measurements are available in the literature. In Table <ref> we list the literature values alongside our own measurements. Several of the velocity dispersions measured from SDSS spectra are below the instrumental resolution, which we deem less reliable, especially for low SNR observations. Three sources can be reliably compared: TDE1, D23H–1 and iPTF–15af. We quote the measurements performed by <cit.> as these authors also use ppxf to measure σ, although they use a different set of templates and a different wavelength regime (4500 – 6500 Å). For TDE1 these authors find σ = 137 ± 12 , while we find a slightly smaller value of σ = 126 ± 7 . The measured values for D23H–1 and iPTF–15af are consistent within the errors with the SDSS measurements of <cit.>. The velocity dispersion of D3–13 was measured using a similar template fitting procedure by <cit.>, and was measured to be 120 ± 10 km s^-1. Using our resampling approach we find σ = 133 ± 6 , slightly higher but consistent within the mutual uncertainties. We also note that for iPTF–16fnl there is a discrepancy between our measured value (55 ± 2 km s^-1) and that of <cit.> (89 ± 1 km s^-1), who fit Gaussian lines to the Mg i b and Ca ii triplet simultaneously.Furthermore, we have WHT and Keck spectra of 4 sources, providing another opportunity for independent measurements. For ASASSN-14ae we measure 56 ± 7 and 53 ± 2 km s^-1 using the ISIS and ESI spectra, respectively, while for ASASSN–14li we measure 72 ± 3 and 81 ± 2 km s^-1. We use the inverse-variance weighted average of these independent measurements as the best estimate of the velocity dispersion: σ_ avg = 53 ± 2and σ_avg = 78 ± 2for ASASSN–14ae and ASASSN–14li, respectively. For PTF–09ge, we calculate an inverse-variance weighted mean of σ_avg = 81 ± 2 . Regarding PTF–09djl, there appears to be an inconsistency of ∼ 40 km s^-1 between the Keck (64 ± 7 km s^-1) and WHT (104 ± 13 km s^-1) values. We note that the overlapping wavelength coverage of the WHT spectrum with the templates is small (∼ 500 Å), and a visual inspection of the best-fitting template with the galaxy spectrum reveals that the fit is poor. Moreover, our WHT spectra use a 11 arcsec slit width, while the bulge half-light radius of this galaxy is 03 and hence does not satisfy the criterion of <cit.> (see discussion below). On the other hand, the best-fitting solution to the Keck spectrum is satisfactory. We therefore adopt the value as measured from the Keck spectrum as the best representation of the central velocity dispersion of this source.§.§ Potential caveats§.§.§ Signal-to-noise ratio (SNR) and σWe have determined the value and uncertainty of σ by performing Monte Carlo simulations (Table <ref>). We find that, as expected, the accuracy with which σ can be recovered is strongly dependent on the SNR and the wavelength coverage of the data.For the spectrum of D23H–1, the relatively low SNR of the spectra causes a degeneracy in the best-fitting velocity dispersion. Due to the large errors in the observed spectrum, the χ^2 minimization is not able to resolve the shallow, narrow absorption lines. Instead, the minimization procedure finds a good fit with larger values of σ ∼ few hundred , essentially fitting only a few broad absorption lines instead of the myriad of low SNR, low equivalent width absorption lines present in the spectrum. In this case, we use the trials corresponding to a limited (but conservatively large) range of σ to determine the best-fitting velocity dispersion. We perform Monte Carlo simulations until this limited range contains at least 1000 trials, to robustly estimate the uncertainty induced by the measurement errors.An illustration is shown in Figure <ref> for the WHT spectrum of D23H–1, including a fit to all the trials (top) and a fit to only the trials in the range σ = [0,130] km s^-1 (bottom). We adopt σ = 77 ± 18in this case. We also note that extracting the central region of the host galaxy to D23H–1 results in a low SNR spectrum. The model fitting becomes less constrained and we are no longer able to robustly measure the central velocity dispersion of this galaxy.For the Keck spectra, the large wavelength coverage makes it is possible to accurately determine the velocity dispersion even with a relatively low SNR per pixel because of the large number of small lines in the spectrum. The large wavelength coverage (hence large number of degrees of freedom), combined with the fact that no very deep absorption lines are present (such as the Ca H+K lines in the WHT spectra) also makes the template selection procedure more prone to make non-optimal choices. This can lead to a divergence in the fitting process if there are only a few absorption lines in common between the science spectrum and the selected templates to determine σ. This is observed as a tail of outliers at high velocity dispersion values; we therefore use a similar procedure as outlined above for D23H–1 to fit a Gaussian to a restricted range in the velocity dispersion distribution. §.§.§ Comparison of luminosity-weighted and central LOSVDsWe find no significant differences between the luminosity-weighted LOSVDs and the central velocity dispersion values. In all cases the measurments yield results that are consistent within the mutual errors. In Table <ref> we provide the host galaxy half-light radius, as determined by SDSS <cit.> from a de Vaucouleur profile fit to the galaxy light. We note that for all sources except for the WHT spectrum of PTF–09djl, our observations are within the regime where the slit width is less than two times the galaxy half-light radius, for which <cit.> have shown that the luminosity-weighted LOSVD is a good tracer of the central velocity dispersion. For the WHT spectra of PTF–09djl we are not in this regime (as discussed in Section <ref>). We therefore adopt the value obtained from the Keck spectrum, obtained with a slit width of 05, as the most reliable measurement. For the other sources, we do not find significant differences between the luminosity-weighted and central LOSVDs, implying as expected that our long-slit data, even when extracting the full galaxy light, are not strongly influenced by the disk of the galaxy. We note that using an optimal extraction for the spectra will have helped in this respect.§.§.§ Choice of M – σ relation The particular choice of M – σ relation and which version is the best version is still a matter of debate, with many versions published in the literature <cit.>. Each of these works has its particular sample selection that comes with advantages and disadvantages. In this work we have chosen to use the relation based on the sample of <cit.>, who included only galaxies for which the sphere of influence had been resolved. If we compare these values with those obtained with the recent <cit.> relation, valid for early-type galaxies, we find that the (non-systematic) difference is less than 0.1 dex for the sources in our sample. Therefore we do not expect the particular choice of the M – σ relation to influence our conclusions. In Figure <ref> we show the original (resolved) sample used by <cit.> to derive the M – σ relation (Eq. <ref>; dashed line). We have overplotted the relation by <cit.> (dotted line) and <cit.> (solid line) to illustrate the effect on the derived masses. We note that the latter relation was explicitly derived for elliptical galaxies and is most likely not appropriate for our sample.Another issue that arises from using the M – σ relation for our sample is that several host galaxies harbour black holes with masses that are lower than the mass range for which the relation was originally derived (see also Figure <ref>). Simulations have shown that the (currently unknown) black hole seed formation scenario has an impact on the validity of theM – σ relation at the low mass end. For example, <cit.> showed that in the case of high-mass seeds the relation should show an increased scatter, possibly combined with a flattening at low σ. However, there is at present no conclusive evidence that corroborates these predictions. For example, <cit.> measure black hole masses for less than 10^6 M_⊙ BHs and find that they lie on the extrapolation of the M – σ relation to lower masses. <cit.> found that the relation derived for quiescent massive ellipticals can also be extrapolated to active galaxies, with masses as low as 2 × 10^5 M_⊙. These authors did not find evidence for an increased scatter in the correlation at the low end of the mass range. We remark that direct mass measurements for these systems are needed to resolve this issue beyond doubt. §.§ A black hole mass distribution for TDE host galaxies Recent theoretical work has used the observed sample of TDE candidates to analyze flare demographics <cit.>, to constrain the SMBH occupation fraction in low mass galaxies <cit.>, and to try to constrain optical emission mechanisms <cit.>. The BH/bulge mass estimates used in these works are inhomogeneous, but are generally based on the M – L relation, and the bulge mass of these galaxies is subject to large uncertainties. Here we present a new and updated black hole mass distribution based on spectroscopic measurements of our host galaxy sample. Our mass distribution, presented in Figure <ref>, contains black hole masses ranging from 3 × 10^5 M_⊙ to 2 × 10^7 M_⊙. It is dominated by low mass black holes in the range ∼ 10^6 M_⊙. The absence of black holes with masses lower than 3 × 10^5 M_⊙ could be explained by the increasingly smaller volume in which TDEs can still be detected around low mass black holes (assuming that the peak luminosity is Eddington-limited or otherwise scales with the black hole mass). Alternatively, this could be a consequence of the black hole occupation fraction in low mass galaxies <cit.> or because of a lower flare luminosity due to inefficient circularization <cit.>. On the high mass end, the lack of SMBHs in excess of 10^7.5 M_⊙ could be explained by the direct capture of stars <cit.>. Testing this hypothesis requires a careful treatment of the survey completeness due to both the host and TDE flux limits, and will be explored in detail in van Velzen et al. (in prep.).We remark that our mass distribution is in contrast with masses taken from the literature (e.g. figure 12 of ). These authors found a more top-heavy M_ BH distribution peaked just below 10^7 M_⊙, with SMBH masses mostly derived using the M – L relation. We list a few potential explanations for this difference below. To start, <cit.> did not apply B/T corrections for most galaxies, implying that the resulting masses are upper limits. A second potential caveat is that many TDE host galaxies are rare E+A galaxies <cit.>, which are thought to possess a central overdensity of stars due to a recent merger <cit.>. These galaxies are observed to have very centrally peaked light profiles (see e.g. ), and therefore they could be overluminous with respect to the galaxies used to derive the scaling relation (typically massive ellipticals). This was also noted by <cit.> as a caveat to their analysis, and may explain why we find lower BH masses for 3 sources (ASASSN–14ae, ASASSN–14li and PTF–09ge) with M_ BH estimated from M_ bulge using stellar population fitting (; their table 2). Finally, <cit.> have shown that the M – L relation may be a broken power law rather than applicable to the whole mass range; they find that it should have a steeper slope (M ∝ L^2 instead of M ∝ L^1) below ∼ 10^8 M_⊙. This would lead to an overestimate of M_ BH for masses below ∼ 10^8 M_⊙. Based on numerical simulations, <cit.> identified that stellar feedback due to star formation may lead to a change of slope in the M – L scaling relation. <cit.> also suggest that a steeper relation can explain the presence of samples of low mass AGNs with seemingly undermassive BHs. §.§ Correlations with other observablesRecent studies investigating potential correlations between the black hole mass and other TDE observables such as peak luminosity and e-folding timescale are reported by <cit.> and <cit.> respectively. Despite some suggestive evidence, no strong correlations were observed. However, this could be a consequence of the heterogeneous mass measurements available in the literature, motivating us to re-investigate potential correlations. In Figure <ref> we plot our black hole masses against other observables. We provide the plotted data in Table <ref>. We search for correlations between the observables using the Spearman rank correlation metric. Similar to previous work, we do not find statistically significant (95 per cent confidence interval) correlations. This could be a consequence of the small sample size, in combination with the degeneracy of different parameters such as the mass of the star and the impact parameter. Nevertheless, it is instructive to discuss some suggestive evidence for correlations with the host black hole mass or derived Eddington luminosity. It is important to note that our galaxy sample is drawn from flux-limited surveys, and we do not consider the effects of a flux limit for the flare itself. We will find that the qualitative trends corroborate the tidal disruption interpretation of these events, and moreover can provide input and constraints for viable TDE emission models.§.§.§ RedshiftFigure <ref> (panel a) suggests that TDEs found at lower redshift are associated with lower mass black holes. The dearth of TDEs found in low mass black holes at higher redshifts may be a consequence of the flux limited nature of our sample. The lack of higher black hole masses for TDE hosts at low redshifts could be explained by the relative rarity of higher mass black holes, as the log(N) – log(M) distribution of black hole masses rises towards lower masses (e.g.). The exponential tail of the black hole mass function implies that a large volume is needed to include enough high mass black holes. As a result, in a flux limited sample the observed black hole mass distribution is expected to correlate with redshift as long as it does not contain a representative sample of galaxies.§.§.§ Peak absolute magnitudeIn panel b) of Fig. <ref> we show the (K-corrected,) peak absolute g-band magnitudes, i.e. the peak luminosity measured at 6.3 × 10^14 Hz in the rest frame, plotted as a function of the black hole mass. We use the peak flux in the filter with the best temporal sampling in the literature, together with the blackbody temperature (taken from the literature, see Table <ref>) to calculate the peak g-band magnitude in the rest frame of the host. Because we correct to the rest frame of the host galaxy, the specific filter choice is irrelevant. We note that for several TDEs we can only determine upper limits as the peak of the lightcurve was not observed. However, a visual comparison of the lightcurves of these events with other well sampled lightcurves of TDEs suggests that the peak was probably missed only by a few days and therefore the difference should be small. We do not observe a statistically significant trend of peak absolute magnitude with black hole mass. The observations suggest that current optical/UV surveys are already probing the fainter end of the TDE luminosity function (illustrated by the spread of optically/UV discovered TDEs between –17 ≤ M_ peak ≤ –21) although it is likely that this luminosity function extends to even fainter sources. The bimodality in peak absolute magnitude is not significant and can be explained by small sample statistics. §.§.§ Eddington ratioUsing the blackbody temperature and the peak absolute magnitude, we calculate the integrated blackbody peak luminosity L_ BB. We determine the uncertainties by varying the temperature of the blackbody function within its errors. In panel c) of Fig. <ref>, we compare L_ BB to the Eddington luminosity implied by our black hole masses. The lines represent constant Eddington ratios, where the solid line represents the Eddington limit (i.e. where L_ BB = L_ Edd). The peak luminosity of all TDEs is consistent with being at the Eddington limit except for the two events with the highest black hole masses, which have Eddington ratios of ∼ 0.02 for TDE1 and 0.07 for D3–13. These properties are in agreement with simple dynamical predictions for the peak mass fallback rate Ṁ_ peak, which give (e.g. )Ṁ_ peak/Ṁ_ Edd≈ 130 η/0.1( M_ BH/10^6M_⊙)^-3/2( M_⋆/M_⊙)^2 ( R_⋆/R_⊙)^-3/2Here η ≤ 1 is the radiative efficiency of the accretion flow produced by the tidal disruption of a star with mass M_⋆ and radius R_⋆ (Ṁ_ Edd≡ L_ Eddη^-1c^-2). In this scenario, the initial fallback rate is super-Eddington for low mass SMBHs and most stars on the main sequence. Nevertheless, if this simple fallback picture holds, the blackbody luminosity is limited to the Eddington luminosity. For a typical lower main sequence star (M_⋆ = 0.3 M_⊙, R_⋆ = 0.38 R_⊙), the initial fallback rate following disruption will be sub-Eddington when M_ BH ≥ 10^7.13 M_⊙, as is probably the case for TDE1 and D3-13. In these flares, the fallback rate is likely sub-Eddington, and assuming that the luminosity tracks the fallback rate, so is the optical emission. If emission mechanisms other than blackbody operate, and depending on if these involve the emission of higher energy (e.g. X-ray) radiation, this picture could change drastically.§.§.§ Photometric evolutionIn Fig. <ref> panel d), we plot the decay rate from the peak of the lightcurve as a function of M_ BH. Because of the heterogeneity of the available data, we use the best sampled lightcurve, which is either the Swift NUV filter or the optical r or g filters. The temperature evolution is observed to be near constant during the evolution of the flares <cit.>. This means that the choice of filter should not impact these measurements significantly. The slope and its associated uncertainty are estimated using the standard formalism of linear regression. Although this may not be the model that best fits the data, it ensures that we can obtain a homogeneous set of measurements for all events. We also correct for the effect of time dilation in the observer's frame by scaling the measured decay rates with (1 + z) to obtain the decay rates in the rest frame of the host galaxies <cit.>. The lowest mass black hole (iPTF–16fnl) hosted the fastest decaying TDE (see ), and the most massive black hole (D3–13) has the slowest decay timescale. The qualitative trend of a faster decay timescale with lower black hole mass as observed here is predicted by theory from the assumption that the peak optical luminosity traces only the peak mass fallback rate, which scales as Ṁ_ peak∝ M_ BH^-1/2 <cit.> and is plotted as a dashed line to guide the eye (note that this is not a fit to the data). However, the actual mechanism producing the optical emission is unknown and therefore it is unclear if a tight correlation should be expected. Other parameters such as the depth of the encounter (e.g. ), the properties of the star <cit.> or the spin of the black hole <cit.> may all influence the photometric evolution of the flare. §.§ The blackbody emission mechanismWe use the blackbody temperatures and luminosities to estimate the blackbody radius where the emission is produced. If no uncertainty on the blackbody temperature is given in the literature, we assume it to be 10 per cent, similar to observed values (Table <ref>). Uncertainties for the blackbody radius are obtained by standard error propagation, and do not include systematic errors. Because we have accurate constraints on the black hole masses, we investigate whether the estimated blackbody radii can discriminate between two current theoretical models for the optical emission.We consider a model where the emission arises directly from a compact accretion disk, which forms at ∼ 2 × R_ p (e.g. ).Alternatively, we consider a class of models where the power source of the flare is dissipation of orbital energy in the circularization process <cit.>, and the blackbody emission originates in shocks at the stream self-intersection radius <cit.>. Stream self-intersection is caused by general relativistic apsidal precession, and scales steeply with the ratio of R_ p to the gravitational radius R_ g=GM_ BH/c^2. For this reason, <cit.> argue that shallow encounters (at low β = R_ T / R_ p, the penetration factor of the fatal orbit) circularize relatively far from the BH, leading to optical/UV emission, while high β encounters produce X-ray TDEs.We estimate the self-intersection radius R_ SI by considering the orbits of test particles around a SMBH. Averaged over one orbit, general relativistic apsidal precession causes the argument of pericenter ω to advance by an amountδω = A_ S-2A_ Jcosι,at leading post-Newtonian order. In this equation, the contributions to apsidal precession from the black hole mass and spin-induced frame dragging are A_ S and A_ J, respectively, and are given by <cit.>A_ S=6π/c^2GM_ BH/R_ p(1+e)≈ 11.5^∘( R̃_ p/47.1)^-1A_ J=4π a_ BH/c^3( GM_ BH/R_ p(1+e))^3/2≈ 0.788^∘( R̃_ p/47.1)^-3/2 a_ BHIn the above equations, the orbital pericenter, eccentricity, and inclination (with respect to the BH equatorial plane) are R_ p, e, and ι, respectively. The BH possesses a mass M_ BH and a spin a_ BH. Likewise, R̃_ p is the orbital pericenter normalized by the gravitational radius R_ g, and R̃_ p= 47.1 for a 10^6M_⊙ SMBH. The approximate equalities on the right assume highly eccentric orbits (1+e ≈ 2).We now limit ourselves to the case of coplanar orbits, i.e. we assume that the orbital plane of the star is perpendicular to the spin axis of the BH. If we assume apsidal precession occurs impulsively at pericenter, we find that the debris stream will self intersect at a distance <cit.>R_ SI=R_ p(1+e)/1+ecos(π + δω/2 )Stream self-intersection may be greatly complicated by inclined orbits undergoing nodal precession <cit.>, but this is primarily due to small vertical offsets between debris streams; the projected radius of self-intersection will not deviate greatly from Eq. <ref> unless R̃_ p∼ 1. In computing the depth β of each encounter, we take the tidal radius R_ T≡R_⋆(M_ BH/M_⋆)^1/3. Here M_⋆ and R_⋆ are the mass and radius of the victim star, respectively, and we assume the lower main sequence relationship R_⋆∝M_⋆^0.8 <cit.>.In Figure <ref> we show the expected emission region in the case of the compact accretion disk model (dotted lines), while the solid (dot-dashed) lines represent Schwarzschild (Kerr) stream self-intersection radii. The shaded areas illustrate the effect of increasing black hole spin (a_ BH), while the different colours represent different impact parameters, with β ≈ 1 being the most common type of event <cit.>. The shaded areas below the solid lines represent retrograde spin values (a_ BH ≤ 0), while the area above the solid line corresponds to prograde spins (a_ BH ≥ 0). A retrograde spin increases the amount of apsidal precession, which decreases the stream self-intersection radius <cit.>. Conversely, a prograde spin diminishes the apsidal precession, forcing a self-intersection at larger radius. Our mass and radius measurements are overplotted as black dots.The dotted lines in Figure <ref> are the same as in Figure <ref>, while the dashed and solid lines illustrate the effect of stellar mass; here the mass of the disrupted star is M_⋆ = 0.1 M_⊙ and M_⋆ = 1 M_⊙, respectively. In this case we have assumed a non-spinning (Schwarzschild) BH. Our inferred blackbody radii, which can be interpreted as the location from which the blackbody emission (at peak brightness) originates, are consistent with the self-intersection radius of shallow impact encounters (β ∼ 1–2), regardless of the BH spin or mass of the disrupted star. A scenario involving an accretion disk which extends to a few tens of gravitational radii from the black hole can be ruled out as the origin of the blackbody luminosity at peak brightness by our measurements. It is clear from Figures <ref> and <ref> that the stream self-intersection radius (at fixed M_ BH) is more sensitive to the mass of the disrupted star than it is to increasing black hole spin. While the degeneracy between a_ BH and M_⋆ precludes us from inferring the specific combination of black hole spin, impact parameter and stellar mass of the TDEs in our sample, it does allow us to conclude that the most likely region of origin for the blackbody emission for all optical/UV TDEs is at the stream self-intersection radius of low β encounters, lending empirical support to the stream-stream collision model for the power source of optical TDEs at peak brightness. However, we note that – while this data is deeply inconsistent with simple models of compact accretion disks – accretion-powered reprocessing models may still be able to explain the observed optical photospheres provided that the reprocessing layer is formed near the stream self-intersection point. The circularization process is still poorly understood, but our results suggest that accretion-powered reprocessing models will only remain viable explanations for TDE optical emission if debris circularization naturally produces optically thick photospheres on self-intersection scales.The shock-powered model of <cit.> predicts that for a circularization powered flare the peak luminosity should depend only weakly on M_ BH, in agreement with our observations (panel b, Fig. <ref>). This model also naturally explains the shrinking of the observed blackbody radius over time <cit.> as an inward drift of the shock after debris that has passed through pericenter settles into more circular orbits <cit.>. However, we do not find a clear correlation between the blackbody temperature and black hole mass as predicted by the same model.It is important to keep in mind that the precise value of the stream self-intersection radius depends on the combination of parameters β, a_ BH and mass of the disrupted star. We note that a complete disruption requires β ≳ 1.85 for low mass stars, and β ≳ 0.95 for Sun-like stars <cit.>. Although all the sources in Figure <ref> are consistent with this criterion, the figure suggests that some TDEs are due to low β encounters of stars near the high mass end of the stellar mass function (M_⋆ ≈ 1 M_⊙) rather than due to 0.3 M_⊙ stars, as expected from the initial mass function <cit.>. It is unclear if a selection bias in the current TDE sample could cause this tension. On the other hand, we remark that a non-zero, prograde BH spin can increase the self-intersection radius at given β and disrupted stellar mass. We speculate that the discrepancy could decrease if some of the SMBHs in our sample have non-zero prograde spins.§.§ Implications for the TDE rate Based on theoretical arguments, it has been proposed that the rate of TDEs should be dominated by the lowest mass galaxies hosting black holes <cit.>. It is unclear how this theoretical TDE rate translates into a observed TDE rate. At present, there is a strong tension between the observed (∼ 10^-5 Mpc^-3 yr^-1, e.g. , , ) and theoretical (∼ 10^-4 Mpc^-3 yr^-1, e.g. , ) TDE rates. <cit.> study the effect of a number of parameters and assumptions that go into the theoretical and observational rate calculations, and conclude that there is no straightforward way to bring the two closer together. Our mass distribution (Figure <ref>) shows that the observations qualitatively agree with the theoretical expectation that the sample of optical TDEs should be dominated by disruptions in galaxies hosting low mass (∼ 10^6 M_⊙) black holes (see e.g. fig. 6 in). The fact that we observe TDEs in lower mass black holes than previously assumed has important consequences for the inferred TDE rate. In particular, there are a number of physical mechanisms that can act to reduce the TDE luminosity (and thus observed rate) for BH masses below ∼10^6.5 M_⊙. For example, <cit.> argue that inefficient circularization affects the TDE energy output for M_ BH ≤ 10^6 M_⊙, while <cit.> suggest that adiabatic losses in a slow and dense outflow may reduce the blackbody luminosity of TDEs around 10^6 M_⊙ black holes. However, our work illustrates that the current TDE sample is dominated by ∼ 10^6 M_⊙ black holes and contains several BHs with lower masses. Therefore, the current rate estimates apply to this low mass regime and cannot be invoked to explain the discrepant TDE rates. In other words, we find the possibility of a hidden population of TDEs around low mass (10^5-6 M_⊙) BHs as an explanation for the rate discrepancy unlikely. Moreover, his is further supported by the fact that we do not observe a strong correlation between the TDE peak luminosity and black hole mass, which implies that any selection effect due to the low volume probed by TDEs around low mass BHs does not significantly affect the current sample (at least down to M_ BH ∼ 10^6 M_⊙).§.§ Intermediate mass black holes Our TDE selected host galaxy sample suggests that there is a large, hidden population of low mass black holes lying dormant in the centers of galaxies. Low mass black holes are notoriously hard to find, even when they accrete from a steady reservoir of gas. Some searches exploit the short timescales of X-ray variability to separate low from high mass black holes (e.g.). Alternatively, scaling relations based on optical emission lines <cit.> or virial based techniques can be used to estimate M_ BH in active galaxies <cit.>. <cit.> show that the AGN fraction in low mass galaxies in the local Universe (0.02 ≤ z ≤ 0.3) does not rise above a few per cent, while <cit.> find that the AGN fraction decreases with increasing SMBH mass. The large majority (≥ 95 %) of black holes in low mass galaxies are therefore currently hidden from our view, and TDEs can be a powerful tool to find and study the demographics of low mass galaxies and their low mass central SMBHs.If the mass distribution of our sample of TDE hosts is representative for the population of all optical/UV TDE host galaxies, this holds exciting prospects for finding intermediate mass black holes in the local universe. In the near future, optical surveys such as performed by the Zwicky Transient Factory (ZTF), Gaia and the Large Sky Synoptic Telescope (LSST) are expected to uncover thousands of TDEs and thus large numbers of low mass black holes. This can open up a new avenue for the systematic study of IMBH formation and evolution, and the galaxies in which they reside. Using TDEs as an independent probe for BHs in low mass galaxies, mass measurements on this future sample of TDE host BHs will shed light on the validity of the M – σ relation at the low end (see Figure <ref>), and will help constrain the black hole occupation fraction at the low mass end. The existence and masses of IMBHs in low mass galaxies are an important tool to differentiate between SMBH formation scenarios (e.g.), and can enable the study of the main mechanisms for low mass SMBH growth and evolution as well as their formation. For example, different seed models leave different (and observable) imprints on the current (z = 0) MBH mass function <cit.>. § CONCLUSIONS We present the first systematic black hole mass measurements for a sample of TDE host galaxies in the Northern sky using the M – σ relation. Our host galaxy sample of optically/UV selected TDEs encompasses 12 sources, and is complete down to g_ host = 22 mag, spanning a redshift range between 0.016 and 0.37. We use medium resolution spectroscopic observations in combination with the penalized pixel fitting routine to extract the line of sight velocity distributions, and in particular the velocity dispersions. Care is taken to correct for the instrumental broadening, and we study the effect of using the luminosity-weighted LOSVD as a proxy for the central velocity dispersion. We find that the luminosity-weighted LOSVD agrees well with the central velocity dispersions.Using the M – σ relation from <cit.> we convert the velocity dispersion measurements into black hole masses. Our galaxies host black holes with masses ranging between 3×10^5 M_⊙ ≤ M_ BH ≤ 2×10^7 M_⊙. Our mass distribution agrees with theoretical estimates; the optical TDE population is dominated by low mass (∼ 10^6 M_⊙) black holes. We find suggestive evidence for a correlation between the black hole mass and redshift, which is expected for a flux-limited sample. Furthermore our observations reveal tentative evidence for a correlation between the photometric evolution timescale (decay rate) and the mass of black hole: TDEs around lower mass black holes evolve faster. We note that these correlations are not statistically significant, potentially due to both the uncertainties on the observables and the small sample size. The blackbody emission of our sources is consistent with being at the Eddington limit at peak brightness, except for the two sources with M_ BH ≥ 10^7.1 M_⊙ for which the Eddington ratio is ≤ 0.1. These properties corroborate the standard TDE picture as a satisfactory explanation for these events.Regarding the origin of the blackbody emission, we compare the blackbody radii of the flares with models proposed to explain the origin of the emission, including a compact accretion disk and shocks due to stream self-intersections. We find that the emission region at peak brightness is located more than ∼100 R_ g from the black holes, and is consistent with the stream self-intersection radius of disruptions at low β ∼ 1 – 2. This rules out a compact accretion disk as the direct origin of the blackbody emission, and suggests that at peak luminosity, TDEs are powered by shocks due to stream-stream collisions rather than directly by accretion power.Finally, our finding that TDEs frequently occur in low mass (∼ 10^6 M_⊙) black holes implies a worsening of the rate discrepancy between theoretical and observational rates. This follows by noting that several mechanisms predict a lower flare brightness for TDEs in low mass ≤ 10^6.5 M_⊙ BHs, while our observations show that the current TDE sample is dominated by such events. This may not be true if the currently observed TDE rate is only a small fraction of the true TDE rate (e.g. due to other selection effects). Our results suggest that there is a large population of dormant, low mass black holes hidden at the centres of local galaxies. TDEs could provide an opportunity to uncover this population through (near-) future time domain surveys, which are expected to find thousands of TDEs per year. The sample of TDE host galaxies may be useful to constrain the properties of low mass black holes, as well as the formation channels and dominant growth and feeding mechanisms of SMBHs.§ ACKNOWLEDGEMENTSWe would like to thank the anonymous referee for suggestions that improved the manuscript. TW wishes to thank D. Lena for assisting in the observing runs, and is grateful to D. Lena and M. Torres for useful discussions. SvV is supported by NASA through a Hubble Fellowship (HSTHF2-51350). PGJ acknowledges support from European Research Council Consolidator Grant 647208. NCS acknowledges support through NASA from Einstein Postdoctoral Fellowship Award Number PF5-160145. The research leading to these results has received funding from the European Union’s Horizon 2020 Programme under the AHEAD project (grant agreement n. 654215). SG is supported in part by NSF CAREER grant 1454816 and NASA Keck Grant 1568615. Part of this work was inspired by discussions within International Team #371 Using Tidal Disruption Events to Study Super-Massive Black Holes at the International Space Science Institute in Bern, Switzerland. We thank Tom Marsh for developing the software package molly. The WHT is operated on the island of La Palma by the Isaac Newton Group of Telescopes in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. The ISIS spectroscopy was obtained as part of programmes W15BN010 and W16AN007. Based on observations made with ESO Telescopes at the La Silla Paranal Observatory with the director's discretionary time, programme ID 297.B-5062(A) (P.I. Jonker). Some of the data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. We wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.mnras.bst | http://arxiv.org/abs/1706.08965v1 | {
"authors": [
"Thomas Wevers",
"Sjoert van Velzen",
"Peter G. Jonker",
"Nicholas C. Stone",
"Tiara Hung",
"Francesca Onori",
"Suvi Gezari",
"Nadejda Blagorodnova"
],
"categories": [
"astro-ph.GA",
"astro-ph.HE"
],
"primary_category": "astro-ph.GA",
"published": "20170627175959",
"title": "Black hole masses of tidal disruption event host galaxies"
} |
Exact functionals for correlated electron-photon systems Angel Rubio December 30, 2023 ========================================================§ ABSTRACTThis paper introduces Gabor scattering, a feature extractor based on Gabor frames and Mallat's scattering transform. By using a simple signal model for audio signals, specific properties of Gabor scattering are studied. It is shown that, for each layer, specific invariances to certain signal characteristics occur. Furthermore, deformation stability of the coefficient vector generated by thefeature extractor is derived by using a decoupling technique which exploits the contractivity of general scattering networks.Deformations are introduced as changes in spectral shape and frequency modulation. The theoretical results are illustrated by numerical examples and experiments. Numerical evidence is given by evaluation on a synthetic and a “real” dataset, that the invariances encoded by the Gabor scattering transform lead to higher performance in comparison with just using Gabor transform, especially when few training samples are available.§ INTRODUCTIONDuring the last two decades, enormous amounts of digitally encoded and stored audio have become available.For various purposes, the audio data, be it music or speech, need to be structured and understood. Recentmachine learning techniques, known as (deep) convolutional neural networks (CNN), have led to state of the art results for several tasks such as classification, segmentation or voice detection, cf. <cit.>. CNNs were originally proposed for images <cit.>, which may be directly fed into a network. Audio signals, on the other hand, commonly undergo some preprocessing to extract features that are then used as input to a trainable machine. Very often, these features consist of one or several two-dimensional arrays, such that the image processing situation is mimicked in a certain sense. However, the question about the impact of this very first processing step is important and it is not entirely clear whether a short-time Fourier transform (STFT), here based on Gabor frames, the most common representation system used in the analysis of audio signals, leads to optimal feature extraction. The convolutional layers of the CNNs can themselves be seen as feature extractors, often followed by a classification stage, either in the form of one or several dense network layers or classification tools such as support vector machine (SVM). Stéphane Mallat gave a first mathematical analysis of CNN as feature extractor, thereby introducing the so called scattering transform, based on wavelet transforms and modulus nonlinearity in each layer <cit.>. The basic structure thus parallels the one of CNNs, as these networks are equally composed of multiple layers of local convolutions,followed by a nonlinearity and, optionally, a pooling operator, cp., Section <ref>. In the present contribution, we consider an approach inspired by Mallat's scattering transform, but based on Gabor frames, respectively, Gabor transform (GT). The resulting feature extractor is calledGabor scattering (GS).Our approach is a special case of the extension of Mallat's scattering transform proposed by Wiatowski and Bölcskei <cit.>, which introduces the possibility to use different semi-discrete frames, Lipschitz-continuous nonlinearities and pooling operators in each layer. , invariance and deformation stability properties of the scattering transform with respect to operators defined via some group action were studied.In the more general setting of <cit.>, vertical translation invariance, depending on the network depth, and deformation stability for band-limited functions have been proved. In this contribution, we study the same properties of the GS and a particular class of signals, which model simple musical tones (Section <ref>). Due to this concrete setting, we obtain quantitative invariance statements and deformation stability to specific, musically meaningful, signal deformations.Invariances are studied considering the first two layers, where the feature extractor extracts certain signal features of the signal model (i.e., frequency and envelope information), cp., Section <ref>. By using a low-pass filter and pooling in each layer, the temporal fine structure of the signal is averaged out. This results in invariance with respect to the envelope in the first and frequency invariance in the second layer output. To compute deformation bounds for the GS feature extractor, we assume more specific restrictions than band-limitation and use the decoupling technique, first presented in <cit.>. Deformation stability is proven by only computing the robustness of the signal class w.r.t spectral shape and frequency modulation, see Section <ref>. The robustness results together with contractivity of the feature extractor, which is given by the networks architecture, yields deformation stability.To empirically demonstrate the benefits of GS time-frequency representation for classification, we have conducted a set of experiments. In a supervised learning setting, where the main aim is the multiclass classification of generated sounds, we have utilized a CNN as a classifier. In these numerical experiments, we compare the GS to a STFT-based representation. We demonstrate the benefits of GS in a quasi-ideal setting on a self implemented synthetic dataset, and we also investigate if it benefits the performance on a real dataset, namely, GoodSounds <cit.>.Moreover we focus on comparing these two time-frequency representations in terms of performance on limited sizes of training data, see Section <ref>. §.§ Convolutional Neural Networks (CNNs) and Invariance CNNs are a specific class of neural network architectures which have shown extremely convincing results on various machine learning tasks in the past decade. Most of the problems addressed using CNNs are based on, often, big amounts of annotated data, in which case one speaks about supervised learning. When learning from data, the intrinsic task of the learning architecture is to gradually extract useful information and suppress redundancies, which always abound in natural data. More formally, the learning problem of interest may be invariant to various changes of the original data and the machine or network must learn these invariances in order to avoid overfitting. As, given a sufficiently rich architecture, a deep neural network can practically fit arbitrary data, cp. <cit.>, good generalization properties depend on the systematic incorporation of the intrinsic invariances of the data. Generalization properties hence suffer if the architecture is too rich given the amount ofavailable data. This problem is often addressed by using data augmentation.Here,we raise thehypothesis that using prior representations which encode some potentially useful invarianceswill increase the generalization quality, in particular when using a restricted size of dataset.The evaluation ofthe performance on validation data in comparison to the results ontest data strengthens our hypothesis for the experimental problem presented in Section <ref>.To understand the mathematical construction used within this paper, we briefly introduce the principal idea and structure of a CNN. We shall see that the scattering transforms, in general, and the GS, in particular, follow a similar concept of concatenating various processing steps, which ultimately leads to rather flexible grades of invariances in dependence on the chosen parameters.Usually, a CNN consists of several layers, namely, an input, several hidden (as we consider the case of deepCNN the number of hidden layersis supposed to be ≥2) and one output layer. A hidden layer consists of the following steps: first the convolution of the data with a small weighting matrix, often referred to as a kernel[We point out that the term kernel as used in this work always means convolutional kernels in the sense of filterbanks. Both the fixed kernels used in the scattering transform and the kernels used in the CNNs, whose size is fixed but whose elements are learned, should be interpreted as convolutional kernels in a filterbank. This should not be confused with the kernels used in classical machine learning methods based on reproducing kernel Hilbert spaces, e.g., the famous support vector machine, c.f. <cit.>], which can be interpreted aslocalization of certain properties of the input data.The main advantage of this setup is that only the size and number of these (convolutional) kernels are fixed, but their coefficients are learned during training.So they reflect the structure of the training data in the best way w.r.t the task being solved. The next building block of the hidden layer is the application of a nonlinearity function, also called activation function, which signals if information of this neuron is relevant to be transmitted. Finally, to reduce redundancy and increase invariance, pooling is applied. Due to these building blocks, invariances to specific deformations and variations in the dataset are generated in dependence on the specific filters used, whether they are learned, as in the classical CNN case, or designed, as in the case of scattering transforms <cit.>. In this work, we will derive concrete qualitative statements about invariances for a class of music signals and will show by numerical experiments that these invariances indeed lead to a better generalization of the CNNs used to classify data. Note that ina neural network, in particular inCNNs, the output, e.g., classification labels,is obtained after several concatenated hidden layers. In the case of scattering network, the outputs of each layer are stacked together into a feature vector, and further processing is necessary to obtain the desired result. Usually, after some kind of dimensionality reduction, cf. <cit.>, this vector can be fed into a SVM or a dense NN, which performs the classification task. §.§ Invarince Induced by Gabor Scattering In this section, we give a motivationfor the theory andunderlying aim of this paper.In Figure <ref>, we see sound examples fromdifferent classes, where Class 0 is a pure tone with 5 harmonics, Class 1 is an amplitude modulated version thereof, Class 2 is the frequency modulated version, and Class 3 contains the amplitude and frequency modulated signal. So, we haveclasses with different amplitudes, as is clearly visible inthe waveforms shown in the left-most plots. In this paper, weintroduce GS, as a new feature extractor that introduces certaininvariances. GS has several layers,denoted byOutA, OutB, and OutC, and each layer is invariantwith respect to some features. The first layer, here OutA, is the spectrogram of the waveform. So, we see thetime-frequency content of the four classes. OutB can be seento beinvariant with respect toamplitude changes, whereas the lastlayer, OutC, is invariant with respect to tofrequency content while encoding the amplitude information. With GS it is therefore possible to separate different qualities ofinformation contained in a spectrogram.We introduce GS mathematically in Section <ref> and elaborate on the resulting invariances in different layers in Section <ref>.Numerical experiments, showing the benefit of GS, are discussedin Section <ref>.§ MATERIALS AND METHODS §.§ Gabor Scattering As Wiatowski and Bölcskei used general semi-discrete frames to obtain a wider class of window functions for the scattering transform (cp. <cit.>), it seems natural to consider specific frames used for audio data analysis. Therefore, we use Gabor frames for the scattering transform and study corresponding properties. We next introduce the basics of Gabor frames, refer to<cit.> for more details. A sequence (g_k)_k=1^∞ of elements in a Hilbert space ℋ is called frame if there exist positive frame bounds A, B>0 such that for all f ∈ℋAf^2 ≤∑_k=1^∞ |⟨ f, g_k⟩ |^2≤ Bf ^2 . If A=B, then we call (g_k)_k=1^∞ a tight frame. In our context, the Hilbert space ℋ is either L^2(ℝ) or ℓ^2(ℤ).To define Gabor frames we need to introduce two operators, i.e., the translation and modulation operator.* The translation (time shift) operator:* for a function f ∈ L^2(ℝ) and x ∈ℝ is defined as T_x f(t):=f(t-x) for all t ∈ℝ.* for a function f ∈ℓ^2(ℤ) and k ∈ℤ is defined as T_k f(j):=(f(j-k))_j ∈ℤ. * The modulation (frequency shift) operator:* for a function f ∈ L^2(ℝ) and ω∈ℝ is defined as M_ω f(t):=e^2π iω tf(t) for all t ∈ℝ. * for a function f ∈ℓ^2(ℤ) and ω∈[-1/2,1/2] is defined as M_ω f(j):=(e^2π i ω jf(j))_j ∈ℤ. We use these operators to express the STFT of a function f∈ℋ with respect to a given window function g∈ℋ as V_gf(x,ω)=⟨ f,M_ω T_x g ⟩. To reduce redundancy, wesample V_gf on a separable lattice Λ= αℤ×ℐ,where ℐ= βℤ in case of ℋ = L^2(ℝ), and ℐ= { 0, ..., (M-1)/M} with β = 1/M in case ℋ = ℓ^2(ℤ). The sampling is done in time by α>0 and in frequency by β>0.The resulting samples correspond to the coefficients of f with respect to a “Gabor system”.(Gabor System) Given a window function 0≠ g ∈ℋ and lattice parameters α, β>0, the set of time-frequency shifted versions of g𝐺(g,α,β)={ M_β jT_α kg : (α k, β j)∈Λ} is called a Gabor system. This Gabor system is called Gabor frame if it is a frame, see Equation (<ref>). We proceed to introduce a scattering transform based on Gabor frames. We base our considerations on<cit.> by using a triplet sequence Ω = ( (Ψ_ℓ,σ_ℓ,S_ℓ))_ℓ∈ℕ, where ℓ is associated to the ℓ-th layer of the network. Note that in this contribution, we will deal with Hilbert spaces L^2(ℝ) or ℓ^2 (ℤ); more precisely in the input layer, i.e., the 0-th layer, we have ℋ_0 = L^2(ℝ) and, due to the discretization inherent in the GT, ℋ_ℓ =ℓ^2 (ℤ)∀ℓ>0.We recall the elements of the triplet:* Ψ_ℓ := { g_λ_ℓ}_λ_ℓ∈Λ_ℓ with g_λ_ℓ = M_β_ℓ j T_α_ℓ k g_ℓ, λ_ℓ = (α_ℓ k,β_ℓ j),is a Gabor frame indexed by a lattice Λ_ℓ. * A nonlinearity function (e.g., rectified linear units, modulus function, see<cit.>) σ_ℓ: ℂ→ℂ, is applied pointwise andis chosen to be Lipschitz-continuous, i.e., σ_ℓ f-σ_ℓ h_2 ≤ L_ℓ f-h_2 for all f,h ∈ℋ. In this paper we only use the modulus function with Lipschitz constant L_ℓ = 1 for all ℓ∈ℕ.* Pooling depends on a pooling factor S_ℓ >0, which leads to dimensionality reduction.Mostly used are max- or average-pooling, some more examples can be found in<cit.>. In our context, pooling is covered by choosing specific lattices Λ_ℓ in each layer.To explain the interpretation of GS as CNN, we writeℐ(g)(t) = g(-t) and have|⟨ f, M_β j T_α kg⟩ | =|f ∗(ℐ(M_β j(g)))| (α k).Thus, the Gabor coefficients can be interpreted as the samples of a convolution.We start by defining “paths” on index sets q := (q_1,...,q_ℓ ) = ( β_1 j_1 , ...,β_ℓ j_ℓ ) ∈β_1ℤ× ... ×β_ℓℤ =: ℬ^ℓ, ℓ∈ℕ. (Gabor Scattering) LetΩ = ( (Ψ_ℓ,σ_ℓ,Λ_ℓ))_ℓ∈ℕ be a given triplet sequence. Then, the components ofthe ℓ-th layer of the GS transform aredefined to bethe output of the operatorU_ℓ[q_ℓ]: ℋ_ℓ-1→ℋ_ℓ, q_ℓ∈β_ℓℤ:f_ℓ^(q_1,...,q_ℓ)(k)=U_ℓ[β_ℓ j_ℓ] f_ℓ-1^(q_1,...,q_ℓ-1) (k):= σ_ℓ(⟨ f_ℓ-1^(q_1,...,q_ℓ-1), M_β_ℓ j_ℓT_α_ℓ kg_ℓ⟩_ℋ_ℓ-1) j_ℓ,k ∈ℤ,where f_ℓ-1 is some output-vector of the previous layer and f_ℓ∈ℋ_ℓ ∀ℓ∈ℕ. The GS operator is defined asU[q]f = U[(q_1,...,q_ℓ)]f := U_ℓ[q_ℓ]··· U_1[q_1]f. Similar to<cit.>, for each layer, we use one atom of the Gabor frame in the subsequent layer as output-generating atom, i.e., ϕ_ℓ-1 := g_ℓ. Note that convolution with this element corresponds to low-pass filtering [In general, one could take ϕ_ℓ-1 := g_λ_ℓ^*, λ_ℓ^* ∈Λ_ℓ. As this element is the ℓ-th convolution, it is an element of the ℓ-th frame, but because it belongs to the (ℓ-1)-th layer, its index is (ℓ-1).].We next introduce a countable set 𝒬 : =⋃_ℓ=0^∞ℬ^ℓ,which is the union of all possible paths of the net and the space (ℓ^2(ℤ))^𝒬 of sets of 𝒬 elements fromℓ^2(ℤ). Now we define the feature extractor Φ_Ω (f) of a signal f ∈ L^2(ℝ) as in (<cit.>, Definition <ref>) based on chosen (not learned) Gabor windows.(Feature Extractor) LetΩ = ( (Ψ_ℓ,σ_ℓ,Λ_ℓ))_ℓ∈ℕ be a triplet sequence and ϕ_ℓ the output-generating atom for layer ℓ. Then the feature extractor Φ_Ω: L^2(ℝ) → ( ℓ^2(ℤ))^𝒬is defined asΦ_Ω (f) : = ⋃_ℓ=0^∞{ ( U[q]f) ∗ϕ_ℓ} _q∈ℬ^ℓ. In the following section we are going to introduce the signal model which we consider in this paper.§.§ Musical Signal Model Tones are one of the smallest units and simple models of an audio signal, consisting of one fundamental frequency ξ_0, corresponding harmonics nξ_0, and a shaping envelope A_n for each harmonic, providing specific timbre.Further, as our ears are limited to frequencies below 20 kHz, we develop our model over finitely many harmonics, i.e., {1,...,N}⊂ℕ.The general model has the following form,f(t) = ∑_n=1^N A_n(t) e^2 π i η_n(t),where A_n(t) ≥ 0∀ n ∈{1,...,N} and ∀ t. For one single tone we choose η_n(t) = nξ_0 t.Moreover, we create a space of tones 𝒯 = {∑_n=1^N A_n(t) e^2 π i n ξ_0 t | A_n ∈𝒞^∞_c (ℝ)} and assume A_n _∞≤1/n.§ THEORETICAL RESULTS§.§ Gabor Scattering of Music Signals§.§.§ Invariance In<cit.>, it was already stated that due to the structure of the scattering transform the energy of the signal is pushed towards low frequencies, where it is then captured by a low-pass filter as output-generating atom. The current section explains how GS separates relevant structures of signals modeled by the signal space 𝒯.Due to the smoothing action of the output-generating atom, each layer expresses certain invariances, which will be illustrated by numerical examples in Section <ref>.In Proposition <ref>, inspired by <cit.>, we add some assumptions on the analysis window in the first layer g_1: |ĝ_1(ω)| ≤ C_ĝ_1 (1+|ω|^s)^-1 for some s>1 and t g_1(t)_1= C_g_1 < ∞. [Layer 1]Let f ∈𝒯 with A'_n _∞≤ C_n< ∞ ∀ n∈{1,...,N}.For fixed j, for whichn_0 =n∈{1,...,N} |β_1 j - ξ_0 n| such that|β j -ξ_0 n_0|≤ξ_0/2, can be found, we obtain U [β_1 j](f)(k)=|⟨ f, M_β_1 jT_α_1 kg_1⟩| = A_n_0(α_1 k)| ĝ_1(β_1 j- n_0ξ_0)| + E_1( k) E_1( k) ≤C_g_1∑_n=1^NA'_n · T_kχ[-α_1; α_1]_∞ + C_ĝ_1∑_n = 2-n_0^N-n_01/n_0+n-1(1+ |ξ_0|^s |n-1/2|^s )^-1,where χ is the indicator function. Equation (<ref>) shows that for slowly varying amplitude functions A_n, the first layer mainly captures the contributions near the frequencies of the tone's harmonics. Obviously, for time sections during which the envelopes A_n undergo faster changes, such as during a tone's onset, energy will also be found outside a small interval around the harmonics' frequencies and thus the error estimate Equation (<ref>) becomes less stringent. The second term of the error in Equation (<ref>) depends only on the window g_1 and its behavior is governed by the frequency decay of g_1. Note that the error bound increases for lower frequencies, as the separation of the fundamental frequency and corresponding harmonics by the analysis window deteriorates.Step 1: Using the signal model for tones as input, interchanging the finite sum with the integral and performing a substitution u = t - α_1 k, we obtain⟨ f, M_β_1 jT_α_1 kg_1⟩ = ⟨∑_n =1^N M_n ξ_0A_n , M_β_1 jT_α_1 kg_1⟩= ∑_n=1^N ⟨ A_n, M_β_1 j- n ξ_0 T_α_1 k g_1 ⟩ = ∑_n =1^N ∫_ℝ A_n(u+α_1 k) g_1(u) e^-2 π i (β_1 j-n ξ_0 )(u+α_1 k) du. After performing a Taylor series expansion locally around α_1 k:A_n(u+α_1 k) = A_n(α_1 k) + u R_n(α_1 k,u), where the remainder can be estimated by |R_n(α_1 k,u)| ≤ A'_n · T_kχ[-α_1; α_1]_∞, we have⟨ f, M_β_1 jT_α_1 kg_1⟩ =∑_n=1^N [ e^-2 π i(β_1 j-n ξ_0 )α_1 kA_n(α_1 k) ∫_ℝ g_1(u) e^-2 π i(β_1 j-n ξ_0 )u du+∫_ℝ u R_n(α_1 k,u)g_1(u) e^-2 π i(β_1 j-n ξ_0 )(u+α_1 k) du]. Therefore, we choose n_0 =n |β_1 j- ξ_0 n|, set ℰ_n(k) =∫_ℝ u R_n(α_1 k,u)g_1(u) e^-2 π i(β_1 j-n ξ_0 )(u+α_1 k) du E( k) = ∑_n=1 nn_o^N e^-2 π i(β_1 j-n ξ_0 )α_1 kA_n(α_1 k) ĝ_1(β_1 j- nξ_0)and split the sum to obtain⟨ f, M_β_1 jT_α_1 kg_1⟩ = A_n_0(α_1 k) e^-2 π i( β_1 j-n_0 ξ_0 )α_1 kĝ_1(β_1 j- n_0ξ_0) +E( k) + ∑_n=1^Nℰ_n(k).Step 2: We bound the error terms, starting with Equation (<ref>): |∑_n=1^N ℰ_n( k)| = | ∑_n=1^N ∫_ℝ u R_n(α_1 k,u)g_1(u) e^-2 π i( β_1 j-n_0 ξ_0 )(u+α_1 k) du|. Using triangle inequality and the estimate for the Taylor remainder, we obtain, together with the assumption on the analysis window,|∑_n=1^N ℰ_n( k)| ≤∑_n=1^NA'_n · T_kχ[-α_1; α_1]_∞∫_ℝ| u g_1(u) |du≤ C_g_1∑_n=1^NA'_n · T_kχ[-α_1; α_1]_∞. For the second bound, i.e., the bound of Equation (<ref>), we use the decay condition on ĝ_1, thus| E(k)| ≤ C_ĝ_1∑_n=1 nn_o^N |A_n(α_1 k)| (1+|β_1 j -ξ_0 n|^s)^-1. Next we split the sum into n>n_0 and n< n_0. We estimate the error term for n>n_0:∑_n=n_0+1^N |A_n(α_1 k)| (1+|β_1 j -ξ_0 n|^s)^-1 = ∑_n=1^N-n_0 |A_n_0+n(α_1 k)| (1+|β_1 j -ξ_0 n_0-ξ_0n|^s)^-1.As n_0 =n |β_1 j - ξ_0 n|, we have|β_1 j- ξ_0n_0 | ≤ξ_0/2 and, also, using A_n _∞≤1/n, we obtain∑_n=1^N-n_0 |A_n_0+n(α_1 k)| (1+|ξ_0/2-ξ_0 n |^s)^-1≤∑_n=1^N-n_01/n_0+n( 1+|ξ_0|^s |n-1/2|^s)^-1. Further we estimate the error for n<n_0: ∑_n=1^n_0-1 |A_n(α_1 k)| (1+|β_1 j -ξ_0 n|^s)^-1≤∑_n=1^n_0-1 |A_n(α_1 k)| (1+|β_1 j -ξ_0 n_0+ξ_0 n_0-ξ_0 n|^s)^-1,where we added and subtracted the term ξ_0 n_0. Due to the reverse triangle inequality and |β_1 j- ξ_0n_0 | ≤ξ_0/2, we obtain|β_1 j - ξ_0n_0 - ξ_0 (n-n_0)|≥|ξ_0 (n_0-n) - ξ_0/2|. For convenience, we call m = n- n_0 and perform a little trick by adding and subtracting 1/2, so |ξ_0 (n_0-n) - ξ_0/2| =|ξ_0||-( m+1) + 1/2|. The reason for this steps will become more clear when putting the two sums back together. Now, we have∑_n=1^n_0-1 |A_n(α_1 k)| (1+|β_1 j -ξ_0 n|^s)^-1≤∑_m=1-n_0^-1 |A_n_0+m(α_1 k)| (1+|ξ_0|^s|(m+1)-1/2|^s)^-1. Shifting the sum, i.e., taking n = m+1, and using A_n _∞≤1/n, we get ∑_m=1-n_0^-1 |A_n_0+m(α_1 k)| (1+|ξ_0|^s|(m+1)-1/2|^s)^-1≤∑_n=2-n_0^01/n_0+n-1(1+|ξ_0|^s|n-1/2|^s)^-1. Combining the two sums Equations (<ref>) and (<ref>) and observing that 1/n_0+n<1/n_0+n-1, we obtain| E(k)| ≤ C_ĝ_1∑_n = 2-n_0^N-n_01/n_0+n-1(1+ |ξ_0|^s |n-1/2|^s )^-1. Summing up the error terms, we obtain Equation (<ref>). To obtain the GS coefficients, we need to apply the output-generating atom as in Equation (<ref>). [Output of Layer 1]Let ϕ_1 ∈Ψ_2 be the output-generating atom, then the output of the first layer is (U_1[β_1 j]f ∗ϕ_1) (k)= |ĝ_1(β_1 j- n_0ξ_0)|(A_n_0∗ϕ_1)(k)+ ϵ_1(k),where ϵ_1(k) ≤E_1_∞^2 ϕ_1_1^2. Here E_1 is the error term of Proposition <ref>.Note that we focus here on an unmodulated Gabor frame element ϕ_1, and the convolution may be interpreted as a low-pass filter.Therefore, in dependenceon the pooling factor α_1,the temporal fine-structure of A_n_0 corresponding to higher frequency content is averaged out.For this proof, we use the result of Proposition <ref>. We show that the calculations for the first layer are similar to those of the second layer:|∑_k (| ⟨ f, M_β_1 j T_α_1 kg_1⟩|-|ĝ_1(β_1 j -ξ_0 n_0)|A_n_0(k))·ϕ_1(l-k)|^2 = |∑_k E_1( k) ϕ_1(l-k) |^2 ≤E_1_∞^2 ϕ_1_1^2where E_1( k) ≤C_g_1∑_n=1^NA'_n · T_kχ[-α_1; α_1]_∞ + C_ĝ_1∑_n = 2-n_0^N-n_01/n_0+n-1(1+ |ξ_0|^s |n-1/2|^s )^-1.We introduce two more operators, first the sampling operator S_α(f(x)) = f(α x) ∀ x∈ℝ and second the periodization operator P_1/α(f̂(ω))=∑_k∈ℤf̂(ω-k/α)∀ω∈ℝ. These operators have the following relation ℱ(S_α(f))(ω) =P_1/α(f̂(ω)) . In order to see how the second layer captures relevant signal structures, depending on the first layer, we propose the following Proposition <ref>. Recall that g_ℓ∈ℋ_ℓ∀ℓ∈ℕ.[Layer 2]Let f ∈𝒯, ∑_k ≠ 0 | Â_n_0(.-k/α_1)| ≤ε_α_1 and |ĝ_2(h)|≤ C_ĝ_2 (1+|h|^s)^-1.Then the elements of the second layer can be expressed asU_2[β_2 h] U_1[β_1j]f(m) =|ĝ_1(β_1 j -ξ_0 n_0)||⟨M_-β_2 h A_n_0, T_α_2 m g_2⟩|+ E_2( m),whereE_2( m)≤ε_α_1 C_ĝ_2 |ĝ_1(β_1 j -ξ_0 n_0)|∑_r ( 1+|β_2h-r|^s)^-1+ E_1_∞·g_1.Note that, as the envelopes A_n are expected to change slowly except around transients, their Fourier transforms concentrate their energy in the low frequency range. Moreover, the modulation term M_-β_2 h pushes the frequencies of A_n_0 down by -β_2 h, and therefore they can be captured by the output-generating atom ϕ_2 in Corollary <ref>. In Section <ref>, we show, by means of the analysis of example signals, how the second layer output distinguishes tones that have a smooth onset (transient) from those that have a sharp attack, which leads to broadband characteristics of A_n around this attack. Similarly, if A_n undergoes an amplitude modulation, the frequency of this modulation can be clearly discerned, cf. Figure <ref> and the corresponding example. This observation is clearly reflected in expression Equation (<ref>). Using the outcome of Proposition <ref>, we obtainU_2[β_2 h] U_1[β_1j]f(m)=|⟨ S_α_1(A_n_0)|ĝ_1(β_1 j -ξ_0 n_0)| +E_1, M_β_2 h T_α_2 m g_2 ⟩_ℓ^2(ℤ)|≤|⟨ S_α_1(A_n_0)|ĝ_1(β_1 j -ξ_0 n_0)| , M_β_2 h T_α_2 m g_2 ⟩_ℓ^2(ℤ)|+|⟨ E_1, M_β_2 h T_α_2 m g_2 ⟩_ℓ^2(ℤ)|. Forthe error E_1(k), we use the globalestimate |⟨ E_1, M_β_2 h T_α_2 m g_2 ⟩_ℓ^2(ℤ)| ≤E_1_∞·g_1. Moreover, using the notation above and ignoring the constant term |ĝ_1(β_1 j -ξ_0 n_0)|, we proceed as follows,[⟨ S_α_1(A_n_0) , M_β_2 h T_α_2 m g_2 ⟩_ℓ^2(ℤ) =∑_k∈ℤ S_α_1(A_n_0(k)) T_α_2 m g_2( k) e^-2π i β_2 hk=;ℱ( S_α_1(A_n_0) · T_α_2 m g_2)(β_2 h)= ℱ( S_α_1(A_n_0))∗ℱ(T_α_2 m g_2)(β_2 h)=;P_1/α_1(Â_n_0)∗( M_-α_2 mĝ_2)(β_2 h)=(∑_k ∈ℤÂ_n_0(.-k/α_1))∗( M_-α_2 mĝ_2)(β_2 h). ]As ĝ is concentrated around 0, the right-hand term in Equation (<ref>) can only contain significant values if A_n_0 has frequency-components concentrated around β_2 h, therefore we consider the case k=0 separately and obtain [⟨ S_α_1(A_n_0) , M_β_2 h T_α_2 m g_2 ⟩_ℓ^2(ℤ)=(Â_n_0∗ M_-α_2 mĝ_2)(β_2 h );+( ∑_k ∈ℤ∖{0}Â_n_0(.-k/α_1)) ∗( M_-α_2 mĝ_2)(β_2 h). ] It remains to bound the sum of aliases, i.e., the second term of Equation (<ref>):[ |(∑_k ∈ℤ∖{0}Â_n_0(.-k/α_1))∗( M_-α_2 mĝ_2)(β_2 h)|=; |∑_r(∑_k ∈ℤ∖{0}Â_n_0(r-k/α_1) )·( M_-α_2 mĝ_2)(β_2 h-r)|≤; ∑_r∑_k ∈ℤ∖{0}|Â_n_0(r-k/α_1)|·|ĝ_2(β_2h-r)| ] Using the assumption ∑_k ∈ℤ∖{0} | Â_n_0(.-k/α_1)| ≤ε_α_1 and also the assumption on the analysis window g_2, namely, the fast decay of ĝ_̂2̂, we obtain[ ∑_r∑_k ∈ℤ∖{0}|Â_n_0(r-k/α_1)|·|ĝ_2(β_2h-r)|≤ε_α_1∑_r|ĝ_2(β_2h-r)|; ≤ε_α_1 C_ĝ_2∑_r ( 1+|β_2h-r|^s)^-1. ] We rewrite the first term inEquation (<ref>) and make use of the operator ℐ introduced in Equation (<ref>):[ (Â_n_0∗ M_-α_2 mĝ_2)(β_2 h )= ∑_r Â_n_0(r) (M_-α_2 mĝ_2)(β_2 h -r)=; ⟨Â_n_0, T_β_2 hℐ M_-α_2 mĝ_2 ⟩= -⟨ A_n_0, M_β_2 hT_α_2 m g_2 ⟩. ] The last Equation (<ref>) uses Plancherl's theorem. Rewriting the last term, we obtain -⟨ A_n_0, M_β_2 hT_α_2 m g_2 ⟩= -⟨ M_-β_2 hA_n_0, T_α_2 m g_2 ⟩. For sufficiently big s the sum ∑_r ( 1+|β_2h-r|^s)^-1 decreasesfast, e.g., taking s = 5the sum is approximately 2. The second layer output is obtained by applying the output-generating atom as in Equation (<ref>).[Output of Layer 2]Let ϕ_2 ∈Ψ_3, then the output of the second layer is (U_2 [β_2 h]U_1 [β_1 j]f∗ϕ_2)(m) =(|ĝ_1(β_1 j -ξ_0 n_0)| |⟨M_-β_2 h A_n_0, T_α_2 m g_2⟩|∗ϕ_2)(m) +ϵ_2(m)whereϵ_2(m)≤E_2_∞^2ϕ_2_1^2. Here E_2 is the error of Proposition <ref>. Note that in the second layer, applying the output-generating atom ϕ_2 ∈Ψ_3 removes the fine temporal structure, and thus the second layer output reveals information contained in the envelopes A_n.Proof is similar to the first layer output, see Corollary <ref>.§.§.§ Deformation Stability In this section, we study the extentto which GS is stable with respect to certain, small deformations. This question is interesting, as we may often intuitively assume that the classification of natural signals, be it sound or images, is preserved under mild and possibly local deformations. For the signal class 𝒯, we consider musicallymeaningful deformations and show stability of GSwith respect to these deformations.We consider changes in spectral shape as well as frequency modulations. Note that, as opposed to the invariance properties derived in Section <ref> for the output of specific layers, the derived stability results pertain to the entire feature vector obtained from the GS along all included layers, cp. the definition and derivation of deformation stability in <cit.>. The method we apply is inspired bythe authors of <cit.> and uses the decoupling technique, i.e., to prove stability of the feature extractor we first take the structural properties of the signal class into account and search for an error bound of deformations of the signals in𝒯. In combination with the contractivity property Φ_Ω(f)-Φ_Ω(h)_2≤f-h_2of Φ_Ω, see (<cit.>Proposition 4),which follows from B_ℓ≤ 1∀ℓ∈ℕ, where B_ℓ is the upper frame bound of the Gabor frame G(g_ℓ, α_ℓ, β_ℓ), this yields deformation stability of the feature extractor.Simply deforming a tone would correspond to deformations of the envelope A_n, n = 1,...,N.This corresponds to a change in timbre, for example, by playing a note on a different instrument. Mathematically this can be expressed as 𝔇_A_τ (f)(t)= ∑_n=1^N A_n(t+τ(t))e^2 π i n ξ_0 t. [Envelope Changes]Let f ∈𝒯 and |A'_n(t)| ≤ C_n (1+|t|^s)^-1, for constants C_n>0,n=1,...,N and s>1. Moreover let τ_∞ < 1/2. Thenf- 𝔇_A_τ(f)_2 ≤ D τ_∞∑_n=1^N C_n,for D>0 depending only on τ_∞. Settingh_n(t) = A_n(t) -𝔇_A_τ(A_n(t)), we obtainf- 𝔇_A_τ(f)_2 ≤∑_n=1^N h_n(t)_2. We apply the mean value theorem for a continuous function A_n(t) and get|h_n(t)| ≤τ_∞y ∈ B_τ_∞(t)|A'_n(y)|.Applying the 2-norm on h_n(t) and the assumption on A_n'(t), we obtain ∫_ℝ|h_n(t)|^2 dt≤∫_ℝτ_∞^2 (y ∈ B_τ_∞(t)|A'_n(y)|)^2 dt≤ C_n^2τ_∞^2 ∫_ℝy ∈ B_τ_∞(t)(1+|y|^s)^-2 dt. Splitting the integral into B_1(0) and ℝ\ B_1(0),we obtainh_n(t)_2^2 ≤ C_n^2τ_∞^2 (∫_B_1(0) 1 dt + ∫_ℝ\ B_1(0)y ∈ B_τ_∞(t)(1+|y|^s)^-2 dt). Using the monotonicity of (1+|y|^s)^-1 and in order to remove the supremum, by shifting τ_∞,we haveh_n(t)_2^2 ≤ C_n^2τ_∞^2 (∫_B_1(0) 1 dt + ∫_ℝ\ B_1(0) (1+||t|-τ_∞|^s)^-2dt ). Moreover for t ∉ B_1(0)we have|(1-τ_∞)t|^s ≤ |(1-τ_∞/|t|)t|^s. This leads toh_n(t)_2^2 ≤ C_n^2τ_∞^2 ( 2 + ∫_ℝ\ B_1(0)(1+|(1-τ_∞)t|^s)^-2dt ). Performing a change of variables, i.e., x = ( 1-τ_∞) t with dx/dt =1-τ_∞>1/2 we obtainh_n(t)_2^2 ≤ C_n^2τ_∞^2 ( 2+2 ∫_ℝ(1+|x|^s)^-2dx)=C_n^2τ_∞^2 ( 2+2 1/1+|x|^s^2_2). Setting D^2 := 2( 1+1/1+|x|^s^2_2) and summing up we obtain f- 𝔇_A_τ(f)_2 ≤ D τ_∞∑_n=1^N C_n. Harmonics' energy decreases with increasing frequency, hence C_n ≪ C_n-1, hence the sum ∑_n=1^N C_n can be expected to be small. Another kind of sound deformation results from frequency modulation of f ∈𝒯. This corresponds to, for example,playing higher or lower pitch, or producing a vibrato. This can be formulated as𝔇_τ: f(t) ↦∑_n=1^N A_n(t)e^2 π i (n ξ_0 t+τ_n(t)). [Frequency Modulation]Let f ∈𝒯. Moreover let τ_n _∞< arccos(1-ε^2/2)/2π. Then,f- 𝔇_τ(f)_2 ≤ε∑_n=1^N 1/n. We havef-𝔇_τf_2≤∑_n=1^Nh_n(t)_2,with h_n(t) = A_n(t) (1-e^2 π i τ_n(t)). Computing the 2-norm of h_n(t), we obtain∫_ℝ |h_n(t)|^2dt= ∫_ℝ |A_n(t) (1-e^2 π i τ_n(t))|^2dt ≤1-e^2 π i τ_n(t)^2_∞A_n(t)^2_∞. We rewrite |1-e^2 π i τ_n(t)|^2 = |1-(cos(2 πτ_n(t))+isin(2 πτ_n(t)))|^2 = 2(1-cos(2πτ_n(t))). Setting 1-e^2 π i τ_n(t)^2_∞≤ε^2, this term gets small if τ_n(t)_∞≤arccos(1-ε^2/2)/2π. Using the assumptions of our signal model on the envelopes, i.e., A_n_∞ < 1/n, we obtainf- 𝔇_τ(f)_2 ≤ε∑_n=1^N 1/n.[Deformation Stability] LetΦ_Ω: L^2(ℝ) → ( ℓ^2(ℤ))^𝒬, f ∈𝒯 and |A'_n(t)| ≤ C_n (1+|t|^s)^-1, for constants C_n>0,n=1,...,N and s>1. Moreover let τ_∞ < 1/2 and τ_n _∞< arccos(1-ε^2/2)/2π. Then the feature extractor Φ is deformation stable with respect to * envelope changes 𝔇_A_τ(f)(t)= ∑_n=1^N A_n(t+τ(t))e^2 π i n ξ_0 t:Φ_Ω(f)-Φ_Ω(𝔇_A_τ(f))_2≤D τ_∞∑_n=1^N C_n,for D>0 depending only on τ_∞.* frequency modulation 𝔇_τ(f)(t)= ∑_n=1^N A_n(t)e^2 π i (n ξ_0 t+τ_n(t)):Φ_Ω(f)-Φ_Ω(𝔇_τ(f))_2≤ε∑_n=1^N 1/n.The Proof follows directly from a result of (<cit.>Proposition 4), called contractivity property Φ_Ω(f)-Φ_Ω(h)_2≤f-h_2of Φ_Ω, which follows from B_ℓ≤ 1∀ℓ∈ℕ, where B_ℓ is the upper frame bound of the Gabor frame G(g_ℓ, α_ℓ, β_ℓ) and deformation stability of the signal class in Lemmas <ref> and <ref>. §.§ Visualization Example In this section, we present some visualizations based on two implementations, one in Matlab, which we call the GS implementation, and the other one in Python, which is the channel averaged GS implementation. The main difference between these implementations is an averaging step of Layer 2 in the case of the Python implementation; averaging over channels is introduced in order to obtain a 2D representationin each layer. Furthermore, the averaging step significantly accelerates the computation ofthe second layer output.Referring to Figure <ref>, the following nomenclature will be used. Layer 1 (L1) is the GT, which, after resampling to the desired size, becomes Out A. Output 1 (O1) is the output of L1, i.e., after applying the output-generating atom. Recall that this is done by a low-pass filtering step. Again, Out Bis obtained byresampling to the desired matrix size. Layer 2 (L2) is obtained by applying another GT for each frequency channel. In the Matlab code,Output 2 (O2) is then obtained by low-pass filtering the separate channels of each resulting spectrogram. In the case of Python implementation (see Figure <ref>), we average all the GT of L2 to one spectrogram (for the sake of speed) and then apply a time averaging step in order to obtain O2. Resampling to the desired size yields Out C.As input signal for this section we generate single tones following the signal model from Section <ref>. §.§.§ Visualization of Different Frequency Channels within the GS ImplementationFigures <ref> and<ref> show two tones, both having a smooth envelope but different fundamental frequencies and number of harmonics. The first tone has fundamental frequency ξ_0 = 800 and 15 harmonics, and the second tone has fundamental frequency ξ_0 = 1060 and 10harmonics. Content of Figures<ref> and<ref>: * Layer 1: The first spectrogram of Figure <ref> shows the GT. Observe the difference in the fundamental frequencies and that these two tones have a different number of harmonics, i.e., tone one has more than tone two.* Output 1: The second spectrogram of Figure <ref> shows Output 1, which is is time averaged version of Layer 1. * Output 2: For the second layer output (see Figure <ref>),we take a fixed frequency channel from Layer 1 and compute another GT to obtain a Layer 2 element. By applying an output-generating atom, i.e., a low-pass filter,we obtain Output 2. Here, we show how different frequency channels of Layer 1 can affect Output 2. The first spectrogram shows Output 2 with respect to, the fundamental frequency of tone one, i.e., ξ_0 = 800 . Therefore no second tone is visible in this output. On the other hand, in the second spectrogram, ifwe take as fixed frequency channel in Layer 1 the fundamental frequency of the second tone, i.e., ξ_0 = 1060 , in Output 2, the first tone is not visible.If we consider a frequency that both share, i.e., ξ = 3200, we see that for Output 2 in the third spectrogram both tones are present.As GS focuses on one frequency channel in each layer element, the frequency information in this layer is lost; in other words, Layer 2 is invariant with respect to frequency.§.§.§ Visualization of Different Envelopes within the GS Implementation Here, Figure <ref> shows two tones, played sequentially, having the same fundamental frequency ξ_0 = 800 and 15 harmonics, but different envelopes. The first tone has a sharp attack, maintains and goes softly to zero, the second starts with a soft attack and has some amplitude modulation. An amplitude modulated signal would, for example, correspond to f(t) = ∑_n=1^N sin(2π 20t)e^2π i n ξ_0 t; here, the signal is modulated by 20Hz. The GS output of these signals are shown in Figure <ref>. * Layer 1: In the spectrogram showing the GT, we see the difference between the envelopes and we see that the signals have the same pitch and the same harmonics.* Output 1: The output of the first layer is invariant with respect to the envelope of the signals. This is due to the output-generating atom and the subsampling, which removes temporal information of the envelope.In this output, no information about the envelope (neither the sharp attack nor the amplitude modulation) is visible, therefore the spectrogram of the different signals look almost the same.* Output 2: For the second layer output we took as input a time vector at fixed frequency of 800 Hz (i.e., frequency channel 38) of the first layer. Output 2 is invariant with respect to the pitch, but differences on larger scales are captured. Within this layer we are able to distinguish the different envelopes of the signals. We first see the sharp attack of the first tone and then the modulation with a second frequency is visible.The source code of the Matlab implementation and further examples can be found in <cit.>. §.§.§ Visualization of How Frequency and Amplitude Modulations Influence the Outputs Using the Channel Averaged ImplementationTo visualize the resampled transformation in a more structured way, we created an interactive plot (see Figure <ref>), which shows 25 different synthetic audio signals side by side, transformed into Out A, Out B, and Out C with chosen GS parameters. Each signal consists of one or more sine waves modulated in amplitude and frequency with 5 Hz steps.The parameters can be adjusted by sliders and the plot is changed accordingly. The chosen parameters to be adjusted were number of frequency channels in Layer 1, number of frequency channels in Layer 2, sampling rate, and number of harmonics of the signal.The code for the interactive plot is available as a part of the repository <cit.>.§ EXPERIMENTAL RESULTS In the numerical experiments, we compare the GS to a GT representation, which is one of the standard time-frequency representations used in a preprocessing phase for training neural networks applied to audio data. We compare these two time-frequency representations with respect to the performance on limited size of the training dataset.To convert the raw waveform into the desired representations (GS and GT), we have used the Gabor-scattering v0.0.4 library <cit.>, which is our Python implementation of the GS transform based on the Scipy v1.2.1 <cit.> implementation of STFT.To demonstrate the beneficial properties of GS, we first create synthetic data in which we have the data generation under a full control. In this case, we generate four classes of data that reflect the discriminating properties of GS. Second, we investigate whether the GS representation is beneficial when using a “real” dataset for training. For this purpose, we have utilized the GoodSounds dataset <cit.>.§.§ Experiments with Synthetic DataIn the synthetic dataset, we created four classes containing 1 s long signals, sampled at 44.1 kHz with 16 bit precision. All signals consist of a fundamental sine wave and four harmonics. The whole process of generating sounds is controlled by fixed random seeds for reproducibility. §.§.§ Data We describe the sound generator model for one component of the final signal by the following equation,f(t) = A ·sin(2π(ξ t + cw_fm(t, A_fm, ξ_fm, φ_fm)) + φ) · cw_am(t, A_am, ξ_am, φ_am),wherecw_fm(t, A_fm, ξ_fm, φ_fm) = A_fm·sin(2πξ_fm t + φ_fm) is the frequency modulation and cw_am(t, A_am, ξ_am, φ_am) = {[A_am·sin(2πξ_am t + φ_am) ifA_am > 0 (φ_am>0 ξ_am>0);1 else ]} is the amplitude modulation. Here, A is the amplitude, ξ denotes the frequency and φ denotes the phase.Furthermore, the amplitude, frequency, and phase of the frequency modulation carrier wave is denoted by A_fm, ξ_fm, and φ_fm, respectively, and for the case of amplitude modulation carrier wave we have A_am, ξ_am, and φ_am. To generate five component waves using the sound generator described above, we needed to decide upon the parameters of each component wave. We started by randomly generating the frequencies and phases of the signal and the carrier waves for frequency and amplitude modulation from given intervals. These parameters describe the fundamental sine wave of the signal. Next we create harmonics by taking multiples (from 2 to 5) of the fundamental frequency ξ, where A of each next harmonic is divided by a factor.Afterwards, by permuting the two parameters, namely, by turning the amplitude modulation and frequency modulation on and off, we defined four classes of sound. These classes are indexed starting from zero. The 0^th class has neither amplitude nor frequency modulation. Class 1 is just amplitude modulated, Class 2 is just modulated in frequency, and Class 3 is modulated in both amplitude and frequency, as seen in Table <ref>. At the end, we used those parameters to generate each harmonic separately and then summed them together to obtain the final audio file.The following parameters were used to obtain GS; n_fft = 500—number of frequency channels, n_perseg = 500—window length, n_overlap = 250—window overlap were taken for Layer 1, i.e., GT, n_fft = 50, n_perseg = 50, n_overlap = 40 for Layer 2, window_length of the time averaging window for Output 2 was set to 5 with mode set to “same”. All the shapes for Output A, Output B, and Output C were 240 × 160. Bilinear resampling <cit.> was used to adjust the shape if necessary. The same shape of all of the outputs allows the stacking of matrices into shape 3×240×160, which is convenient for CNN, because it can be treated as a 3-channel image. Illustration of the generated sounds from all four classes transformed into GT and GS can be seen in Section <ref> and Figure <ref>. With the aforementioned parameters, the mean time necessary to compute the GS was 17.4890 ms, whereas the mean time necessary to compute the GT was 5.2245 ms, which is approximately 3 times less. Note that such comparison is only indicative, because the time is highly dependent on chosen parameters, hence the final time depends on the specific settings.§.§.§ Training To compare the discriminating power of both GS and GT, we have generated 10,000 training samples (2500 from each class) and 20,000 (5000 from each class) validation samples. As the task at hand is not as challenging as some real-world datasets, we assume these sizes to be sufficient for both time-frequency representations to converge to very good performances.To compare the performance of GS and GT on a limited set of training data, we have altogether created four scenarios in which the training set was limited to 400, 1000, 4000,10,000 samples. In all of these scenarios, the size of the validation set remained at its original size of 20,000 samples and we have split the training set into smaller batches each containing 100 samples with the same number of samples from each class. Batches were used to calculate the model error based on which the model weights were updated.The CNN consisted of the batch normalization layer, which acted upon the input data separately for each channel of the image (we have three channels, namely Out A, Out B, and Out C), followed by four stacks of 2D convolution with average pooling. The first three convolutional layers were identical in the number of kernels, which was set to 16 of the size 3×3 with stride 1×1. The last convolutional layer was also identical apart from using just 8 kernels. Each convolutional layer was initialized by a Glorot uniform initialization <cit.>, and followed by a ReLu nonlinearity <cit.> and an average pooling layer with a 2×2 pool size. After the last average pooling the feature mapswere flattened and fully connected to an output layer with 4 neurons and a softmax activation function <cit.>. For more details about the networks architecture, the reader should consult the repository <cit.>. There one also finds the exact code in order to reproduce the experiment. The network's categorical cross-entropy loss function was optimized using the Adam optimizer <cit.> with lr = 0.001, β_1 = 0.9, and β_2 = 0.999. To have a fair comparison, we limit each of the experiments in terms of computational effort as measured by a number of weight updates during the training phase. One weight update is made after each batch. Each experiment with synthetic data was limited to 2000 weight updates. To create the network, we used Python 3.6 programming language with Keras framework v2.2.4 <cit.> on Tensorflow backend v1.12.0 <cit.>. To train the models, we used two GPUs, namely, NVIDIA Titan XP and NVIDIA GeForce GTX 1080 Ti, on the OS Ubuntu 18.04 based system. Experiments are fully reproducible and can be obtained by running the code in the repository <cit.>.§.§.§ Results The results are shown in Table <ref>, listing the accuracies of the model's best weight update on training and validation sets. The best weight update was chosen based on the performance on the validation set. More detailed tables of the results can be found in the aforementioned repository. In this experiment, we did not use any testing set, because of the synthetic nature of the data. Accuracy is computed as a fraction of correct predictions to all predictions.The most important observation is visible in Figure <ref>, where it is shown that in the earlier phases of the training, GS reaches higher accuracies after less weight updates than GT. This effect diminishes with bigger training sets and vanishes completely in case of 100 training batches. In case of very limited data, i.e., with only 400 training samples, the results show that GS even outperformed GT. With more training samples, i.e., 1000 and 4000, the best performances of GT and GS are nearly the same. In this case we could hypothesize that the prior knowledge of the intrinsic properties of a time series signal shown by GS (in the invariances of Layer 1 and Layer 2) is not needed anymore and the network is able to learn the necessary transformation itself. §.§ Experiments with GoodSounds DataIn the second set of experiments, we used the GoodSounds dataset <cit.>. It contains monophonic audio recordings of single tones or scales played by 12 different musical instruments. The main purpose of this second set of experiments is to investigate whether GS shows superior performance to GT in a classification task using real-life data.§.§.§ DataTo transform the data into desired form for training, we removed the silent parts using the SoX v14.4.2 library <cit.>; next, we split all files into 1 s long segments sampled at a rate of 44.1 kHz with 16 bit precision. A Tukey window was applied to all segments to smooth the onset and the offset of each with the aim to prevent undesired artifacts after applying the STFT.The dataset contains 28.55 h of recordings, which is a reasonable amount of audio data to be used in training of Deep Neural Networks considering the nature of this task. Unfortunately, the data are distributed into classes unevenly, half of the classes are extremely underrepresented, i.e., half of the classes together contain only 12.6% of all the data. In order to alleviate this problem, we decided upon an equalization strategy by variable stride. To avoid extensive equalization techniques, we have discarded all classes that spanned less than 10% of the data. In total we used six classes, namely, clarinet, flute, trumpet, violin, sax alto, and cello. To equalize the number of segments between these classes, we introduced the aforementioned variable stride when creating the segments. The less data a particular class contains, the bigger is the overlap between segments, thus more segments are generated and vice versa. The whole process of generating sounds is controlled by fixed random seeds for reproducibility. Detailed information about the available and used data, stride settings for each class, obtained number of segments and their split can be seen in Table <ref>. As seen from the table, the testing and validation sets were of the same size comprising the same number of samples from each class. The remaining samples were used for training. To prevent leaking of information from validation and testing sets into the training set, we have excluded all the training segments originating from the audio excerpts, which were already used in validation or testing set. More information can be found in the repository <cit.>.The following parameters were used to obtain GS; n_fft = 2000—number of frequency channels, n_perseg = 2000—window length, n_overlap = 1750—window overlap were taken for Layer 1, i.e., GT, n_fft = 25, n_perseg = 25, n_overlap = 20 for Layer 2, window_length of the time averaging window for Output 2 was set to 5 with mode set to `same'. All the shapes for Output A, Output B and Output C were 480 × 160. Bilinear resampling <cit.> was used to adjust the shape if necessary. The same shape of all the outputs allows the stacking of matrices into shape 3×480×160. Illustration of the sounds from all six classes of musical instruments transformed into GT and GS can be found in the repository <cit.>.§.§.§ TrainingTo make the experiments on synthetic data and the experiments on GoodSounds data comparable, we again used the CNN as a classifier trained in a similar way as described in Section <ref>. We have also preprocessed the data, so the audio segments are of the same duration and sampling frequency. However, musical signals have different distribution of frequency components than the synthetic data, therefore we had to adjust the parameters of the time-frequency representations. This led to a change in the input dimension to 3 × 480 × 160. These changes and the more challenging nature of the task led to slight modification of the architecture in comparison to the architecture in the experiment with synthetic data:The number of kernels in the first three convolutional layers was raised to 64. The number of kernels in the last convolutional layer was raised to 16. The output dimension of this architecture was set to 6, as this was the number of classes. The batch size changed to 128 samples per batch. The number of weight updates was set to 11,000. To prevent unnecessary training, this set of experiments was set to terminate after 50 consecutive epochs without an improvement in validation loss as measured by categorical cross-entropy. The loss function and optimization algorithm remained the same as well as the used programming language, framework, and hardware. Experiments are fully reproducible and can be obtained by running the code in the repository <cit.>. Consider this repository also for more details about the networks architecture.In this set of experiments, we have trained 10 models in total with five scenarios with limited training set (5, 11, 55, 110, and 550 batches each containing 128 samples) for each time-frequency representation. In all of these scenarios, the sizes of the validation and testing sets remained at their full sizes each consisting of 188 batches containing 24,000 samples.§.§.§ Results Table <ref> shows the accuracies of the model's best weight update on training, validation, and testing sets. The best weight update was chosen based on the performance on the validation set. As before, more details can be found in the aforementioned repository. In this experiment using GoodSounds data, a similar trend as for the synthetic data is visible. GS performs better than GT if we are limited in training set size, i.e., having 640 training samples, the GS outperformed GT.In Figure <ref>, we again see that in earlier phases of the training, GS reaches higher accuracies after less weight updates than GT. This effect diminishes with bigger training sets and vanishes in case of 550 training batches.§ DISCUSSION AND FUTURE WORKIn the current contribution, a scattering transform based on Gabor frames has been introduced, and its properties were investigated by relying on a simple signal model. Thereby, we have been able to mathematically express the invariances introduced by GS within the first two layers. The hypothesis raised in Section <ref>, that explicit encoding of invariances by using an adequate feature extractor is beneficial when a restricted amount of data is available, was substantiated in theexperiments presented in the previous section. It was shown that in the case of a limited dataset the application ofa GS representationimproves the performance in classification tasks in comparison to using GT.In the current implementation and with parameters described in Section <ref>, the GS is approximately 3 times more expensive to compute than GT. However, this transformation needs to be done only once—in the preprocessing phase. Therefore, the majority of computational effort is still spent during training, e.g., in the case of the GoodSounds experiment, the training with GS is 2.5 times longer than with GT. Note that this is highly dependent on the used data handling pipeline, network architecture, software framework, and hardware, which all can be optimized to alleviate this limitation. Although GS is more computationally-expensive, the obtained improvement justifies its use in certain scenarios; in particular, for classification tasks which can be expected to benefit from the invariances introduced by GS. In these cases, the numerical experiments have shown that by using GS instead of GT a negative effect of a limited dataset can be compensated. Hypothetically, with enough training samples, both GS and GT should perform equally assuming sufficient training, i.e., performing enough weight updates. This is shown in the results of both numerical experiments presented in this article (see Tables <ref> and <ref>). This is justified by the fact that GS comprises exclusively the information contained within GT, only separated into three different channels. We assume it is easier for the network to learn from such a separated representation. The evidence to support this assumption is visible in the earlier phases of the training, where GS reaches higher accuracies after less weight updates than GT (see Figures <ref> and <ref>). This effect increases with smaller datasets while with very limited data GS even surpasses GT in performance.This property can be utilized in restricted settings, e.g., in embedded systems with limited resources or in medical applications, where sufficient datasets are often too expensive or impossible to gather, whereas the highest possible performance is crucial.We believe that GT would eventually reach the same performance as GS, even on the smallest feasible datasets, but the network would need more trainable parameters, i.e., more complex architecture to do the additional work of finding the features that GS already provides. Unfortunately, in such a case, it remains problematic to battle the overfitting problem. This opens a new question—whether the performance boost of GS would amplify on lowering the number of trainable parameters of the CNN. This is out of the scope of this article and will be addressed in the future work.In another paper <cit.>, we extended GS to mel-scattering (MS), where we used GS in combination with a mel-filterbank. This MS representation reduces the dimensionality, and therefore it is computationally less expensive compared to GS.It remains to be said that the parameters in computing GS coefficients have to be carefully chosen to exploit the beneficial properties of GS by systematically capturing data-intrinsic invariances.Future work will consist of implementing GS on the GPU, to allow for fast parallel computation. At the same time, more involved signal models, in particular, those concerning long-term correlations, will be studied analytically to the end of achieving results in the spirit of the theoretical results presented in this paper. § ACKNOWLEDGMENT This work was supported by the Uni:docs Fellowship Programme for Doctoral Candidates in Vienna, by the Vienna Science and Technology Fund (WWTF) project SALSA (MA14-018), by the International Mobility of Researchers (CZ.02.2.69/0.0/0.0/16 027/0008371), and by the project LO1401. Infrastructure of the SIX Center was used for computation. unsrt | http://arxiv.org/abs/1706.08818v4 | {
"authors": [
"Roswitha Bammer",
"Monika Dörfler",
"Pavol Harar"
],
"categories": [
"cs.SD",
"cs.LG"
],
"primary_category": "cs.SD",
"published": "20170627123910",
"title": "Gabor frames and deep scattering networks in audio processing"
} |
1Graduate School of Science, Hokkaido University, Kita 10 Nishi 8, Kita-ku, Sapporo 060-0810, Japan 2Institute of Space and Astronautical Science (ISAS), Japan Aerospace Exploration Agency (JAXA), Sagamihara, Kanagawa 252-5210, [email protected], [email protected], [email protected] ISM: clouds— Stars: formation — Methods: numerical— Hydrodynamics FORMATION OF MASSIVE, DENSE CORES BY CLOUD-CLOUD COLLISIONS KenTakahira1, Kazuhiro Shima, 1*,Asao Habe1*, andElizabeth J. Tasker2* December 30, 2023 ================================================================================ We performed sub-parsec (∼ 0.014pc) scale simulations of cloud-cloud collisions oftwo idealized turbulent molecular clouds (MCs) with different masses in the range of 0.76 - 2.67 × 10^4M_ and with collision speeds of 5 - 30 km/s.Thoseparameters are larger thanTakahira, Tasker and Habe (2014)(paper I) in whichthe colliding system showed a partial gaseous arc morphology that supportsthe NANTEN observations of objects indicated to be collidingMCsby using numerical simulations.Gas clumps with density greater than 10^-20 gcm^-3were identified as pre-stellar cores and tracked through the simulation toinvestigatethe effect of mass of colliding clouds and collision speedson the resulting core population.Our results demonstrate that the smaller cloud property is moreimportant for the results of cloud-cloud collisions.Themass function of formed cores can be approximated bya power law relation with index γ = -1.6 in slower cloud-cloud collisions (v ∼ 5 km/s),in good agreement with observation of MCs.A faster relative speed increases the number of cores formed in the earlystage of collisions and shortensthe gas accretion phase of coresin the shocked region,leading to the suppression ofcore growth.Thebending point appears in thehigh mass part of the core mass function and the bending point mass decreases with increasing of the collision speed for the same combination of colliding clouds. The high mass part of thecore mass function than the bending point mass can be approximated by a power law with γ = -2∼ -3 that issimilar to the power index of the massive part of the observed stellar initial mass function. We discuss implications of our results for themassive star formation in our Galaxy. § INTRODUCTIONMassive star formation is very important in astrophysics, since massive stars play critical roles in galaxy formation and evolution.However, the processes of massive star formation are not wellunderstood. Recent observations show possible connection betweenmassive star formation and cloud-cloud collisions <cit.>. Molecular lines observations ofthe two Super Star Clusters,Westerland 2 and NGC 3603 <cit.> andtheTrified Nebula<cit.>showtwo molecularclouds of relative velocities of 10 -20km/s associated with these objects. Notably,the relative velocities observed in these three events are too high for the clouds to be gravitationally bound to one another.Bridge features that appearin the position velocity diagramsin these observationsare clear evidence of interaction by cloud-cloud collisions<cit.>.The resulting shock waves between the clouds compress the gas and form dense gas that is possible to form the massive stars.<cit.> report two molecular clouds with relative velocitiesin the Spitzer bubble,RCW 120,as an evidence of cloud-cloud collision. They also noted thatthe ring structure seen in RCW 120 is very similar to the product of cloud collisionsby theoretical calculations by<cit.>,<cit.>, and<cit.> (paper I).Similar arc like structures are observed in galactic central molecular clouds <cit.>. <cit.> performed two dimensional,axisymmetric simulations of a head-on collision between non-identical smoothed clouds. They found that the larger cloud was disrupted by thebow-shock caused by the colliding smaller cloudwhich wasalsocompressed by the same bow-shocked region. This compression caused the post-shock gas in the smaller cloud to become gravitationally unstable,even in the case where the cloud was initially below its Jeans mass. The geometric structure of the collision shows thering-like morphology similar to thegas ringobserved in RCW 120,with dense cores for the expected star formation forming in the compressed shockat the edge of the ring.Studies of cloud collision frequency byperforming numerical simulation of a Milky Way-type disk showthatmultiple collisions can occur per one orbital period<cit.>. Recent numerical studies report the similar collision frequency <cit.>.This rate agrees with theanalytical estimationmade by<cit.> who studied the rate of cloud collision thatcan explain the star formation rate in the galaxy,producing the empirical Kennicutt-Schmidt relation between gas surface density and star formation rate <cit.>.Sincethe observational studies and theoretical studies of cloud collision frequencysupport an idea of animportant role of cloud-cloud collisions in the massive star formation, theoretical studies onthe possible connection between cloud-cloud collisions and massive star formation are very interesting.In paper I,we have studied cloud-cloud collisions of ratherless massive clouds (417 M_ and1635 M_) by using numerical simulations, assuming hydrodynamic, turbulent internal motions in the colliding clouds before their collisions.We have shownthat many clumps are formed by shock compression induced by the cloud-cloud collision, and a dense and massive clump as high as 100 M_ is finally formedforcollision speeds, 3km/s and 5km/s.Mass growth of dense clumps is mainlymass accretion of surrounding gas on the clump. Inhigher collision speed case, 10km/s,clouds are highlycompressed by shock wave induced by the cloud-cloud collision,but the duration time of collision is not longenoughfor growing ofthe clump mass by the gas accretion and nomassive, dense clump isformed.The numerical results of paper I indicatethat we should simulatemore large and massive cloudscollision cases with highercollision speeds.In this paper,we study a cloud-cloud collision of more massive clouds with larger collision speedsthan paper I to examineformation of massive, dense clumpsand to investigate the impact ofcollision speed on properties and evolution of these clumps.We assumethatthe initial mass of the clouds are0.76 - 2.67× 10^4 M_.Formore massive cloudsthan this mass range, the cloud shape should be farfromspherical.We will study collisions of such massive cloudsby usingnumerical simulation,by picking upthemfromthe numerical results of galaxy scale simulations by <cit.> and<cit.>in aforthcoming paper. In order to resolve internal turbulent motionof dense clumpsformed by cloud-cloud collision,we improve our numerical resolution from 0.06pc (paper I)to 0.014pc (this paper). We concentrate onthe study ofmass function of densegas clumpsformed by the cloud-cloud collisions, since the core mass function is very important for study of the stellarinitial mass function (e.g. <cit.>).In 2 we describeour simulation method.In 3 we show our numerical results and we givediscussionsin 4and summaryin 5.§NUMERICAL METHODWe use the same simulation methodas in paper I. Since the detailed information of the simulation method is already describedin paper I, we briefly summarize it.We used Enzo; a three-dimensional hydrodynamical adaptive mesh refinement code <cit.>.A simulation boxsize of120pc ×120pc × 120pc, aroot grid size of 128 ×128 × 128and an additional sixlevels of refinement were used,giving a limiting resolution (smallest cell size) of ∼0.014pc.We used the refinement criteriabased on gas mass andthe resolution of the Jeans length,which must be refined by at least four cells as suggested by <cit.>. By using this resolution, wecan resolve internal motions in a dense core of which size islarger than ∼ 0.21 pc <cit.>.The hydrodynamics were solved by using the Zeusmethod <cit.>. In order to prevent gas undergoing unrefined self-gravitational collapse on the finest grid level,we impose a pressure floor as in paper I.Gas cools radiatively down to 10 K using acooling table created by using the CLOUDY cooling codewith thesolar metallicity and a density of 100 cm^-3 <cit.>. For the densities achieved in our simulation,the cooling function remains relatively constant,allowing us to use this simplification. Dense cores are identified in the simulation via a constant density contour-finding algorithm <cit.>. In the finding algorithm, we use fourdensity thresholds,ρ_crit = 10^-20,5× 10^-20, 10^-19, and 5 × 10^-19 gcm^-3 andcomparethe time evolutions of core number definedby these density thresholds.In order to analyze the evolution of the cores,we track their motion over the time of the simulation. This process is performed in a similar manner to the cloud tracking scheme presented in <cit.> and paper I. §.§ The initial conditionWe assume initialclouds withhigher masses and larger sizes than paper I.The density profile of each cloud is assumed to be a Bonner-Ebert sphere <cit.>; a hydrostatic isothermal self-gravitating gas sphereconfined by its external pressure. The maximum mass of the Bonner-Ebertsphereis given by: M_BE = c_BEc_s^4/P ^1/2_ext G^3/2= 4600( c_s/1km/s)^4(P _ext/4000k_B)^-1/2 M_⊙ ,where c_s is the isothermal sound speed,c_BE = 1.18 is a constant,P_ext is the external gas pressure, k_B is the Boltzmann constant and G is the gravitational constant.A cloud withless than M_BE is dynamically stable.The simulated clouds initially fulfill this stability requirement. Their properties are summarizedin Table <ref>.Once cooling begins,the cloud becomes self-gravitationally unstable and, without additional supportwill start to collapse by the self-gravity of the cloud. Adding to the Bonner-Ebertprofile clouds,we usethe constant density cloudsthat have a higher gas density andmass than the Bonner-Ebert mass. The clouds are not in free-fall collapsedue to additional support by internal turbulent motions. In Table <ref>, T_BE is temperature of theclouds and σ_v is thevelocity dispersion corresponding to T_BE. Initial clouds' radius and velocity dispersionare roughly consistent with theLarson relation <cit.>. The internal turbulent motions ininitial clouds are assumed to havethe power spectrum v_k^2∝ k^-4, corresponding to the Larson relation <cit.>. Since the focus of this paper is on the impact of the cloud collision,we selected turbulence modes that would initially stabilize the isolated clouds,preventing collapse prior to collisional contact from the gas cooling. We summarize k range in Table <ref>. The amplitude of the turbulence is givenby the Mach number,M=σ_v/c_s,where σ_v is the velocity dispersion inside the cloud and c_s is the sound speed.In the initial cloud, we assume that M = 1.As shown in paper I, we confirmed that clouds are stabilized more than their free fall time by the turbulent motions. When the turbulence is applied,the clouds remain in their initial positions for 0.5Myrs toreachthe turbulent supported cloud state, as measured bythe probability density distribution function (PDF) evolving to the expected lognormal profile for super-sonic isothermal turbulent gas <cit.>.PDFs of the Small,Medium and Large clouds with the turbulence are shown in Figure <ref> andclosely follow a lognormal distributionf(x;μ, σ )=A/σ√(2π)e^-1/2(x-μ/σ)^2where x = ln (ρ /ρ̅) and the constants have values A ∼1.2,μ = 0.0 and σ= 1.2, above a density of ∼ 10.0 cm^-3 ,as shown in paper I.We note that spikes on PDF lines near 10^-24 g/cm^3 correspond to gasnear the cloud surfaces affected by the turbulent motions in the clouds. These spikes do not affect dense core formation, sincethe spikescorrespond to rather low gas density regions.The collisions between these clouds in Table <ref> were performed under a variety of (1)different collision speeds,(2) different combination of colliding cloud sizes, and (3) different cloud density profiles.§RESULTS §.§ The effect ofcollision speeds In this subsection, we show the simulation results of cloud-cloud collisionswiththe collisionspeeds, 5,10, 20 and 30km/s, for the same combination of clouds, Medium and Large clouds. These collision speedsare largerthan paper I and are in the range ofcloud collision velocities found in global numerical simulation of cloud formation in a barred galaxy <cit.>.We foundformation of dense cores by cloud-cloud collisions in this speed range formore massive clouds than paper I. §.§.§ Core number evolution We showthe time evolution of the core numberas a fractionof thefree-fall time of the Medium Cloud,t_ff, Min Figure <ref> for the four collision speeds, 5,10,20 and 30 km/s.The initial time in the plots,t=0,corresponds to the time when the clouds surfaces touch.In this figure, we showthe core number evolutions for the four density thresholds,ρ_crit= 10^-20(black solid line),5 × 10^-20 (red dashed line), 10^-19(gray dot line) and 5 × 10^-19 gcm^-3 (blue densely dotted line).We restrict the cores plotted to thosecontaining more than27 cells to ensure the best resolution.Figure <ref> shows thatthe core numberfor ρ_crit=10^-20 gcm^-3(black solid line) increases to its maximum,thendecreasesand finally attains a nearlysteady-state.The time and valueof the maximum core number depend on the collision speed,with the higher relative speed creating more numerous coresquickly.The decrease of core number after the peaksmeans that a large part of thecoresselected by this density thresholdisnot tightly bound and hard to keeptheir structures in the later stage. The early increase of the dense core number forρ _crit =10^-20 gcm^-3is mainly causedby fragmentation of filamentary structures formed in the shocked cloud medium.The epoch of the first peaks of core numberforthis density threshold is roughly equal to the shock crossing time of the smaller (Medium) cloud, 2r_c,Medium/v_sh.After the shock crossing time,the shock compression becomes weaker due to decreasing of the velocity of the shockedmedium andgravitationally unbound cores are destroyed bythe internal irregular motions of the cores. This is the reason ofthe core number decreasingafter the first peaks of the core numbers. We show an example ofdense core formation byfragmentationin more denserfilamentsand core mergingin middle and left panels of Figure <ref>.The number of coresselected by more denser threshold, ρ_crit= 5× 10^-19 gcm^-3 (blue densely dotted line),monotonically increases with time for 5 - 30 km/s. This means that ρ_crit= 5 × 10^-19 gcm^-3is largeenough to select tightly, gravitationally boundcores in this collision speed range.Thecore numbers for the various core density thresholdsconverge in the later stage. This convergence means that surviving cores are very dense andcompact. We call this stage the converging point.§.§.§ morphology of colliding cloudsThe time evolutionsof these four simulations are shown inFigure <ref>.Each image shows slicesof the Mediumand Large clouds. Each vertical line of panels corresponds to the same event in each simulation: the left-handpanels show the first touching,the middle panels show the colliding clouds withthemaximum numberof cores. The final right-hand panels showthe converging point, wherewe shift each x-origin of the coordinate as,x (5 km/s),x+20 pc (10 km/s),x+60 pc (20 km/s) and x+80 pc (30 km/s). In themiddle panels,the Medium Clouds are compressed by shock waves andthe partial arcs of dense gas are formed.These arcs haveirregular ripple structures due to the Rayleigh-Taylor instability <cit.> and thethin shell shock instability <cit.>.Since theinitial size of the Medium cloud issmaller than the size of Large cloud byonly a factor 1.5, the arcs are rather widely open.As shown in subsection <ref>,the partial arcs aremore close forcollisions between the Smalland Large clouds . At theconverging points (the right panels)the elongated filament structures areformed and,in higher collision speed case,the filamentary structure appears more clearly and becomes oscillatorywith alarger amplitude.§.§.§ Probability distribution function The probability distribution functions (PDF) is very usefulto examine a turbulence property and self-gravitatingstructures of turbulent clouds <cit.>.PDFs are shown for the four simulation results of the different collision velocitiesat the converging pointsin Figure <ref>. PDF is obtainedfor gaswithin asphere of 30 pc radius centered on the Large Cloud.The blue dashed line in all panels isthe log-normal profile shown in Figure <ref>. In Fig. <ref>, the PDF tails extend from the log-normalto the higher gas density. The PDF tails consist of a power-law tail with a shallower extension.Power index of the power-law tailis approximately -1.5 thatwell agrees with the simulation results of a self-gravitationally collapsing,turbulent isothermal cloud<cit.>. <cit.> discuss thattheindex of the power-law tail can be explained bythe self-gravitationally collapsing structures ofisothermal gas. The power-law tail is evidence ofself-gravitational collapse of dense gasin colliding clouds in our simulations.We note thatthe power-law tail agreeswith the observations by <cit.>,who show power-lawtailsin the PDFs of the column density in the giant molecular clouds with the star formation activity, while thegiant molecularclouds with nostar formation activity show the log-normal distribution.§.§.§ Cumulative core mass distributionThe core mass distribution at the converging point is shown bythe cumulative core mass distribution (CMD)in Figure <ref>.The CMD is given bythe core mass function,ϕ_core,asN_core (>M)=∫ _Mϕ_core(M) dM.We plot the red dashed line in Figure <ref> to show a power-lawrelation N_core(> M) = 300M^-0.6 whichis given bydN_core/dM= ϕ_core∝ M^γ with γ = -1.6 and this relationfits the CMD for M > 3 M_in the 5km/s case.The power law fit with the same power index isappliedtothe CMDs in thesmaller clouds collisions with the low collision speedsin paper I.It is very interesting that the power indexagrees well with the observational results of <cit.>,who show that molecular cores foundin the Orion A molecular cloudcan be fitted bya value of γ = -1.6 ± 0.3 for M > 50M_.In the 5km/s case, the CMD is nearly flat forM < 3 M_.The nearly flat CMD means thatalmost all coresare limitedin the mass range of M > 3 M_.The flat part of the CMD shifts to more massive core mass with increasing the collision speed. For 10 km/s, another steeper power law part appearsinM > 300 M_ in the CMD.We call thebreak point of the power law fit the bending point.The power index values of the steeperpart increasewiththe collision speed asγ = -2 (20 km/s) andγ = -3 (30 km/s).The maximum mass of core and thetotal core number at the converging pointdecreases withincreasing ofthe collision speed. This result can be understood by acore accretion growth argument as showninthefollowing.Figure <ref> shows the time evolution of three cores for10 (top-row) and 20 km/s (bottom-row),plotted over the fraction of the Medium Cloud's free-fall time. The black solid line marks the core mass.The blue dashed line shows the mass expected fromthe accretion rate defined asṀ =π r_acc^2σ_eff,thρ_acc ,where σ_eff, th is σ_eff, th=√(c_s^2+ σ_turb^2) which includes the turbulent velocity dispersion of the internal gas motion, σ_turb, and ρ_acc is the mean gas density in a sphere surroundingthe core with radius,r_acc,given by the modified Bondi radius,r_acc = 2GM /σ_eff, th^2+ r_core.Note that we do not include the core mass and the others core mass within the modifiedBondi radius in ρ_acc calculation and we add the core radius to the Bondi radius to ensure a reasonable sample of gas outside the core.Here,we call this accretion region the surrounding region. The red dotted line shows the mean gas density of the surrounding region.The green dot-dashed line shows the sumof mass of the surrounding region and the core mass when the core mass reaches the core's effective Jeans mass,M_J, eff=π/6(c_s^2+σ̅_turb^2)^3/2/G^3/2ρ̅^1/2,where c_s,σ̅_turb, and ρ̅are the averaged thermal sound speed,the mean turbulent gas velocity dispersionand the mean density of the core, respectively. Figure <ref> shows that the core mass (black solid line) is well fitted with the accretion mass (blue dashed line). This shows that these cores growth is predominantly by accretion. In the left and centre panels,after the core mass begins to increase,the surrounding gas density is rapidly decreasingafter the core mass exceeds the green dot-dashed line. This suggests that the core mass grows rapidly until the core eats almostthe all surrounding region mass and,during this stage,the surrounding gas density is decreasing.After the core eats almost thesurrounding mass,the rate of core mass growth is decreasing and the slow growth of the core mass corresponds to thesteeper part inthe CMD plots.In the right panels, the growth of core mass is predominantly by accretion,but a jumpseen after ∼ 0.7 t_f f , M (10 km/s) and ∼0.42 t_f f , M (20 km/s)indicatesmerger events afterthe surrounding density becomeslow.Based on the core mass evolution as shown in Figure <ref>, the single power law fit of the CMD means thatthe accretion growth of cores continues to the converging point in 5 km/s.Sincedensity of the shock compression layerislow in 5 km/s,it takes long time to form the high density cores in the layer as shown in Fig. <ref> andduring core formation,the surrounding regions of cores also get gas.This means that the cores cannot consume all surrounding gas mass and the cores mass can grow continuously. Higher collision speed forms a denser layer.This process causes the rapid core formation and,due to short formation time,the surrounding regions cannot get gas enough to feed the cores continuously. This effect is more clearly in20 and 30 km/s,there is no γ = -1.6 power-law slopeas in Fig. <ref>.Note that,in these two cases,there are the mass excesses in the high mass range M >20 M_.These excesses are the results of the core merging events as shown in the right panel in Figure <ref>.§.§.§ Mass supply rate inside the dense cores The maximum mass of a star formingdense core will be limited by the radiation pressure from a protostar <cit.>.<cit.> showed that thehigh accretion rate, Ṁ>10^-4 M_ /yr, can overcome the radiation pressure in themassive star formation ofM>8 M_. Figure <ref> shows time evolution ofthe core mass and the mass supply rate inthe dense cores shown in Fig.<ref>, where the mass supply rate is the rate of gas accretionto a protostarin a collapsing core andmay be given by ṁ_*∼ m_core/t_ff,core,where m_core ismass of the dense core and t_ff,core isthe free fall time of the dense core. We usehalf of the core's mass or the effective Jeans mass,whichever is larger, for m_core, ṁ_* ∼max(0.5M_core,M_J.eff)/ t_ff, core.We also show the effective Jeans mass in Figure<ref>.The core mass reaches the effectiveJeans mass at approximately 1- 2 M_.If stars form at this point, 1 - 2M_ stars will be formed, since the typical free fall time is 2.5 × 10^5 yr for ρ=10^-19 g cm ^-3.However,if the cores are prevented from collapsing (for instance, by magnetic fields, turbulence or core rotation unresolved in our simulation) then the core mass may increase as shown in Figure <ref>and their mass supply rate will be larger than 10^-4 M_ yr^-1.In this situation,the cores will collapse to form massive stars. §.§ The effect of the cloud size To explore the effect of sizes of colliding cloudsonproperties ofthe formedcores,we compare the numerical results of thefour combinations,Small-Medium,Small-Large,Medium-Mediumand Medium-Largecloudsfor the same collision speed, 10 km/s.The evolution of these four simulations is shownin Figure <ref>. For the Small-Large clouds case, the arc-like structure is more clearly formedthanthe Medium-Large clouds case. We analyze the core number evolution,PDF and CMD as shown in Figures <ref>,<ref>, and <ref>. The evolution of the core numbershown in Figure <ref> show thatthemaximum core number, thecore number at theconverge point and the epoch of the converging point,strongly depend on smaller clouds.The PDFsshown in Figure <ref> showthe power-law tail in thedensergasthan 10^-22 g/cm^-3. In theMedium-Mediumclouds,extension of thepower-law tail is smaller than the other case.In this case,the shockedlayer shows no arc-like structureat the converging point as shown in Fig. <ref>.This may be the reason of the small extension of the power-law tail in theMedium-Mediumclouds, since the arc structures formed in the non-identical clouds help the core mass increase.In Figure <ref>, we show the CMDs and the power-law fit of γ = -1.6 with the red dashed line. Same as the core number evolution,each CMD shape mainly depends on thesmaller cloud property. For the Small cloud,the massive part of the CMDscan be fitted by the power-law ofγ = -2. This means thatgas accretion does notcontinue long enough for the core mass growthin the Small cloud case with the collision speed of 10 km/s .§.§ The effect of initial density distribution - a compact cloud with constant density To explore the effect of the initial cloud structure,we simulate the collision of a compact cloud with an initial constant density (Constant cloud) and the Large cloud. In Table 1 we showparameters of the Constant cloud.We assume the initial temperature forthe Constantcloudto beinthevirial equilibriumby its thermal energy. We also add turbulent motion of Mach number = 1 at t=0.The turbulent motion can support the Constant cloud for more than t= 0.5 Myr.Same as theprevious simulation,we give the collision speed to the Constantcloudat t = 0.5Myr. The evolution of this simulation is shownin Figure <ref>. The evolution of the core number plotted as a fraction of the Constantcloud's free-fall time,t_ffc, is shown in Figure <ref> for collisional speed, 10 km/s (left) and 20 km/s (right). In each plotlines show core numbers definedby the fourthreshold density valuesas in Figure<ref>. As in the Bonnor-Ebert cloud case, the time for the maximum number of coresdepends on collision speed, with high relative speed creating cores more rapidly.The cumulative core mass distributions are shown in Fig. <ref>. We also plot a line of the powerindex γ =-1.6. The bending point appears atM=100 M_in 20 km/s.The mass of the bending point is much larger than the Medium-Large case for 20 km/s in which the bending point is at M= 30 M_. This may be due to the fact thatsincethe surrounding mass of coresinthe Constant cloud case is much larger than the Medium cloud, the gas accretion to the core cancontinue and increase the core massmore than in the Medium cloud case.The power index of thecumulative core mass distribution intherangemore than the bending point isγ = -2.5. § DISCUSSION Mass fraction of theformed massivedense cores to total mass of the colliding cloudsis very interesting fordiscussion of a possible role ofcloud-cloud collisions in massive star formation in our Galaxy. Since the number of dense coresforρ_crit=5× 10^-19 g/cm^3isalmost constantfor more than 1 Myr after the converging point,such massive dense coreshave enough time for the massive star formation as discussedin section <ref>.We summarize the totalmass of massive dense cores, M_core,tot, anda mass fractionof M_core,tot to the total mass of colliding cloudsin Table <ref>, where M_core,tot is the total massof massive dense cores morethan 10 M_ andM_cl,tot is the total mass of the colliding clouds.M_core,total/M_cl,tot is0.046 - 0.288 for the collision speeds, 10 - 30 km/s,that are inthe speed range of the observedcolliding clouds <cit.>. In numerical simulation of cloud formation and evolution in a weak barred galaxy by<cit.>,they showed thatthe collision events ofv < 5 km/sis much smaller than that of10 - 30 km/s. Thelow speed collisions with v < 5 km/sshouldhave a minor role for the star formation than the10 - 30 km/s collisions. We discuss the role of cloud-cloud collisions in the star formation in the Galaxy by using our numerical results andthe formula of<cit.> who estimated thestar formation rate by cloud-cloud collisions as Σ_SFR=ϵ f_sfN_AM_c/t_coll,where ϵ is thefraction of gravitational unstable mass produced by a cloud-cloud collision, f_sf is the mass fraction of newly formed stars to the gravitational unstable mass,N_A is the cloud number, M_c isthe typicalcloud massand t_coll is the mean cloud collision time scale.<cit.>showed that star formation rate in our Galaxy can be explained for ϵ∼ 0.2 and f_sf∼ 0.5. The value of ϵ is comparable to our numerical results of the mass fraction of total core mass for the collision speeds,10 - 30 km/s. The typical collision time scale obtained by<cit.>is comparable to thevalue adopted by <cit.>. If these results can be used inthe formula,our simulation results indicate thatcloud-cloud collisions well contribute tothemassive star formation in the Galaxy,although mass of colliding clouds in our simulation is smaller than the typical mass of molecular clouds adopted in <cit.>. The mass fraction of massive dense cores to the total mass of colliding clouds decreases with collision speeds as shown Table <ref>.This result indicatesthat inveryhigh speed collision environments efficiency of massive dense core formation becomes smaller thanlower speed collision environments. <cit.> has shown that large part of colliding clouds in the bar region in the barred galaxy have high cloud collision speeds of much more than 30 km/s by their simulation. With this result,the low star formation efficiency observed in the bar regions <cit.>is possible to be explainedby the low core formation efficiency by thehigh velocity cloud-cloud collisionsin the bar region. For more detailed discussion of the role of cloud-cloud collision in star formation in the Galaxy, we should extend cloud-cloud collision simulations to more general cases, e.g. off-center cloud-cloud collisions, more general shape of colliding clouds, andmore massive clouds.We should consider magnetic field effectand feedback effects by newly formed massive stars. These can affectthe core formation and evolutionduring the cloud-cloud collision.In an off-center collision, a large shear flow willdominate in the interface layer of colliding clouds than the face-on collision cases andwill affectthe core formation and evolution in the interface layer. Massive clouds have internal substructures andtheir shapes are far from a spherical shape. Although MHD simulations are already madeby <cit.>, and <cit.> and radiative feedback effects are simulated by <cit.>,more extensive studies are needed for thedetailed study of the role of cloud-cloud collision in star formation in the Galaxy. Massive star formation process in dense cores formed in our simulation is beyondour numerical simulation ability in this paper. <cit.> have shown that dense cores formed inmagnetohydrodynamic colliding flowshave strong internalturbulent motions and strong magnetic fields. Bothcanlead tohigh accretion rate in the coresthat favors tohigh mass star formation, since protostars can grow via accretion againstthe radiation pressure barrier to massive stars.We obtain the CMDs from our numerical results. The CMDscan be fitted by the power law with the index γ =-1.6 in the lowcollision speedandthe steeper power law part appears with the bending points in the higher collision speed for the same combination of colliding clouds as shown in section <ref>. The presence of the bending points of the CMDs depends on the collision speeds and the smaller cloud property. We suggest thatthe steeper power law parts with bending points in the observed CMDs arethe evidence of cloud-cloud collisions.The steeper power law fit with the bending pointis reported in the CMDin the observed candidate of the cloud-cloud collision region in theGalactic center 50 km/s molecular cloud<cit.>.Their result supportour suggestion. § SUMMARYWe have explored the formation and evolution of dense gas cores incollisions of non-identical clouds with the Bonner-Ebert density profiles and the constant density profile using hydrodynamical simulations. Mass range of clouds is 0.76 - 2.67 × 10^4 M_ that is larger than paper I.The limiting resolution was 0.014pc that is small enough to resolve formation of molecular cores.We have shown the effect of collision speeds,cloud sizesand initial density distributions of cloudson thecore formation andevolution.Our numerical results show that the smaller cloud property is important for the core formation and evolution.Collision speeds are also important for the early formation ofdense cores and the late mass evolution ofmassive dense cores.As a result of core evolution, the shape of core mass function changes withthe cloud collision speeds for the same combination of colliding clouds. Collision betweentwo clouds produces a shocked gas layer that contains dense gas filaments, since thepre-collision clouds have filamentary structures produced by the internal turbulent motions. Many dense cores are formed by fragmentation ofthe dense gas filaments. Due to the size difference between the two clouds,the shocked gas region becomes oblique as the clouds merge andforms a partial arc-like structure that is commonly observed in observations of candidates ofcloud-cloud collision events <cit.>.The CMDs of cores formed by cloud-cloud collisions show that most of the dense cores have massmore than two or three solar massfor the collision speeds more than 5 km/s in our simulation results.The CMDs arewell approximatedbythepower lawforthe high mass cores. Forhigher collision speeds, the CMDs canbe described by the two power law fitswith the bending point that is the break point of the two power law fitsof the CMDs. The bending point shifts to the lower masswith the highercollision speed because the surrounding gas mass rapidly decreases in the higher collision speed case.The bending point also depends onthe smaller cloud property.In thesmaller cloud with lower mass, the bending point will appear in more earlier stage of cloud cloud collisionthan the more massive smaller cloud case. The bending points in the CMDs can be evidence of cloud-cloud collisions withhighercollision speeds. The power law index of CMDs is similar to the stellar initial mass function (Salpeter 1955; Chabrier2003).§ ACKNOWLEDGMENTS The authorsthank Yasuo Fukui,Takashi Okamoto,Kazuo Sorai, Kazufumi Torii and Tsuyoshi Inoue for their fruitful discussions.The authors also thank an anonymous referee for constructive comments. Thanks to the yt development team <cit.> for support during the analysis of these simulations. 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"authors": [
"Ken Takahira",
"Kazuhiro Shima",
"Elizabeth J. Tasker",
"Asao Habe"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170627031849",
"title": "Formation of Massive, Dense Cores by Cloud-Cloud Collisions"
} |
1Center for the Exploration of the Origin of the Universe (CEOU), Astronomy Program, Department of Physics & Astronomy, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 151-742 Korea 2LOCOOP, Inc., 311-1, 108 Gasandigital2-ro, Geumcheon-gu, Seoul, Korea3Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA 4School of Space Research, Kyung Hee University, 1732 Deogyeong-daero, Giheung-gu, Yongin-si, Gyeonggi-do 446-701, Korea 5SongAm Space Center, 103, 185 Gwonnyul-ro, Jangheung-myeon, Yangju-si, Gyeonggido 482-812 Korea 6Astronomy Program, Department of Physics & Astronomy, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 151-742 Korea 7Arizona State University, School of Earth and Space Exploration, PO Box 871404, Tempe, AZ 85287-1404, U.S.A. 8Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon, Republic of Korea 9Subaru Telescope, National Astronomical Observatory of Japan, 650 North A'ohoku Place, Hilo, HI 96720, U.S.A. ⋆Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme 091.A-0878. †Visiting Astronomer, Kitt Peak National Observatory, National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under cooperative agreement with the National Science Foundation. E-mail: [email protected], [email protected] We present our first results of the survey for high redshift quasars at5 ≲ z≲ 5.7. The search for quasars in this redshift range has been known to be challenging due to limitations of filter sets used in previous studies. We conducted a quasar survey for two specific redshift ranges, 4.60 ≤ z ≤ 5.40 and 5.50 ≤ z ≤ 6.05, using multi-wavelength data that include observations using custom-designed filters, is and iz. Using these filters and a new selection technique,we were able to reduce the fraction of interlopers.Through optical spectroscopy, weconfirmed seven quasars at 4.7 ≤ z ≤ 5.4 with -27.4 < M_1450 < -26.4 which were discovered independently by another group recently.We estimated black hole masses and Eddington ratios of four of these quasarsfrom optical and near-infrared spectra, and found that these quasars areundergoing nearly Eddington-limited accretion which is consistent with the rapid growth of supermassive black holes in luminous quasars at z ∼ 5. § INTRODUCTION Observations have shown that large numbers of quasars are found at z ∼ 4.5 and at z > 6 <cit.>. They harbor supermassive black holes (SMBHs) as massive as ∼ 10^10 M_⊙<cit.> andappear to be vigorously evolving<cit.>. However, thereis a dearth of quasars with measured black hole masses that makes it difficultto investigate how they evolved at 5 < z < 6<cit.>. Measuring the black hole masses for a significant number of objects at this redshift range allows us to: (1) derive the Eddington luminosities, and consequently, the Eddington ratios, tounderstand the growth of these quasars.One simply expect the growth to slow down toward lower redshifts in comparison to z ∼ 6; (2) constructthe black hole mass functionto understand the cosmic emergence of the most massive quasars;(3) investigate the spectral energy distributions (SEDs) of quasars toexplore whetherquasars with very massive black holes have a lower accretion disk temperature <cit.>.The redshift gap at 5 < z < 6 mentioned above is partlydue to the inefficiency of quasar selection techniques at 5.2 < z < 5.7 in previous studies<cit.>. This low efficiency is due to limitations of current filter systems employed by these studies: the colors of z ∼ 5.5 quasars using conventional filters are similar tothose of late type stars or brown dwarfs.Figure <ref> shows two color-color diagrams generally used for high redshift quasar selection. The black solid lines with asterisks are quasar tracks redshifted from the SDSS composite quasar template from <cit.> including the intergalactic medium (IGM) attenuation <cit.>, the triangles are model colors of brown dwarfs from <cit.>, the squares are model colors of stars by <cit.> from the Bruzual-Persson-Gunn-Stryker (BPGS) atlas, and the crosses are point-like sources from the Sloan Digital Sky Survey (SDSS) Star Catalog.<cit.> used the r-i vs. i-z color-color diagram to identify quasars at z > 4.5 (Figure <ref>a; r-dropout quasars) and <cit.> used the i-z vs. z-J color-color diagram for quasars at z ∼ 6 (Figure <ref>b; i-dropout quasars). The solid boxes indicate their quasar selection criteria.We see that r-dropout quasars at z > 5.1 (Figure <ref>a) and i-dropout quasars at z < 5.7 (Figure <ref>b) are mixed with the late type stars or brown dwarfs on these color-color diagrams. Therefore, the r-dropout technique alone cannot be used for z ∼ 5.5 quasar selection. As can be seen from above, any configuration of colors from SDSS ugriz or the Two Micron All Sky Survey (2MASS) JHK filters cannot separate quasars at 5.1 < z < 5.7 from stars effectively; a new filter system that exploits the wavelength range between conventional filters is necessary to find these quasars.Thus, we searched for and studied high redshift quasars at 5 < z < 6 by using new, additional datasets and performing follow-up observations.First, we designed a new filter set, is and iz, to supplement the previous filter systems for selecting quasars at this redshift range. Since the central wavelengths of these filters are located between r and i, and between i and z, respectively, we can select high redshift quasars at this redshift gap, where the SDSS or other filter sets cannot explore. Second, we needed a special optical detector which has better sensitivity than previous CCDs at longer wavelengths, leading to more efficient observations with the is and iz filters. Considering these requirements, we developed a CCD camera system, the Camera for QUasars in EArly uNiverse <cit.>.Equipping a deep-depletion CCD chip to provide high quantum efficiency (QE) at 0.7 – 1 μm, we conducted follow-up imaging observations of quasar candidates with the is and iz filters and narrowed down the quasar candidates. CQUEAN was installed on the 2.1-m Otto Struve Telescope at McDonald Observatory in 2010 August, and it has since been used to obtain photometric data for many scientific programs, including our high redshift quasar survey.In Figure <ref>, we plot the filter transmission curves of is and iz (black solid lines), and the SDSS gri and the Large Synoptic Survey Telescope zY bands (colored dashed lines) installed on CQUEAN, with the QE of the CCD taken into consideration (gray solid line). The green line represents the SDSS composite quasar template redshifted to z ∼ 5 and IGM attenuation taken into consideration.Note that a similar survey of z ∼ 5 luminous quasars is being conducted by <cit.> and <cit.>. Their method relies on the archived multi-wavelength dataset only, while our method includes the use of the custom is and iz filters. Section <ref> describes our quasar selection algorithm including color cuts, multi-wavelength data used, and imaging and spectroscopic follow-up observations.The photometric and spectroscopic analysis of our discovered quasars are shown in Section <ref>. We discuss our quasar selection efficiency and expected number of quasars in Section <ref>. Section <ref> presents physical properties of the newly discovered quasars from the spectroscopy. We summarize this survey in the final section (Section <ref>). Throughout this paper, we use a cosmology with Ω_M=0.3, Ω_Λ=0.7 <cit.>, and H_0 = 70 km s^-1Mpc^-1. We use the AB magnitude system.§ QUASAR SELECTION AND OBSERVATION§.§ Quasar Candidate SelectionTo select quasars at5 < z < 6 , we employed multi-wavelength data that cover a large area: SDSS DR8, and the United Kingdom Infra-Red Telescope Infrared Deep Sky Survey Large Area Survey<cit.> DR10;the full overlapping area between the two surveys is ∼3,400 deg^2. Ther, i, z, J, and K magnitudes are used.Since the contamination rate using these filters is still high,we adoptedis and iz-band photometry to discriminate brown dwarfs from r-dropout objects.Then we set additional criteria to assign priorities for follow-up observations. No stellarity cut is made to avoid missing quasars that are classified to be extended objects (e.g., due to host galaxy or noise in stellarity calculation), although we used the stellarity as a way to set priorities for follow-up observation. Figure <ref> shows a main quasar candidate selection algorithm. §.§.§ r-i-z-J-K, is and iz-band Selections To select quasar candidates from broadband photometry, we used the dropout feature at the Lyman α (Lyα) emission line that are common in high redshift objects.The Lyα dropouts can be identified using the r-i color for quasars at z > 3.6, and r-i > 1.5 for quasars at z > 4.6.To discriminate high redshift quasars from red, low mass stars, we usedthree color cuts, r-i, z-J, and J-K: r-i to select dropout objects, z-J to removebrown dwarfs, and J-K to eliminate other stars.Figure <ref> shows two color-color diagrams with model brown dwarfs from <cit.> (green triangles), observed brown dwarfs from <cit.> and <cit.> (green squares), and stellar sources from the SDSS catalog (gray circles, ∼10,000 randomly selected sources). The black dots indicate SDSS stellar sources with r-i > 1.5. To verify the position of quasars at 4 < z < 6, we plotted previously discovered quasars from the SDSS DR7 quasar catalog and <cit.>(crosses; the color indicates its redshift, as shown on the color bar in Figure <ref>b). The quasar redshift track at 4 < z < 6 is plotted with the black solid line by assumingthe redshifted and IGM-attenuated SDSS composite quasar template. The thick solid lines indicate the selection cuts for our quasar selection and the dotted box in (a) is the selection box from <cit.> for comparison. The selection boxes from SDSS and UKIDSS LAS datasets are defined as below:) r-i > 1.5 ) [0 < J-K < 1] ∩ [-1 < z-J < 0.5] ∩ [(z-J) < (J-K)+0.2] is for selecting the r-dropout objects andis for weeding out late type stars and brown dwarfs.Sincedoes not adopt the i-z cut, unlike <cit.>, quasar candidates at z ∼ 5.5 can be selected with this color cut. However, since the selection box ofis close to the stellar locus (gray circles) and part of the stellar sources selected fromis still located inside(black circles inside ), the selected sample is still significantly contaminated by stars (more than 99% of the selected objects are expected to be stars; see Section <ref>).To reduce stellar contamination in our sample, we impose magnitude cuts in the shorter wavelength data, as well as in the z-band.We set the magnitude cuts as below: ) u, g fainter than the 3σ detection limits (u > 22.85 and g > 23.55 mag)) z < 19.5 magFrom the cross-matched sourcesfrom SDSS DR8 and UKIDSS LAS DR10, 98.4% of sources are rejected via the above four criteria, and, after visual inspection for false detection, about 3,600 candidates are finally listed. We checked the sources which were classified as quasars at z > 4.6 from the SDSS DR7 quasar catalog and found that14 quasars at 4.69 < z < 5.29 and 2 quasars at 5.50 < z < 6.05 were already spectroscopically identified. These ∼3,600 candidates still contain a significant fraction of contaminants considering that the expected number of quasars at z ∼ 5 in 3,400 deg^2 is ∼30 (Section <ref>),showing that about 99% of these sources will be interlopers. This is because the selected candidates from these two color-color diagrams are still contaminated by stellar sources, which are shown as the black circles insidein Figure <ref>b. To eliminate these contaminants, we employed an additional selection method:photometry from is/iz-bands. We now apply selection cuts using the is and iz-bands of CQUEAN.The color cuts were defined using quasar redshift tracks. We optimized our quasar selectionusing(r-is-iz: selection method A) or(is-iz-J: selection method B) on color-color diagrams, which explore the redshift ranges of 4.60 ≤ z ≤ 5.40 and 5.50 ≤ z ≤ 6.05, respectively (Section <ref>). The criteria for the selections are:(r-is-iz for 4.60 ≤ z ≤ 5.40): selection method A[r-is > 1.2] ∩ [is-iz < 1.2] ∩ [is-iz < 1.5 × (r-is)-1.2] (is-iz-J for 5.50 ≤ z ≤ 6.05): selection method B[s-iz > 1.8] ∩ [iz-J < 1.5].Figure <ref> shows these two color-color diagrams withquasar redshift tracks(black lines with asterisks; from the redshifted and IGM-attenuated SDSS composite quasar template),model brown dwarfs<cit.>, stars from <cit.> (green squares),star forming galaxy redshift tracks (blue line; model colors from M51), passive galaxy redshift tracks (red line; model colors from the <cit.> modelof a passively evolving 5 Gyr-old galaxy with spontaneous burst, metallicity of Z = 0.02, andthe Salpeter initial mass function),and SDSS quasars with is and iz observations for comparison (blue square).The two color cuts are denoted.About 1,400 among ∼3,600 quasar candidates were imaged with CQUEAN (gray crosses)and among them,about 500 candidates satisfy these color cuts. However, selected candidates instill show a high contamination ratebecause the stellar locus is found near the quasar redshift track.After considering the spectral shape of quasars, we selected about 60 targets as promising candidatesvia visual inspection of SEDs, because quasars at 5 < z < 6 tend to have H-K colors redderthan those of dwarf stars (H-K ≳ 0) due to the power-law continuum of quasars. During the visual inspection, SEDs that show a turn down in flux toward longer wavelengths (Figure <ref>a) are rejected in comparison to those that are retained as candidates (Figure <ref>b). §.§.§ Ancillary SelectionWe set additional selection criteria for assigning prioritiesfor imaging and spectroscopic follow-up observations.WISE Selection: The WISE catalogprovides 3.4, 4.6, and 12 micron data (W1, W2, and W3-bands) that are useful for quasar candidate selection: due to the nature of quasar continua, we expect quasars at z ∼ 5 to have -0.6 < K-W1 < 2.0 and W1-W2 > -0.6 while about 60% of brown dwarfs do not. The cut of W1-W3 > -0.6 is also adopted to remove the brown dwarf outliers, although this cut is not as powerful as the other WISE cuts. We selected red sources in WISE bands and assigned high priorities to these sources for follow-up observations.Figure <ref> shows our ∼3,600 candidates with WISE detections (gray crosses), 9 previously discovered quasarswith WISE detections (blue squares), and model brown dwarfs (green triangles). Since the model brown dwarf templates from <cit.> do not extend to the W3-band, only Figure <ref>a shows the colors of model brown dwarfs (green triangles). We do not consider the quasar redshift track since the rest-frame optical region of the quasar template from <cit.>, which are sampled by WISE bands, are affected by host galaxy.Therefore, based on the observed quasar colors,we defined(purple boxes).We adopt the following selections: : [W1-W2 > -0.6] ∩ [-0.6 < K-W1 < 2] and/or [W1-W3 > -0.6] ∩ [-0.6 < K-W1 < 2] Candidates detected in WISE bands were assigned higher priorities and some of them showing strong power law continuum at the rest-frame ultraviolet spectral regionwere followed-up with optical spectroscopy.53 candidates were given higher priorities due to the WISE criteria (see Table <ref>).c|c|c|c|rPriorities for CQUEAN imaging follow-up observations0ptPriority Stellarity WISE McGreer+13 or Polsterer+13Number0 yes yes yes 81 yes no yes 242 yes yes no 453 yes no no 1,0394 3c|yes or yes or yes 1,1425 noyes 12310 3c|others 1,105 Color cuts from <cit.>: <cit.> discovered a number of quasars at 4.7 < z < 5.1over the area covered by SDSS, including Stripe 82. From the cross-matched sources from SDSS DR8 and UKIDSS LAS DR10,148 candidates with z < 19.5 magsatisfy the these conditionsand 9 of them are included in our ∼3,600 quasar candidates. We gave high priorities to our candidates that satisfied the color cuts used in their work.Sources selected from these color cuts with WISE selection, but not included in the r-i-z-J-K color cuts, are also added to our candidate list.Candidates from <cit.>: <cit.> provide a quasar candidate catalog containing 121,909 sources with their photometric redshifts at 2.558 ≤ z ≤ 6.131. 10 sources are included in our candidate list and we gave higher priorities to these sources.Stellarity: We usefor UKIDSS LAS andfor SDSS to distinguish point sources from extended sources. We defined that a source with= -1 or -2, or= 6, is a point source, and gave higher priorities to these sources. We did not exclude the extended sources because 17% of the discovered quasars from <cit.> are classified as extended sources in their i-band, meaning that some quasars may be classified as extended sources.§.§.§ Selection SummaryThe selection method used in this paper can be summarized as the following. We begin with an adjoint sample of SDSS DR8 and UKIDSS LAS DR10. We select objects showing Lyα drops between r and i, and remove brown dwarfs and stars using the r-i-z-J-K color-color diagrams (). To decrease the number of stellar contaminants, we adopt magnitude cuts in the u, g, and z bands (). These four criteria decrease the sample to ∼3,600 objects.Among them, sources with strong WISE detection and WISE selection () are listed as promising candidates.Objects not included in the r-i-z-J-K selection, but selected from the <cit.> cuts with WISE selection (), are added to the candidate list. Among the ∼3,600 candidates, to reduce contamination, we utilized two color-color diagrams, r-is-iz and is-iz-J, employing our new filter system and selected quasar candidates at two redshift ranges (). For the CQUEAN imaging follow-up observations, we set priorities of our candidates considering the stellarity, the WISE detection, the color cuts from <cit.>, and candidates from <cit.>.Objects showing point-like shapes with WISE detections as well as satisfying the color cuts from <cit.> or candidate list from <cit.> were classified as the important candidates.Table <ref> lists the priority for each case, with smaller numbers indicating higher priorities. We have been conducting the is and iz imaging for the high priority objects and about half of the sample was imaged in these two filters. Finally, via visual inspection of the SEDs, ∼60 targets were selected to be our main samples for spectroscopy. §.§ Optical Imaging Follow-up Observations with CQUEAN Follow-up observations of our high redshift quasar candidates using CQUEAN began in 2010 August and are still on-going. About 1,400 among ∼3,600 candidates with high priorities have been observed with CQUEAN until now. We used short single exposure times of 30 sec for iz and 60 sec for is filters, respectively.Number of frames varieddepending on the sky conditions, such as seeing conditions and extinction. If the peak value of a target was greater than 80 ADU after a 30 sec exposure with iz, 2.5 (30 sec × 5) and 5 (60 sec × 5) minutes were used as the integration times for theiz and is filters, respectively. If the signal was lower than the criterion, we exposed 5 (30 sec × 10) and 10 (60 sec × 10) minutes with iz and is, respectively, or more.Preprocessing including bias subtraction, dark subtraction and flat fielding, were preformed using the usual data reduction procedures in the IRAF[IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy,Inc.,under cooperative agreement with the National Science Foundation.]package. Since the bias values may change with time <cit.>, we used bias images thatwere taken closest to the object frames, time-wise.We combined images of each field andfilter in average. We used thetask of IRAF and SCAMP <cit.> to derive astrometric solutions. SExtractor<cit.> was used for the source detection and photometry. We derived auto-magnitudes which are taken as the total magnitudes. For the photometric calibration, we used SDSS photometry of stellar objects inside each target field. We performed χ^2 fitting to the SDSS r, i, z magnitudes of stellar sources, to determine best-fit stellar spectral types. For this, we used the SEDtemplates from <cit.>, containing 175 spectra of various stellar types.The model is and iz magnitudes were calculated from the best-fit templates and these are used to define thezero-points (Zp) of each filter image of each field.The Zp values were calculated for each star, and we took the average of these values asZp and the standard deviation of the scatters as its Zp error. The average Zp error is about 0.05 mag. During the calculation, objects with large reduced χ^2 values (χ^2_ν > 5) were rejected for the estimation.Note that this photometric calibration method is described in more detail in <cit.>. clccccccccccccSpectroscopic observation summary of IMS quasars 0ptSpectroscopy Date Telescope Target Integration Time (min) Slit Width ()Optical2013 Jan. 16 KPNO 4-m IMS J1022+0801 80 3.02013 May 6 NTT IMS J1437+0708 40 1.22013 May 6 NTT IMS J2225+0330 90 1.02013 May 7 NTT IMS J1437+0708 60 1.02013 Sep. 27 KPNO 4-m IMS J0122+1216 45 1.52013 Sep. 28 KPNO 4-m IMS J0155+0415 60 1.52013 Sep. 28 KPNO 4-m IMS J0324+0426 45 1.52013 Sep. 29 KPNO 4-m IMS J2225+0330 60 1.52013 Sep. 29 KPNO 4-m IMS J0122+1216 45 1.5NIR 2014 Oct. 6 Magellan IMS J0122+1216 60 1.0 2014 Oct. 7 Magellan IMS J0155+0415 30 1.02014 Oct. 6 Magellan IMS J0324+0426 60 1.02015 Aug. 30 Gemini-N IMS J2225+0330 53 0.675 §.§ Optical Spectroscopic Follow-up Observations We observed 47 candidates using the Kitt Peak National Observatory (KPNO) 4-m Mayall telescope and the European Southern Observatory (ESO) New Technology Telescope (NTT). The KPNO 4-m observations were performed over three runs for 10 nights from 2013 January to September, and the NTT observation was done for 3 nights in 2013 May. For the observations at KPNO, we used the Ritchey-Chrétien Focus Spectrograph in a longslit mode (RCSPL[http://www-kpno.kpno.noao.edu/manuals/l2mspect/index.html]) with a LB1A CCD, the BL400 grating of R ∼ 500 for a 2 slit, and OG400 filter.LB1A uses a thick CCD chip, therefore it does not suffer much from fringing. The wavelength coverage is 5,000Å – 10,000Å. For the observation at the ESO NTT, we used the ESO Faint Object Spectrograph and Camera v.2 <cit.>. The EFOSC2 was used with Gr#2 that has a wavelength coverage of 5,100Å – 11,000Å and R ∼ 135 for a 1 slit. We took calibration frames including bias, dark, flat, and arc. Standard stars such as G191B2B, GD153, CD-32d9927, LTT7379, LTT3864, Feige110, and HR7596 were observed for the flux calibration. The slit widths varied from 10 to 30, depending on the seeing conditions.Table <ref> shows thesummary of the optical spectroscopic observations of the discovered quasars, namely the total integration time and the slit width for each target.We followed the typical steps for preprocessing,including bias subtraction, dark subtraction, and flat fielding, for each science image, standard star image and arc image, using thepackage in IRAF. The spectra were extracted using theor thepackages in IRAF for each single image. We used an optimal aperture size for each image where the S/N is highest. After this, wavelength and flux calibrations were conducted. The spectra were flux-calibrated using spectra of the standard stars. Considering the light loss due to variable seeing conditions, we scaled the spectra using broadband photometry.We chose i-band for this calibration, since we get the highest S/N in this bandfor the observed spectra.The flux-calibrated spectra were combined in median using thetask of IRAF and were corrected for Galactic extinction using values from <cit.> and <cit.>.We observed 47 candidates and 6 of them turned out to be high redshift quasars at 4.7 ≤ z ≤ 5.4: these are referred to as Infrared Medium-deep Survey (IMS) quasars. Table <ref> lists the names, coordinates, and redshifts (Section <ref>) of the 6 quasars.The naming convention of our quasars is IMS JHHMMSS.SS±DDMMSS.S in J2000.0 coordinates(IMS JHHMM±DDMM for brevity). cclcccccGeneral information of IMS quasars0ptName R.A. and Dec. (J2000.0)Redshift M_1450IMS J032407.70+042613.3 03:24:07.70+04:26:13.3 4.70(Lyα)a, 4.68( ), 4.73( )-27.21±0.29IMS J012247.33+121623.9 01:22:47.33+12:16:23.9 4.83(Lyα)b, 4.81( )-26.47±0.68IMS J143704.82+070808.3 14:37:04.82+07:08:08.3 4.94(Lyα)c -27.14±0.09IMS J222514.39+033012.6 22:25:14.39+03:30:12.6 5.35(Lyα)d, 5.26( )-26.47±0.29IMS J102201.90+080122.2 10:22:01.90+08:01:22.2 5.36(Lyα)-27.38±0.10IMS J015533.28+041506.8 01:55:33.28+04:15:06.8 5.35(Lyα)e, 5.27( )-26.85±1.09az_spec=4.72 from <cit.> bz_spec=4.76 from <cit.> and z_spec=4.79 from <cit.> cz_spec=4.93 from <cit.> dz_spec=5.24 from <cit.> ez_spec=5.37 from <cit.> z_spec from other papers are all derived from Lyα §.§ NIR Spectroscopic ObservationTo measure their black hole masses and Eddington ratios, we observed four of the six newly discovered quasarsusing the Folded-port InfraRed Echellette (FIRE[http://web.mit.edu/∼rsimcoe/www/FIRE/index.html]) spectrograph on the Magellan telescope (IMS J0324+0426, IMS J0122+1216, and IMS J0155+0415) and using the Gemini Near Infra-Red Spectrograph (GNIRS)on the Gemini North (Gemini-N) telescope (IMS J2225+0330; program GN-2015B-Q-77).Table <ref> shows the summary of the Magellan and Gemini-N observations. In the Magellan/FIRE observation,we used a slit width of 100 with the Echelle mode (R = 3,600). The ABBA pointing method was used for the sky subtraction between exposures. We observed standard stars for each target. Data for the flat fielding and the wavelength calibration were also taken.The data reduction was conducted using the IDL suite, . This pipeline conducts the preprocessing, object extraction, telluric correction, flux calibration, and spectra combining. In the Gemini-N/GNIRS observation,we used the cross-dispersed (XD) mode with the 32 line mm^-1 grating,the short blue camera, and its SXD prism. Adopting the slit of 0675 width,we obtained R ∼ 800. We also used the ABBA pointing method and observed standard stars and calibration data.For the data reduction, we use the Gemini IRAF package following the reduction scripts in the Gemini web page[https://www.gemini.edu/sciops/instruments/gnirs/data-format-and-reduction/reducing-xd-spectra].The steps include pattern noise cleaning using thescript, reducing the science data using flatfield images, combining images, wavelength calibration,extracting spectra, and flux calibration using standard stars. We scaled the flux of the combined spectra using broadband photometry.After that, the spectra were corrected for Galactic extinction using <cit.> and <cit.>.§ HIGH REDSHIFT QUASARS §.§ Photometric Properties cccccccccccccccc Optical photometric information of IMS quasars 0ptName g r i z is izIMS J0324+0426 23.95±0.39 20.39±0.04 19.03±0.03 19.15±0.06 IMS J0122+1216 24.29±0.37 22.35±0.14 19.37±0.03 19.27±0.06 IMS J1437+0708 25.02±0.72 20.71±0.04 19.20±0.02 19.10±0.06 19.17±0.11 19.01±0.09IMS J2225+0330 25.67±0.68 22.01±0.14 20.02±0.05 19.47±0.10 IMS J1022+0801 25.23±0.64 21.27±0.06 19.74±0.02 19.07±0.05 19.74±0.13 19.20±0.18IMS J0155+0415 24.07±0.38 21.81±0.10 19.98±0.03 19.26±0.06 cccccccccccccccc NIR photometric information of IMS quasars 0ptName W1 W2 W3 W4 Y J H K IMS J0324+0426 18.47±0.05 18.45±0.09 16.74±0.31 15.27±0.38 19.39±0.05 19.23±0.05 18.96±0.05 18.83±0.05IMS J0122+1216 18.28±0.05 18.36±0.09 16.67±0.17 99.00±99.00 19.12±0.04 18.92±0.04 18.56±0.04 18.50±0.04IMS J1437+0708 18.99±0.07 19.12±0.13 18.10±0.46 99.00±99.00 19.40±0.05 19.39±0.08 19.01±0.06 18.99±0.08IMS J2225+0330 19.44±0.12 19.28±0.22 99.00±99.00 99.00±99.00 19.48±0.06 19.33±0.06 19.04±0.10 18.99±0.08IMS J1022+0801 18.23±0.05 18.26±0.10 17.01±0.36 99.00±99.00 19.21±0.05 19.06±0.05 18.82±0.06 18.71±0.05IMS J0155+0415 18.98±0.08 18.68±0.11 99.00±99.00 99.00±99.00 19.66±0.07 19.28±0.06 19.00±0.06 18.91±0.06We used a dummy value of 99.99 for non-detections. cccccccccccccccccc Selection methods of IMS quasars0ptName WISEa WISEb McGreer+13c Polsterer+13dr-is-ize is-iz-Jf (K-W1-W2) (K-W1-W3) IMS J0324+0426 yes yes yes no IMS J0122+1216 yes yes yes no IMS J1437+0708 yes yes yes yes yes IMS J2225+0330 yes yes no no IMS J1022+0801 yes yes no no yes IMS J0155+0415 yes yes no no a Does it satisfy the color cut of K-W1-W2?b Does it satisfy the color cut of K-W1-W3?c Does it satisfy the color cuts from <cit.>?d Is it contained in the candidate list from <cit.>?e Does it satisfy the color cut of r-is-iz?f Does it satisfy the color cut of is-iz-J? We list the photometric information from SDSS, UKIDSS LAS, WISE, and CQUEAN of our newly discovered quasars inTables <ref> and <ref>.Table <ref> shows their selection properties.All six of them have WISE detections and are located inside the WISE color cuts (Figure <ref>; K-W1-W2 or K-W1-W3). IMS J0324+0426, IMS J0122+1216, and IMS J1437+0708 also satisfy the color cuts of <cit.> that are aimed at selectingz < 5.1 quasars. <cit.> provided a photometric redshift for IMS J1437+0708 of z=4.961±0.127, which is in agreement with our redshift measured from the Lyα emission line (Section <ref>).For the two IMS quasars with is and iz photometry, Figure <ref>a shows their colors in the r-is-iz color-color diagram.Only two quasars among ∼1,400 sources with is and iz photometry were newly identified as high redshift quasars in the r-is-iz color-color diagram, and none of our candidates werediscovered in the is-iz-J color-color diagram. The other quasars were selected as candidates using the WISE photometry or the color cuts from <cit.>.The expected numbers of quasars for each selection method from 3,400 deg^2 are 24.4^+67.7_-17.9 for 4.60 ≤ z ≤ 5.40 and 5.6^+15.4_-4.1 for 5.50 ≤ z ≤ 6.05 (Section <ref>). For 4.60 ≤ z ≤ 5.40, the number of quasars that we found is 6; including 14 previously discovered quasars in the literature, the total number of quasars is 20, which is in agreement with the expected number. The selection for 5.50 ≤ z ≤ 6.05 identified two quasarsthat were published in previous studies, while we were unable to discover new quasars so far (see Section <ref>), this number is also as expected.§.§ Spectroscopic PropertiesFirst, we present the optical spectra of the 6 quasars at 4.7 ≤ z ≤ 5.4 in Figure <ref>. We plotted spectra smoothed to the resolution of each instrument (black lines) together with the original spectra (gray lines). The blue lines denote the errors of the spectra. Second, we present NIR spectra of four objects,IMS J0324+0426, IMS J0122+1216, IMS J2225+0330 and IMS J0155+0415 in Figure <ref>. The reduced spectra were binned to the spectral resolution of each instrument using the median statistics.Errors of the smoothed spectra were calculated from the errors of the original spectra via standard error propagation.For the spectrum from Gemini-N/GNIRS, the gray bars show regions ofstrong atmospheric absorption,where the spectra shows low S/N. We find diverse Lyα shapes for the six quasars. IMS J0324+0426, IMS J0122+1216, and IMS J1437+0708 show strong Lyα emission, while the other three show smoother shapes.These weak Lyα lines are fairly common at high redshift.<cit.> and <cit.> show that a significant fraction of quasars at high redshift have weak Lyα <cit.>. Most of the emission lines with the exception of Lyα are difficult to verify due to the imperfectsky line subtraction and low QE of the detector at wavelengths longer than 0.8 μm.IMS J0122+1216 shows significant deep absorptionfeatures and we classify it asa broad absorption line (BAL) quasar.This property can be noticed in its NIR spectrum more clearly.Table <ref> lists the redshifts and absolute magnitudes of the continua at rest-frame 1450Å (M_1450) of the quasars.The redshifts of IMS J2225+0330 and IMS J1022+0801 were measured from the Lyα emission lines by fitting Gaussian profiles.However other spectra show a sharp drop bluewards of Lyα.In these cases, their redshifts were measured by fitting the spectra(the orange line in Figure <ref>) from the redshifted and IGM-attenuated SDSS composite quasar template.The redshift errors estimated from these optical spectra contain the uncertainties from the spectral resolution of each instrument (typically ∼0.05), becauseone of the most dominant uncertainties of the redshift measurement is caused by the low spectral resolution.Also we list the redshifts estimated using the or emission linesfrom the NIR spectra (see section <ref>) in Table <ref>. The redshift error estimated from the NIR spectra due to the spectral resolutionis about 0.002 for Magellan/FIRE and about 0.007 for Gemini-N/GNIRS. The redshifts estimated from the optical spectra and the NIR spectra show discrepancies, and we believe that this is caused by the ambiguous Lyα shapes,which can be heavily affected by the Lyα forest and the blending with the emission line. <cit.> provide photometric redshifts (z_ phot) for three out of the six quasars, IMS J0122+1216, IMS J1437+0708, and IMS J2225+0330. Their estimate for IMS J0122+1216<cit.> does not agree with our spectroscopic redshift (z_ Lyα = 4.83),while IMS J1437+0708(z_ phot=5.075^+0.505_-0.455 from Richards et al. 2009 or 5.265^+0.115_-0.505 from Richards et al. 2015) and IMS J2225+0330<cit.>are in agreement with our estimates (z_ Lyα = 4.94 and 5.35, respectively). The discrepancy between z_ phot and z_ spec for IMS J0122+1216 islikely because the object is a BAL quasar. We calculated the M_1450 values using the average flux at 1440Å – 1460Å from the optical spectra in Table <ref>.The uncertainties were estimated from the rms continuum flux density. For z = 5.0 quasars, the observed wavelength of the rest-frame 1450Å is located at 8700Å, where the sky emission lines are significant. Due to the difficulty of subtracting the sky from the relatively low S/N spectra, these values are crude and the actual magnitude uncertainties could be higher than our error estimates. Our IMS quasars are within the M_1450 range of -27.4 – -26.4. §.§ Individual Properties of Quasars IMS J0324+0426 (z_ Lyα=4.70, z_ CIV=4.68, z_ MgII=4.73): This quasar has a strong Lyα emission line. It also shows relatively strong Lyman β (Lyβ),, +, and emission lines, and a weak emission line.In the NIR spectrum,,, and emission lines are prominent.<cit.> reported z=4.72.IMS J0122+1216 (z_ Lyα=4.83, z_ CIV=4.81): We classify this as a BAL quasar because of deep absorption features bluewards of Lyα,, +, and lines. It has a strong Lyα emission line, and a weak Lyβ emission line.We are not able to identify other emission lines due to these deep absorptions.The NIR spectrum has strong,, and emission lines.The left side (shorter wavelengths) of these lines are severely absorbed.<cit.> analyzed this quasar and derived a redshift of z=4.76 while <cit.> reported z=4.79. IMS J1437+0708 (z_ Lyα=4.94): Its spectrum was obtained from NTT/EFOSC2 with R ∼ 130 and it has the highest S/N ratio among the optical spectra.However it does not show any prominent emission lines except the Lyα. <cit.> reported z=4.93. IMS J2225+0330 (z_ Lyα=5.35 and z_ MgII=5.26): This source was observed by two telescopes, the KPNO 4-m telescope and NTT, andthe two spectra were combined in average.It has a smooth Lyα emission line and does not show any other emission lines. In the NIR spectrum, the,, and emission lines are strong but the emission line has a rough shape due to the strong atmospheric absorption. <cit.> reported z=5.24.IMS J1022+0801 (z_ Lyα=5.36): This quasar has the weakest Lyα emission line among the six observed quasars. No other emission lines are visible due to low S/N. This quasar was recently discovered independently by <cit.>, reporting the spectroscopic redshift of z=5.30.IMS J0155+0415 (z_ Lyα=5.35, z_ CIV=5.27): The optical spectrum shows a weak Lyα emission line and other emission lines are not detected.In the NIR spectrum, it has prominent +,, and emission lines.Theemission line is hidden due to telluric absorption. <cit.> reported z=5.37.§ SELECTION COMPLETENESSTo calculate the expected number of quasars for each selection method, we derived the quasar selection completeness, which can be affected by various effects.The completeness from color selection is defined as the fraction of quasars inside specific color cuts among all quasars within specific redshift and magnitude bins. First, applying various quasar templates, we calculated the completeness using the fraction of quasars that fall within each selection box, as a function of redshift and M_1450 (Section <ref>). Then we apply this completeness to our quasar surveys and predict the expected quasar number of each selection method in Section <ref>.§.§ Completeness from Color Cuts To measure the fraction that a quasar with a given redshift, M_1450, and intrinsic SED meets our selection criteria,we follow approaches from previous studies<cit.>. The composite quasar template from <cit.> is redshifted to various values, assuming that the spectral properties of quasars do not evolve significantly with redshift<cit.>, except wavelengths blueward of the Lyα line.Fluxes in these shorter wavelengths are absorbed by neutral hydrogen () in the IGM, and the absorption becomes stronger toward higher redshift because the fraction of increases with redshift<cit.>. We applied this attenuation effect to our redshifted spectra using the IGM attenuation model of <cit.>. We redshifted the spectrum to 4 ≤ z ≤ 8 with steps of Δz = 0.05 and adopted M_1450 in the range -30 < M_1450 < -20 with steps of Δ M_1450=0.5. Then we calculated model magnitudes for each band.The most important factor in the observed color distributionis the continuum slope of quasars.We considered 13 cases of models for each redshifted spectrum with continuum slopes of -1.3 ≤α_ν≤ -0.1 (where F(ν) ∝ν ^ α_ν) with steps of Δα_ν = 0.1.This range was derived based on the range of α_ν values from the SDSS DR12 quasar catalog <cit.>that includes about 230,000 quasars with a mean value of α_ν=-0.7 anda 1σ dispersion of 0.6 (68.3% confidence level).<cit.> analyzed a sample of four quasars at z > 6.5 and three of these four quasars (75%) fall in this α_ν range.We also considered variable rest-frame equivalent widths (EW_0) of the Lyα emission line: 8 cases of 50 ≤EW_0 ≤ 85 with steps of ΔEW_0 = 5 <cit.>.In total, we generate a database of 104 model quasarsof which the continuum slopes and Lyα EWs are uniformly sampled within given ranges and calculate the average selected fraction as a function of redshift and M_1450.Figure <ref>a shows the completeness distribution as a function of redshift and M_1450, for the selection using the r-i-z-J-Kand r-is-iz color-color diagrams (selection method A), and Figure <ref>c shows the completeness distribution when using r-i-z-J-K and is-iz-J color-color diagrams (selection method B). In Figures <ref>b and <ref>d, we plot the completeness as a function of redshift for the two methods,for the case of M_1450 = -29. The completeness in Figure <ref>b rises steeply from 0% to 100% between z = 4.60 and z = 4.70, remains at 100% up to z = 5.15, and drops below 80% for z > 5.35. In the case of Figure <ref>d, the slopes of the completeness distribution at the borderline redshift values are more gradual than those in Figure <ref>b. The redshift ranges of the completeness greater than 80% are 4.60 ≤ z ≤ 5.40 for method A and 5.50 ≤ z ≤ 6.05 for method B, which representthe expected redshift ranges of quasars selected from the two color-color diagrams.The completeness of both selection methods drop to below 50% at M_1450 > -27.0 when z = 4.90 and z = 5.80,where the M_1450 limit corresponds to our magnitude cut, z < 19.5 mag. We also plot the redshifts and M_1450 of our six newly discovered quasars (Table <ref>) with red boxes in Figure <ref>a. §.§ Expected Quasar Number from Our Surveys c|c|c|c|cc|cExpected number of quasars from our survey 0ptSelection Method Area (deg^2) Redshift Range M_1450Limit 2c|Expected NumberSelected Number(1) (2) (3) (4) (5)a (6)b(7)r-is-iz 3,400 4.60 –5.40 -27.0 24.4^+67.7_-17.9 47.3^131.2_-34.7 20 is-iz-J 3,400 5.50 – 6.05 -27.0 5.8^+15.9_-4.3 6.9^+19.0_-5.1 2a For k=-0.47b For k=-0.71We calculated the expected number of quasars from our survey by extrapolatingthe luminosity function of z ∼ 6 quasars from <cit.>. We considered the 10^kz factor that accounts for the decline in number densityas a function of redshift.We adopted two values of k: k=-0.47 from <cit.>and k=-0.71 from <cit.>.Then, we extrapolated the luminosity function of z ∼ 6 to our redshift range, and derived the expected number of quasars from our survey.Table <ref> shows our quasar selection with different selection methods (column 1), survey area (column 2), redshift range (column 3), and M_1450 limit (column 4). The expected number of quasars for each quasar selection are listed in columns 5 and 6 for the case of k=-0.47 and k=-0.71, respectively,with the 1σ errors caused by the uncertainties in break magnitude M_1450 ^* and bright end slope β from <cit.>.We only considered the completeness from our color cuts, and assumed that the efficiency of each selection in its redshift range (column 3)and the M_1450 limit (column 5) is 100%.Our quasar survey discovered 20 quasars including 6 new quasars at 4.60 ≤ z ≤ 5.40. This number is consistent with that from the luminosity function at 4.60 ≤ z ≤ 5.40. However we could not find any new quasars at 5.50 < z < 6.05,except two previously discovered quasars. We believe that the absence of any new quasars at 5.50 < z < 6.05 is due to the lack of WISE photometry (they are fainter than quasars at 4.60 ≤ z ≤ 5.40),resulting in a lower priority for the CQUEAN imaging. We expect to uncover more promising candidates as we build up the CQUEAN follow-up imaging sample. § PHYSICAL PROPERTIES OF QUASARS In this section, we present the physical properties of four IMS quasars,IMS J0324+0426, IMS J0122+1216, IMS J2225+0330 and IMS J0155+0415, based on the data obtained with optical and NIR spectroscopy. In our NIR spectra, we identified both the and lines for IMS J0324+0426 and IMS J0122+1216,only the line for IMS J2225+0330,and only the line for IMS J0155+0415. After modeling the continuum and emission lines of and, we estimatedcontinuum slopes α_ν(where α_νisfor F(ν) ∝ν ^α_ν),line widths (full width at half maximum; FWHM),continuum luminosities at the rest-frame wavelengths of 1350Å and 3000Å (λ L_λ(1350) and λ L_λ(3000)) for each emission line (Section <ref>).From these measurements, we calculated the black hole mass (M_ BH) from the emission line (M_ BH,CIV) or from the emission line (M_ BH,MgII) through different relations from <cit.>, <cit.>, and <cit.> (Section <ref>).For the virial factor in these black hole mass estimators, we adopted f=5.1±1.3 from <cit.>.In Section <ref>, we compare theEddington ratios of our quasarsto lower redshift quasars.§.§ Analysis of NIR Spectra We modeled the quasar NIR continuum assuming two components, a power law component and a component thatdescribes the pseudo-continuum due to the blended forest ofemission lines as given below:F(λ) = a ×λ ^α_λ +b × FeII (λ, v)where α_λ is the continuum slope (in this case, α_ν=-α_λ - 2 for F(λ) ∝λ ^α_λ), andv and b are the width and strength oftemplates. We used two templates from <cit.> and <cit.>.A scaled and broadened template was used for modeling the emissions from our spectra.In the case of the emission line, only <cit.> provide the template in this wavelength range. We modeled the two components simultaneously. The quality of the continuum subtraction depends on the determination of the continuum fitting ranges. We selected narrow fitting windows which minimize the contributions from other components. Since the qualities of the emission line in the IMS J0324+0426 spectrumand the emission line in the IMS J2225+0330 spectrum arenot sufficient to constrain the emissions, we failed to find the component. Since IMS J0122+1216 shows significant broad absorption features bluewards of the and emission lines,we narrowed the fitting window ranges to exclude the absorption part.Since most of the uncertainties in the continuum slope or the line width result from the fitting range of the continuum modeling, we adopted 36 different fitting ranges within the given wavelength windowsand performed model fitting for each different sub-wavelength rangeto calculate the uncertainties.Since we cannot vary the continuum fitting range of of IMS J0122+1216, we set the uncertainty of this line width as 5% of the line width instead of the uncertainty derived from the various continuum ranges. This fraction is identical to the ratio of line widths and their uncertainties,for all other lines.After subtracting the best-fit continuum from each spectrum,we fit the and emission lines. We used single and double-Gaussian profiles considering the presence of asymmetric profiles characterized by red or blue wings.For the fitting ranges, we set1500Å – 1600Å for the lineand 2700Å – 2900Å for the line, except for the of IMS J0122+1216, which is affected by broad absorption.In this case, we set the fitting range to 1530Å – 1590Å. The lines of IMS J0324+0426 and IMS J0122+1216 are well fit by double-Gaussian profiles due to their asymmetric shapes, whereas the other lines can be fit using a single-Gaussian profile.One of the double-Gaussian components of IMS J0324+0426 is a narrow line (violet line in Figure <ref>b) with FWHM = 800±40 km s^-1. To obtain the line width FWHM, the measured FWHM_obs was corrected for the instrumental resolution FWHM_ins: FWHM = √((FWHM_obs)^2 - (FWHM_ins)^2).We used an IDL procedure,to find the best-fit models to the observed spectra that uses the χ^2 minimization method for both the continuum and the emission line.We included 1σ errors of the spectra for each fitting.From the best-fit model,we obtained the best-fit estimates for each parameter, such as the power law slope and the line width.The uncertainties for each parameter were calculated as follows. The error for each parameter is dominated by the scatter of the various best-fits whenaltering the fitting range for the continuum.We compared the best-fit parameters for each trialand we set the average and standard deviation of the values as the best-fit parameter and its error. §.§ Ultraviolet Luminosity and M_ BH crrccccccccccccPower-law slopes, line widths, and continuum luminosities estimated from the NIR spectra 0ptName α_ν, CIV α_ν, MgII FWHM_ CIV FWHM_ MgII λ L_λ(1350) λ L_λ(3000) (km s^-1) (km s^-1) (10^46 erg s^-1) (10^46 erg s^-1) IMS J0324+0426 1.34±0.60 -0.42±0.78 6070±300 2660±280 6.93± 2.22 3.69± 0.35IMS J0122+1216-1.57±0.31 6240±310 4210±160 5.91± 0.08 6.11± 0.64IMS J2225+03300.49±0.382750±4904.08±0.22IMS J0155+0415 -0.71±0.858140±8006.44± 0.48In Figure <ref>, the best-fit continuum and emission line models are shown for each emission line.In Table <ref> we list the best-fit estimates of the power law slope (α_ν,CIV and α_ν,MgII) and the line width (FWHM_ CIV and FWHM_ MgII)and their errors for each emission line. There is no significant difference in the derived power law slope and line width parameterswhen using different templates from <cit.> and <cit.>. Note that the IMS 2225+0330 spectrum has low S/N and the uncertainty of the line width estimated using the method in Section <ref> is underestimated.The 1σ error from the Gaussian fitting is about 15%. The power law slopes of quasars vary significantly between sources. For example, <cit.> found -1.5 < α_ν < 0.5 for quasars at 0.76 < z < 1.26 and 1.67 < z < 2.07.At high redshift,quasars at 4 < z < 6.5 from <cit.> showed -4 < α_ν < 0.7,and quasars at z > 6.5 from <cit.> showed -0.67 < α_ν < 0.56. The slope coefficients from our results are in agreement with these values at high redshift. The λ L_λ(1350) and λ L_λ(3000) in Table <ref> are also calculatedfrom the optical and NIR spectra.For IMS J0324+0426, we used the optical and NIR spectra for the λ L_λ(1350) and λ L_λ(3000), respectively. The λ L_λ(1350)of IMS J0155+0415 and the λ L_λ(3000) of IMS J0122+1216 were estimated from their NIR spectra. Since the continuum spectra near the 3000Å of IMS J2225+0330 show low S/N due to the strong atmospheric absorption, we used fit spectra usingthe redshifted SDSS composite quasar template. In the case of the λ L_λ(1350) of IMS J0122+1216, the continuum near 1350Å shows deep drops in its optical spectrum.Therefore, we used the fit spectrum when we estimated the redshift in Section <ref>. The λ L_λ(1350) and λ L_λ(3000) were calculated from the average flux in the 1340Å – 1360Å and 2950Å – 3050Å, respectively. The uncertainty in the continuum luminosity was estimated from the scatter on the continuum flux in each window. ccccccccccccccc M_ BH, L_ Bol,L_ Edd, and Eddington ratios 0ptName M_ BH,CIV M_ BH,MgII L_ Bol(1350)L_ Bol(3000) L_ Edd( CIV) L_ Edd( MgII)Edd. ratioEdd. ratio (10^9 ) (10^9 ) (10^47 erg/s) (10^47 erg/s) (10^47 erg/s) (10^47 erg/s) (1350, CIV) (3000, MgII)IMS J0324+04267.60± 1.55 1.17±0.31 2.6± 0.81.9± 0.2 9.6±2.01.5± 0.4 0.28±0.16 1.29±0.60IMS J0122+12167.38±0.784.76±0.52 2.3± 0.13.1± 0.3 9.3± 1.06.0± 0.7 0.24±0.04 0.53±0.16IMS J2225+03301.35±0.592.1±0.11.7±0.71.24±0.91IMS J0155+0415 13.53±2.872.5± 0.217.1±3.60.14±0.05In Table <ref>, we list the virial black hole mass estimates obtained from and emission lines (M_ BH,CIV and M_ BH,MgII)using relations presented in <cit.>. The uncertainties of the masses propagate from the uncertainties of the FWHM and the continuum luminosity. The Eddington luminosities (L_ Edd) estimated from the two mass estimators are listed in Table <ref>. Comparing the two mass estimates (M_ BH, CIV and M_ BH, MgII) for IMS J0324+0426 and IMS J0122+1216, M_ BH,CIV is larger than M_ BH,MgII by 0.8 dexand 0.2 dex, respectively.We note that M_ BH values from show a larger scatter with respect to those from or Hβ/Hα <cit.>.For example, the intrinsic scatters of the M_ BH, CIV and M_ BH, MgII from <cit.> is 0.40 dex and 0.09 dex, respectively. Therefore the large discrepancy between M_ BH, CIV and M_ BH, MgII can be understood as a result of the large scatter in M_ BH, CIV estimators. Hence, we take the based values to be more reliable.The M_ BH values are roughly consistent with each other, when using different estimates(e.g. <cit.> or <cit.>) that use the same emission line, within the error bars and the intrinsic scatter in the M_ BH estimators.§.§ Accretion Rate of Newly Discovered QuasarsBolometric luminosities (L_ Bol) and Eddington ratios are given in Table <ref>, where L_ Bol are computed from λ L_λ(1350) and λ L_λ(3000) by multiplying 3.81 and 5.15, respectively <cit.>. For IMS J0122+1216, the L_ Bol values that are calculated from λ L_λ(1350) and λ L_λ(3000) do not agree with each other. Since theλ L_λ(1350) is estimated from the best-fit model spectrum,we adopt λ L_λ(3000) as more reliable. In the case of IMS J0324+0426,theL_ Bol(1350) has a larger uncertainty due to significant contamination from sky emission lines.The Eddington ratios from M_ BH,CIV and λ L_λ(1350) are smaller by a factor of a few than those using M_ BH,MgII and λ L_λ(3000). The discrepancy is most likely caused by the difference in the derived M_ BH values. As we mentioned earlier, -based M_ BH values are in general more uncertain than -based values, and therefore we consider -based Eddington ratios to be more reliable.Figure <ref> shows M_ BH as a function of L_ Bol.To compare our sources with low redshift quasars, we used the SDSS samples of quasars <cit.>. Quasars with M_ BH,MgII information were selected and they cover a redshift range of 0.35 < z < 2.25 (gray points and black contours).We also includequasars atz ∼ 5<cit.>, z ∼ 6<cit.>, andz ∼ 7<cit.>. All M_ BH values are derived using estimators. The Eddington ratios,L_ Bol /L_ Edd = 0.01, 0.1, and 1,are indicated with black solid lines. Our sources are plotted with the red filled circles from M_ BH,MgII and L_ Bol(3000) except IMS J0155+0415.We can see that the high redshift sample occupies a region of the parameter space different from that of the low redshift sample with similar L_ Bol: the Eddington ratios of these high redshift quasars are significantly larger than those of the low redshift sample.In particular, our high redshift quasars have Eddington ratios around 1,suggesting that these quasars are growing vigorously.The Eddington ratio of IMS J0155+0415 is an exception, because it was estimated from M_ BH,CIV andL_ Bol(1350), which are less reliable than M_ BH,MgII andL_ Bol(3000), respectively. <cit.> show similar results that the luminosity-matched quasar samples at z = 2 and z = 6 have different Eddington ratio distributions. However, to compare the Eddington ratio distribution of low redshift quasars to their high redshift counterparts, less luminous samples will be needed.Intrinsic Eddington ratios of normal high redshift quasars can be studied by discovering quasars from deeper surveys <cit.> and Eddington ratio distributions at high redshift when less luminous quasars are included can be different (e.g., Kim et al. in preparation). § SUMMARYWe conducted a quasar survey at 5 ≲ z≲ 5.7using multi-wavelength data with new selection techniques. First, candidates were selected from our r-i-z-J-K color cuts,then we exploited the WISE colors to narrow down the candidates.The candidates were also observed with the CQUEAN is and iz filters that overcome the limitations of previous filter systems. We then carried out optical spectroscopic observationsto confirm our high redshift quasar candidates and discovered six new quasars.Four of themwere observed by NIR spectroscopy to measure their physical properties (M_ BH, L_ Bol, L_ Edd, and Eddington ratio) via spectral modeling of their continuum and emission lines.We compared Eddington ratios of our sources to those of low and high redshift quasars,and found that the Eddington ratio of our quasars at z ∼ 5 have values close to 1.These results,characterized by high luminosities (M_1450 < -27 mag), larger black hole masses of >10^9, and near-Eddington limit luminosities, support the scenario of rapid growth of supermassive black holes in the early universe. This work was supported by the National Research Foundation of Korea (NRF) grant, No. 2008-0060544, funded by the Korean government (MSIP). The Gemini data were taken through the K-GMT Science Program (PID: KR-2015B-005) of Korea Astronomy and Space Science Institute (KASI). Based on observations obtained at the Gemini Observatoryacquired through the Gemini Observatory Archive and processed using the Gemini IRAF package, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina), and Ministério da Ciência, Tecnologia e Inovação (Brazil). This paper includes data taken at The McDonald Observatory of The University of Texas at Austin. Based on observations at Kitt Peak National Observatory, National Optical Astronomy Observatory(NOAO Prop. ID: 2012B-0537, 2013A-0506, 2013B-0534; PI: Y. Jeon),which is operated by the Association of Universities for Research in Astronomy (AURA) undercooperative agreement with the National Science Foundation. The authors are honored to be permitted to conduct astronomical research on Iolkam Du'ag (Kitt Peak), a mountain with particular significance to the Tohono O'odham. This paper includes data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory, Chile. M.H. acknowledges the support from Global Ph.D. Fellowship Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013H1A2A1033110). H.D.J is supported by an appointment to the NASA Postdoctoral Program atthe Jet Propulsion Laboratory, administered by Universities Space Research Association under contract with NASA. D.K. acknowledges the fellowship support from the grant NRF-2015-Fostering Core Leaders of Future Program, No. 2015-000714 funded by the Koreangovernment. We thank the anonymous referee for useful comments, which improved the content of this paper. 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"authors": [
"Yiseul Jeon",
"Myungshin Im",
"Dohyeong Kim",
"Yongjung Kim",
"Hyunsung David Jun",
"Soojong Pak",
"Yoon Chan Taak",
"Giseon Baek",
"Changsu Choi",
"Nahyun Choi",
"Jueun Hong",
"Minhee Hyun",
"Tae-Geun Ji",
"Marios Karouzos",
"Duho Kim",
"Jae-Woo Kim",
"Ji Hoon Kim",
"Minjin Kim",
"Sanghyuk Kim",
"Hye-In Lee",
"Seong-Kook Lee",
"Won-Kee Park",
"Woojin Park",
"Yongmin Yoon"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170626160614",
"title": "The Infrared Medium-deep Survey. III. Survey of Luminous Quasars at 4.7 $\\leq$ z $\\leq$ 5.4"
} |
firstpage–lastpage Near-Earth Object Orbit Linkingwith the Large Synoptic Survey Telescope Steven R. Chesley December 30, 2023 =========================================================================Stars with convective envelopes display magnetic activity, which decreases over time due to the magnetic braking of the star. This age-dependence of magnetic activity is well-studied for younger stars, but the nature of this dependence for older stars is not well understood. This is mainly because absolute stellar ages for older stars are hard to measure. However, relatively accurate stellar ages have recently come into reach through asteroseismology. In this work we present X-ray luminosities, which are a measure for magnetic activity displayed by the stellar coronae, for 24 stars with well-determined ages older than a gigayear. We find 14 stars with detectable X-ray luminosities and use these to calibrate the age-activity relationship. We find a relationship between stellar X-ray luminosity, normalized by stellar surface area, and age that is steeper than the relationships found for younger stars, with an exponent of -2.80 ± 0.72. Previous studies have found values for the exponent of the age-activity relationship ranging between -1.09 to -1.40, dependent on spectral type, for younger stars. Given that there are recent reports of a flattening relationship between age and rotational period for old cool stars, one possible explanation is that we witness a strong steepening of the relationship between activity and rotation.X-rays: stars – stars: activity – stars: late-type – stars: coronae § INTRODUCTION`Magnetic activity' is a collective term to describe a plethora of magnetically-driven phenomena observed mainly for cool stars, such as flares, coronal mass ejections, starspots, and the very existence of a chromosphere and corona, which are much hotter than the underlying stellar photosphere. These phenomena are caused by the highly localised and time-variable magnetic fields, which are in turn driven by the (radial and longitudinal) differential rotation of these stars. Over time, these stars spin down and magnetic activity becomes less pronounced; observationally, this has led to many studies concerning the evolution of stellar rotation with age <cit.> and stellar activity with age <cit.>. <cit.> first proposed that angular momentum was removed from stars via magnetic braking where the material lost from the stellar surface (i.e the stellar wind) is kept in corotation with the stellar surface by the magnetic field up to a critical distance. Beyond this the material is lost from the star and carries away some angular momentum. Over time, the angular momentum lost decreases the rotational velocity of the star resulting in a longer stellar rotation period. Magnetic braking needs a substantial magnetic field, which is found in low mass main sequence stars that have a radiative core surrounded by a convective envelope. This causes a dynamo effect which is driven by the interplay between convection and differential rotation <cit.>. This generates the necessary magnetic fields. There have been many studies on how stellar rotation varies with age, also known as gyrochronology <cit.>. Some studies have used stars with known ages to calibrate the relationship (e.g. ), while others report on stellar data to validate certain gyrochronology relations (e.g. ). Semi-physical spin-down models have also been presented in the literature <cit.>. Studies that calibrate the relationship generally construct a power law that incorporates the age, rotation and some mass dependent parameter such as colour or convective turnover time. However, the majority of these studies concentrate on measuring the period of rotation of stars in clusters with well known ages and using these to calibrate the relationship. Recently, <cit.> measured the rotation periods of older stars with ages determined by asteroseismology, and presented models that incorporated dramatically weakened magnetic braking for these older stars. This reported result may have significant implications for gyrochronology as it calls into question the use of stellar spin-down as an age diagnostic past a given age and would limit the sample of stars that the method could be applied to.The relationship between magnetic activity and stellar age was first presented by <cit.>. Skumanich plotted the Calcium 2 emission (a proxy for magnetic activity), average equatorial velocities, and lithium abundance as a function of age and noted that the calcium emission and average equatorial velocities followed a similar relationship. Skumanich suggested that this was a consequence of the stellar dynamo; over time the slower stellar rotation leads to a reduced velocity shear at the tachocline between the radiative and convective zones, resulting in reduced magnetic field generation by the stellar dynamo <cit.>. Several proxies for magnetic activity have been studied in the literature. Among them is the emission in chromospheric lines such as Ca II H & K <cit.> as well as H-α. A study of the chromospheric emission usually needs to include an analysis of the so-called basal part of the emission in the relevant lines, i.e. the part of the line emission that is not driven by magnetic activity <cit.>. However, the analysis of the basal flux does not account for metallicity or surface gravity, which has an effect on the shape of the Ca II profiles <cit.>. This makes calibrating the age-activity relationship using Ca II emission more complex (e.g. ). Calcium emission is also particularly difficult to study for M dwarfs due to the low level of continuum present in the region around the Ca II H & K lines. Consequently, calcium emission studies tend to focus on FGK stars and the H-α line is widely used as an activity indicator in M dwarfs (e.g. ).Another proxy for magnetic activity is the emission from the stellar corona <cit.>, which is mainly observed in the soft X-ray band (0.1-10 keV). This emission stems from the hot (several million Kelvin) plasma in the stellar corona, which is collisionally excited and cools through emission at X-ray wavelengths <cit.>. Since the X-ray luminosity does not contain a photospheric component, it means that the X-ray luminosity is unambiguously associated with magnetic heating unlike Ca II H & K emission. In this work we use the X-ray luminosity as a magnetic activity indicator in order to study the full range of stars with convective envelopes (mid-F to mid-M).To study the evolution of stellar activity for old stars, it is crucial to obtain a good estimate for the stellar age in the first place. This can be difficult for stars older than a gigayear, as most age-determination methods work best for younger stars. However, recent observational advances have made it possible to study ages for a larger number of stars through asteroseismology. Asteroseismology provides critical information about the interior of stars through observations of stellar oscillations. This has become a valuable tool since the launch of the CoRoT and Kepler missions, which have provided higher quality photometry <cit.> enabling accurate and precise measurements offundamental properties of stars, including ages <cit.>. Indeed, asteroseismology has proved to be the most accurate age-dating method for old field stars – and opens up the possibility of stellar age investigations for stars older than a gigayear.The age-activity relationship is not only useful for inferring ages for stars with L_x measurements, but it can also give us insight into the high-energy environment that stars provide their exoplanets <cit.>, and how this changes over time. This is important when considering the effects of high-energy radiation on the habitability of exoplanets.In this work, we use ages from recent asteroseismology studies coupled to X-ray luminosities for these stars to investigate the age-activity relationship for stars older than a gigayear. In Section <ref> we present the data used in the analysis followed by Section <ref> which details the analysis performed on the data. Section <ref> presents the results and Section <ref> presents the discussion. Finally, Section <ref> summarises the conclusions from this work.§ OBSERVATIONS §.§ Sample selectionOur target stars with well-determined old ages were selected from different sources. The majority stems from asteroseismology where we chose stars with precisely determined stellar properties, including ages, obtained by modelling the individual oscillation frequencies in the spectrum observed by the Kepler satellite <cit.>. For stars where the Kepler observations are not of sufficient signal-to-noise ratio to extract the individual oscillation modes, we combined the asteroseismic detections reported bywith the spectroscopic effective temperatures and metallicities derived byand determined asteroseismic ages using the BAyesian STellar Algorithm (BASTA, ). Specifically, this new age determination was performed for the stars KIC 10016239, KIC 12011630, KIC 3123191, KIC 5309966, and KIC 7529180, yielding asteroseismically determined stellar ages and radii. Near-by stars were selected for dedicated X-ray observations with XMM-Newton and Chandra (PI Poppenhaeger). The second source was a sample of G and K type stars with archival X-ray observations that are located in wide binaries containing a white dwarf companion. Targets with existing X-ray observations were identified from the samples in <cit.> and <cit.>. These are particularly useful systems as the ages of the white dwarfs are reasonably well known through their cooling times and therefore can act as a stellar chronometer for the system that is independent of spin-down. We have calculated the ages of those systems from the white dwarf parameters, which we explain in more detail in Section <ref>. The third source were individually selected stars with archival X-ray observations and relatively well known ages determined through various methods. These methods included asteroseismology (16 Cyg A and B, ; the α Cen/Proxima Cen system, ), isochrone fitting (61 Cyg A and B, ) and association with a sub-population of stars in the galaxy (HR7703, ). Proxima Centauri is a fully convective star, so one might wonder if it is appropriate to include in this sample of otherwise cool stars with radiative cores; however, a recent study <cit.> found that fully convective stars exhibit a rotation-activity relationship that is indistinguishable to that of solar-type stars, which is why we chose to include Proxima Cen in our analysis. Also included in our sample is the Sun; its X-ray luminosity was adapted using the model parameters in <cit.> and Xspec to encompass only the 0.2 - 2.0 keV energy range. The solar age is well constrained by meteorite studies and is adopted to be 4.57 ± 0.02 Gyr <cit.>.For stars with exoplanets in close orbits, effects on the stellar activity through star-planet interaction are expected from theoretical considerations <cit.> and have been observed for some systems with high-mass exoplanets in very close orbits (see for example ). However, in our sample there are no stars with Hot Jupiters present, therefore such effects are not expected to play a role in our investigation. Two X-ray observatories are used in our study, XMM-Newton and Chandra. Their main characteristics and the basics of our data reduction are shortly explained in the following paragraphs. The details of the observations (obtained from both XMM-Newton and Chandra) that we analysed are presented in Section <ref> and listed in Table <ref>.The XMM-Newton X-ray Telescope <cit.> is equipped with the European Photon Imaging Camera (EPIC) which consists of three X-ray CCD cameras, one PN and two metal oxide semi-conductor (MOS) cameras. EPIC allows imaging over the telescope's 30 arc-minute field of view and in the energy range of 0.2 to 15 keV making it suitable to study the X-ray emission from late-type stars. Archival observations were obtained through the XMM-Newton Science Archive and analysed using SAS (Science Analysis System) version 15.0. Using SAS, we filtered the data to remove any bad pixels or bad events by setting criteria to limit the probability of double photon impact events. These events occur when two photons hit the same or neighbouring pixels in the same readout time frame and cause a slightly different pattern on the chip compared to a single photon event. In our data analysis of the observations from XMM-Newton we used a standard source extraction radius of 20" and chose a source-free background region with a radius of 70". Data analysis was preferably performed with PN observations due to the higher signal to noise obtained with this instrument.The Chandra X-ray telescope <cit.> has two focal plane instruments, the Advanced CCD Imaging Spectrometer (ACIS) and the High Resolution camera (HRC). The ACIS provides images alongside spectral information on the object in the energy range 0.2 - 10 keV. The HRC only provides images and no spectrally resolved data; we therefore restricted our analysis to observations conducted with ACIS. Observation files from Chandra were obtained through the Chandra X-ray Centre (CXC) public archive and were analysed with CIAO (Chandra Interactive Analysis of Observations) <cit.>. We used a standard source extraction radius of 1.5" and a source-free background region with a radius of 15". §.§ Distances and spectral typesSince the rotational evolution and the X-ray luminosity of stars no longer on the main sequence differs from those still on it, it was important to ensure only main sequence stars were considered. For several stars from the asteroseismic part of the sample no luminosity classes were given in the literature. We therefore compared their surface gravity to the relation of B-V colour and surface gravity for main-sequence stars, as given by <cit.>, and excluded stars which differed by more than 0.2 dex from our sample. For one of the stars from the asteroseismic part of our sample, no distances or parallaxes were found in the literature. Therefore, for this star the Barnes-Evans method was used. This method was used to calculate the angular diameter of the star using V-K <cit.>. Since the radius of the star is known from asteroseismology <cit.> it was then possible to calculate the distance to the star. As the stars in this study are all located relatively nearby, reddening is expected to be insignificant and was not taken into account. The Barnes-Evans method has been used previously to obtain stellar radii for extra-solar planet host stars and found good agreement with published values <cit.>. The X-ray luminosity was then calculated using the flux (see Section <ref>) and distance determined from the appropriate method.Finally it was necessary to determine the spectral type of the candidates since this influences both the stellar activity and its evolution with time. The stellar spectral types were collected from the SIMBAD database, or estimated from the stellar effective temperatures as published in . §.§ Ages of systems with white dwarfsCool stars located in a wide binary system with a white dwarf provide the opportunity to infer the age of the system and therefore the age of the cool star from the physical properties of the white dwarf. When the effective temperature and surface gravity of the white dwarf are known, one can infer its mass and cooling time, and estimate the mass of the progenitor star and its main sequence lifetime. The sum of the main sequence lifetime and the white dwarf cooling time is then the total age of the system.<cit.> have performed such age estimates for several systems; however, investigations into 3D-model atmospheres of white dwarfs revealed that previous fits of spectra to 1D-models overestimated the true effective temperatures T_eff and surface gravities log g of cool (<13000 K) white dwarfs <cit.>. We have therefore re-calculated the ages for the systems we use in this work, specifically the systems 40 Eri A/40 Eri B, CD -3710500/L481-60, and NLTT 7887/NLTT 7890, where the second object in each listed pair is the white dwarf. Specifically, we used the published T_eff and log g values from <cit.> and <cit.> for the white dwarfs and corrected them according to the formulae (7) and (8) from <cit.>. We then used the Montreal model grids for white dwarfs with hydrogen atmospheres[Available at <http://www.astro.umontreal.ca/ bergeron/CoolingModels/>] to estimate the masses and cooling ages of the white dwarfs <cit.>; progenitor masses were estimated using the initial-final mass relation by <cit.>, and the progenitor main-sequence lifetimes were estimated using the Padova stellar evolution model grids from <cit.>. In this manner, we estimate the system ages for the 40 Eri A/40 Eri B and CD -3710500/L481-60 systems to be 3.70^+3.57_-1.34 Gyr and 1.77^+0.65_-0.27 Gyr, respectively. NLTT 7890 has large error bars on its surface gravity given by <cit.>, therefore the estimated system age for NLTT 7887/NLTT 7890 has larger errors with 4.97^+8.8_-3.0 Gyr.§ DATA ANALYSISIn the following section we will provide details on the methods used to determine the X-ray luminosity for our sample of stars. In addition, Appendix <ref> provides supplementary information on the analysis of individual stars that are not included in this section. §.§ Source detectionsFor each of our targets we tested if the source was significantly detected in X-rays. We extracted X-ray counts in the energy band from 0.2-2 keV, as this is where weakly active cool stars display most of their X-ray emission, for the background and source regions. For XMM-Newton, there is typically a significant background signal observed in any observation. We therefore tested for XMM-Newton targets if the number of source counts exceeded the number of counts expected form a pure background signal (estimated from the larger background region) by at least 3σ, with σ being estimated as the square root of the number of expected background counts. If so, we counted the source as detected and proceeded with a flux determination (see Section <ref>); otherwise, we chose the 3σ level over the background signal as the upper limit for the source.For Chandra, the background signal is typically very low, meaning that an approximation of using the square root as the error on the expected background counts is invalid. We therefore used full Poisson statistics and calculated the inverse percent point function at which, for a given expected number of background counts, the number of counts in the source region had a probability of less than 0.3% of occurring as a random fluctuation. If the number of observed source counts was at this number or larger, we counted the source as detected; otherwise we used that number as the upper limit on the X-ray counts for the source.§.§ Stellar Variability Magnetic activity can vary on several timescales, including timescales shorter than the typical exposure time for an X-ray observation. Therefore, for detected target stars, a light curve was extracted from each observation to check for short term magnetic phenomena such as flares. In the case of Proxima Centauri, the light curve showed several rapid increases in count rate over the observation timescale, indicating several flares. Flares increase the temperature and emissionmeasure of the corona, resulting in a significantly higher X-ray luminosity compared to the quiescent emission level of the star. Therefore, the quiescent value for the X-ray luminosity of 4.9 × 10^26 ergs s^-1 was taken from <cit.> and used in this work. An inspection of the light curve for HR 7703 indicated that a flare had occurred towards the end of the observation. In this case, the time interval associated with this flare was excluded from the data analysis. §.§ Determining the X-ray Flux For X-ray sources with significant detections and where the source region contained ≈ 90 or more counts, we chose to model the spectra of the source with a coronal plasma model using the following method. A spectrum of the star was extracted from the observation using the relevant analysis tools for each telescope. The extracted spectra were fitted with an optically thin thermal plasma model (APEC model) using the Xspec fitting software. Since all of the stars in this study are located nearby the redshift was fixed at zero; the abundances were assumed to be solar using abundances from <cit.>. The two variables that were fitted were coronal temperature and emission measure; in some cases, two temperature components were required to find a fit that describes the coronal emission well. From the best fit model to each object the flux was then calculated in a fixed energy band from 0.2-2 keV. The details and plots of the best fit models are shown in Appendix <ref> and Appendix <ref> respectively.In the case where the source region contained less than ≈ 90 counts then typically there were not enough data points to fit a spectrum accurately. Under these circumstances an estimate of the X-ray flux was obtained through WebPIMMS[<https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/w3pimms.pl>] using the mean count rate of the source region. A typical spectrum was assumed for the stellar corona and WebPIMMS calculated the source flux using the instrument characteristics. The X-ray flux was calculated in the 0.2 keV and 2.0 keV energy range assuming an Apec model of solar abundance and logT value of 6.5 (T≈ 3 MK), appropriate for inactive cool stars (see for example ). The conversion factor from counts to flux used for each of the instruments are shown in Table <ref>; note that the sensitivity of Chandra changes significantly over the years of operation, and therefore the correct conversion factors need to be chosen for the observing cycle in which a given observation took place. For XMM-Newton observations, the encircled energy fraction factor needs to be applied as well, since its PSF is significantly larger than typical practical source extraction radii.The statistical error on the values of X-ray counts were calculated from the square root of the number of source counts (C_s) (as the distribution is described by Poisson statistics) and divided by the number of source counts to obtain a fractional error. This fractional error on the number of counts was also used as the error on the X-ray flux and the X-ray luminosity. However, the X-ray luminosity of a star is known to vary on various timescales, including timescales much shorter than the star's main sequence lifetime, therefore a minimum physical error of 0.1 dex in log L_x was applied to the data to account for this variability, even if the statistical error was smaller. This value was determined from the long-term X-ray monitoring of 61 Cyg B <cit.>, a star without an apparent activity cycle, where the standard deviation of the X-ray luminosity was at 0.1 dex over several years of observations.§ RESULTS §.§ Magnetic activity across spectral typesIt is known that there is a mass dependence on the rotational spin down of late type stars due to the varying depth of the convection zone. F type stars have a thinner convection zone resulting in rotational spin down that occurs on a different timescale than for M type stars that have much thicker convection zones. This mass dependency seen in the rotational spin down is also present in the X-ray activity across varying spectral types. Since the sample of our stars is relatively small, we wish to avoid splitting the sample by spectral type and perform an activity-age analysis of the whole sample instead.When dealing with X-ray activity, some studies normalize L_X by the stellar bolometric luminosity and then split the sample into different mass bins <cit.>. A different approach was demonstrated by <cit.>: in their volume-complete sample of cool stars in the solar neighbourhood, they found that when the X-ray luminosity is divided by the stellar surface area 4π R_∗^2 with R_∗ being the stellar radius, i.e. when one considers the X-ray flux through the stellar surface as a quantity of interest, then stars of all spectral types from F to M show the same spread of this quantity. We visualise the relevant quantities in Fig. <ref>. We took the X-ray luminosities of near-by stars reported by <cit.>, giving preference to data collected with the PSPC detector without the Boron filter if several detections were reported, and plot them against absolute stellar brightness as a proxy for spectral type. We also show the X-ray flux through the stellar surface, as well as the X-ray luminosity divided by the bolometric luminosity. As can be clearly seen, a flat distribution of the X-ray surface flux versus spectral type is present, as has been reported by <cit.>.We therefore chose to follow this approach and use a normalization by stellar surface in our data in order to perform a combined analysis of all spectral types present in our sample. Stellar radii were either taken from the asteroseismic studies as mentioned in Section <ref>, or calculated from absolute brightnesses and stellar effective temperatures.§.§ Fitting the DataIn total, twenty four stars were fully analysed and are shown in Figure <ref>; the full details of the result for each star and stellar properties are listed in Appendix <ref> and <ref> respectively.The majority of the sample stars have asteroseismic ages, only eight of the samplehave ages determined from other methods such as isochrone placement, white dwarf cooling times or association with a sub-population of stars in the galaxy. Ten stars in our sample have upper limits to their X-ray luminosities. The small number of X-ray detections demonstrates how difficult both the asteroseismic and X-ray measurements are. High-precision light-curves sampled at short cadences do not necessarily guarantee a precise measurement of age from asteroseismology. Even when a well-constrained age is determined, it does not guarantee an X-ray detection as resources are limited. Figure <ref> shows that the X-ray luminosity decreases with age as expected, but to gain more insight into what timescale this decrease occurs on the best fit relationship was found. Measurement errors are present in both the stellar age and the X-ray luminosity; we therefore performed an orthogonal distance regression (ODR) to fit the logarithmic X-ray luminosity (normalized by stellar surface) against the stellar age. In order to have normalized X-ray luminosities with values familiar to the reader, we chose to normalize by the stellar surface not in units of centimetres, but relative to the solar surface. This leads to normalized X-ray luminosities around 10^27 erg s^-1 R_⊙^-2. Only the 14 stars with X-ray detections or known X-ray luminosities were considered in this fit; for fitting purposes, we used symmetric errors in age and in log L_X. We obtained as the best fit relationship: logL_x/(R_∗/R_⊙)^2 = 54.65 ± 6.98 - (2.80 ± 0.72)log t We display this result visually in Figure <ref> where the fitted relationship is displayed as the solid black line. Upper limits are shown only for reference in the plot.§ DISCUSSION<cit.> (henceforth MH08) also derived an age-activity relationship for chromospheric activity derived from Ca II H&K emission. They also transform this into a relationship between X-ray luminosity and age, using a scaling relationship between chromospheric Ca II H&K and coronal X-ray emission, but no actual X-ray measurements of the sample stars are used. When we compare the L_x ages derived from the age-activity relationship from MH08 to the literature ages for our sample, we find that the MH08 L_x ages tend to be somewhat younger than the literature ages. Two factors may come into play here: the relationship between R'_HK and X-ray luminosity in MH08 used very few stars which are less active than log R'_HK≈ -5. Our sample contains very old stars whichwe would expect to have chromospheric activity levels less thanlog R'_HK≈ -5, therefore probing a different part of the age/activity range than considered in the MH08 sample. Additionally, the use of their scaling relation between chromospheric and X-ray emission, which had been derived by <cit.> for stars with activity levels log R'_HK > -5 , may not fully catch the actual relation of those emissions for very old and inactive stars.Figure <ref> shows the cluster data from <cit.> for ages below one gigayear, which we normalized by stellar surface area based on spectral type, alongside the sample of stars from this research for ages above one gigayear with our best fit age-activity relationship shown in black. As has been reported in many studies <cit.>, for very young stars there is a saturation of the X-ray luminosity until approximately 100Myr when the X-ray luminosity starts to decay. <cit.> quantitatively investigated the age-activity relationship using clusters as calibrators, normalizing by the bolometric luminosity and splitting into several mass bins. They found slope values for the age-activity relationship ranging from -1.09 ± 0.28 to -1.40 ± 0.11, considering seven spectral bins across the range from F-type stars to early M-type stars. Comparing these values to the slope value found in this work of -2.80 ± 0.72, we find a steeper slope for the age-activity relationship at old ages than what is reported for any of the spectral bins for the younger stars. This steepening indicates a more rapid decay of stellar activity with age for cool stars older than a gigayear than for younger stars.We will now discuss the implications of this steepening in relation to stellar spin-down and activity decrease in general. As mentioned previously the rotational velocity of a star will decrease over time as a result of magnetic braking where the rotation is related to the time (or age) by v_rot∝ t^-α where α = 0.5 <cit.>. The first study of the relationship between rotation and activity was bywho found that L_x∝ (vsin(i))^1.9. Observations of solar-like stars confirmed that the relationship between activity and rotation takes the form of L_x∝ v_rot^β where β≈ 2 <cit.>. From these two relationships one can predict how X-ray luminosity varies with time as shown in Equation (<ref>). L_x∝ t^-αβwhereαβ≈ 1 Some previous studies have investigated the value of αβ. For example,studied nine solar-like G stars with ages ranging from 70 Myr to 9 Gyr (however they were constrained to rotation-inferred ages for most stars with ages beyond one gigayear) and found the value to be 1.5 for ages greater than 100 Myr. Later studies includedthat studied the 1.5 Gyr NGC 752 cluster and presented results that were consistent with a value for αβ of 1.5, but also found evidence for a steepening of the X-ray luminosity scaling law after the age of the Hyades cluster (625 million years). However,found an excellent fit for their data with a value for αβ of 2 but also could not rule out the predicted value due to the small sample and systematic uncertainties.The results from this research indicate that the value of αβ for stars older than a gigayear is 2.80 ± 0.72, which is larger than the expected value of unity, and more in line with the direct investigations of L_X versus age in the studies discussed in the previous paragraph. This leaves the challenge of explaining why the decay of magnetic activity is faster than predicted. One possibility is that the rotational spin down could be more rapid than expected from constant magnetised stellar winds <cit.>, i.e. α has a value greater than 0.5. <cit.> also postulated that the coronal mass ejections may contribute to stellar angular deceleration, changing the alpha exponent. But a recent study by <cit.> reports that there is weakened magnetic breaking for older late-type stars. Unfortunately, their sample and ours do not have sufficient overlap to compare age, rotation, and activity all together. A recent theoretical model by <cit.> predicts weakened magnetic braking for older late-type stars; their model suggests that conductive losses are more important for these stars than wind losses which would imply a reduced angular momentum loss. Other theoretical work includes <cit.> and <cit.>, which show that the rotational spin down of a star may depend on the magnetic field geometry. If the rotational spin down was the only factor to affect the age-activity relationship then it should also show weakened magnetic braking and the exponent of the relationship should decrease and not increase as found in this work. From this evidence one would disfavour the more rapid rotational spin down as the cause of the more rapid decay of the magnetic activity.Another possible explanation for the increased decay in magnetic activity for stars older than a gigayear is that the relationship between the X-ray luminosity and rotational velocity is not constant, i.e. the β term changes as the star ages. There is some evidence for the steepening of the activity-rotation relationship, <cit.> considered a small, unbiased subset of their large sample of solar and late-type stars and found that a value for β of 2.7 was a better fit for their data than the generally accepted value of 2. This was in agreement with <cit.> who found a value for β of 2.64 for a sample of nine solar analogs and <cit.> who found an un-expectedly steep decay of X-ray emission as a function of age which could indicate a steepening of the activity-rotation relationship. However, more research is needed into the activity-rotation relationship to confirm if there is a steepening of the relationship as one of the lowest values considered in the <cit.> subset of data was of the Sun. In this research older stars have been considered that have lower X-ray luminosities than the Sun therefore the activity-rotation relationship needs to be extended to lower X-ray luminosities.§ CONCLUSIONSIn this work we have presented new X-ray detections of several old cool stars together with an analysis of archival data to form a sample of 24 cool stars with ages beyond one gigayear. Most stellar ages in our sample have been determined by asteroseismology, providing more accurate ages for old stars than most other studies were able to provide. We have investigated the age-activity relationship of these stars using observations from the Chandra and XMM-Newton X-ray telescopes. X-ray luminosities were determined for fourteen stars primarily, and spectral modelling was performed for eight of those stars; upper limits to the X-ray luminosity were determined for a further ten stars. We normalized the X-ray luminosity of the sample stars by the stellar surface, in order toperform an analysis across varying spectral types. We find an age-exponent of αβ = 2.80 ± 0.72, which represents a steepening of the age-activity relationship compared to what is seen for stars in clusters with ages below one gigayear.A possible explanation for this steepening of the age-activity relationship is that rotational spin down is more rapid than previously thought. However, a recent observational study <cit.> indicates that there is weakened magnetic braking for older cool stars. If this is indeed true, our data presents evidence that there is a strong steepening of the rotation-activity relationship at old stellar ages instead of the age-rotation relationship itself. In either case, the data we have presented here demonstrates that the relationship between stellar age and activity steepens towards old stellar ages. Combined studies of age, rotation, and activity will be able to shed light on which components of the relationship are responsible for this.§ ACKNOWLEDGEMENTSWe thank the anonymous reviewer for their detailed and insightful comments, which added significantly to the clarity of this paper. The scientific results reported in this article are based on observations made by the Chandra X-ray Observatory and by XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. Support for this work was provided by the National Aeronautics and Space Administration through Chandra Award Number GO5-16101X issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. This work has made use of data from the European Space Agency (ESA) mission Gaia (<http://www.cosmos.esa.int/gaia>), processed by the Gaia Data Processing and Analysis Consortium (DPAC, <http://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This research made use of public databases hosted by SIMBAD, maintained by CDS, Strasbourg, France. RB acknowledges funding from DE. CAW acknowledges support from the STFC grant ST/L000709/1. Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation (Grant agreement no. DNRF106). V.S.A. acknowledges support from Villum Fonden (research grant 10118). mnras§ ADDITIONAL INFORMATION ON INDIVIDUAL ANALYSIS OF STARSSection <ref> details the methodology used to determine the X-ray luminosities for our sample of stars. Since the sample is made up of a number of different observations from both the XMM-Newton and Chandra telescopes, we present additional details about the individual observations and the data reduction here.16 Cyg A: This star was detected in combining two Chandra observations on a front-illuminated CCD, which provides energy sensitivity only above 0.6 keV. We extrapolated the flux for the full 0.2-2 keV band, assuming a coronal temperature of log T = 6.5. The star 16 Cyg B was covered by the same observations, but is undetected and we report the corresponding upper limit. There is also an earlier, shorter XMM-Newton observation covering both stars, but they were both undetected in that observation, which is consistent with the Chandra data.40 Eri A: This star is in a wide binary system with the white dwarf 40 Eri B, with a projected distance of 83^'' <cit.>, translating to a physical distance of ca. 400 AU. Using the techniques outlined in section <ref>, we use its reported T_eff and log g values <cit.> to calculate an age of 3.70^+3.57_-1.34 Gyr for 40 Eri B, which we adopt as the age of the system. 40 Eri A was observed with a back-illuminated Chandra CCD, which provides energy sensitivity above 0.245 keV. We performed a spectral fit and extrapolated the stellar X-ray flux for the full 0.2-2 keV energy range. There is also a third star, 40 Eri C, present in the system; however, it is close enough to the white dwarf (ca. 35 AU) that its rotation and activity properties may have been affected during the evolution of the white dwarf progenitor, which is why we do not include 40 Eri C in our analysis. 61 Cyg A and B: These stars have been monitored in X-rays with XMM-Newton over several years. We show one exemplary X-ray spectrum for each star in Fig. <ref>. 61 Cyg A has been found to display an activity cycle <cit.>, and we adopt the full range of observed X-ray luminosities as the error bar for 61 Cyg A's normalized X-ray luminosity in our analysis. 61 Cyg B has been found to have a flat activity profile <cit.>, and we adopt its mean X-ray luminosity and the standard deviation over all X-ray observations as the error bar for our analysis. CD -3710500: This star is in a wide binary system with the white dwarf L481-60. Using the techniques outlined in section <ref>, we use its reported T_eff and log g values <cit.> to calculate an age of 1.77^+0.65_-0.27 Gyr for L481-60, which we adopt as the age of the system. The star CD -3710500 was observed with a front-illuminated Chandra CCD, which only provides energy sensitivity above 0.6 keV. We performed a spectral fit and extrapolated the stellar X-ray flux for the full 0.2-2 keV energy range.GJ 176: This star was detected with a back-illuminated Chandra CCD, which provides energy sensitivity above 0.245 keV. We performed a spectral fit and extrapolated the stellar X-ray flux for the full 0.2-2 keV energy range.GJ 191:This star was detected with a back-illuminated Chandra CCD, which provides energy sensitivity above 0.245 keV. We performed a spectral fit and extrapolated the stellar X-ray flux for the full 0.2-2 keV energy range. Our result is consistent within errors with the values reported by <cit.>.HR 7703: This source was detected with XMM-Newton, and we performed a spectral fit for the full energy range of 0.2-2 keV.KIC 12011630, KIC 3123191, KIC 5309966: These stars were in the field of view of Chandra during observations of other targets, and are all undetected in X-rays. The sources are located on front-illuminated CCDs and are far from the centre of the field of view. Since Chandra's PSF becomes large at the edges of the field of view, large extraction regions had to be used, which led to quite high upper limits for the X-ray luminosities for these stars.KIC 10016239, KIC 7529180: These stars were detected with XMM-Newton. KIC 7529180 was detected with a sufficient number of source counts so that a spectral fit could be performed; for KIC 10016239, the excess source counts were used to calculate the X-ray flux using a coronal temperature of log T = 6.5.KIC 6116048, KIC 6603624, KIC 8292840, KIC 9025370, KIC 9410862: These stars were observed with XMM-Newton, but undetected in X-rays.KIC 9955598: This star was observed and detected with XMM-Newton. Since there is another X-ray source at close projected distance (ca. 20^''), we chose an extraction region with a radius of 10^'' instead of 20^'', and applied the correct encircled energy fraction factor to account for the smaller extraction region when calculating the flux.NLTT 7887: This star is in a wide binary system with the white dwarf NLTT 7890. Since the reported surface gravity of the white dwarf has large errors <cit.>, the age we derive has large errors as well with 4.97^+8.8_-3.0 Gyr. The star NLTT 7887 was covered by an XMM-Newton observation, but is undetected.α Cen A, α Cen B, Proxima Cen: The age of this triple system has been derived from asteroseismic observations of α Cen A and α Cen B using different underlying models <cit.>; we adopt the mean of the asteroseismic age estimates as the age of the system. α Cen A and α Cen B have been monitored in X-rays with XMM-Newton <cit.>. α Cen B was found to display an activity cycle, and we use the full range of observed X-ray luminosities as the error bar on its X-ray luminosity for our analysis. α Cen A is reported by the same authors to potentially be in an activty cycle as well, however only the low-activity part has been observed so far, and there is no information on what its X-ray luminosity might be during the high-activity part of the cycle. We have therefore chosen not to include α Cen A in our analysis. Proxima Cen has been observed with XMM-Newton several times as well, including multiple stellar flares; a detailed analysis is given by <cit.>. We adopt their quiescent X-ray luminosity of log L_X = 26.69 for Proxima Cen in our analysis. § X-RAY LUMINOSITY AND AGE RESULTS 1.4 Name of Star / White Dwarf Age / Gyr log(L_x / ergs s^-1) log(L_x/R_⊙^2 / ergs s^-1 R_⊙^-2) Spectral Type Age DeterminationAge Reference 16 Cyg A 6.67^0.81_0.7726.89^0.10_0.10 26.71^0.10_0.10 G1.5Vb Asteroseismology 1 16 Cyg B 7.39^0.89_0.91 <25.85 <25.77 G3V Asteroseismology1 40 Eri A / 40 Eri B 3.70^3.57_1.34 26.81^0.10_0.10 26.97^0.10_0.10 K0.5V White Dwarf 4 61 Cyg A [X-ray luminosity adopted from <cit.>] 6.00^1.00_1.00 27.08^0.23_0.23 27.43^0.23_0.23 K5Ve Isochrone Fitting 5 61 Cyg B [X-ray luminosity adopted from <cit.>] 6.00^1.00_1.00 26.88^0.10_0.10 27.33^0.10_0.10 K7V Isochrone Fitting 5 CD -3710500 / L481-60 1.77^0.65_0.27 28.18^0.10_0.10 28.22^0.10_0.10 G7IV White Dwarf 4 GJ 176 2.00^0.80_0.80 27.03^0.10_0.10 27.80^0.10_0.10 M2.5V Member of HR 1614 6 GJ 191 11.00^1.00_1.00 26.41^0.10_0.10 27.02^0.10_0.10 sdM1.0 Galatic Halo population 7 HR 7703 10.00^2.00_2.00 26.80^0.10_0.10 27.04^0.10_0.10 K2.5V Old disk star 8 KIC 10016239 2.47^0.67_0.61 27.91^0.10_0.10 27.71^0.10_0.10 F6IV Asteroseismology 2 KIC 12011630 8.48^1.53_1.42 <30.06 <30.05 G1.5V Asteroseismology 2 KIC 3123191 4.26^0.80_0.75 <29.12 <28.84 F4.5V Asteroseismology 2 KIC 5309966 3.51^1.23_0.42 <29.44 <29.02 F6V Asteroseismology 2 KIC 6116048 9.58^2.16_1.90 <26.89 <26.74 F9IV-V Asteroseismology 1 KIC 6603624 7.82^0.94_0.86 <27.08 <26.96 G8IV-V Asteroseismology 1 KIC 7529180 1.93^0.35_0.30 28.98^0.10_0.10 28.66^0.10_0.10 F5IV-V Asteroseismology 2 KIC 8292840 3.85^0.81_0.75 <28.14 <27.88 F7V Asteroseismology 3 KIC 9025370 6.55^1.26_1.13 <27.24 <27.23 F8 Asteroseismology 1 KIC 9410862 6.93^1.49_1.33 <27.98 <27.86 F7V Asteroseismology 1 KIC 9955598 6.98^0.40_0.50 26.90^0.10_0.10 27.01^0.10_0.10 K0V Asteroseismology 3 NLTT 7887 / NLTT 7890 4.97^8.80_3.00 <26.92 <27.13 K2 White Dwarf 4 Proxima Centauri[X-ray luminosity taken from <cit.>] 6.13^0.55_0.55 26.69^0.10_0.10 28.23^0.10_0.10 M5.5Ve Asteroseismology 9 Alpha Centauri B [X-ray luminosity adopted from <cit.>] 6.13^0.55_0.55 27.06^0.36_0.36 27.19^0.36_0.36 K1V Asteroseismology 9 Sun [X-ray luminosity adapted from model values given in <cit.>] 4.57^0.02_0.02 26.77^0.72_0.72 26.77^0.72_0.72 G2V Isotopic Dating 10 References: (1) <cit.>; (2) this work, age recalculated with BASTA algorithm <cit.> using values from <cit.> and<cit.>; (3) <cit.>; (4) this work, see Section <ref>; (5) <cit.>; (6) <cit.>; (7) <cit.>; (8) <cit.>; (9) <cit.>; (10) <cit.> § STELLAR PROPERTIES OF SAMPLE1.5 h]l l l l l l Name of Star / White Dwarf Radius / R_⊙ T_eff / K V_mag Distance / pc References 16 Cyg A 1.22 5825 5.95 21.29 1,6,8,10 16 Cyg B 1.10 5720 6.20 21.22 1,6,8,10 40 Eri A / 40 Eri B 0.835225 4.43 4.98 5,6,8,9 61 Cyg A 0.67 4450 5.21 3.49 4,6,8,9 61 Cyg B 0.60 4050 6.03 3.50 4,6,8,9 CD -3710500 / L481-60 0.96 5530 6.01 15.25 5,6,8,10 GJ 176 0.41 3475 9.95 9.27 5,6,8,9 GJ 1910.49 3680 8.85 3.91 5,6,8,9 HR 7703 0.77 4940 5.32 6.02 5,6,8,9 KIC 10016239 1.26 6482 9.81 175.93 3,7,8,10 KIC 12011630 1.01 5817 10.27 114.40 3,7,8,11 KIC 3123191 1.37 6568 9.90 167.67 3,7,8,10 KIC 5309966 1.62 6356 10.62 288.60 3,7,8,10 KIC 6116048 1.19 6072 8.47 75.15 1,7,8,10 KIC 6603624 1.15 5612 9.19 83.56 1,7,8,10 KIC 7529180 1.45 6682 8.49 108.53 3,7,8,10 KIC 8292840 1.35 6239 10.51 250.69 2,2,8,9 KIC 9025370 1.00 5659 8.95 87.42 1,7,8,10 KIC 9410862 1.16 6230 10.78 202.27 1,7,8,10 KIC 9955598 0.88 5434 9.64 69.39 2,7,8,10 NLTT 7887 / NLTT 7890 0.78 5040 9.84 41.16 5,6,8,10 Proxima Centauri 0.17 2925 11.13 1.30 5,6,8,9 Alpha Centauri B 0.86 5316 1.33 1.35 12,12,8,12 Sun1.00 5777 -26.74 1 AU 13 References: (1) <cit.>; (2) <cit.>; (3) this work, radius recalculated with BASTA algorithm <cit.> using values from <cit.> and<cit.>; (4) <cit.>; (5) Radius calculated from T_eff and absolute brightness; (6) T_eff estimated from Spectral Type using Table 5 from <cit.>; (7) <cit.>; (8) V_mag from SIMBAD; (9) Parallax from SIMBAD; (10) Parallax from Gaia DR1 <cit.>; (11) Distance from Barnes-Evans method (see Section <ref> for details); (12) <cit.>; (13) Table 1 in <cit.> § SPECTRAL MODELLING RESULTS h]l l l l Name of Star Model kT Model Emission MeasureReduced chi-squared (keV) (4π d^2/10^-14 cm^-3) 40 Eri A 0.19 9.9× 10^-5 1.59[0.3cm] CD -3710500 0.44 2.2× 10^-4 0.32 GJ 176 0.37 3.4× 10^-5 1.49 GJ 191 0.30 5.4× 10^-5 2.52 HR 7703 0.17 5.31× 10^-5 1.34 0.76 1.45× 10^-5 KIC 7529180 0.22 9.8× 10^-6 1.70 0.91 1.7× 10^-5 61 Cyg A 0.21 4.09× 10^-4 1.99 0.79 1.75× 10^-4 61 Cyg B 0.19 1.71× 10^-4 1.17 0.67 5.06× 10^-4 | http://arxiv.org/abs/1706.08979v1 | {
"authors": [
"R. S. Booth",
"K. Poppenhaeger",
"C. A. Watson",
"V. Silva Aguirre",
"S. J. Wolk"
],
"categories": [
"astro-ph.SR",
"astro-ph.EP"
],
"primary_category": "astro-ph.SR",
"published": "20170627180007",
"title": "An Improved Age-Activity Relationship for Cool Stars older than a Gigayear"
} |
First detection of methanol towards a post-AGB object, HD 101584 H. Olofsson1 W.H.T. Vlemmings 1 P. Bergman 1 E.M.L. Humphreys 2 M. Lindqvist 1 M. Maercker 1 L. Nyman 3,4 S. Ramstedt 5 D. Tafoya1Received 14 May 2017; accepted 19 June 2017 =======================================================================================================================================================================================================================================§ ABSTRACTGiven a finite Borel measure μ on ^n and basic semi-algebraic sets _i⊂^n, i=1,…,p, we provide a systematic numerical schemeto approximate as closely as desired μ(⋃_i_i), when all moments of μ are available (and finite). More precisely, we provide a hierarchy of semidefinite programs whose associated sequence of optimal values is monotoneand converges to the desired value from above. The same methodology applied to the complement ^n∖ (⋃_i_i) provides a monotone sequence that converges to the desired value from below. When μ is the Lebesgue measure we assume that :=⋃_i_i is compact and contained in a known box :=[-a,a]^n and in this case the complement is taken to be ∖. In fact, not only μ() but also every finite vector of moments of μ_ (the restriction of μ on )can be approximated as closely as desired, and so permits to approximate the integral onof any given polynomial. Keywords: Lebesgue and Gaussian measure; semi-algebraic sets; moment problem and sums of squares; semidefinite programming; convex optimizationMSC: 44A60 28A75 90C05 90C22§ INTRODUCTION Given a set ⊂^n and a finite Borel measure μ on ^n, computing μ() is a very challenging problem. In fact even approximating the Lebesgue volume of a convex body ⊂^n (e.g. a polytope) is difficult; see e.g.Bollobás <cit.> and Dyer and Frieze <cit.>. However, in the latter case some efficient (non deterministic) algorithms with probabilistic guarantees are available and for more details the interested reader is referred to e.g. Dyer et al. <cit.>, Cousins and Vempala <cit.> and the references therein.In the non convex case no such algorithm isavailable and one is left with approximatingμ() with Monte Carlo (or Quasi-Monte-Carlo) type methods as described in e.g. Niederreiter <cit.>. That is, one first generates a sample of N points infollowing the distributionμ onand then one counts the number N_ of points that fall into . This realization of the random variable N_/N provides an estimate of μ() but by no means an upper bound or a lower bound on μ(). Of course this method is quite fast, especially is small dimension. Yet, as μ()is indeed very difficult to compute exactly, a less ambitious but still useful goal would be to provide upper and/or lower bounds on μ(). Even better, aconverging sequence of upper (or lower) bounds would be highly desirable. This is the strategy proposed in Henrion et al. <cit.> whenis a compact basic semi-algebraic set and μ is the Lebesgue measure. In <cit.> the authors have provided a (deterministic) numerical scheme which yields a monotone sequence of upper bounds converging to μ(). It consists of solving a hierarchy of semidefinite programs of increasing size.By repeating the procedure but now with the complement ∖, one also obtains a monotone sequence oflower bounds converging to μ(). However, even on typical 2 or 3-dimensional examples, the convergence was rather slow and the authors proposed a slight modification which turned out to be much more efficient; the convergence was much faster but unfortunately not monotone anymore. §.§ ContributionThe purpose of this paper is to introduce a deterministic method to approximate (in principle as closely as desired) the measure μ() of the union=⋃_i_i of finitely many basic semi-algebraic set. The finite Borel measure μ is anymeasure whose all moments are finite, e.g., the Lebesgue measurewhenis compact, the Gausssian measure dμ=exp(-‖‖^2)d for non-compact set .The method is similar in spirit to the one in <cit.> for a compact basic semi-algebraic set and the one in <cit.> for computing Gaussian measures of basic closed semi-algebraic sets (not necessarily compact), but with two important novelties.∙ In contrast to <cit.> and <cit.>, we consider a finite unionof (non disjoint) basic semi-algebraic sets, which complicates matters significantly.∙ We include a technique to accelerate the convergence different from the one described in <cit.>. Indeed in contrast to <cit.>, it has the highly desirable feature to maintain the monotone convergence to μ() which is essential if one wishes to obtain upper and lower bounds. It consists of using moments constraints coming from a particular application of Stokes' theorem. In fact this numerical schemeallows to approximatenot only μ() but also any fixed finite sequence of moments of the measure μ_ (where μ_ is the restriction of μ to ). One might invoke the inclusion-exclusion principle which states thatμ (⋃_i=1^p _i) = ∑_j=1^p (-1)^j+1∑_1 ≤ i_1 < ... < i_j ≤ pμ (_i_1∩ ... ∩_i_j),so that in principle it suffices to compute (or approximate) μ (_i_1∩ ... ∩_i_j) for all possibleintersections of the _j's, e.g.by the approach of <cit.> or <cit.>. But this approach has two major drawbacks. First there are possibly 2^p such setsand secondly, to compute an upper bound one has to compute an upper boundforsuch intersections with an odd number of elementary sets _i_j, and a lower boundforsuch intersections with an even number of elementary sets. The latter lower bound in turn is obtainedby computing an upper bound for the complement. This makes the whole procedure tedious and complicated. Finally, Bonferroni's inequalities also provide a (finite) sequence of upper and lower bounds on μ() but computing those bounds involves sums similar to the right-hand-side of (<ref>), hence with the same drawbacks just mentioned. Our proposed technique is direct with no partial computation on intersections of elementary sets _i_j. Of course, the technique described in this paper is computationally expensive. In particular, its applicabilityis limited by theperformance of the state-of-the-art semidefinite solvers because the size of the semidefinite programs increases fast with the rank in the hierarchy. Therefore it makesits application limited to small dimensional problems (n≤ 3,4). For higher dimensions only a few steps in the hierarchy can be implemented and therefore only upper and lower bounds (possibly crude) are expected.But the reader should keep in mind that the problem is very difficult and to the best of our knowledgewe are not aware of an algorithm (at least at this level of generality) which provides certified upper and lower bounds with such convergence properties (even for convex sets and in particular for non compact sets ). In our opinionthis methodology should be viewed as complementary to (rather than competing with) probabilistic methods. § NOTATION, DEFINITIONS AND PRELIMINARY RESULTS §.§ Notation and definitionsLet [] be the ring of polynomials in the variables =(x_1,…,x_n). Denote by []_d⊂[] the vector space of polynomials of degree at most d, which has dimension s(d):=n+dd, with e.g., the usual canonical basis (^γ)_γ∈^n_d of monomials, where ^n_d := {γ∈^n: |γ|≤ d},is the set of natural numbers including 0 and |γ| := ∑_i=1^nγ_i. Also, denote by Σ[]⊂[] (resp. Σ[]_d⊂[]_2d) the cone of sums of squares (s.o.s.) polynomials (resp. s.o.s. polynomials of degree at most 2d).If f∈[]_d, we write f()=∑_γ∈^n_df_γ^γ in the canonical basis and denote by f=(f_γ)_γ∈^s(d) its vector of coefficients.Finally, let S^n denote the space of n× n real symmetric matrices, with inner product ⟨,⟩ = trace. We use the notation ≽ 0 (resp. ≻ 0) to denote thatis positive semidefinite (definite). With g_0:=1, the quadratic module Q(g_1,…,g_m)⊂[] generated by polynomials g_1,…,g_m, is defined by:Q(g_1,…,g_m) := {∑_j=0^mσ_j g_j : σ_j∈Σ[] }.The quadratic module Q(g_1,…,g_m) is Archimedean if there exists M>0 such that the quadratic polynomial ↦ g_m+1:=M-‖‖^2 belongs to Q(g_1,…,g_m). Notice that g_m+1∈ Q(g_1,…,g_m) is an algebraic certificate that the set :={: g_j()≥0, j=1,…,m} is compact. If the set:{:g_j()≥0, j=1,…,m} is compactthen ‖‖^2≤ M for some M>0, and one may always include the redundant quadratic constraint θ():=M-‖‖^2≥0 in the definition ofwithout changing . Then the quadratic module Q(g_1,…,g_m+1) is Archimidean.§.§ Moment and localizing matrixWith a real sequence =(y_γ)_γ∈^n_d, one may associate the (Riesz) linear functional L_:[]_d→ defined byf (=∑_γ f_γ ^γ)↦ L_(f) := ∑_γ f_γ y_γ,Denote by _d() the moment matrix associated with , the real symmetric matrix with rows and columns indexed in the basis of monomials (^γ)_γ∈^n_d, and with entries:_d()(α,β) := L_(^α+β) = y_α+β,∀ α,β∈^n_d.Next, given g∈[], denote by _d(g ) the localizing moment matrix associated withand g, the real symmetric matrix with rows and columns indexed in the basis of monomials (^γ)_γ∈^n_d, and with entries:_d(g )(α,β) := L_(g() ^α+β) = ∑_γg_γ y_α+β+γ,∀ α,β∈^n_d.If =(y_γ)_γ∈^n is the sequence of moments of some Borel measure μ on ^n then _d()≽0 for all d∈.However the converse is not true in general and it is related to the well-known fact that there are positive polynomials that are not sums of squares. Similarly, if the support of μ is contained in {:g()≥0} then _d(g )≽0 for all d∈. For more details the interested reader is referred to e.g. <cit.>.Given a Borel set ⊂^n let ℳ() be the space of finite signed Borel measures onand let ℳ()_+⊂ℳ() be the convex cone of finite Borel measures on . §.§ The measure of a basic semi-algebraic set Let ,⊂^n with ⊃ and let μ be a finite Borel measurewhose support is .(Typically μ is the Lebesgue measure on a boxand one wishes to compute the Lebesgue volume vol(); alternatively =^n,μ is the Gaussian measure dμ=exp(-‖‖^2)d and one wishes to compute μ().) §.§ An infinite-dimensional linear program Let f∈ℝ[] be positive almost everywhere onand consider the following infinite-dimensional LP problem ::f^*=ϕ{∫_ fdϕ : λ≤μ; ϕ∈ℳ()_+ }The measure ϕ^*=μ_ (the restriction of μ to ) is the unique optimal solution of . In particular, if f()=1 for all , thenf^*=μ().§.§ Semidefinite relaxationsOf course problemin (<ref>) is infinite-dimensional and cannot be solved directly. However, whenis a basic semi-algebraic set then Theorem <ref> can be further exploited.So given (g_j)_j=1^m⊂[], let ⊂^n be the basic semi-algebraic set= { ∈^n: g_j() ≥ 0, j=1,…,m },assumed to nonempty and compact. Let ⊃ and let μ be a finite Borel measure whose all moments=(z_α) with z_α = ∫_^α dμ(),α∈^n,are available in closed form or can be computed.To approximate f^* as closely as desired in <cit.> the authors propose to solve the following hierarchy (_d)_d∈ ofsemidefinite programs[A semidefinite program (SDP) is a conic convex optimization problem with a remarkable modeling power. It can be solved efficiently(in time polynomial in its input size) up to arbitrary precision fixed in advance; see e.g. Anjos and Lasserre <cit.>] indexed by d∈: _d:[ρ_d=sup_{ L_(f):; _d() ≽0;_d(-) ≽ 0; _d-r_j(g_j ) ≽ 0, j = 1,…,m}. ]Observe that _d is a relaxation of P, and so ρ_d≥μ() for all d. In addition, the sequence (ρ_d)_d∈ is monotone non increasing. The dual of (<ref>) is the semidefinite program:^*_d:ρ^*_d=inf_p∈[]_2d { ∫_ p dμ:p-f≥0; p∈Σ[]_d },and by weak duality, ρ_d≤ρ^*_d≤ f^* for all d. Assume that Q(g_1,…,g_m) is Archimedean. Then ρ_d→ f^* as d→∞. Ifhas nonempty interior thenρ^*_d=ρ_d and (<ref>) has an optimal solution p^*∈[]_2d.So when f=1, (ρ_d)_d∈ provides us with a monotone sequence of upper bounds on f^*=μ(). Unfortunately the convergence is rather slow as observed on several numerical examples. This is because in the dual (<ref>) the optimal solution p^*∈[]_2d tries to approximate from above (in L_1(,μ)) the discontinuous function 1_, which implies an annoying Gibb's phenomenon[The Gibbs' phenomenon appears at a jump discontinuity when one approximates a piecewise C^1 function with a continuous function, e.g. by its Fourier series.]. To remedy this problem the authors in <cit.> propose to use a polynomial f, nonnegative onand which vanishes on ∂. In this case the convergence ρ_d→∫_ f dμ as d→∞ is still monotone and if ^d=(y^d_α)_α∈^n_2d denotes an optimal solution of (<ref>) then y^d_0→μ() as d→∞. However, while faster than with f=1, the latter convergence of y^d_0 to μ() is not monotone anymore, a rather annoying feature which prevents from obtaining a non increasing sequence of upper bounds.§ MAIN RESULT§.§ The context Let ⊂^n be a box, and for every i=1,...,p, let _i := { ∈^n :g_ij(x) ≥ 0, j=1,…,m_i}, for some polynomials (g_ij)⊂[]. Assume thathas been chosen so as to satisfy::= ⋃_i=1^p _i ⊂. The goal is to provide a numerical scheme to approximate as closely as desired the Lebesgue volume μ(). (We will see how to adapt the methodology to also approximate as closely as desired μ() whenis not necessarily compact and μ is a Gaussian measure.) One possible approach described below is to use the powerful inclusion-exclusion principle and/or the associated Bonferroni inequalities. §.§ The inclusion-exclusion principle and Bonferroni InequalitiesLet :S_k:= ∑_1 ≤ i_1 < ... < i_k ≤ pμ (_i_1∩ ... ∩_i_k), k=1,…,p.By the inclusion-exclusion principle, μ (⋃_k=1^p _k) = ∑_k=1^p (-1)^k+1S_k,which allows us to work with intersections of the _k's only. In addition, the Bonferroni inequalities state thatμ (⋃_i=1^p _i)≤∑_j=1^2k+1 (-1)^j+1S_j∀ 2k+1 ≤ p≥∑_j=1^2k (-1)^j+1S_j∀ 2k ≤ pwhich provides sequences of (increasingly tighter) upper and lower bounds. Therefore to compute μ() we only have to compute the measure of the intersectionΘ_i_1,…,i_k:=⋂_j=1,…,k_i_j, for all 1 ≤ i_1 < ... < i_k ≤ p.Notice that there are 2^p such sets. As each Θ_i_1,…,i_k⊂ is a compact basic semi-algebraic set, one may apply the methodology described in <ref>, to obtain a sequence (ρ^(i_1,…,i_k)_d)_d∈ which converges to μ(Θ_i_1,…,i_k) as d→∞, and thereforelim_d→∞(∑_i=1^p(-1)^k+1∑_1≤ i_1<…<i_k≤ pρ_d^i_1,…,i_k) = μ().Notice that the convergence is not monotone non increasing even if one solves (<ref>) with f=1 becausewe sum up negative and positive terms. To maintain the monotone convergence (when f=1) it suffices to compute a lower bound on the complement ∖Θ_i_1,…,i_k when k is even. However as already mentioned the convergence is expected to be rather slow. To accelerate the convergence one may use f=∏_j=1^k∏_ℓ=1^m_i_jg_i_jℓ when one solves (<ref>) with =Θ_i_1,…,i_k as f≥0 on Θ_i_1,…,i_k andf=0 on ∂Θ_i_1,…,i_k. But then the convergence lim_d→∞(∑_i=1^p(-1)^k+1∑_1≤ i_1<…<i_k≤ py_d,0^i_1,…,i_k) = μ(),(where ^i_1,…,i_k_d=y^i_1,…,i_k_d,α is an optimal solution of (<ref>) with =Θ_i_1,…,i_k) is not monotone anymore. §.§ A direct approach In this section we describe a direct approach with two distinguishing features: * It does not use the inclusion-exclusion principle and the need to approximate μ(⋂_j=1^k_i_j) for all 2^p such sets.* The convergence to μ() (and also to μ(∖)) is monotone non increasing, that is, we can compute two sequences (ω_d)_d∈ and (ω_d)_d∈ such that:ω_d ≤ μ() ≤ ω_d, d∈;μ() = lim_d→∞ω_d = lim_d→∞ω_d. Recall that any finite number of moments μ_α = ∫_𝔹x^α dμ(x),α∈^n,are either available in closed-form or can be obtained numerically. §.§ A multi infinite-dimensional linear program As in <ref> we first introduce an infinite-dimensional LP problemwhose unique optimal solution is the restriction of μ on(and whose dual has a clear interpretation). Let f∈ℝ[] be positive almost everywhere onand consider the following infinite-dimensional LP problem ::f^*=sup_ϕ_1,…,ϕ_p{∑_i=1^p∫__i f dϕ_i : ∑_i=1^p ϕ_i ≤μ; ϕ_i ∈ℳ(_i)_+, i=1,…,p}.Problemhas an optimal solution (ϕ^*_1,…,ϕ^*_p) andevery optimal solution satisfies ∑_i=1^p ϕ^*_i = μ_, where μ_ is the restriction of μ to .We first prove thathas an optimal solution. The set Δ_μ:={ϕ∈ℳ(^n)_+:ϕ≤ μ} is weakly sequentially compact; see e.g. Dunford & Schwartz <cit.>. Therefore let (ϕ^k_1,…,ϕ^k_p)_k∈ be a maximizing sequence of feasible solutions of . There exists a subsequence (k_ℓ)_ℓ∈ such that for every i=1,…,p, ϕ^k_ℓ_iw→ϕ^*_i for some ϕ^*_i∈ℳ(^n)_+. The above weak convergence and ∫__i^c dϕ_i^k_ℓ=0 implies ∫__i^c dϕ_i^*=0, that is, ϕ^*_i∈ℳ(_i)_+ for all i=1,…,p. Weak convergence again implies ∑_i=1ϕ^*_i≤μ and so (ϕ^*_1,…,ϕ^*_p) is a feasible solution of . Finally weak convergence also implies f^* = lim_ℓ→∞∑_i=1^p∫ f dϕ^k_ℓ_i = ∑_i=1^plim_ℓ→∞∫ f dϕ^k_ℓ_i = ∑_i=1^p∫ f dϕ^*_i,which proves that (ϕ^*_1,…,ϕ^*_p) is an optimal solution of .We next prove that f^*=∫fdμ_. Indeed, firstly observe that for every feasible solution (ϕ_1,…,ϕ_p) of , ∑_i=1^p∫ f dϕ_i ≤∫_ fdμ= ∫ fdμ_. On the other hand, for every i=1,…,p, denote by θ_i the measurable function defined onby :↦θ_i() = 1/|{j ∈{1,…,p} : ∈_j }| 1__i(),∈. The (discontinuous) functions (θ_i)_i=1,…,p form a partition of unity subordinate to the open cover ⋃_i int(_i). For every i=1,…,p, let ϕ^*_i∈ℳ(_i)_+ be the finite Borel measure defined by:ϕ^*_i(C):= ∫_C θ_i() dμ(),∀ C∈ℬ(^n).Hence, ∑_i=1^pϕ^*_i(C)=∫_C∑_i=1^p θ_i()dμ() = ∫_C 1_()()dμ() = μ_(C) ≤μ(C). Therefore (ϕ^*_1,…,ϕ^*_p) is a feasible solution ofsuch that ∑_i=1^pϕ^*_i=μ_, and so ∑_i=1^p∫ fdϕ^*_i=f^*, i.e., (ϕ^*_1,…,ϕ^*_p) is an optimal solution of . In fact, every optimal solution (ϕ^*_1,…,ϕ^*_p) ofsatisfies∑_i=1^p ∫ fdϕ^*_i=f^*, and therefore ϕ^*:=∑_i=1^pϕ^*_i∈ℳ()_+ is an optimal solution of sup_ϕ{∫ fdϕ:ϕ≤μ; ϕ∈ℳ()_+}. By Theorem <ref> this solution ϕ^* is unique, which yields the desired result.§.§ A hierarchy of semidefinite relaxations Let = (z_α)_α∈^n be the sequence of all moments of μ , that is,z_α := ∫^α dμ(),α∈^n. Let ⊂^n be a box and ⊂ be a compact semi-algebraic as in (<ref>). With no loss of generality and possibly after scaling, we may and will assume that ⊂ [-1,1]^n and μ is a probability measure. Therefore | z_α|≤ 1 for all α∈^n.Let r_ij= ⌈ deg(g_ij)/2 ⌉ and let f∈ℝ[] be a given polynomialpositive almost everywhere on(and define r_00 := ⌈ deg(f)/2 ⌉). For d ≥ d_0:=_i,j r_ij, consider the followinghierarchy of semidefinite programs (_d) indexed by d∈ :_d:[ρ^f_d=sup_^1,…,^d{ ∑_i=1^p L_^i(f);_d( - ∑_i=1^p ^i) ≽ 0; _d(^i)≽0, i=1,…,p;_d-r_ij(g_ij ^i) ≽ 0,j=1,…,m_i; i=1,…,p }. ] Observe that ρ^f_d≥ f^* for all d∈. Indeed, if (^1,…,^p) is the sequence of moments of an optimal solution (ϕ^*_1,…,ϕ^*_p) of in (<ref>) then (^1,…,^p) is also a feasible solution of _d.Consider the semidefinite programs (_d), d≥ d_0. Then :(i) _d has an optimal solution and the associated sequence of optimal values (ρ^f_d)_d∈ is monotone non increasing and converges to f^*, that is:ρ^f_d ↓f^* = ∫_ f dμ,(ii) Let (^1,d,…,^p,d) be an optimal solution of _d. Then for each α∈^n:lim_d→∞∑_i=1^py^i,d_α =z^*_α = ∫_^α dμ.and in particular lim_d→∞∑_i=1^py^i,d_0=μ().For a sequence = (y_α), let τ_d() = max_i=1,…,nL_(x_i^2d) and recall that if _d()≽0 then | y_α|≤max[y_0,max_iL_(x_i^2d)] for every α∈^n_2d; see <cit.>. Next, observe that from _d( -∑_i=1^p^i,d) ≽ 0 and _d(^i) ≽ 0, _d( - ^i,d)≽ _d(∑_j ≠ i^j) ≽ 0, i=1,…,n.Hence the diagonal elements z_2α - y^i,d_2α are all nonnegative which in turn implies τ_d(^i,d) ≤τ_d()≤ 1 for all i=1,…,n.As z_0=1 thenby <cit.> | y^i,d_α|≤ 1for every α∈ℕ^n_2d, and so the feasible set of semidefinite program _d is closed, bounded, hence compact, and therefore _d has an optimal solution.Next, let (^1,d,…,^p,d) be an optimal solution of _d and by completing with zeros, make (^1,d,…,^p,d)an element of the unit ball of(ℓ_∞)^p (where ℓ_∞ is the Banach space of bounded sequences, equipped with the sup-norm).As (ℓ_∞)^p is the topological dual of (ℓ_1)^p, by the Banach-Alaoglu Theorem, there exists (^1,*,..,^p,*) ∈ (ℓ_∞)^p and a subsequence {d_k} such that (^1,d_k,…,^p,d_k) → (^1,*,…,^p,*) ask→∞, for the weak ⋆ topology σ((ℓ_∞)^p, (ℓ_1)^p). In particular,lim_k →∞y_α^i,d_k=y_α^i,*,∀α∈ℕ^n, ∀ i ∈{1,..,p}.Next let d∈ℕ be fixed arbitrary. From the pointwise convergence (<ref>) we also obtain _d(^i,*) ≽ 0 and _d(- ∑_i=1^p^i,*) ≽ 0 for every i=1,…,p. Similary, _d-r_ij(g_ij^i,*) ≽ 0 for every i and j.As d was arbitrary, by Putinar's Positivistellensatz <cit.>, ^i,* has a representing measure ϕ_i supported on_i for all i=1,…,p, and ∑_i=1^p ϕ_i ≤μ.In particular from (<ref>), as k →∞,f^* ≤ρ^f_d_k = ∑_i=1^p L_^i,d_k(f) ↓∑_i=1^p L_^i,*(f) = ∑_i=1^p ∫ f dϕ_i.Therefore (ϕ_1,…,ϕ_p) is admissible for problemwith value ∑_i=1^p ∫ fdϕ_i ≥ f^*, and so (ϕ_1,…,ϕ_p) is an optimal solution of .Finally, by Theorem <ref>, ∑_i=1^p ϕ_i = μ_. And so for each α∈^n: lim_k→∞ ∑_i=1^py^i,d_k_α=∑_i=1^py^i,*_α= z^*_α = ∫_^α dμ().As the converging subsequence (d_k)_k∈ was arbitrary, it follows that in fact the whole sequence (∑_i=1^py^i,d_α)_dconverges to z_α, for all α∈^n, that is, (<ref>) holds.§.§ The dual of _dLet g_i0()=1 for all ∈^n, i=1,…,p. The dual of the semidefinite program _d is the semidefinite program:^*_d:[ (ρ^f _d)^*=inf_q,σ_ij {∫_ q dμ:; q-f = ∑_j=0^m_iσ_ij g_ij, i=1,…,p; σ_ij∈Σ[]_d-r_ij, j=0,…,m_i; i=1,…,p; q∈Σ[]_d}. ]Assume that for every i=1,…,p, both _i and ∖_i have nonempty interior. Then there is no duality gap between (<ref>) and its dual (<ref>), that is, ρ^f _d=(ρ^f _d)^*for all d≥ d_0. Moreover (<ref>) has an optimal solution (q^*,(σ_ij^*).Let (ϕ^*_1,…,ϕ^*_p) be the measures defined in (<ref>) the proof of the Theorem <ref> and let^i_d be the sequence of moments up to degree d of ϕ^*_i, i=1,…,p. As every_i has nonempty interior, then clearly _d(^i) ≻ 0 and _d-r_ij(g_ij^i) ≻ 0 for everyj= 1,…,m_i and i = 1,…,p. As ∖ also has nonempty interior then _d( - ∑_i=1^p^i) ≻ 0.Therefore Slater's condition holds for _d. In addition, the set of admissible solution of _d^* is nonempty (set q=f and σ_ij=0 for all i,j), and therefore a standard result in conic convex optimization yields the desired result[In fact as the set of optimal solutions of (<ref>) is compact, the absence of a duality gap between (<ref>) and (<ref>) also follows from <cit.> without the conditions int(_i)≠∅ and int(∖_i)≠∅.].As in the case of a basic closed semi-algebraic set, when f is the constant function 1 the convergence ρ^f_d→ f^* =μ() is monotone non increasing, a highly desirable feature. Howeverin typical examples this convergence is rather slow. Again one may take for f a function that is nonnegative onand which vanishes on ∂. This accelerates the convergenceboth ρ^f_d→ f^* and ∑_i y^id_0→μ() as d→∞, but if by construction the former is monotone non increasing,the latter is not monotone anymore, a rather annoying feature if the goal is to obtain aconverging sequence of upper bounds. In the next section we describe a technique that allows to accelerate significantly the convergence ∑_i y^id_0→μ() as d→∞, while maintaining its monotone non increasing character.§.§ Convergence improvement using Stokes' formula In this section we show how to improve significantly the monotone non increasing convergence of ρ_d^1 (i.e. ρ^f_d with f=1) to μ(). To do this we will use Stokes' theorem for integration and in the sequel, to avoid technicalities we assume that ⊂ℝ^n is the closure of its interior, i.e., = int(). The basic idea is simple to express in informal terms.Since we know in advance that(ϕ^*_1,…,ϕ^*_p) in(<ref>) is an optimal solution of problem , every additional information in terms of linear constraints on the moments of ϕ^*_i can be included inwithout changing its optimal value. BUT when included in the relaxation _d it will provide useful additional constraints that restrict the feasible set of _d and somake its optimal value necessarily smaller.Suppose for the moment thatis compact with smooth boundary, and assume that the measure μ has a density h with respect to Lebesgue measure d of the form q()exp(r())1_() for some polynomial r,q∈[].Let X be a given vector field and f∈[]. Then Stokes' theorem states:∫_ Div(X) f()h() d + ∫_⟨ X, ∇ (f() h())⟩ d = ∫_∂ K⟨ X,n⃗_⟩ f()h() dσ(),where n⃗_ is outward pointing normal at ∈∂, and σ is the(n-1)-dimensional Hausdorff measure on ∂. In particular if f vanishes on ∂and =e_k∈^n (where e_k(j)=δ_k=j) Stokes' formula becomes∫_∂/∂ x_k (f() h())d = 0.To exploit (<ref>) in our particular context whereis defined in (<ref>),let g = ∏_i=1^p ∏_j=1^m_i g_ij and let ↦ f() = ^α g()q() with α∈^n arbitrary. Then f vanishes on ∂ andon ∂_i_1,…,i_s:=_i_1∩⋯∩_i_s for all 1≤ i_1<…<i_s≤ p, s=1,…,p. Hence by (<ref>):∫_ p_α,k () q() exp(r()) d_dμ() = 0∫__i_1,…,i_s p_α,k () q() exp(r()) d_dμ() = 0,for all 1≤ i_1<…<i_s≤ p, s=1,…,p, where p_α,k () = q()∂/∂_x_k (^α g()) + 2^α g() ∂/∂_x_kq() + ^α g()q()∂/∂_x_k r().Recalling how ϕ^*_i is defined in (<ref>), it can be written as ϕ^*_i = ∑_s=1^p∑_1≤ i_1<⋯<i_s≤ pϕ^*_i, i_1,…,i_s,where each ϕ^*_i, i_1,…,i_s is supported on _i∩_i_1,…,i_s and has a constant density w.r.t. μ. Therefore, for every i=1,…,p:∫__ip_α,k () dϕ^*_i = 0,∀α∈^n; k=1,…,n. Hence (<ref>) provides additional useful information on the optimal solution (ϕ^*_1,…,ϕ^*_p) ofdefined in (<ref>). Namely it translates into L_^i(p_α,k) = 0,∀α∈^n; k=1,…,n;i=1,…,p,i.e., linear constraints on the moments of ϕ^*_i, for every i=1,…,p. Plugging this additional linear constraints on the moments of ϕ^*_i into the relaxation _d, yields the following new hierarchyof SDP-relaxation (_d^ stokes)_d≥ d_0:[ ρ^ Stokes_d=sup_^1,…,^d{ ∑_i=1^p ^i_0;_d( - ∑_i=1^p ^i) ≽ 0; _d(^i)≽0, i=1,…,p; _d-r_ij(g_ij ^i) ≽ 0,j=1,…,m_i; i=1,…,p; L_^i(p_α,k) = 0, k=1,…,n; |α|≤ 2d- deg(p_α,k); i=1,…,p}. ]By construction ρ^1_d≥ρ^ Stokes_d≥μ() holds for every d≥ d_0, and the analogue of Theorem <ref> (with f=1) reads:Consider the semidefinite programs (_d^ Stokes), d≥ d_0, defined in (<ref>). Then :(i) _d^ Stokes has an optimal solution and the associated sequence of optimal values (ρ^ Stokes_d)_d∈ is monotone non increasing and converges to μ(), that is:ρ^ Stokes_d ↓ μ(),(ii) Let (^1,d,…,^p,d) be an optimal solution of ^ Stokes_d. Then for each α∈^n:lim_d→∞∑_i=1^py^i,d_α =z^*_α = ∫_^α dμ.The proof being almost a verbatim copy of that of Theorem <ref>, is omitted.The important feature of Theorem <ref> is that we now havethe monotone non increasing convergence ρ^ Stokes_d↓μ() (compare with (<ref>) (with α=0) in Theorem <ref>). §.§ Gaussian measure of non compact sets So far Theorem <ref> and Theorem <ref> have been given for μ supported on a box , and so only for setsin (<ref>) that are compact.It turns out that for a Gaussian measure μ of (possibly non-compact) sets =⋃_i_i, Theorem <ref> (resp. Theorem <ref>) is still valid with exactly the same statement and exactly the same semidefiniterelaxations (<ref>) (resp. (<ref>)), except that now=(z_α) is the vector of moments of the Gaussian measure μ (instead ofthe moments of the Lebesgue measure onpreviously).However in the gaussian case the proof of Theorem <ref>(i)-(ii) and Theorem <ref>(i)-(ii) uses quite different arguments (some already used in <cit.> for a basic semi-algebraic set). Indeed asis not necessarily compact :- The uniform bound sup_α|_α|≤1 is not valid any more for the relaxations _d and ^Stokes_d.- One cannot invoke Putinar's Positivstellensatz <cit.> any more.- The standard version of Stokes' theorem whereis compact cannot be invoked anymore either.The new arguments that we need are the following:∙ A crucial fact is that μ satisfies Carleman's condition∑_k=1^∞(∫_^n x_i^2k dμ())^-1/2k = +∞, i=1,…,n.Then a sequence =(y_α)_α∈^n such that _d()≽0 for all d∈, and ∑_k=1^∞ L_(x_i^2k)^-1/2k = +∞, i=1,…,n,has a unique representing measure ϕ on ^n which is moment determinate; see for instance <cit.>.∙ If in addition _d(h )≽0 for all d∈ (where h∈[]), and as ϕ satisfies (<ref>), then h()≥0 for allin the support of ϕ; see Lasserre <cit.>. This argument is used to show that ϕ is supported on .∙ To obtain a version of Stokes fornon-compact setwith boundary ∂, we invokea limiting argument that uses (the standard) Stokes's theorem on the compact ∩(0,M) (where (0,M)={:‖‖≤ M}). LettingM→∞ and using the Monotone and Bounded Convergence theorems yields the desired result. For more details the reader is referred to <cit.> where such arguments have been used in the case of a basic semi-algebraic set.Finally it is worth emphasizing that this methodology also works for any measure μ that satisfies (<ref>) (andwhose moments are known or can be computed); an important spacial case is the exponential measure on the positive orthant ^n_+. As mentioned above, in <cit.> the first author has already used Stokes' formula to accelerate the convergence of a hierarchy of semidefinite relaxations to approximate the Gaussian measure μ() of a basic semi-algebraic set , not necessarily compact.The important and non trivial novelty here is that(i) =⋃_i=1^p_i is now a union of basic semi-algebraic sets, and (ii) even if this complicates matters significantly, we are still able to work with measures ϕ_i, each supported on _i (a basic semi-algebraic set).It turns out that μ()=∑_i=1^pϕ_i^* where iseach ϕ^*_i has a piecewise constant density w.r.t. μ (constant on each of the possible intersections _i∩_i_1,…,i_p).By using a family of polynomials that all vanish on the boundary of each _i∩_i_1,…,i_p, we can exploit Stokes' Theorem on each piece and sum up to obtain a family of linear constraints on the moments of ϕ^*_i. § NUMERICAL EXPERIMENTS AND DISCUSSIONFor illustration purposes we have applied the methodology on a few (simple) examples. We report some numerical experiments carried out in Matlab and GloptiPoly3 <cit.>, a software package for manipulating and solving generalized problems of moments. The SDP problems were solved with SeDuMi 1.1R3.§.§ Lebesgue volume of a union of two ellipsoids We first consider a simple example of two ellipsoids in ^2 where the exact value μ() can be computed exactly so that we can compare with our upper bounds. So we want to compute the Lebesgue measure of= _1 ∪_2 with _1 = {(x_1,x_2) ∈ℝ^2 : x_1^2/4 + x_2^2 ≤ 1} and _2 = {(x_1,x_2) ∈ℝ^2 :x_2^2/4 + x_1^2 ≤ 1}. In this examplewe take :=[-2,2]^2.The results are displayed inthe Figure <ref> with:in orange the approximation of the Lebesgue volume μ() without using Stokes' formulas, in red the approximation when using Stokes' formulas and in blue theexact value of μ(). We next consider a union of two ellipsoids in dimension n=3. Let _1 = {∈ℝ^3 : x_1^2 + 4x_2^2 + 4x_3^2≤ 1}, _2 = {∈ℝ^3 : x_2^2 + 4x_1^2 + 4x_3^2≤ 1}, =_1∪_2 and 𝐁 = [-1,1]^3.Results are displayed in Figure <ref>. In both examples one can check that the convergence is much faster when usingStokes' formula.§.§ Lebesgue measure a union of three ellipsoids We nextconsider a union of three ellipsoid in dimension n=2, with: _1 = {∈ℝ^2 : (x_1,x_2).[[ 16/90;04 ]]([ x_1; x_2 ]) ≤ 1}, _2 = {(x_1,x_2) ∈ℝ^2 : 1/9(x_1-0.1,x_2-0.1).[[31 5√(3); 5√(3)21 ]]([ x_1-0.1; x_2-0.1 ]) ≤ 1}, _3 = {(x_1,x_2) ∈ℝ^2 : 1/9(x_1+0.1,x_2-0.1).[[ 31 -5√(3); -5√(3) 21 ]]([ x_1+0.1; x_2-0.1 ]) ≤ 1},and = [-1,1]^2. In Figure <ref> we also compare our results with those obtainedwhen using Bonferroni inequalities.In red the upper bounds obtained by solving Q^Stokes, in orange the lower bounds obtained by solving Q^Stokes for the complement, and in blue the upper bounds obtained by using Bonferroni inequalities. (For a fair comparison, for each relaxation in Bonferroni case we also use appropriate Stokes' constraints.)§.§ Examples for the Gaussian measureIn this section we consider the Gaussian measure dμ = exp(-x^2/σ^2)dxwith variance σ^2 = 0.8. For each example we have computed two upper-bounds and two lower-bounds for μ(). The first(resp. second) upper-bound ρ_d (resp. ρ^Stokes_d)is obtained bysolving the semidefinite relaxationQ_d (resp. Q^Stokes_d). Similary, the lower-bounds ρ_d (resp. ρ^Stokes_d) are obtained fromupper boundsfor the complement ^n∖. The respective relative error-gap are denoted by ϵ_d= ρ_d - ρ_d/ρ_d and ϵ^Stokes_d = ρ^Stokes_d - ρ^Stokes_d/ρ^Stokes_d. In this exampleis the union of two ellipsoids. Let := _1 ∪_2 with _1 = {x∈ℝ^2 : (x - u)^T A_1 (x - u) ≤ 1} and _2 = {x∈ℝ^2 : (x - v)^T A_2 (x - v) ≤ 1} for the values[u = (0,0), (0.1,0.5),(0.5,0.5) ]v = (1,0),A_1 = [[ 1 0; 0 1/4 ]] A_2 = [[ 1/4 0; 0 1 ]]In this case ρ:=μ( can be computed exactly and sowe have displayed the values of the relative errors denoted by ϵ_d = ρ_d - ρ/ρ and ϵ^Stokes_d = ρ^Stokes_d - ρ/ρ respectively, depending on whether or not we have used Stokes' formula. As one can see in Table <ref> fora reasonable value d=10 the relative error (when using Stokes' formula) is quite good. The respective behaviors are displayed in Figure <ref>. Consider = _1 ∪_2 with _1 = {x∈ℝ^2 : (x - u)^T A_1 (x - u) ≤ 1} and _2 = {x∈ℝ^2 : (x - v)^T A_2 (x - v) ≤ 1} for the values u = (0,0), v = (-2,0),A_1 = [[ 1/160;01 ]] A_2 = [[ 1/4 1/2; 1/2-1 ]]In this caseis not a compact set as it is unbounded. The results for d=9 displayed in Table <ref> show that a good value is already obtained when using Stokes' formula. The respective behaviors of ϵ_d and ϵ^Stokes_d displayed in Figure <ref> also show that using Stokes' formula yields a significant improvement. Consider = _1 ∪_2 with _1 = {x∈ℝ^2 : (x - u)^T A_1 (x - u) ≤ 1} and _2 = {x∈ℝ^2 : (x - v)^T A_2 (x - v) ≤ 1} for the values u = (0,0), v = (-2,0),A_1 = [[ -1/16 0; 0 1 ]] A_2 = [[ 1/4 1/2; 1/2-1 ]]Againis not compact. The results in Table <ref> and the respective behaviors of ϵ_d and ϵ^Stokes_d displayed in Figure <ref> confirm that using Stokes' formula yields a significant improvement. We next consider an example in dimension n=3. Let _1 = {x∈ℝ^3 : (x - u)^T A_1 (x - u) ≤ 1} and _2 = {x∈ℝ^2 : (x - v)^T A_2 (x - v) ≤ 1} for the values u = (0,0,0), v = (-2,0,-1),A_1 = [[ -1/16 0 0; 0 1 0; 0 0 1/4 ]] A_2 = [[ 1/4 1/2 0; 1/2-1 0; 1/4 1/4 1/2 ]] Results for d=6 are displayed in Table <ref> and the relative errors ϵ_d and ϵ^Stokes_dare displayed inFigure <ref>. The quality of results is comparable to that in Examples <ref> and <ref> for d=6. Still in dimension n=3, let= _1 ∪_2 with _1 = {x∈ℝ^3 : x^Te ≤ 1} and _2 = {x∈ℝ^3 : x ^T Ax ≤ 1}, wheree = (1,1,1) andA = [[ 1/4 1/2 0; 1/2-1 0; 1/4 1/4 1/2 ]].The relative errors ϵ_d and ϵ^Stokes_dare displayed inFigure <ref>.One can see that in all examples quite good approximations are obtained with relatively few moments (up to order 2d≤ 18 for n=2 and2d≤ 14 for n=3) provided that we use the hierarchy (<ref>) with the additionalmoments constraints induced by Stokes' formula. The convergenceof the hierarchy (<ref>) (without those Stokes constraints) is indeed much slower. For all the examples that we have treated, the (crucial) moment and localizing matrices involved in (<ref>) and in (<ref>)have been expressed in the canonical basis (^α)_α∈^n of monomials forsimplicity and easyness of implementation of the SDP relaxations. But this choice is in fact the worst from a numerical point of view (numerical stability and robustness) which prevented us from solving (<ref>) and (<ref>) for d≥ 7 when n=3. It is very likely that the basis of orthonormal polynomials w.r.t. μ (Legendre for the Lebesgue measure μ on [-1,1] and Hermite for the Gaussian measure μ) is a much better (and recommended) choice. Such a more sophisticated implementation was beyond the scope of this paper.§ CONCLUSION In this paper we have provided a numerical scheme to approximate as closely as desired the measureμ() of a finite union =∪_i=1^p_i of basic semi-algebraic sets (the case of a single basic semi-algebraic set was treated in <cit.>)). Surprisingly, even though the case of a union of semi-algebraic sets complicates matters significantly we are still able to adapt the methodology developed in <cit.> and provide a monotone non-increasing (resp. non-decreasing) sequence of upper (resp. lower)bounds that converges to μ() as the number of moments considered increases. In addition we are also able to use additional moment constraints induced by an appropriate application of Stokes' Theoremwhich permits to improve significantly the convergence. In fact those additional moment constraints are crucial to obtain good bounds rapidly as theypermit strongly attenuate a Gibbs' phenomenon that otherwise appears.Our current implementation could be significantlyimproved by using a basis for polynomials more appropriate than the usual canonical basis of monomials(the worst choice from a numerical stability point of view). For instance in doing so it should be possible to implement step d=8,9 of the hierarchy in dimension n=3, and step d=7 for n=4. As the convergence seems to be fast, each additional step of the hierarchy can yield a significant improvement.The methodology was presented for the Lebesgue measure μ whenis compact and the Gaussian measure for non-compact sets , but in fact and remarkably, the same methodology works for any measureμ that satisfies Carleman's condition and provided that all its moments are available (or can be computed easily). Of course the methodology proposed in this paperis computationally expensive, especially when compared with Monte-Carlo type methods.But the latter provide only an estimate of μ() and by no means an upper or lower bound on μ() and therefore these two types of methods should be seen as complementary rather thancompeting. In its present form it is also limited to small dimension problems (typically n≤ 3,4) because since each upper (or lower) bound requires to solve a semidefinite program whose size increases fast in the hierarchy, one islimited by the current efficiency of state-of-the-art semidefinite solvers. However to the best of our knowledge this is the first method thatprovides a sequence ofupper and lower bounds with strong asymptotic guarantees, at least at this level of generality. §.§ AcknowledgementResearchfunded by by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement ERC-ADG 666981 TAMING)" las handbook Anjos M. , Lasserre J.B. (Eds.). Handbook of Semidefinite, Conic and Polynomial Optimization, Springer, New York, 2012. Bollobas Bollobás B. Volume estimates and rapid mixing. In: Flavors of Geometry,MSRI Publications 31, 1997, pp. 151–180. cousins1 Cousins B., Vempala S. A cubic algorithm for computing gaussian volume. Proceedings of the 2014 ACM-SIAM Symposium on Discrete Algorithms (SODA14), Portland, January 2014. cousins2 Cousins B., Vempala S. A Practical Volume Algorithm, Math. Program. Comput. 8, pp. 133–160, 2016. curto1 Curto R.E., Fialkow L.A. Flat extensions of positive moment matrices: recursively generated relations,Memoirs. Amer. Math. Soc. 136, AMS, Providence, 1998. curto2 Curto R.E.,Fialkow L.A. The truncated K-moment problem in several variables, J. Operator Theory 54, pp.189–226, 2005. dunford Dunford N., J. Schwartz. Linear Operators. Part I: General Theory, John Wiley & Sons, Inc., New York, 1958. dyer1 Dyer M.E., Frieze A.M. The complexity of computing the volume of a polyhedron, SIAM J. Comput. 17, pp. 967–974, 1988. dyer2 Dyer M.E., Frieze A., Kannan R. A random polynomial-time algorithm for approximating the volume of convex bodies, J. ACM 38, pp. 1–17, 1991. sirev Henrion D., Lasserre J.B., Savorgnan C. Approximate volume and integration for basic semialgebraic sets, SIAM Review 51,pp. 722–743, 2009. gloptipoly Henrion, Lasserre J.B., Lofberg J. Gloptipoly 3: moments, optimization and semidefinite programming, Optim. Methods & Softwares 24, pp. 761–779, 2009. lass-decomp Lasserre J.B.. Lebesgue decomposition in action via semidefinite relaxations, Adv. Comput. Math. 42,pp. 1129–1148, 2016. lass-gaussian Lasserre J.B.. Computing gaussian and exponential measures of semi-algebraic sets, arXiv:1508.06132, 2015. submitted. lass-book-icp Lasserre J.B. Moments, Positive Polynomials and Their Applications, Imperial College Press, London, 2010 newlook Lasserre J.B. A new look at nonnegativity on closed sets and polynomial optimization, SIAM J. Optim. 21,pp. 864–885, 2011.vempala1 Lovász L., Vempala S. Simulated annealing in convex bodies and anO^*(n^4)volume algorithm. J. Comput. Syst. Sci., 72, pp. 392–417, 2006. niederreiter Niederreiter N. Random Number Generation and Quasi-Monte Carlo Methods, Society for Industrial and Applied Mathematics, Philadelphia, 1992. putinar Putinar M. Positive polynomials on compact semi-algebraic sets, Ind. Univ. Math. J. 42, pp. 969–984, 1993. strongTrnovská M. Strong duality conditions in semidefinite programming. J.Elec. Eng. 56, pp. 1–5, 2005. | http://arxiv.org/abs/1706.08253v1 | {
"authors": [
"Jean Lasserre",
"Youssouf Emin"
],
"categories": [
"math.OC"
],
"primary_category": "math.OC",
"published": "20170626071432",
"title": "Lebesgue and gaussian measure of unions of basic semi-algebraic sets"
} |
A. Martín-Ruiz and L. F. Urrutia Gravitational waves propagation in nondynamical Chern-Simons gravityInstituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, 04510 México, Distrito Federal, México^*[email protected] ^†[email protected] waves propagation in nondynamical Chern-Simons gravity A. Martín-Ruiz^*and L. F. Urrutia^† December 30, 2023 ==================================================================== Day Month Year Day Month Year We investigate the propagation of gravitational waves in linearized Chern-Simons (CS) modified gravity by considering two nondynamical models for the coupling field θ: (i) a domain wall and (ii) a surface layer of θ, motivated by their relevance in condensed matter physics. We demonstrate that the metric and its first derivative become discontinuous for a domain wall of θ, and we determine the boundary conditions by realizing that the additional contribution to the wave equation corresponds to one of the self-adjoint extensions of the D'Alembert operator. Nevertheless, such discontinuous metricsatisfies the area matching conditions introduced by Barrett. On the other hand, the propagation through a surface layer of θ behaves similarly to the propagation of electromagnetic waves in CS extended electrodynamics. In both cases we calculate the corresponding reflection and transmission amplitudes. As a consequence of the distributional character of the additional terms in the equations that describe wave propagation, the results obtained for the domain wall are not reproduced when the thickness of the surface layer goes to zero, as one could naively expect.PACS numbers: 04.50.Kd, 04.30.-w, 04.60.Bc § INTRODUCTION One of the possible low-energy consequences of string theory is the addition of a Chern-Simons (CS) term coupled to an scalar field θ to the Einstein-Hilbert action of general relativity. Recently it has been suggested that this model provides the correct low-energy effective action to describe the thermal (gravitational) response of topological superconductors (TSCs) and superfluids, with the coupling field θ characterizing its topological nontriviality. CS gravity is an appealing topological extension of general relativity, where the Einstein-Hilbert action is supplemented with the parity-violating Pontryagin invariant coupled to an scalar field θ, which can be considered either as a dynamical field or as an external prescribed quantity of spacetime <cit.>. In much of the work on CS gravity, θ is assumed to be spatially homogeneous but time varying. This assumption can be motivated by arguments analogous to those that have been made suggesting that the quintessence field should be coupled to the electromagnetic CS term <cit.>.Recently, CS theories have acquired great interest in condensed-matter (CM) physics, since they describe the topological response theories of topological insulators (TIs) and TSCs. TIs display nontrivial topological order and are characterized by a fully insulating bulk and gapless edge or surface states, which are protected by time-reversal symmetry <cit.>. TSCs are analog of TIs in superconductors, which have a full superconducting gap, and gapless edge states propagating on the boundary. In the case of the three-dimensional (3D) time-reversal invariant TIs, the topological response theory is described by the additional term in the electromagnetic action: (α / 32 π ^2 ) ∫θϵ _μναβ F ^μν F ^αβ d ^4 x, in which θ is quantized to be 0 (topologically trivial state) or π mod 2 π (topologically nontrivial state). Such a topological term has the same form as the coupling of axions with gauge fields proposed in high-energy physics <cit.>. However, in a TI, θ is a constant determined by the bulk topology rather than a dynamical field. Many interesting properties for a domain wall of θ have been highlighted. For example, the image magnetic monopole effect <cit.>, the topological Kerr and Faraday rotations <cit.> arising from electromagnetic waves propagating through the θ boundary, and the quantized Hall effect <cit.>. A general technique to analyze the electromagnetic response of TIs has been elaborated in terms of Green's functions <cit.>.In the case of TSCs, since charge and spin are not conserved, the electromagnetic response is not adequate for defining the topological response theory. However the coupling to gravitational field, which describes the quantized thermal response, can be a good probe for topological nontriviality because energy is still conserved. It has been suggested that, in 3D time-reversal invariant TSCs and superfluids, one possible candidate for the topological response theory is the gravitational analog of the θ term for a gauge field <cit.>, i.e. a term proportional to the Pontryagin invariant of the Riemann curvature: 𝒮 _ = 1/1536 π ^2∫θ _CMϵ ^μναβ R ^σ _σταβ R ^τ _τσμν d ^4 x ,with the topological order parameter being θ _CM = 0 or π mod 2 π. It has been argued that the microscopic source of this gravitational response is the energy flow at the surfaces of a topological phase <cit.>. This is based upon the fact that temperature couples to the local energy density in the same way as an applied gravitational potential <cit.>.Motivated bythe steps followed in the study of the electromagnetic CS coupling, the main goal of this paper is to analyze the propagation of gravity waves in two models ofnondynamical CS gravity which have been shown to be relevant in CM physics. Since in this scheme θ characterizes the topological nontriviality of the media, we consider two nondynamical θ-models which interpolate between two constant values of θ: (i) a domain wall and (ii) a surface layer of θ. In these scenarios, the gravitational field equations acquire additional contributions arising from the coupling between the Pontryagin density and the topological order parameter θ. For a domain wall of θ, these new distributional contributions have support only on the interface, while the usual Einstein field equations hold in the bulk. In this way, we have to deal with the problem of determining the right junction conditions to be imposed on the bulk metrics, such that their union form a valid solution to the field equations. This problem has been extensively discussed in the literature <cit.> and arises in physical situations like the study of line sources, shock waves and thin shells of matter, for example. In this work the boundary conditions are not imposed by hand but they are dictated by the singular contributions to the field equations. Indeed, in the linear approximation, we find that the additional contribution to the gravitationalwave equation, induced by a delta distribution together with its derivative, corresponds to one of the self-adjoint extensions (SAE) of the D'Alembert operator. The general distributional analysis of this problem provides the junction conditions on the metric perturbation and its derivative at the domain wall <cit.>. We find that even though the metric and its first derivative become discontinuous at the interface, the area matching condition introduced by Barrett <cit.> is satisfied. This amounts to replace the standard junction condition requiring the continuity of the metric at a given null hypersurface by the weaker condition that the area of any two-surface gives a unique result when measured from each side of the hypersurface. On the other hand, the case for a surface layer of θ is simpler since the additional contribution appearing in the field equation depends only on delta distributions supported at the interfaces. This case has a close resemblance with the propagation of electromagnetic waves in CS electrodynamics. We show that the results obtained for a surface layer do not reduce to the ones obtained with a domain wall model of θ . This is because, even when the surface layer become a domain wall when its thickness goes to zero, the limit in the field equations does not reproduces those of the domain wall.This paper is organized as follows. We begin in Sec. <ref> by reviewing the basics of nondynamical CS gravity following closely Ref. Jackiw. We also discuss the linear theory and we demonstrate its consistency. In Sec. <ref>, we introduce the nondynamical models for the coupling field θ, we work with. Section <ref> is devoted to the analysis of gravitational waves propagating across our θ-models. We derive the boundary conditions by interpreting the field equations in a distributional sense. See also Sec. <ref> and the <ref>. In Sec. <ref> we provide some phenomenological estimations of the parameters in our model. We include the comparison with two condensed matter representatives which are particular cases of our general discussion of wave propagation, once the scales are properly adjusted <cit.>. Finally, our summary and conclusions are given in Sec. <ref>. Our conventions are those of Ref. Schutz. We use the signature ( - , + , + , + ), the Riemann tensor is R ^μ _μναβ = ∂ _αΓ ^μ _νβ + Γ ^μ _σαΓ ^σ _νβ - (α↔β ), the Ricci tensor is R _μν = R ^α _αμαν and R = R ^μ _μμ is the Ricci scalar.§ NONDYNAMICAL CS GRAVITY§.§ Basics Let us consider the spacetime ( ℳ , g ), with ℳ a four-dimensional manifold and g a metric on ℳ. The action of the nondynamical CS modified gravity reads𝒮 = 𝒮 _ + 𝒮 _ + 𝒮 _ ,where𝒮 _ = 1/2 κ∫ _𝒱 d ^4 x √(g) R, 𝒮 _ = 1/2 κ∫ _𝒱 d ^4 x θ/4𝒫 .Here, κ = 8 π G, 𝒱 is a four-volume in ℳ with boundary ∂𝒱, g is the determinant of the spacetime metric g _μν, R is the Ricci scalar, θ is the CS scalar field, and 𝒮 _ (Ψ) is the matter action which does not depend on θ. The field Ψ collectively denotes the nongravitational matter fields. The quantity 𝒫 in the CS action is the Pontryagin invariant defined as𝒫 = ^∗ R _στ^στμν R ^τ _τσμν,where ^∗ R^στμν _στ = 1/2ϵ ^μναβ R ^σ _σταβ is the dual of the Riemann tensor and ϵ ^μναβ is the Levi-Cività symbol, with the convention ϵ ^0123 = +1.The Pontryagin density can be expressed as a divergence ∇ _μ K ^μ = 𝒫, whereK ^μ = ϵ ^μναβΓ _νσ^λ( ∂ _αΓ _βλ ^σ +2/3Γ _αξ ^σΓ _βλ ^ξ) ,is thethe CS topological current and Γ is the Christoffel connection. Accordingly, if θ is globally constant at all spacetime points, the CS action is a topological term not contributing to the field equations since its variation is a boundary term that can be dropped under the usual boundary conditions.The field equations of the nondynamical CS gravity are obtained by varying the action (<ref>) with respect to the metric. One findsG ^μν + C ^μν = κ T ^μν ,where G ^μν = R ^μν - 1/2 g ^μν R is the covariantly conserved Einstein tensor and T ^μν is the matter stress-energy tensor. The expression for the symmetric traceless second rank tensor C ^μν, a four-dimensional (4D) generalization of the 3D Cotton-York tensor, isC ^μν = - 1/2 √(g)[ υ _λϵ ^λμαβ∇ _α R ^ν _β + υ _λσ^∗ R ^σμλν + ( μ↔ν) ],where υ _λ = ∇ _λθ is called the embedding coordinate and υ _λσ = ∇ _συ _λ is its covariant derivative.By construction, the Einstein tensor G _μν is divergenceless. If θ is treated as an external prescribed quantity, then general covariance, which requires ∇ _μ T ^μν =0, leads to the constraint ∇ _μ C ^μν = 0. However, C ^μν is not covariantly conserved. Rather, we have∇ _μ C ^μν = ∂ ^νθ/2 √(g)𝒫 ,and thus the consistency requirement on Eq. (<ref>) demands 𝒫 = 0. Alternatively, if we treat θ as a dynamical field, then the variation of the action with respect to θ will lead to the same constraint on 𝒫 <cit.>. In this paper, we are interested in the propagation of gravitational waves, where T _μν = 0, and the constraint ∇ _μ C ^μν = 0 is satisfied regardless we view θ as a dynamical field or a fixed, externally specified quantity, because the Pontryagin density is identically zero for GW spacetimes <cit.>. §.§ Linear theoryWe work in the weak field approximation of the nondynamical CS gravity. In the linear approximation, g _μν= η _μν + ε h _μν, the source free field equation to first order in ε≪ 1 is G ^μν _ + C ^μν _ = 0, whereG ^μν _ = 1/2[ -h ^μν + ∂ _α∂ ^μ h ^να + ∂ _α∂ ^ν h ^μα - ∂ ^μ∂ ^ν h - η ^μν( ∂ ^α∂ ^β h _αβ -h ) ] , C ^μν _ = 1/4[ υ _λϵ ^λμαβ∂ _α(h ^ν _νβ - ∂ ^κ∂ ^ν h _βκ) + υ _σλϵ ^σμαβ∂ _α( ∂ ^λ h ^ν _νβ - ∂ ^ν h ^λ _λβ)+ ( μ↔ν ) ] ,are the linearized Einstein and Cotton-York tensors, respectively. Here, = ∂ _μ∂ ^μ is the flat-space d'Alembertian, h = h ^μ _μμ and indices are moved with η ^μν. The consistency condition requiring the vanishing divergence of the field equation can be directly verified. On the one hand, the Einstein tensor is naturally divergenceless, i.e. ∂ _μ G ^μν _ = 0. On the other hand, using the antisymmetry of the Levi-Cività symbol, the divergence of the Cotton-York tensor becomes∂ _μ C ^μν _ = 1/4[ υ _λμϵ ^λναβ∂ _α(h ^μ _μβ - ∂ ^κ∂ ^μ h _βκ)+ υ _σλϵ ^σναβ∂ _α∂ _μ( ∂ ^λ h ^μ _μβ - ∂ ^μ h ^λ _λβ) + υ _σλμϵ ^σναβ∂ _α( ∂ ^λ h ^μ _μβ - ∂ ^μ h ^λ _λβ) ] ,with υ _σλμ = ∂ _μυ _σλ = ∂ _σ∂ _λ∂ _μθ. The sum of the first two terms in the right-hand sideis zero since = ∂ _μ∂ ^μ, and the last term identically vanishes by symmetry considerations.Next, we simplify the field equations by using the gauge freedom of the linearized approximation. We work with the trace-reversed metric perturbation, h _μν = h _μν - 1/2η _μν h, and impose the Lorentz gauge condition, ∂ ^μh _μν = 0 to obtain the linearized tensorsG ^μν _ = - 1/2h ^μν, C ^μν _ = 1/4[ υ _λϵ ^λμαβ∂ _αh ^ν _νβ + υ _σλϵ ^σμαβ∂ _α( ∂ ^λh ^ν _νβ - ∂ ^νh ^λ _λβ) + ( μ↔ν ) ] .This choice reduces the number of degrees of freedom from 10 to 6. A further gauge transformation can be made. In the next section, we will show that the TT gauge can be imposed to analyze the propagation of gravitational waves in our models, reducing the number of degrees of freedom from 6 to 2, as it should be.§ NONDYNAMICAL Θ-MODELSOne of the main ingredients of nondynamical CS gravity is the CS coupling scalar θ = θ (x ^μ), which is a prescribed function of spacetime. In the nondynamical framework, the functional form of the CS scalar field is usually taken to be dependent only of time, θ (t) = τ t, where τ is a constant. The assumption being that θ is a quintessence field or some other field that somehow echoes the arrow of time associated with the cosmic expansion. In this paper, we are concerned with nondynamical θ-models inspired by CM systems. We begin by recalling the emergence of the CS term in the theory of TIs and TSCs. The magnetoelectric polarization θ characterizing the momentum-space topology of band insulators gives rise to CS electrodynamics, in whichθ couples with the electromagnetic field via α/16 π ^2θ ^∗ F ^μν F _μν. A domain wall of θ leads to the quantized Hall effect and the image monopole effect. In the case of TSCs, the scalar field θ, associated with the topological nontriviality of these states, gives rise to CS gravity, in which θ couples to the gravitational field via 1/768 π ^2θ𝒫, where 𝒫 is the gravitational Pontryagin density (<ref>). The microscopic source of this gravitational response is the energy flow at surfaces of a topological phase. According to the Luttinger derivation of the thermal transport coefficients, in a near-equilibrium system, the effect of a thermal gradient is equivalent to that obtained from a gravitational potential <cit.>. Consequently, since the surface Majorana mode that exists in this phase does not carry charge, but it does carry energy, it leads to a thermal Hall effect for a domain wall of θ. In analogy with the electromagnetic case, the propagation of gravitational waves in the models described above is a topic which deserves to be studied on its own. In this work, we shall consider that the CS scalar field depends only of one spatial coordinate, which we choose to be z, i.e. θ = θ (z). Therefore, the embedding coordinate and its derivative become υ _μ = θ ^' n _μ and υ _μν = θ ^'' n _μ n _ν, respectively. Here, n _μ=(0,0,0,1) is the unit vector pointing in the direction of the coordinate z and the primes denote derivatives with respect to z.The linearized field equations can be further simplified imposing an additional gauge. Hereafter, we work in the transverse-traceless gauge, defined by the (previously imposed) Lorentz gauge together with the conditions h ^μ _μμ = 0 and U ^μh _μν = 0, withU^μ being any constant timelike four vector. With the appropriate choice of the four vector U ^μ we can make n ^μh_μν = 0. For example, for monochromatic plane waves with four momentum k ^μ = ωk̂ ^μ = ω (1, k̂), the choice U ^μ = k̂ ^μ - (k̂· n) n ^μ implies n ^μh_μν = 0, when k· n ≠ 0. Then, the field equations reduce toh ^μν = 1/2ϵ ^3 μαβ( θ ^' + θ ^''∂ _z) ∂ _αh ^ν _νβ + ( μ↔ν ) .Since the contributions of the θ-term to the field equations can be seen as a source of energy-momentum, which nevertheless arises from the fields themselves, it is important to verify that there are no obstructions to attain the TT gauge. Let us recall that in the standard case of gravitational wave propagation in the presence of matter characterized by an energy-momentum tensor T^μν, it is always possible to implement the Lorentz gauge (since ∂_μ T^μν=0), but not the subsequent TT gauge conditions. This is because, in general, we have T^μ_μμ≠ 0 and n_μ T^μν≠ 0, which will make the equation of motion inconsistent after imposing the TT gauge <cit.>. In our case, this does not happen because the effective source of energy-momentum is given by h ^μν. In fact, Eq. (<ref>) is consistent with the TT gauge conditions h^μ_μμ=0,h^3ν=0 ,∂_μh^μν=0. One can verify that: (i) The traceless condition arises from the symmetry properties in the indices μ=ν, β. (ii) Equation (<ref>) vanishes when projected along n _μ. The first term in the right-hand side is zero due to ϵ^3 3 αβ=0 and the second term contains h^3_3β=0. (iii) When applying ∂_μ upon Eq. (<ref>), the Lorentz condition is satisfied because the first term in the right-hand side is zero, since ϵ^3μαβ restricts both μ and α to live in the subspace orthogonal to n_μ. In this way, ∂_μ commutes with the operator in square brackets (which depends only on z)yielding∂_μ∂_αh^ν_νβ, which is zero due tothe antisymmetry of the Levi-Cività symbol. For analogous reason, the second termyield the contribution ∂_μh^μ_μβ=0.Next we define the particular nondynamical θ-models we deal with: (i) a domain wall and (ii) a surface layer of θ.§.§ Domain wall model We first consider a domain wall between two different constant values of θ, lying along the xy-plane at z=0, as shown in Fig. <ref>. More specifically, we assume that the region z < 0 is characterized by a constant θ=θ _1, while the region z > 0 has a different constant value θ=θ _2. The invariance of the CS term in the action under shifts of θby any constant, θ = θ ^' + C, can be used to set θ _1 to zero and θ _2 to θ̃≡θ _2 - θ _1. In this way, the derivatives of the CS scalar field appearing in Eq. (<ref>) become θ ^' = θ̃δ (z) and θ ^'' = θ̃δ ^' (z), and the field equation (<ref>) readsh ^μν = θ̃/2ϵ ^3 μαβ[ δ (z)+ δ ^' (z) ∂ _z] ∂ _αh ^ν _νβ + ( μ↔ν ) .The presence in the above equation of the Dirac delta distribution and its derivative implies that the right-hand side is supported only at the interface z=0 and vanishes in the bulk. Therefore, the metric perturbation satisfies the usual wave equation in the bulk regions, z<0 and z>0, and the domain wall acts as a localized source depending on the gravitational field itself. The δ ^' distribution appearing in Eq. (<ref>) is a consequence of the third derivatives of the metric tensor arising from the CS contribution, and it implies that both, (or at lest some components of) the metric perturbation and its first derivative become discontinuous at the interface. Here, the field equations dictate the junction conditions for the metric at the interface. §.§ Surface layer model The second model we shall consider consists of a surface layer where θ changes linearly across a thickness d. More precisely, we assume that θ is constant in the region z < 0, then changes linearly in z within the surface layer (0 < z < d) and finally becomes constantagain outside the layer (z>d), as shown in Fig. <ref>.Again, we can use the symmetry of the CS term under shifts of θ(z) by any constant to set θ_1 = 0 for z<0 and θ̃≡θ_2-θ_1 for z>d. In this model, the derivatives of the CS scalar field become θ ^' = ( θ̃ / d ) Π (z) and θ ^'' = (θ̃ / d ) [ δ (z) - δ (z - d) ], where Π (z) = H(z) H (z-d) is the box function within the layer. The field equations now take the formh ^μν = θ̃/2 dϵ ^3 μαβ{Π (z)+ [ δ (z) - δ (z - d) ] ∂ _z}∂ _αh ^ν _νβ+ ( μ↔ν ) .We observe that the first term in the right-hand side is nonzero within the layer, but vanishes outside it. The second term has support onlyat the surfaces of the layer, that is to say at z=0 and z=d. Therefore, the above equation implies that the metric perturbation satisfies the usual wave equation h ^μν =0 outside the layer, and h ^μν = (θ̃ / 2 d ) ϵ ^3 μαβΠ (z) ∂ _αh ^ν _νβ in the interior region. In other words, the solution inside the layer can also be taken as satisfying the usual wave equation. § PROPAGATION OF GRAVITATIONAL WAVES In this section, we examine the effects of our nondynamical θ-models in the propagation of gravitational waves. To begin with let us consider an incident monochromatic plane wave impinging from the left towards the θ-interface at z=0, at an angle α with respect to the normal n ^μ = (0,0,0,1), with wave four-vector k ^μ = ωk̂ ^μ = ω( 1 , sinα , 0 , cosα). The constant timelike vector which fixes the TT gauge is U ^μ = ( 1, sinα , 0 , 0), with 0 ≤α≤π /2. In this gauge, the metric perturbation h _μν has the formh _μν( t , x , z ) = ẽ _μν ^(+) h _+( t , x , z ) + ẽ _μν ^(×) h _×(t , x ,z ) .Here h _+ and h _× are the amplitudes of the two independent components with linear polarizations, andẽ _μν ^(+) = ( [ sin ^2α- sinα 0 0;- sinα 1 0 0; 0 0 - cos ^2α 0; 0 0 0 0 ]), ẽ_μν ^(×) = ( [ 0 0 -sinα 0; 0 0 1 0; -sinα 1 0 0; 0 0 0 0 ]) ,are the corresponding polarization tensors. The modified field equations (<ref>) and (<ref>) can be written in the generic formh _μν = - i θ̃𝒪̂[ ẽ _μν ^(+) h _× - cos ^2α ẽ _μν ^(×) h _+] ,where the differential operator 𝒪̂ depends on the model under consideration. For the domain wall model, it is𝒪̂ _ dw = i [ δ (z)+ δ ^' (z) ∂ _z] ∂ _t ,while for the surface layer model it becomes𝒪̂ _ sl = i/d{Π (z)+ [ δ (z) - δ (z - d) ] ∂ _z}∂ _t .Equation (<ref>) couples the polarization modes + and ×, which are detached by introducing the right- and left-handed circularly polarized modesh _R/L = cosαh _+± i h _× ,that satisfyh _R/L = ∓θ̃cosα 𝒪̂ h _R/L .The plus and minus signs correspond to the right- and left-handed modes, respectively. Now we have to solve Eq. (<ref>) for the different models.As discussed before, the differential operator in the right hand side of Eq. (<ref>) is singular, in the sense that it is supported at z=0 for the domain wall model, and at the surfaces (at z=0 and z=d) of the surface layer model, but vanishes in the bulk. This situation resembles the case of point interactions in nonrelativistic quantum mechanics, which are usually described by the so-called Fermi pseudopotentials. Point interactions are modeled by an idealized localized singular interaction with zero range occurring at one point on the space. One of these point interactions is the familiar δ( z ) pseudopotential, which is well defined and has well-known solutions. This kind of interaction can be described by a free system on the line without the singular point, i.e. in the region ℝ∖{0}, in which case the interaction is encoded in the boundary conditions rather than in a formal Hamiltonian operator. For the special case of the δ-interaction, the Schrödinger equation can be integrated twice for obtaining the boundary condition at z=0, which demand: (1) continuity of the wave function and (2) discontinuity of its first derivative. However, attempts to consider more general interactions have been known to be plagued with difficulties associated with the definition of the interaction term. For example, there is some controversy on the meaning of the δ ^'( z )-potential, as different regularization produce different reflection and transmission coefficients <cit.>. The origin of such difficulties lies in the fact that point interactions are represented by pseudopotentials which are not ordinary functions but distributions. A mathematically rigorous approach to deal with generalized point interactions in nonrelativistic quantum mechanics is due to Kurasov and coworkers in Refs. Kurasov-Albeverio. In this approach, both the interaction and the discontinuous wave function are considered as distributions, precluding the naive definition of the interaction term as the usual product between the wave function and the potential energy, since such a product is generally ill-defined in distribution theory. Also, Kurasov deals with the problem of point interactions from the perspective that they are defined by SAE of the operator -d^2/dz^2, which is a well studied problem in the literature. This point of view has the advantage that the definition for the singular potentials depends only on the matching conditions at the singularity and not on the choice of a particular model of the delta function and its derivatives.In the problem at hand, we have to deal with a second-order differential equation for the metric, subject to the singular differential operator 𝒪̂ defined in Eqs. (<ref>) and (<ref>). The goal of the rest of this section is to analyze the one-dimensional (1D) differential equation (<ref>) following Kurasov's prescription, which is rephrased in terms of pseudopotentials and discontinuous functions in the <ref>. In this way, following Refs. Kurasov-Albeverio, we are able to determine the boundary conditions for Eq. (<ref>) by standard manipulations of second-order linear differential equations.To begin with we recall the expressions (<ref>), arising from the approach of Ref. Kurasov, for the product of a function ψ (x) which is discontinuous at x=0 times δ (x) and δ ^'(x):ψ( x ) δ( x )= ψ( 0 )δ( x ) , ψ( x ) δ ^'( x )= ψ( 0 )δ ^'( x ) - ψ ^'( 0 )δ( x ) ,withχ( 0 ) = 1/2[ χ( 0 ^+) + χ( 0 ^-) ] .Here χ denotes either ψ or ψ ^' and χ( 0 ^+) and χ( 0 ^-) are the limits of χ( x ) when x approaches 0 from the positive and negative sides, respectively. In the next sections, we will tackle each of our models independently. First, we start with the modified field equations from the distributional point of view. Next, we establish the corresponding boundary conditions for the metric perturbation and its derivative, and finally we compute the reflection and transmission coefficients by using an appropriate ansatz for the solutions. §.§ Gravitational waves propagating through a domain wall We begin with the distributional differential equation (<ref>) describing the propagation through a domain wall. This is obtained by introducing the relations (<ref>) into the modified wave equation (<ref>) with the point interaction (<ref>). We obtainh _R/L( z )= ∓ i θ̃cosα[ δ( z ) ∂ _t ( - ∂ _z ^2) h _R/L( 0 ) + δ ^'( z ) ∂ _z∂ _t h _R/L( 0 )] ,where the overline denotes average at the domain wall, i.e. at z=0. Now we are in position to obtain the boundary conditions.§.§.§ Boundary conditions The first boundary condition for the metric perturbation at the domain wall can be obtained by integrating (<ref>) over the interval [ - ε , + ε] and taking the limit ε→ 0^+. The result ish _R/L ^'( 0 ^+) - h _R/L ^'( 0 ^-) = ∓ i θ̃cosα ∂ _t ( - ∂ _z ^2) h _R/L( 0 ) ,where the relations ∫ _-ε ^+ εδ( x ) dx = 1 and ∫ _- ε ^+ εδ ^'( x ) dx = 0 have been used. Meanwhile, integrating (<ref>) from - L (with L positive) to z yieldsh _R/L ^'( z ) - h _R/L ^'( - L ) - ∂ _t ^2∫ _- L ^z h _R/L( z ^') dz ^' = ∓ i θ̃cosα×[ H ( z ) ∂ _t ( - ∂ _z ^2) h _R/L( 0 ) + δ( z ) ∂ _z∂ _t h _R/L( 0 )] ,where H ( z ) is the Heaviside function. Here, we have used the relations ∫ _- L ^zδ( y ) dy = H ( z ) and ∫ _- L ^zδ ^'( y ) dy =δ( z ). Integrating further (<ref>) with respect to z from - ε to + ε, and taking the limit ε→ 0 ^+, we find thath _R/L( 0 ^+) - h _R/L( 0 ^-) = ∓ i θ̃cosα ∂ _z∂ _t h _R/L( 0 ) ,which is the second boundary condition for the metric perturbation at the domain wall.§.§.§ The transmission and reflection coefficients It is well known that the linearized Einstein field equations in vacuum (T ^μν = 0) far outside the source of the field has a (complex) solution of the formh ^μν = A ^μν e^i k _λ x ^λ,describing plane linearized gravitational waves, where A ^μν are the (complex) constant amplitudes of the wave and k ^μ is the null wave four-vector given at the beginning of Sec. <ref>. Here, we shall use this simple solution to illustrate the effect of a domain wall of θ in the propagation of linearized gravitational waves. Assuming that the source lies in the region z<0, the metric perturbation impinging on the surface z=0 from the left can thus be written ash _R/L (x,z,t) = {[ e ^i k _μ x ^μ + ℛ _R/L e ^i k̃ _μ x ^μ; 𝒯 _R/L e ^i k _μ x ^μ ]. [ ;z < 0; ;z > 0 ]The four-vector k̃ ^μ = ω (1 , sinα , 0 , - cosα) is the wave vector of the reflected wave, and ℛ_R/L and 𝒯_R/L are the corresponding reflected and transmitted amplitudes. By using the simple results ∂ _th_R/L=- i ω h _R/L and ∂ _x h _R/L = i ωsinα h _R/L, the matching conditions (<ref>) and (<ref>) can be written as follows:[ [ h _R/L ( 0 ^+ ); h _R/L ^' ( 0 ^+) ]] = 1/Δ[ [1 + ( ξ / 2) ^2∓ ( ξ / ωcosα ); ∓ ( ξωcosα ) 1 + ( ξ / 2 ) ^2 ]] [ [h _R/L ( 0 ^- ); h _R/L ^' ( 0 ^- ) ]] ,where Δ = 1 - (ξ /2) ^2 and ξ≡θ̃ω ^2cos ^2α≠ 2. As shown in the <ref>, this result corresponds to the choiceX _1 = ∓ξωcosα ,X _4 = ±ξ / ωcosα ,X _2 = X _3=0 ,which shows that the operator 𝒪̂ _ dw, defined by Eq. (<ref>), corresponds to one of the possible SAE of the operator - d ^2 / dz ^2, according to the results in Ref. Kurasov. In other words, we have chosen the boundary conditions (<ref>) for Eq. (<ref>) by demanding the operator- d ^2/dz ^2∓θ̃ωcosαd/dzδ( z) d/dz∓θ̃ω ^3cos ^3αδ( z)to be self-adjoint. We emphasize that our notation is d/dzδ( z ) d/dz = δ( z ) d ^2/dz ^2 + δ ^'( z ) d/dz instead of d/dzδ( z ) d/dz = δ ^'( z ) d/dz as considered in Refs. Langmann and Gadella.Imposing the boundary conditions on h _R/L, we obtain𝒯 _R/L( ξ) = 4 - ξ ^2/4 + ξ ^2 , ℛ _R/L( ξ) = ± i 4 ξ/4 + ξ ^2 ,where ξ = θ̃ω ^2cos ^2α≠ 2. It can be easily verified that |𝒯_R/L| ^2 + |ℛ_R/L| ^2 = 1, which is consistent with energy conservation.Going back to our analogy with quantum mechanics, let us recall that in one-dimensional scattering the fraction of particles that is transmitted (for an arbitrary given potential) in general vanishes at threshold, i.e. as the kinetic energy of the incident particles approaches zero. Intuitively, this occurs because the potential is acutely large as compared with the (small) energy of the incident particles. However, it has been shown that for a potential consisting of two Dirac delta functions of arbitrary strength, a finite portion of the incident particles is transmitted at threshold for certain choices of the set of parameters defining the potential <cit.>. In the problem at hand, the only parameter associated with the point interaction in (<ref>) is θ̃, and clearly our result (<ref>) suggests a threshold anomaly in the sense of Ref. Senn for an arbitrary value of θ̃. In the absence of the domain wall, our results lead to total transmission and consequently to no reflection, as expected. However this result can also be obtained in the limit ω→∞, i.e. when the energy of the incident wave is very large as compared to the potential strength θ̃. As the expressions for the reflection and transmission coefficients indicate, the effect of the domain wall becomes important when ξ=θ̃ω ^2cos ^2α≃ 2, by substantially increasing the reflecting property of the domain wall. Nevertheless, as shown in Eq. (<ref>), the value ξ=2 is strictly forbidden, in such a way that we never have a perfect mirrorfor gravitational waves.§.§.§ The emergence of the area matching condition In this section, we show that the discontinuous metric at z=0 which we found in the previous section,satisfies the area matching condition on the spacelike two-dimensional surfaces arising from the intersection ofthe null hypersurface N describing the incidentwave and the hypersurface Σ describing the domain wall. We illustrate the process for the case of normal incidence (α = 0) in Fig. <ref>, but we discuss the general case in the following. Let us recall that, in a given coordinate system (η ^2 , η ^3),the cross-sectional area 𝒜 of any spacelike two-surfaceiscalculated as 𝒜 = ∫√(σ) d η ^2 d η ^3.In our case, σ is the determinant of the corresponding induced two-metric σ _AB in N ∩Σ, and A , B = 2 , 3. In this way, we need to prove that σ is continuous there even if the metric σ _AB is discontinuous. As we will see in the following, the detailed expression for the discontinuous four-metric (<ref>) is irrelevant for our purposes. Let us recall that the null hypersurface N,determinedby the impinging gravitational wave,is defined by theconstant phase Φ=k̂_μ x^μ,yielding Φ = - t + x sinα + z cosα = C .The normal vector to N is N _μ = ∂Φ / ∂ x ^μ = ( - 1, sinα , 0 , cosα). Following Ref. Poisson, we introduce on the hypersurface N a coordinate system y ^a = ( λ , η ^2 , η ^3), a=1,2,3, in such a way that its parametrization x ^μ = (t , x , y , z) = x ^μ (λ ,η ^2 , η ^3) satisfies∂ x ^μ/∂λ = k ^μ = e _1 ^μ , ∂ x ^μ/∂η ^A = e _A^μ ,A=2,3,with e _A^μ spanning any two-dimensional subspace orthogonal to k ^μ. The following relations t = λ - C,x = λsinα + η ^2cosα , y = η ^3 ,z = λcosα - η ^2sinα,are a parametrization of Φ and lead to ∂ x ^μ/∂λ = ( 1 , sinα , 0 , cosα ) = k ^μ , e _2 ^μ = ( 0 , cosα ,0 , - sinα ),e _3 ^μ = (0,0,1,0).For the case of normal incidence, we have k ^μ = (1,0,0,1) together with e ^μ _2 = (0,1,0,0) and e ^μ _3 = (0,0,1,0), the later being the two basis vectors for the xy plane located atthe domain wall for constant t. The next step is to obtain the induced two-metric σ _AB = g _μν e _A ^μ e _B ^ν, in N. Recalling the general expression g _μν = η _μν + εẽ _μν^(+) h _+ + εẽ _μν ^(×) h _× ,where the polarization tensors are given in Eq. (<ref>), a direct calculation yields [ σ _AB] = [[ 1 + ε h _+cos ^2αε h _×cosα;ε h _×cosα 1 - ε h _+cos ^2α ]] .Since we are considering only the linear approximation in our calculation, we must ignore terms proportional to ε ^2. This leads to σ =[ σ _AB] = 1 + 𝒪 (ε ^2).That is to say, the determinant of the induced two-metric is continuous in N, in particular in N ∩Σ, thus erasing any discontinuous contribution to the calculation of areas of spacelike two-surfaces in N.§.§ Gravitational waves propagating through a surface layerHere, we proceed in a similar fashion to that of the previous section. Introducing the relations (<ref>) into the modified wave equation (<ref>) with the point interaction (<ref>), we obtain h _R/L( z )= ∓ i θ̃cosα/d[ Π( z ) ∂ _t h _R/L (z) + δ( z ) ∂ _z∂ _t h _R/L( 0 ) - δ( z - d ) ∂ _z∂ _t h _R/L( d )] .Next we proceed to the calculation of the boundary conditions.§.§.§ Boundary conditions We first compute the boundary condition for the derivative of the metric perturbation at z=0 by integrating (<ref>) over the interval [- ε , + ε] and taking the limit ε→ 0 ^+, i.e.h ^' _R/L (0 ^+) - h ^' _R/L (0 ^-) = ∓ i θ̃cosα/d[ lim _ε→ 0 ^+∫ _- ε ^+ ε H ( z ) ∂ _t∂ _z ^2 h _R/L dz + ∂ _z∂ _t h _R/L( 0 )] .The remaining integral can be computed by parts. The result ish ^' _R/L (0 ^+) - h ^' _R/L (0 ^-) = ∓ i θ̃cosα/d∂ _z∂ _t h _R/L (0 ^+) ,where we used that H (- 0 ^+) = 0. In order to obtain the boundary condition for the metric perturbation, we start by integrating Eq. (<ref>) from -L (with L>0) to z:h ^' _R/L (z) - h ^' _R/L (- L) - ∂ _t ^2∫ _-L ^z h _R/L (z ^') dz ^' = ∓i θ̃cosα/d× [ ∫ _-L ^z H(z ^')^'∂ _t h _R/L (z ^') dz ^' + H(z) ∂ _z∂ _t h _R/L( 0 )] .The integral in the right-hand side can be computed by parts. The result ish ^' _R/L (z) - h ^' _R/L (- L) - ∂ _t ^2∫ _-L ^z h _R/L (z ^') dz ^' = ∓i θ̃cosα/d H (z) ∂ _z∂ _t h _R/L (z) .Next we integrate the above equation with respect to z from - ε to + ε. Taking the limit ε→ 0 ^+, we obtainh _R/L (0 ^+) - h _R/L (0 ^-)= ∓i θ̃cosα/d[ ∂ _t h _R/L (0 ^+) - ∂ _t h _R/L (0)] .Since the time-dependence of all wave amplitudes is e ^- i ω t, the only solution to Eq. (<ref>) ish _R/L (0 ^+) = h _R/L (0 ^-) .In other words, the metric is continuous at z = 0.The boundary conditions atz = d are obtained from a calculation completely analogous to that for z=0, which we do not reproduce here. The results areh ^' _R/L (d ^+) - h ^' _R/L (d ^-)= ±i θ̃cosα/d∂ _t h ^' _R/L (d ^-) , h _R/L (d ^+)= h _R/L (d ^-) .§.§.§ The reflection and transmission coefficientsAs discussed in Sec. <ref>, the metric perturbation satisfies the usual wave equation both within and outside the surface layer. Assuming that the source lies in the region z < 0, yielding agravitational wave impinging on the surface z = 0 from the left, the metric perturbation can be taken ash _R/L = {[e ^i k _μ x ^μ + ℛ _R/L e ^i k̃ _μ x ^μ; A _R/L e ^i k _μ x ^μ + B _R/L e ^i k̃ _μ x ^μ;𝒯 _R/L e ^i k _μ x ^μ ]. [ ;z < 0; ;0<z<d; ;z > d ]Imposing the previously found boundary conditions on the metric perturbation, we obtain the following reflection and transmission amplitudesℛ _R/L = ±γ[ 2 ±γ] ( 1 - e^ i ϕ)/γ^2 e ^ iϕ - [ 2 ±γ] ^2 , 𝒯 _R/L = -4 [ 1 ±γ] /γ^2 e ^ i ϕ - [ 2 ±γ] ^2,whereγ= ωθ̃cosα / d, ϕ=2ω d cosα.Let us recall that in the units we are working θ̃ has dimensions of length square. Note that the above results naturally depend on the thickness d of the surface layer. One easily verifies that |ℛ _R/L| ^2 + |𝒯 _R/L| ^2 = 1, which is consistent with energy conservation.It is interesting to compute the limit when d → 0. After performing a series expansion in powers of d and taking the limit, we findℛ _R/L = - i θ̃ω ^2cos ^2α/i θ̃ω ^2cos ^2α∓ 2 , 𝒯 _R/L = ∓ 2/ i θ̃ω ^2cos ^2α∓ 2.From the above, the transmission and reflection coefficients in this limiting case are|ℛ _R/L| ^2 = ξ ^2/ 4 + ξ ^2 , |𝒯 _R/L| ^2 = 4/4 + ξ ^2 ,where ξ = θ̃ω ^2cos ^2α. Let us observe that the limiting results in Eqs. (<ref>) and (<ref>) do not coincide with the corresponding values that we obtained in the case of a domain wall of θ. Nevertheless, they have a similar form to those obtained in CS electrodynamics <cit.> with the exception that, contrary to that case, now these coefficients depend on the frequency. This is expected since the gravitational equations of motion have additional derivatives with respect to those in electrodynamics.§ SOME PHENOMENOLOGICAL ESTIMATIONSLet us now discuss some estimations of the parameters in our model, together with their impact upon wave propagation, codified in the reflection and transmission coefficients, by restricting our general approach to some cases already considered in the literature. In this section, we go back to standard units in order to make a smooth transition to the condensed matter case.Let us first recall the general conditions for the applicability of the linearized approximation. To being with, we are dealing with the weak field limit of Einstein equations | h _μν|≪ 1, which means that the Riemann tensor can be estimated to be | R |∼ω ^2 /c ^2≪ M ^2 c ^2 / ħ ^2, where ω is the frequency of the wave which propagates with the maximum attainable velocity c in the medium and M is the mass scale under which the effective theory is valid. In this approach, the CS interaction is considered as an effective theory arising from the integration of fermions in a more fundamental model valid for energies larger than the effective energy M c ^2. As a matter of fact, both Loop Quantum Gravity and String Theory unavoidable yield the CS modified gravity included in their low energy limit <cit.>. Moreover, the CS contribution is expected to be a small perturbation of the Einstein-Hilbert term, which can be codified in the ratio [(θ /4) 1/2^∗ RR] / [R] ∼ω ^2θ / 8 c ^2.In order to have a unified description of the systems to be considered we start from the action (<ref>). The Einstein-Hilbert contribution is 𝒮 _ = c ^3/16 π G∫ d ^4 x √(g) R = c ^3M ^2/16 πħ∫ dt d^3x√(g)R ,where G = ħ c / M^2 is the effective gravitational constant. The CS action in Eq. (<ref>) is parametrized as𝒮 _ =c^3M^2/64πħθ∫ dt d^3 x𝒫where the parameter θ with dimensions [ θ] =^2, is to be identified in each case. §.§ The SECM case We first consider the gravitational case of Ref. Smith (to be called SECM for the initial of the authors), where the effect of the CS theory on bodies orbiting the earth is considered, and apply it to our surface layer model. Here, the parameters take their standard values where M = M _Planck, G is the standard Newton constant and c is the speed of light. Thus, in this section, it is simpler to work in units where ħ =c=1. The identification of the action (<ref>) with the corresponding one in Ref. Smith yields θ = 16 π/3l θ _SECM/M _Planck ^2 .The relevant parameter in Ref. Smith is what they call the CS mass, defined bym _CS = - 3M _Planck ^2/8 π1/lθ̇ _SECM .This yields θ̇ = - 2 / m _CS, where θ̇ = d θ /dt. Besides, the lower bound m_0 = 2 × 10 ^-13 eV, such that m _0 < m _CS, has been established in Ref. Smith. This bound has been improved to m_0 = 4.7 × 10 ^-10 eV in Ref. Qiang. Now we make contact with our surface layer model where the parameter determining the size of the CS corrections to the reflection and transmission coefficients is γ = ω ( θ̃ / d ) cosα, where we can identify θ̃ / d as d θ / dz within the layer. Since Eq. (4) in Ref. Smith tells us that θ _SECM propagates at the speed of light, we can estimate d θ / d z = θ̇ = - 2 / m _CS in such a way that γ = - 2 ω/m _CS cosα .A Taylor expansion in powers of γ produces, for example,| T_R| = 1 - 1/2 (1-cosϕ ) γ ^2 + O (γ ^6) .Taking γ∼ 0.01 generates corrections of the order of 10 ^-4 in | T_R| (for the largest value ϕ = π) which yields ω _min = γ m _0 / 2 = 2. 4 × 10 ^-12= 585.5 Hz. Given that ω _min / ω _Planck∼ 10 ^-40, we can safely increase ω to get a larger value of γ, still remaining within the weak field approximation. Also we notice that ω _min is comparable with the frequencies in the interval 35-250 Hz corresponding to the recently observed gravitational waves in LIGO <cit.>. §.§ The condensed matter case The next case we consider is in the realm of TSCs and superfluids. Here, the part of the effective action which is related with the topology of the band structure of such materials corresponds to an action of the CS type, while the nontopological term is given by the standard Einstein-Hilbert action. Clearly, the effective character of those actions in the CM case must be reflected in choosing the appropriate scales corresponding to the basic parameters G, c and θ of the action (<ref>) <cit.>. Following Ref. GEM-Sekine, we take the maximum attainable velocity in the material to be the Fermi velocity v_F, in such a way that now we have the replacement c → v_F. Also, the maximum energy scale is defined as the energy Δ of the superconductor energy gap (the difference between the ground state of the superconductor and the energy of the lowest quasiparticle excitation). This corresponds to the replacement M →Δ / v _F ^2 and yields G → G _CM = ħ v _F ^5 / Δ ^2.One of the main objectives in this CM situation is to determine the thermal response of such materials. In very general terms, the idea is that a temperature gradient in the z-direction, for example, would produce a thermal current in the x-y plane which effects as a mechanical rotation with angular velocity in the z-direction could be detected experimentally <cit.>. This is completely analogous to the magnetoelectric effect in TIs, whereby an electric field in the z-direction induces an electrical (Hall) current in the x-y plane, which in turn generates a magnetic field in the z-direction. A precise way of dealing with this approach is to push further the electromagnetic analogy by introducing the gravitoelectromagnetic (GEM) approximation in the effective gravitational theory described by the actions 𝒮 _ + 𝒮 _ with the corresponding parameters v _F and Δ. The details are given in Refs. TSC-Ryu and GEM-Furusaki-Luttinger.What is relevant for us is that gravitoelectromagnetism, being a weak field approximation already pertinent to CM physics, serves to motivate the study of the complementary sector which describes wave propagation. To this end, we will restrict our general action <ref>, with the corresponding change of scales, to the proposed actions already considered in the CM literature in order to estimate the impact upon the reflected and transmitted waves in the domain wall case.§.§.§ The purely topological CS action 𝒮 _CM The corresponding action is given by Eq. (<ref>), where θ _CM = π describes a nontrivial topological contribution. We identify the values of θ and ξ (at normal incidence) entering in Eq. (<ref>), for the domain wall model, as θ = 1/12ħ ^2 v _F ^2/Δ ^2 , ξ = 1/12ħ ^2/Δ ^2ω ^2 = 1/12( ω/ω _max) ^2,where we have introduced ω _max = Δ / ħ = 1.1 × 10 ^12 Hz. For a typical topological superconductor such as Cu_xBi_2Se_3, the experimental values of the required quantities can be estimated as <cit.>Δ = 7 × 10 ^-4 = 1.12 × 10 ^-15, v _F = 5 × 10 ^7,at T = 3K. As in the previous case we take ξ = 0.01 (corrections of order 10 ^-4) which yields ω = 0.35ω _max, which is very close to the maximum frequency where the linear approximation ceases to be valid. Taking an upper limit ω / ω _max < 10 ^- 2 and assuming the validity of the linear approximation we obtain ξ = 8.3 × 10 ^- 6, which produces corrections of the order 10 ^- 10 in the reflection and transmission coefficients.§.§.§ The gravitoelectromagnetic case A second possibility to deal with the thermal effects in TSCs is encoded in the modified CS action <cit.>𝒮 _ = k _B ^2 T ^2/24 ħ v _F∫ dt d ^3xθ _ (𝐱,t) 𝐄_g·𝐁_g ,which is adopted in complete analogy with the electromagnetic case, where 𝐄 _g and 𝐁 _g are the gravitoelectric and gravitomagnetic fields, respectively. As a first approximation, we take θ _ (𝐱,t) = π. Let us remark that the above action does not have a precise topological content and that has not been already derived from a corresponding microscopic action by fermion integration <cit.>. Moreover, the GEM limit of the CS action in Eq. (<ref>) produces two additional derivatives with respect to the analogous electromagnetic action. In Ref. Smith, it is shown that the weak field approximation produces, schematically,^∗ RR → - 16 ∂ E _g∂ B _g , [ E _g] = [ B _g] = 1/.For the sake of estimations, we are not considering the explicit form of the limit. In order to compare our action (<ref>) with (<ref>), we need to assess the effect of the two additional derivatives. We do this by taking∂ ^2∼1/ L _CM ^2 ,L _CM = √(ħ G _CM/ v _F ^3) ,in such a way that the action (<ref>) reads𝒮 _ = - Δ ^4/4 πħ ^3 v _F ^3θ∫ dt d ^3x E _g B_g ,which compared with (<ref>) produces, taking the absolute value, the following expression for θ and ξ in the domain wall model:θ = π ^2/6 v _F ^2k _B ^2 T ^2/Δ ^21/ω _max ^2 , ξ = π ^2/6k _B ^2 T ^2/Δ ^2( ω/ω _max)^2 .Choosing ξ = 0.01 and using the experimental values given in (<ref>), we obtain ω = 2.9 ω _max which is certainly beyond the linear approximation. Taking an upper limit ω /ω _max < 10 ^- 2 and assuming the validity of the linear approximation, we have ξ = 2.2 × 10 ^-1 which produces corrections of the order 10 ^- 2 in the reflection and transmission coefficients. Again, the corrections are rather small, though eight orders of magnitude above the previous case.§ SUMMARY AND FINAL COMMENTS In this paper, we have studied the propagation of gravitational waves in nondynamical CS gravity, which is defined in the action (<ref>) by the coupling of a scalar field θ to the gravitational Pontryagin invariant ^∗ R _στ^στμν R ^τ _τσμν. We demonstrate that the resulting modified wave equation (<ref>) is consistent with the choice of the TT gauge corresponding to ∂_μh^μν=0, h^μ_μμ=0 and h ^3 ν=0 for the trace-reversed metric perturbation h _μν. Motivated by their relevance in CM physics, we have considered two nondynamical models for the CS coupling field θ: (i) a planar domain wall defined by z=0 and (ii) a planar surface layer where θ changes linearly across a thickness d. While the former is employed forthe description of the surface of the ^3He-B phase in the presence of surface magnetization, the latter can be useful to analyze the magnetic dipole response of a topological singlet superconductor.For both models, the field equations couple the amplitudes of the two independent linear polarization modes and include distributional contributions like δ ^'(z) and/or δ (z), as shown in Eq. (<ref>). This equation is decoupled by introducing the right- and left-handed circularly polarized modes h _R and h _L, respectively. Since the Cotton-York tensor is supported only at the interfaces of the θ-models, i.e. (i) at the domain wall and (ii) at the faces of the surface layer, the θ-boundaries act as an effective thin shell of matter depending onthe components of the gravitational field itself. Therefore, the bulk metrics satisfy the standard Einstein equations and they should be properly joined at the interfaces in order to provide a valid solution of the modified field equations.To determine the boundary conditions for the metric perturbation, we have adopted the rigorous distributional approach introduced in Ref. Kurasov to analyze generalized point interactions in nonrelativistic quantum mechanics. In <ref> we review the basics of 1D point interactions considered as self-adjoint extensions of the operator - d ^2 / dz^2. Following the distributional approach, we obtain the wave equations (<ref>) and (<ref>) describing the propagation of gravity waves through a domain wall and a surface layer, respectively. To this end, we need to introduce the relations (<ref>) for the product of a discontinuous function times δ and δ ^', which arise from the approach in Ref. Kurasov. The boundary conditions are then obtained by integrating the equations of motion (<ref>) and (<ref>) over a pill-shaped region across the interfaces. We also demonstrate that the additional contribution to the field equations, i.e. the Cotton-York tensor, corresponds to one of the self-adjoint extensions of the operator - d ^2 / dz^2, allowing us to verify the previously obtained boundary conditions. We find many subtleties when we deal with the propagation of gravity waves through a domain wall. In this case, the linearized field equations include distributional contributions of the type δ (z) and δ ^' (z), which arise from the fact that the CS coupling field θ is piecewise constant and the gravitational Pontryagin invariant contains second-order derivatives of the metric perturbation. Our main finding in this case is that the boundary conditions imply that both the metric and its first derivative become discontinuous at the interface. Nevertheless, as shown in Sec. <ref>, the metric naturally satisfies the area matching condition introduced by Barrett <cit.>, in which the junction condition requiring the continuity of the metric at a given hypersurface is replaced by the weaker condition that the area of any 2-surface gives a unique result when measured from each side of the hypersurface. As the expressions for thereflection and transmission coefficients indicate, the effect of the domain wall would becomes important when ξ = θ̃ω ^2cos ^2α≃ 2, by substantially increasing the reflecting property of the domain wall. Nevertheless, as shown in Eq. (<ref>), the value ξ=2 is strictly forbidden, in such a way that we never have a perfect mirror for gravitational waves due to the domain wall. This case also presents the following bizarre property: the gravitational waves suffer from the threshold anomaly, which means that the transmitted amplitude does not vanish when the frequencygoes to zero. This behavior is also present in some cases of potential scattering in 1D quantum mechanics <cit.>. A similar phenomenon has also been measured in some nucleus-nucleus scattering processes in nuclear physics <cit.>. A deeper understanding of this feature is beyond the scope of the present paper.In the surface layer case, the linearized field equations include distributional contributions of δ-type in each face of the layer. From the distributional point of view, this problem closely resembles the nonrelativistic quantum-mechanical delta potential. In this case, the metric perturbation becomes continuous at the surfaces, with discontinuous first derivatives. This situation is far more similar to the analogous case of propagating electromagnetic waves in Maxwell-CS electrodynamics for a domain wall of θ. This similarity relies in the fact that both the Cotton-York tensor (which contains second-order derivatives of the field θ) in the surface layer case and the Maxwell-CS equations (which contains first-order derivatives of the field θ) depend on the δ distribution, in contrast to the gravitational domain wall, for which the Cotton-York tensor contains distributional contributions of the type δ and δ ^'.As observed at the end of Sec. <ref>, the limiting case d→ 0 of the surface layer model does not reproduce the case of the domain wall, as one could naively expect because we have θ _dw (z) = lim _d → 0θ _sl (z). This can be understood because the operators distinguishing both cases do not match in such limit when acting on the metric perturbation. From Eqs. (<ref>) and (<ref>) we have𝒪̂_ sl h _R/L=i/d{Π (z) + [ δ (z)-δ (z-d) ] ∂ _z}∂ _t h _R/Lfor the surface layer, and 𝒪̂_ dw h _R/L = i [ δ (z) + δ ^' (z) ∂ _z] ∂ _t h _R/L,for the domain wall. Recalling the distributional definitions for the product of a discontinuous function at z=a times the δ (a) and δ ^' (a) distributions, Eq. (<ref>), we conclude thatlim _d → 0𝒪̂_ sl h _R/L≠𝒪̂_ dw h _R/L ,because from the distributional point of viewlim _d → 01/d[ δ (z) ∂ _z h _R/L (0) - δ (z-d) ∂ _z h _R/L (d)] ≠δ ^' (z) ∂ _z h _R/L (0) - δ (z) ∂ _z ^2 h _R/L (0).We gave some estimations of the parameters in our model by restricting our general approach to some cases already considered in the literature. We first recalled the nontopological (Einstein-Hilbert) and the topological (Chern-Simons) terms of the action in standard units. They take the form of Eqs. (<ref>) and (<ref>), respectively, in terms of the maximum attainable velocity c in the medium and the mass scale M under which the effective theory is valid. The surface layer model was successfully compared with the results obtained in Ref. Smith, where the authors study the effect of the CS theory on bodies orbiting the earth. By relating the CS mass defined in Ref. Smith to our parameter γ, and using the lower bound for the former, we found that the minimum frequency required to obtain a deviation of the order of 10^-4 in the reflection and transmission coefficients of the wave is 585 Hz, which is comparable with the frequencies recently observed gravitational waves in LIGO. The next case we considered is in the realm of TSCs and superfluids, where we compared our results of the domain wallmodel withtwo different models of the CS action: (i) the full topological response theory defined by Eq. (<ref>) and (ii) the GEM case defined by Eq. (<ref>). The latter has been systematically used in the literature on CM systems and exhibits a complete analogy with the electromagnetic case which describes the response of TIs. The effective character of those actions in the CM case must be reflected in choosing the appropriate scales corresponding to the basic parameters G, c and θ. Imposing the frequency of the propagating wave in the material to be at least 10^-2 times, the maximum attainable frequency, we find corrections in the reflection and transmission coefficients of the order 10^-10 for the case (i) and 10^-2 for the case (ii). Our results show that the CS corrections in wave propagation become sizable when the frequency ω of thewave satisfy the following requirements: (a) ω is close enoughto the maximum allowed frequency ω_max in each particular case and(b) ω satisfies ω/ω_max < 1 so that the linearized approximation is valid. Let us observe that in the CM system so far considered we have ω _max=1.1× 10^12 Hz, while the corresponding value in standardgravityis ω _max=2.4× 10^42 Hz, the Planck frequency,about thirty orders of magnitude higher. In this way, topological CM systems could afford a realistic possibility to experimentally probe the effects of the CS coupling.To close, we discuss the relevance of the CS term with respect to additional combinations of the Riemann tensor that can be added to the action andwhich can be at least as important as the CS term in the weak field limit, such as the quadratic term in f(R)-gravity <cit.> for example. On one hand, in the standard gravitational case the CS term is singled outas been an unavoidable contribution arising in thelow energy limitof String Theory and Loop Quantum Gravity <cit.>. On the other hand,in order to describe the gravitational response of topological matter, the effective field theory should be topological in nature, thus strongly restricting the possible theories which could be taken into consideration. Therefore, besides the CS gravity defined through the Pontryagin density, other options are to consider the Euler invariant and the torsion dependent Nieh-Yan density. The latter has been recently proposed as an alternative to describe the gravitational response theory of TSCs and superfluids <cit.>. The analysis of such possibilities in the realm of wave propagation would constitute an interesting extension of the present work.§ ONE-DIMENSIONAL POINT INTERACTIONS AS SELF-ADJOINT EXTENSIONS OF THE OPERATOR -D^2/DZ^2 Point interactions in one dimension appear frequently as a method of simplifying the description of various physical situations, making emphasis on the significant features of the problem but leaving aside a detailed description of the interaction, which can be later included to produce a more realistic solution. Point interactions, also known as Fermi pseudopotentials, are associated with a given differential equation and correspond to potentials V_FP(z) that are nonzero only in some specific points in the line, where they become singular. The differential equation we consider here is[ -d^2/dz^2+V_FP(z) ] ψ (z)=Eψ (z),which simplest physical version corresponds to the classical example of the δ-function pseudopotential. On a formal level we can associate to this system the one-dimensional Schrödinger operatorH = - ħ ^2/2md ^2/dz ^2 + λδ( z) ,where λ is a real coupling constant and z ∈ℝ. The resulting eigenvalue equation, Hψ =Eψ, acquires a precise meaning when converting it into a boundary value problem. Heuristically, this equation can be interpreted as consisting of the free Schrödinger equation - ħ ^2/2mψ ^'' = E ψ for z ∈ℝ∖{ 0 }, together with the boundary conditions ψ( 0 ^+) = ψ( 0 ^-) and ψ ^'( 0 ^+) - ψ ^'( 0 ^-) = 2m λ/ħ ^2ψ( 0 ) at z=0. The argument presented above can be phrased in rigorous terms by using the theory of distributions. In doing so, we should consider that observables in quantum mechanics are required to be self-adjoint operators. In our case, the observable is the energy, which is formally represented by the operator H _0 = - ħ ^2/2md ^2/dz ^2. Each function ψ in the domain of H _0 must live in the Hilbert space ℋ = L ^2( ℝ) of functions square-integrable on ℝ and be such that ψ ^' is absolutely continuous at all points z ≠ 0, satisfying also ψ ^''∈ℋ.We require the following two conditions in order to declare that the operator H _0 is self-adjoint. (i) To begin with, H _0 must be hermitian (or symmetric in the mathematical language), which means that we must impose the following additional boundary conditions at z=0, for all ψ and φ in the domain of H _0,∫ _ℝ∖{0}[ ( H _0 ^†ψ( z ) ) ^∗φ( z ) - ψ ^∗( z ) H _0φ( z ) ] dz = 0 ,with H _0 ^† = H _0. By partial integration of this relation translates into the boundary condition[ ψ ^∗φ ^' - ψ ^∗'φ] _z = 0 ^+ = [ ψ ^∗φ ^' - ψ ^∗'φ] _z = 0 ^-.Note that this result does not require either the functions or their first derivatives to be continuous at z=0. Neither it requires that the boundary conditions for the function ψ on the right are exactly the same as the boundary condition for the function φ on the left in Eq. (<ref>). In fact, the functions ψ (z), defined in ℝ∖{0}, which are in the domain of the adjoint H _0 ^† are required to be continuous with continuous bounded first derivatives except at the origin, where they can have arbitrary finite discontinuities both in ψ and ψ ^'. This last property means that the corresponding limits at 0 ^+and at 0 ^- are finite and well defined.In general, boundary conditions involving a function ψ and its derivative ψ ^' are of the form[ [ψ ( 0 ^+ ); ψ ^' ( 0 ^+ ) ]] = [ [ u_11 u_12; u_21 u_22 ] ] [ [ψ ( 0 ^- ); ψ ^' ( 0 ^- ) ]] = U [ [ψ ( 0 ^- ); ψ ^' ( 0 ^- ) ]] ,parametrized by the complex 2 × 2 matrix U ≡[ u_ij]. (ii) The second condition for an operator to be self-adjoint is that both the domain and the action of the operator acting on the right are equal to the domain and the action of the adjoint operator acting on the left. In this case, we have only one matrix U for both type of functions ψ and φ, and the condition (<ref>) translates into <cit.>U ^† J U = J,J = [ [01; -10 ] ] .The above equation implies that (U) is a phase and also reduces the parametrization of U from eight to four real independent numbers. The boundary conditions for the δ-function pseudopotential corresponds to the simple choice u _11 = u _22 = 1, u _12 = 0 and u _21 = 2m λ / ħ ^2. Thus we now interpret the δ-interaction as a SAE of the operator H _0.To summarize, the basic problem posed in general by 1D point interactions is to consistently determine the boundary conditionswhich the solutions ψ (z) of the associated differential equation must satisfy around the points in the line where the pseudopotential diverges. In most of the cases discussed in the literature, this point has raised many controversies which are far from been settled down <cit.>. As mentioned above, in quantum mechanics, the operator in the left-hand side of Eq. (<ref>) is usually the Hamiltonian of the system. This suggests a natural way to define the corresponding boundary conditions by demanding the operator to be self-adjoint. Although we are not dealing with a quantum mechanical problem, in this work we adopt the same approach to define our boundary conditions.General SAE of the operator -d^2/dz^2 have been considered in Refs. Kurasov and Albeverio, among others, leading to a generalized point interactions which depends on both, the δ- and δ ^'-interactions. This method of defining the required boundary conditions has the advantage of producing results which are independent of specific models of the delta distribution and its derivatives.It isknown that the most general SAE of the operator - d ^2/dz ^2 is parametrized by four real independent parameters which we denote by X _i, i=1,2,3 and 4. <cit.> The application of the distributional method in Ref. Kurasov to the problem of the SAE of H _0 leads to the following result for the matrix U:U = e ^-i(D)[ [ (2+X_2)^2-X_1X_4+X_3^2/|D|-4X_4/|D|; 4X_1/|D| (2-X_2)^2-X_1X_4+X_3^2/|D| ] ] ,whereD = 4 + X _1 X _4 - X _2 ^2 - X _3 ^2 - 4 i X _3is in general a complex number. The above equation (<ref>) corresponds to Eq. (19) in Ref. Kurasov and shows the general form of the matrix U, being a phase times a matrix with determinant one. The explicit form of the boundary conditions is- X _1ψ (0 ^+) + [ 2 + ( X _2 - i X _3) ] ψ ^' (0^+) = X _1ψ (0 ^-) + [ 2 - ( X _2 -i X _3) ] ψ ^' (0 ^-) ,[ 2 - ( X _2 + i X _3) ] ψ (0^+) + X _4ψ ^' (0 ^+) = [ 2 + ( X _2 + i X _3) ] ψ (0 ^-) - X _4ψ ^' (0 ^-) .Our problem now is to obtain the matrix U in Eq. (<ref>) starting from the formulation of the problem in terms of pseudopotentials, in order to identify the operator that arises via the Cotton-York tensor modifications to the wave equation as a SAE of H_0, which will lead to the appropriate boundary conditions for the solution. Based upon the results of Kurasov <cit.>, we propose the following interpretation of his distributional operator, in terms of Fermi pseudopotentials defining a second-order differential equation of the standard type for the functions ψ (z) defined aboveH ψ≡ - d ^2/dz ^2ψ - X _4d/dzδ (z) d/dzψ + i X _3( δ ^' (z) + 2 δ (z)d/dz) ψ + X _1δ (z) ψ + X _2δ ^' (z) ψ = E ψWe emphasize that our notation isd/dzδ( z ) d/dz = δ( z ) d ^2/dz ^2 + δ ^'( z ) d/dz ,instead of d/dzδ( z ) d/dz = δ ^'( z ) d/dz as considered in Ref. Langmann. Next, we need to make sense of products like ψ (z)δ (z) and ψ(z) δ ^' (z). For continuous functions φ (z), with continuous first derivatives, at z=0, the following properties are well-known <cit.>φ( z ) δ( z )= φ( 0 ) δ( z ) ,φ( z ) δ ^'( z )= φ( 0 ) δ ^'( z ) - φ ^'( 0 ) δ( z ) .The generalization of the above equations to the case of the functions ψ (z) defined in ℝ∖{0} andarising from the distributional construction of Ref. Kurasov, consists in replacing the value of φ(0) (and of φ ^'( 0)) by their mean value at the originψ( 0 ) = 1/2[ ψ( 0 ^+) + ψ( 0 ^-) ] ,where ψ( 0 ^+) and ψ( 0 ^-) are respectively the limits of ψ( z ) when z approaches 0 from the positive and negative sides. Then, the definitions (<ref>) should be read asψ( z ) δ( z )= ψ( 0 )δ( z ) ,ψ( z ) δ ^'( z )= ψ( 0 )δ ^'( z ) - ψ ^'( 0 )δ( z ) .Inserting the expressions (<ref>) in (<ref>) we obtainH ψ = - d ^2ψ/dz ^2 + [ X _1ψ( 0 ) - ( X _2 - i X _3) ψ ^'( 0 )] δ (z)+[ ( X _2+ i X _3) ψ( 0 ) - X _4ψ ^'( 0 )] δ ^'(z) = E ψ .Note that the dependence on ψ ^''( 0 ^+) and ψ ^''( 0 ^-) has canceled between the contributions arising from (<ref>) and the substitution (<ref>) when ψ( z ) →ψ ^'( z ). Following Ref. Albeverio, we recover the boundary conditions determined by U in Eq. (<ref>) dealing with Eq. (<ref>) in the standard way. First, we integrate the equation from - ε to + ε using∫ _- ε ^+ ε dz δ (z) = 1, ∫ _- ε ^+ ε dz δ ^' (z) =0 .The result is[ ψ ^' (0 ^+) - ψ ^' (0 ^-) ] - X _1ψ( 0 ) + ( X _2 - i X _3) ψ ^'( 0 ) = 0which we can explicitly rewrite as[ - 2 + i X _3 - X _2] ψ ^' (0 ^+) + X _1ψ (0 ^+) + [ 2 + i X _3 - X _2] ψ ^' (0 ^-) + X _1ψ (0 ^-) = 0.The above equationis precisely the boundary condition in Eq. (<ref>). The remaining boundary condition is obtained integrating Eq. (<ref>) from - L to positive z, and further from - ε to + ε. In this way, we haveE ∫ _- L ^zψ(z ^') dz ^' = - [ ψ ^' (z) - ψ ^' (L) ] + [ ( X _2 + i X _3) ψ( 0 ) - X _4ψ ^'( 0 )] δ (z)+ [ X _1ψ( 0 ) - ( X _2 - i X _3) ψ ^'( 0 )] H (z) ,where H (z) is the Heaviside function. Performing the second integration in z from - ε to + ε, we obtain[ 2 - ( X _2 + i X _3) ] ψ (0 ^+) + X _4ψ ^' (0 ^+) = [ 2 + ( X _2 + i X _3) ] ψ (0 ^-) - X _4ψ ^' (0 ^-) ,in the limit, since the discontinuities produced by H (z) are finite. Equation (<ref>) reproduces the boundary conditions in Eq. (<ref>). To summarize, we have regained the conditions for the SAE of the operator - d ^2 / dz ^2 by providing an interpretation in terms of pseudopotentials (given by Eq. (<ref>)) of the distributional operator in Ref. Kurasov, plus the use of the relations (<ref>).Next we apply these general results to our problem. Clearly, there is a close analogy between the quantum-mechanical case and the linearized CS gravity we have presented in Sec. <ref>. In the problem at hand, defined by Eq. (<ref>), the basic hermitian operator is the same as in quantum mechanics, H _0 = - d ^2/dz ^2, with E = ω ^2cos ^2α. From Eq. (<ref>), we read the CS modified wave equation[ -d^2/dz^2∓θ̃ωcosαd/dzδ( z) d/dz∓θ̃ω ^3cos ^3α δ( z ) ] h _R/L = ω ^2cos ^2αh _R/L.Now we show that our CS point interaction is a particular case of the general SAE of the operator H _0. In fact, comparing Eq. (<ref>) with Eq. (<ref>), we identifyX _1 = ∓θ̃ω ^3cos ^3α , X _4 = ±θ̃ωcosα ,X _2 = X _3 = 0 ,in such a way that the matrix U determining the boundary conditions readsU = 1/1- ( ξ / 2 ) ^2[ [ 1 + ( ξ / 2 ) ^2 ∓( ξ / (ωcosα) ); ∓( ξωcosα) 1 + ( ξ / 2 ) ^2 ]] ,where ξ = θ̃ω ^2cos ^2α≠ 2. One can further verify that U ^† J U = J. Therefore, we have demonstrated that, to linear order, the CS contribution to the wave equation for a domain wall of θ, can be described as a SAE of the 1D operator -d^2/dz^2, with boundary conditions determined by the matrix U in Eq. (<ref>), according to the relations (<ref>).§ ACKNOWLEDGEMENTS Useful discussions with M. Cambiaso, L. Huerta, A. Toloza and J. Zanelli at an early stage of this work are warmly appreciated. Thanks are also due to C. Chryssomalakos for providing valuable references and to E. Nahmad for updating our knowledge in differential geometry. 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Alexander and N. Yunes, Phys. Rev. D 77 (2008) 124040.Qiang Li. Qiang and P. Xu, Gen. Rel. Grav. 47 (2015) 26.LIGO B. P. Abbott et al., Phys. Rev. Lett. 116 (2016) 061102.Stone M. Stone, Phys. Rev. B 85 (2012) 184503.FRgravity1 T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82 (2010) 451.FRgravity2 A. De Felice and S. Tsujikawa, Living Rev. Relativ. 13 (2010) 3.NiehYan Y. Hidaka, Y. Hirono, T. Kimura and Y. Minami, Prog. Theor. Exp. Phys. 2013 (2013) 013A02. | http://arxiv.org/abs/1706.08843v1 | {
"authors": [
"A. Martín-Ruiz",
"L. F. Urrutia"
],
"categories": [
"gr-qc",
"hep-th"
],
"primary_category": "gr-qc",
"published": "20170627160346",
"title": "Gravitational waves propagation in nondynamical Chern-Simons gravity"
} |
School of Electronic and Information Engineering & State Key Laboratory for Mechanical Behavior of Materials, Xi'an Jiaotong University, Xi'an 710049, ChinaPhysics Department and Institute for Nanoscience and Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA Physics Department and Institute for Nanoscience and Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA Physics Department and Institute for Nanoscience and Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA Institute of Physics and Physics Department of Southern Federal University, Rostov-na-Donu 344090, RussiaPhysics Department and Institute for Nanoscience and Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA School of Electronic and Information Engineering & State Key Laboratory for Mechanical Behavior of Materials, Xi'an Jiaotong University, Xi'an 710049, ChinaMaterials Research and Technology Department, Luxembourg Institute of Science and Technology, 5 avenue des Hauts-Fourneaux, L-4362 Esch/Alzette, LuxembourgMaterials Research and Technology Department, Luxembourg Institute of Science and Technology, 5 avenue des Hauts-Fourneaux, L-4362 Esch/Alzette, LuxembourgPhysics Department and Institute for Nanoscience and Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA Atomistic effective Hamiltonian simulations are used to investigate electrocaloric (EC) effects in the lead-free Ba(Zr_0.5Ti_0.5)O_3 (BZT) relaxor ferroelectric. We find that the EC coefficient varies non-monotonically with the field at any temperature, presenting a maximum that can be traced back to the behavior of BZT's polar nanoregions. We also introduce a simple Landau-based model that reproduces the EC behavior of BZT as a function of field and temperature, and which is directly applicable to other compounds. Finally, we confirm that, for low temperatures (i.e., in non-ergodic conditions), the usual indirect approach to measure the EC response provides an estimate that differs quantitatively from a direct evaluation of the field-induced temperature change. Electrocaloric effects in the lead-free Ba(Zr,Ti)O_3 relaxor ferroelectric from atomistic simulations L. Bellaiche December 30, 2023 =====================================================================================================§ INTRODUCTION The electrocaloric (EC) effect characterizes the change in temperature induced by a change in electric field <cit.>, with the electrocaloric coefficient being defined as α=.∂ T/∂ E|_S, where T is the temperature, E is the electric field and S is the entropy. It has the potential to be an efficient solid-state refrigeration for a broad range of applications <cit.>. Numerous studies have been recently conducted via measurements, phenomenologies and atomistic simulations (see, e.g., Refs <cit.> and references therein) and have led to a better knowledge of electrocaloric effects in typical ferroelectrics, such as BaTiO_3, LiNbO_3, Pb(Zr_0.4Ti_0.6)O_3, (Ba_0.5Sr_0.5)TiO_3, as well as antiferroelectrics such as La-doped Pb(Zr,Ti)O_3. On the other hand, fewer investigations about EC effects <cit.> have been performed in another class of ferroelectrics, namely the relaxor ferroelectrics. These intriguing materials exhibit unusual features, such as a frequency-dependent and broad dielectric response versus temperature while remaining macroscopically paraelectric down to 0 K <cit.>. They also display several characteristic temperatures (i.e., the T_b Burns temperature, the T^* temperature and the T_m temperature) that are associated with a subtle change in some physical properties <cit.>. For instance, in Ba(Zr_0.5Ti_0.5)O_3 (BZT) relaxor ferroelectrics, simulations <cit.> indicate that the Burns temperature (below which the dielectric response does not obey the Curie-Weiss law <cit.>) is T_b≃450 K , T^* ≃240 K, and T_m≃130 K is the temperature at which the dielectric response exhibits a peak, as also in-line with measurements in BZT compounds <cit.>. The microscopic origin of these features is commonly believed to be the existence of the so-called polar nanoregions (PNRs) below the Burns temperature <cit.>. Interestingly, studies devoted to EC effects in relaxor ferroelectrics have resulted in original findings. One example includes the failure of indirect methods (which are based on thermodynamic equilibrium considerations) in the relaxor ferroelectric PVDF-TrFE-CFE terpolymer to obtain the real change in temperature induced by an electric field for temperatures below which the broad dielectric constant peaks, because of non-ergodicity <cit.>. Another example is the non-monotonic behavior of the EC coefficient with the magnitude of the electric field at the fixed critical point temperature T_CP in Pb(Mg,Nb)O_3 (PMN), (Pb,La)(Zr,Ti)O_3 and Pb(Mg,Nb)O_3–PbTiO_3 relaxors <cit.>; especially intriguing is the existence of a maximum of this coefficient at the specific field E_CP for this T_CP temperature, with (T_CP, E_CP) corresponding to the critical point at which the paraelectric-to-ferroelectric transition changes its nature from first order to second order. It is worthwhile to realize that these latter results were obtained for lead-based relaxor ferroelectrics while there are also (environmentally-friendly) lead-free relaxor ferroelectrics, such as Ba(Zr_1-xTi_x)O_3, that are fundamentally distinct. For instance, the difference in polarizability between Ti and Zr ions in Ba(Zr_0.5Ti_0.5)O_3 was found to be essential to reproduce relaxor behavior via the formation of small Ti-rich PNRs embedded in a paraelectric matrix <cit.>, while the relaxor nature of lead-based PMN was predicted to rather originate from a complex interplay between random electric fields, ferroelectric and antiferroelectric interactions – yielding much larger PNRs touching each other at low temperatures <cit.>. Another striking difference between Ba(Zr_0.5Ti_0.5)O_3 and PMN is that a recent atomistic simulation did not find any trace of a first-order paraelectric-to-ferroelectric phase transition when subjecting Ba(Zr_0.5Ti_0.5)O_3 to electric fields, that is, the polarization seems to always continuously evolve with the magnitude of the dc electric field in this lead-free compound <cit.>.One may therefore wonder about EC effects in lead-free relaxor ferroelectrics, even more when realizing that a recent study done in Ba(Zr_1-xTi_x)O_3 with x=0.20 reported a giant α electrocaloric coefficient <cit.> (note that this system is different from Ba(Zr_0.5Ti_0.5)O_3 in the sense that it possesses a polar ground state in addition to some relaxor features). For instance, many questions remain to be addressed in Ba(Zr_0.5Ti_0.5)O_3: Do indirect and direct methods also provide different results below a specific temperature? How does α behave with the dc electric field for the different temperature ranges in BZT, i.e. above T_b, between T_b and T^*, between T^* and T_m, and below T_m? In particular, can α exhibit a maximum for some intermediate field at any of these temperature ranges? If such maximum exists, what is its microscopic origin? Other natural questions to ask are if and how α depends on temperature for fixed electric fields, and if it is possible to reproduce and understand such (presently unknown) dependency.As we will see below, this manuscript provides an answer to all these open questions, by conducting and analyzing atomistic simulations on Ba(Zr_0.5Ti_0.5)O_3 ferroelectric relaxors. This article is organized as follows. Section II provides details about the methods used here. Results are given, analyzed and explained in Section III. Finally, Section IV concludes this work.§ METHODS We use here a first-principles-based effective Hamiltonian (H_eff) approach that has been recently developed for Ba(Zr_0.5Ti_0.5)O_3 (BZT) solid solutions <cit.>. The total energy of the effective Hamiltonian used here contains two main terms: E_int({𝐮_i},{𝐯_i},η_H,{σ_j})=E_ave({𝐮_i},{𝐯_i},η_H)+E_loc({𝐮_i},{𝐯_i},{σ_j}), where {𝐮_i} is the local soft mode in unit cell i (which is related to the electric dipole of that cell and that is technically centered on the Zr or Ti ions), {𝐯_i} are variables related to the inhomogeneous strain inside each cell, η_H is the homogeneous strain tensor, and {σ_j} represents the atomic configuration of the BZT solid solutions (i.e., how Zr and Ti ions are distributed within the B-sublattice of BZT). E_ave contains five energetic terms: (i) the local-mode self-energy; (ii) the long-range dipole-dipole interaction; (iii) the energy due to short-range interactions between local modes; (iv) the elastic energy; and (v) the energy representing the interaction between local modes and strains <cit.>. E_loc describes how the actual distribution of Zr and Ti cations affects the energetics involving the local soft-modes 𝐮_i and the local strain variables, and therefore depends on the {σ_j} distribution <cit.>. One can also add to E_int an energy given by the dot product between polarization and electric field, in order to mimic the effect of such field on physical properties.This effective Hamiltonian successfully predicted the existence of three characteristic temperatures in BZT, namely the Burns temperature (T_b≃450 K) below which the dielectric response does not follow anymore the Curie-Weiss law <cit.>, the so-called T^* (that is close to ≃240 K), and the T_m temperature at which the dielectric response can exhibit a peak (T_m≃130 K) <cit.>, as consistent with experimental findings for BZT systems <cit.>. This atomistic scheme also yields polar nanoregions inside which the Ti-centered dipoles are aligned parallel to each other, with these PNRs being dynamic in nature between T^* and T_b while, below T_m, they are static and all have a polarization pointing along one of the eight ⟨ 111⟩ pseudo-cubic directions <cit.>. The polarizations of these different PNRs cancel each other, as consistent with the fact that BZT is macroscopically paraelectric down to 0 K <cit.>. This effective Hamiltonian was also successful in reproducing the unusual dielectric relaxation known to occur in relaxor ferroelectrics <cit.>. Here, we implement this H_eff within Monte Carlo (MC) and Molecular Dynamics (MD) simulations, in order to determine and understand EC effects in BZT relaxors – as modeled by 14×14×14 supercells (13720 atoms) in the MC computations and 32×32×32 (32768 atoms) in the MD simulations. Note that this different choice of supercells between the MC and MD simulations originates from the fact that the code we used for the MD computations can handle larger supercells, and that the use of 32×32×32 supercells allows the temperature change in MD simulations to be easily sorted out from the temperature fluctuations. Note also that we numerically checked that the use of 12×12×12, 14×14×14 and 16×16×16 supercells provides similar results, which suggests that our Monte-Carlo simulations are free from significant size effects. These supercells are periodic along the three Cartesian directions, and Zr and Ti atoms are randomly distributed inside them. We also average our physical results over 20 of these random configurations for both MC and MD simulations, in order to mimic well disordered BZT solid solutions.Let us now indicate how we practically compute, from these simulations, the electrocaloric coefficient α=.∂ T/∂ E|_S. One approach we use here is based on the Maxwell thermodynamical relationship .∂ S/∂ E|_T=.∂ P/∂ T|_ E leading to the adiabatic temperature change Δ T=-_ E_1^ E_2T( E)/C_ E(T).∂ P/∂ T|_ Ed E,where P is the macroscopic polarization and C_ E is the heat capacity per unit volume under constant dc electric field. Such latter equation therefore tells us that we can obtain α from MC simulations by computing α=-T/C_ E.∂ P/∂ T|_ E.This way of extracting α is coined MC-1 here.For instance, Fig. <ref>(a) reports the polarization as a function of temperature obtained from MC simulations on Ba(Zr_0.5Ti_0.5)O_3, for dc electric fields all applied along the pseudo-cubic [001] direction and ranging between 2.0×10^7 and 3.0×10^8 V/m in magnitude. Values of .∂ P/∂ T|_ E are then obtained from cubic B-spline fits to these P(T) curves, which allows us to determine α via Eq. (<ref>). Note that the heat capacity at a given electric field E is calculated as: C_ E=(N⟨E_int^2⟩ -⟨E_int⟩ ^2/T^2k_B+15/2k_B)/V, where N is the number of sites in the supercell, E_int is the total internal energy provided by the effective Hamiltonian, ⟨ ⟩ denotes the average over the MC sweeps at every considered T temperature, k_B is the Boltzmann constant, and V is the volume of the unit cell. The factor 15/2 in that formula reflects that there are five atoms in the unit cell of perovskites <cit.>. Moreover, C_ E is computed for different temperatures and electric fields, implying that it can, in principle, depend on T and E. However, we numerically found that these dependencies are rather weak as consistent with measurements <cit.> and that C_ E is always very close to 2.18 MJ/K m^3.Interestingly, there is another way to obtain the EC coefficient from MC runs, that is by taking advantage of the cumulant formula given in Ref. <cit.>: α=- Z^*a_latNT{⟨|𝐮|E_int⟩ -⟨|𝐮|⟩⟨E_int⟩/⟨E_int^2⟩ -⟨E_int⟩ ^2},where Z^* is the Born effective charge, a_lat is the five-atom lattice constant, N is the number of sites in the supercell, T is the considered temperature, 𝐮 is the supercell average of the local mode, E_int is the total energy of the effective Hamiltonian, and ⟨ ⟩ denotes the average over the MC sweeps at every considered temperature. This method will be called MC-2 here. Technically, the computation of α via Eq. (<ref>) is done for a chosen combination of temperature and magnitude of a dc electric field applied along the pseudo-cubic [001] direction, which therefore allows us to determine the effect of temperature and applied electric field on the EC coefficient. In the following, we will also be interested in comparing the predictions of MC-1 and MC-2, mostly because the MC-2 method is less known than MC-1 while being computationally more accurate (since, unlike MC-1, it does not rely on a fit of .∂ P/∂ T|_ E).Regarding the direct approach, we determine the electrocaloric coefficient by using the ramping method of Ref. <cit.> within Molecular Dynamics. First, an Evans-Hoover thermostat <cit.> is used in the MD simulations in order to equilibrate the system at an initial temperature T when no electric field is applied. The electric field is then applied along the pseudo-cubic [001] direction and ramped up (with time) from zero to a specific value, E_f, and then ramped down from E_f to zero. Practically, we chose the time dependence of the applied field ℰ(t) amplitude to beℰ(t)=ℰ_f/2(tanh(t-t_up/τ)-tanh(t-t_down/τ)),where t_up and t_down denote the times when the field magnitude reaches ℰ_f/2 during ramping up and down, respectively. The ramping up/down time frames thus correspond to t_up/down - τ/2≲ t ≲ t_up/down + τ/2,with τ representing the time interval during which the field on/off switching happens. The “hyperbolic tangent” time profile is commonly used in linear response calculations and was chosen to obtain a smooth time dependence of the external field. Notably we observed no significant differences with test calculations where the time dependence of the external field was assumed linear as described in Ref. <cit.>. To test the convergence of results with respect to τ, and the integration time-step Δ t, the test runs were performed for values of τ ranging from 20 ps to 200 ps and values of Δ t from 0.001 fs to 4 fs. All the simulation were performed using the Omelyan second order symplectic integration algorithm <cit.>. Based on the convergence tests, the final chosen value of τ was of 188 ps with Δ t equal to 0.1 fs ensuring the energy conservation for constant field simulation up to the maximum relative error of 10^-6. The inverse rate of the change of the applied field was thus close to 188 fs·cm/kV for the applied field magnitude of 1000 kV/cm. For the chosen simulation parameters, we find that the calculated field induced temperature change upon ramping down Δ T_down is equal in magnitude, but opposite in sign, to the temperature change Δ T_up produced by the switching on the external field for temperatures above T_m — a result that is naturally expected for time-reversible processes. However, for T<T_m, during the ramping down of the applied field the temperature first exhibited a drop which was subsequently followed by an increase (note that this result was also tested for convergence with respect to τ and Δ t). Such behavior, broadly speaking, can be attributed to the loss of ergodicity below T_m. The detailed investigation of the microscopic mechanism responsible for this unusual behavior lies beyond the scope of the current study and, for the purposes of the present work, the EC temperature change Δ T was defined to be equal to Δ T_up, and the α EC coefficient associated with a specific field's magnitude can then be obtained by taking the derivative of Δ T_up with respect to E_f at this specific field's magnitude. Such results will be denoted as “MD” here <cit.>.Note that data from MC-1 and MC-2 approaches can be considered to be associated with the indirect method to obtain EC effects, because they are based on thermodynamic equilibrium. On the other hand, data obtained from MD computations yield the direct EC effects, which may differ from those obtained from the indirect way for systems adopting non-ergodic behavior, as the one that relaxors are known to exhibit below some specific temperature T_m at which the dielectric response peaks <cit.>. Comparisons between our MC and MD results should thus tell us the difference between the indirect and direct ways to extract EC effects in relaxors. Since we are also interested in checking if and how this difference (if any) depends on the investigated temperature region, we decided to focus on four particular representative temperatures. They are: (1) 500 K, which is above the predicted Burns temperature (T_b≃450 K) of BZT <cit.>; (2) 300 K, which is located in-between our critical T^*≃240 K <cit.> and T_b; (3) 200 K, that is now between the computed T_m temperature of BZT (T_m≃130 K) <cit.> and T^*; and (4) 100 K, which is thus below T_m (note that the Supplemental Material <cit.> also shows our results for the EC coefficient in BZT at 600 K).§ RESULTS §.§ EC coefficients Figure <ref> shows the electrocaloric coefficient as a function of electric field, E, for these four different selected temperatures, and as computed from the aforementioned MC-1, MC-2 and MD methods. One can first clearly see that, for any of these temperatures, the (indirect) MC-1 and MC-2 approaches provide nearly identical results. Similarly, α predicted by the (direct) MD scheme agrees very well with those of MC-1 and MC-2 for 200 K, 300 K and 500 K at any field, which demonstrates that indirect methods based on Maxwell thermodynamic relation can be safely used to estimate α above the T_m temperature of relaxors. On the other hand, Fig. <ref>(a) clearly reveals that the EC coefficient of the MD method significantly differs from that predicted by MC-1 and MC-2 at 100 K, as a result of non-ergodicity. In particular, at 100 K, the α deduced from the indirect methods are smaller than that those directly extracted, which is in agreement with previous reports <cit.>. It is also interesting to realize that the EC coefficient of the MD method gets closer to those of MC-1 and MC-2 at 100 K for the highest considered electric fields. This is because, under high electric fields, BZT relaxors can be converted to a normal ferroelectric and thus becomes ergodic <cit.>.Moreover, the results of Fig. <ref>(d) also indicate that α at 500 K is vanishing at small fields and then increases with E, until it very slightly decreases for our highest investigated fields. Interestingly, our values of α for high fields at 500 K are of the order of 0.5×10^-7 K m/V, that is similar to the predicted one of 0.67×10^-7 K m/V in a ferroelectric phase of (Ba,Sr)TiO_3 <cit.>. Figures <ref>(a), <ref>(b) and <ref>(c) also show that, for temperatures below the Burns temperature, α adopts a very clear maximum for an intermediate field (whose value is dependent on temperature) within our investigated range of electric fields. In other words, at temperatures of 300 K, 200 K or 100 K, the EC coefficient first increases with field before noticeably decreasing. Such non-mononotic behavior of α (starting with a vanishing value at small fields and having a peak for an intermediate field before decreasing for larger fields) was indeed measured, as well as reproduced by the so-called phenomenological spherical random bond random field model, in Pb(Mg,Nb)O_3, (Pb,La)(Zr,Ti)O_3 and Pb(Mg,Nb)O_3–PbTiO_3 relaxors in Ref. <cit.>, but only for a specific temperature: namely, the critical temperature at which the discontinuous electric-field-induced ferroelectric transition of these systems becomes continuous (for the value of the electric field associated with the maximum of α). Our results displayed in Fig. <ref> therefore generalize such finding by indicating that, for any temperature, α of BZT can also exhibit a maximum within the investigated field range. Further, note also that BZT differs from the cases of Pb(Mg,Nb)O_3, (Pb,La)(Zr,Ti)O_3 and Pb(Mg,Nb)O_3–PbTiO_3 in the sense that the temperature behavior of the polarization displayed in Fig. <ref>(a) is always continuous for any investigated field. It is worthwhile to know that the maximum of α at a certain field was also predicted to occur in Ba_0.5Sr_0.5TiO_3 <cit.> and defect doped BaTiO_3 <cit.>, and that we also found this non-mononotic behavior of α in the paraelectric phase of BaTiO_3 (BTO) bulk – as evidenced in the Supplemental Material <cit.>. §.§ Analysis of the results via a Landau-like model Let us now try to understand the main results of Fig. <ref>. For that, we start from a simplest Landau free-energy potential describing the behavior of a non-linear dielectric F =F_0(T)+Δ F(T,P, E) =F_0(T)+1/2a(T)P^2+1/4bP^4- EP,where F_0(T) captures the basic temperature dependence of the free energy of the materials, and the other terms account for the variations that involve the development of a polarization or application of an electric field. Note that the temperature dependence of the harmonic a(T) parameter can be a complex one in our BZT compound with various regimes, as inferred from the temperature behavior of the dielectric response under dc field and discussed in Ref. <cit.>: for T>T_b we have a(T)∝(T-T_0), while for T<T_m we have da(T)/dT∼0, and for T_m<T<T_b we have a smooth interpolation between these two regimes (note that (i) T_0 is extracted from the Curie-Weiss behavior of the dielectric response above T_b and can be negative in relaxor ferroelectrics, as predicted and experimentally found in Refs. <cit.> ; and (ii) that the aforementioned behaviors of a(T) implies that it is increasing with temperature above T_m). In the following equations we will work with a generic a(T)>0, noting that the final results have to be interpreted depending on the T region we are in. In particular, the phenomenological equations to be derived here (namely, Eqs. (6)-(16)) can only be safely applied to temperatures above T_m. This is because these equations rely on thermodynamic equilibrium while BZT is non-ergodic below T_m. Finally, the positive parameter b>0 accounts for the saturation of the dielectric response of the material.Let us now discuss the behavior of the EC coefficient as predicted by this simple model. The entropy can be obtained as S=-dF/dT=-dF_0/dT-∂Δ F/∂ T-∂Δ F/∂ PdP/dT.Noting that at equilibrium we have ∂Δ F/∂ P=0, we obtain: S=-dF_0/dT-a'(T)/2P^2,where a'=da/dT. It is then straightforward to derive the following expression for α: α=-.T/C_ E∂ S/∂ E|_T =.Ta'(T)/2C_ E∂ P^2/∂ E|_T =Ta'(T)/C_ EPχ,where χ is the dielectric susceptibility.Interestingly, the behavior of a dielectric for small electric fields can be readily discussed from this expression. Indeed, if P=0 for E=0, then we have P=χ E, which leads to α∝ E, assuming that the dependence of the specific heat C_ E on the electric field can be neglected. This prediction is fully consistent with the null value of α reported in Fig. <ref> at zero field for any temperature, and immediately implies that Δ T∝ E^2 – which shows that the EC effect is null in the limit of small E.To discuss the behavior of α for arbitrary electric-field values, we recall the equilibrium condition ∂ F/∂ P=0 to obtain a(T)P+bP^3= E.Further, if we take the derivative with respect to the electric field on both sides of this equation, we get a(T)χ+3bP^2χ=1,which leads to α=2T/C_ Ea'(T)P/a(T)+3bP^2.This interesting expression implies that, in the limit of large polarizations (or, equivalently, large electric fields), we have α→0. Hence, since we also know that α=0 for E=P=0, it immediately follows that the EC coefficient will present at least one extremum (maximum or minimum) at intermediate values of the electric field, as also consistent with our numerical results of Fig. <ref>. Of course, whether or not such an extremum is experimentally accessible will depend on the breakdown field of a particular material or sample; yet, at least one extremum has to exist in principle. Note also that α will adopt a maximum if a'(T) is positive (which is the case of BZT) while it will possess a minimum if a'(T) is negative.To find the electric field that makes α maximum, we have to solve dα/d E=-2a'(T)/C_ E(χ^2+P_ mχ')=0,where χ'=dχ/d E captures the non-linear dielectric response of the material, and P_ m is the value of the polarization for which α is maximum. The non-linear response χ' is related to P and χ by a(T)χ'+6bPχ^2+3bP^2χ'=0,which we obtain by taking the field derivative of both sides of Eq. (<ref>). From the last two relations, one can show that the condition to have an extremum of α reduces to P_ m^2=a(T)/3b,from which several conclusions can be immediately drawn. First, for stiff materials – i.e., those with a(T)≫0 – the extremum of α will occur at relatively large value of the polarization and applied electric field. Similarly, if the dielectric response is very linear – i.e., for small b>0 –, the extremum of α will also tend to occur for large values of P and E. Finally, using a linear approximation for the polarization as a function of field, P∼χ E, we can write E_ m^2≈a(T)/3bχ^2=4a^3(T)/3b,which provides us with a useful (albeit approximate) expression for the electric field corresponding to α's extremum. For instance, it tells us that E_ m should increase with temperature if a(T) is enhanced with temperature (which is precisely the case for BZT). This increase of E_ m with temperature is indeed confirmed in Fig. <ref> for temperatures above 200 K, and is also consistent with the fact that, at 500 K, the maximum of α occurs for electric fields being close to our highest investigated values.Moreover, the second line of Eq. (9) indicates that α=β T.∂ P^2/∂ E|_T, with β=a'(T)/2C_ E. In other words, assuming that C_ E is independent of temperature and electric field, and that a'(T) is also a constant (which is, e.g., what Curie-Weiss law <cit.> provides), this expression implies that the numerical data of the MC-1 and MC-2 approaches for the EC coefficient should be well fitted by the product of temperature and the derivative of the square of the polarization with respect to electric field, once rescaling this product by a constant <cit.>. Figure <ref> indeed tells us that this is the case for any temperature (especially at and above 200 K, where we are in ergodic equilibrium conditions), since these figures further display the results of such fits by means of solid green curves. In other words, one can safely use Eq. (9) to reproduce and understand the EC coefficients numerically obtained by the indirect methods for any temperature and field (note that the Supplemental Material also shows that Eq. (9) can be accurately used for the α coefficient of typical ferroelectrics, such as BaTiO_3, which further emphasizes its generality). In particular, the second line of Eq. (9) indicates that, for a given temperature, the non-monotonic and unusual behavior of α with fields obtained by MC-1 and MC-2 should be directly related to the dependence of .∂ P^2/∂ E|_T with E. To check such interesting idea, Figs. <ref>(a)-<ref>(d) report the square of the macroscopic polarization as a function of electric field applied along the [001] direction at 100 K, 200 K, 300 K and 500 K, respectively. The central inset of these figures displays the derivative of this quantity with respect to the field, and reveal that, indeed, .∂ P^2/∂ E|_T has the same trend as the indirect EC coefficient of Fig. <ref>. In particular, Figs. <ref>(a)-<ref>(d) reveal that α is very small for low fields at any temperature, simply because the square of the polarization is basically independent of electric fields for small E <cit.>. Such strong connection between α and .∂ P^2/∂ E|_T is reinforced when realizing that the field resulting in a maximum of the α coefficient of the MC-1 and MC-2 methods at 100 K, 200 K, 300 K and 500 K is very close to the field at which .∂ P^2/∂ E|_T is optimal at these temperatures. It is also interesting to realize that the maximal value of the α of the indirect methods increases by a factor of about 3 when increasing the temperature from 100 K to 300 K, while the corresponding maximum of .∂ P^2/∂ E|_T is quite similar between 100 K and 300 K. Such feature can, in fact, be understood by the fact that the second line of Eq. (9) indicates that the EC coefficient is directly proportional to the temperature. In other words, increasing the temperature increases α in case of similar .∂ P^2/∂ E|_T (note that Eq. (9) is also consistent with the computational finding of the enhancement of α with temperature in the ferroelectric phases of (Ba,Sr)TiO_3 in Ref. <cit.>). §.§ Microscopic insights Let us now try to reveal the microscopic origins of the maximum of .∂ P^2/∂ E|_T at 200 K and 300 K (which explains the maximum of the indirect and direct α of these temperatures) as well as the peak of the α obtained by the MD simulations at 100 K (recall that, for temperature below ≃ 130 K, BZT is non-ergodic and thus can not be technically described by Eq. (9)). For that, we focus on the field evolution of the microscopic configurations of BZT at 100 K. Some insets of Fig. <ref>(a) show dipolar snapshots within a given (x, z) plane obtained from MC simulations at 100 K for different electric fields. They reveal that the microscopic dipolar pattern is rather complex and sensitive to electric fields. For instance, there are different polar nanoregions inside which the dipoles centered on Ti ions align along one of the eight ⟨ 111⟩ pseudocubic directions (with this direction varying from one PNR to another, e.g. from [111] to [111̅]), when no external field is applied [see left bottom inset of Fig. <ref>(a)]. Increasing the electric field then leads to the local dipoles of the PNRs rotating towards the field's direction, as well as the formation of rather large PNRs having local dipoles lying along the applied electric field direction [see bottom right inset of Fig. <ref>(a) for a field of 1.2×10^8 V/m]. Finally, Fig. <ref>(a) further indicates that increasing the field up to our considered maximum value E=3.0×10^8 V/m causes nearly all Ti-centered local dipoles to align along the field's direction, which can be seen as indicative that BZT is converting from a relaxor behavior to a normal ferroelectric [see the top right inset of Fig. <ref>(a)].Interestingly, the aforementioned field-induced rearrangement of the local dipoles for fields close to 1.2×10^8 V/m generates a maximal change of the entropy, as evidenced by the fact that Fig. <ref> reveals that the fields associated with maximal values of α obtained by the direct approach at 100 K [see Fig. <ref> (a)] are precisely the fields for which a specific microscopic feature occurs: the number of dipoles pointing along ⟨ 111⟩ pseudocubic directions for which the z-component is positive (i.e., which have a z component parallel to the applied electric field) is maximal for these fields. This microscopic feature was also numerically found (not shown here) for the fields associated with the maximum values of α at 200 K and 300 K (note that BZT does not possess any PNR at 500 K because this latter temperature is above the Burns temperature). §.§ Resulting change in temperature Let us now concentrate on the Δ T change in temperature, associated with the EC coefficient and as computed from Eq. (1), for the four studied temperatures of 100 K, 200 K, 300 K and 500 K. Note that, unlike for 200 K, 300 K and 500 K, this change in temperature will not be the “direct” one for 100 K because the system is non-ergodic at this temperature, while Eq. (1) assumes thermodynamic equilibrium. We nevertheless report in Fig. <ref>(b) the data for Δ T as a function of a change in electric field, Δ E, at 100 K, along with those of 200 K, 300 K and 500 K, for the sake of comparison. Technically, the Δ T of Eq. (1) is computed by integrating the α coefficient calculated by the MC-1 indirect method (see Eq. (2)) from E_1 to E_2, with Δ E being the difference between the magnitude of these two fields and always choosing E_1=2.0×10^7 V/m while varying E_2 when changing Δ E. Two main features can be seen from Fig. <ref>(b): (i) for any temperature, Δ T is not linear with Δ E, as also observed near 310 K in the Ba(Zr_0.2Ti_0.8)O_3 material <cit.> exhibiting relaxor behavior and which is in contrast with, e.g., the cases of the ferroelectric Pb(Zr_0.95Ti_0.05)O_3, Pb(Zr_0.4Ti_0.6)O_3, (Ba_0.5Sr_0.5)TiO_3 and Pb(Mg,Nb)O_3-PbTiO_3 systems reported in Refs. <cit.>; and (ii) for any given electric field above ≃ 1.5×10^8 V/m, Δ T is enhanced when the considered initial temperature increases. Item (i) originates from the fact that α strongly depends on electric field and can even be non-mononotic with E in relaxor ferroelectrics (see Fig. <ref>). Item (ii) can be simply understood by realizing that Eq. (9) provides a dependence of the EC coefficient on temperature. Note that we also numerically checked that our Δ T are not directly proportional to the power 2/3 of the electric field, except for fields above 10^8 V/m at 500 K, which contrasts with the prediction of Ref. <cit.>. Furthermore, our MD predictions for Δ T at 100 K are also given for comparison in Fig. <ref>(b), which demonstrates, once again, that results from direct and indirect approaches differ below T_m. One should also recall that atomic schemes, such as effective Hamiltonians, typically provide an overestimation by one order of magnitude with respect to experiments for electric fields <cit.> while they tend to yield correct values for the EC coefficient (as shown in the Supplemental Material). Experiments are thus called for to determine by which factors the temperatures and fields of Fig. <ref>(b) would have to be rescaled in BZT (if any).§ SUMMARY In summary, we combined an atomistic effective Hamiltonian scheme with Monte-Carlo and Molecular Dynamics techniques to investigate electrocaloric effects in the lead-free BZT systems subject to electric fields of different magnitude and all oriented along the pseudo-cubic [001] direction. It is found that, for any temperature, α exhibits a non-monotonic behavior with field that consists of small values at low fields, followed by an increase up to a maximum before decreasing for larger fields. Below the Burns temperature, this maximum of α is demonstrated to be correlated to a very specific microscopic feature, namely to the largest number of dipoles being oriented along ⟨ 111⟩ directions having positive z-component. Finally, equalities that are derived from a simple Landau model (including one relating α with the product of temperature and the partial derivative of the square of polarization) reproduce and further help to understand the anomalous behavior of α with field and temperature in BZT, for any temperature above T_m (note that we also found that this model can predict EC effects in typical ferroelectrics, such as BaTiO_3, as shown in the Supplemental Material). Our simulations also confirm that indirect and direct approaches yield similar results of the α EC coefficient for any temperature above the T_m temperature but differ from each other for temperature below T_m, because of the non-ergodicity adopted by BZT at these low temperatures <cit.>.We therefore hope that our study leads to a broader knowledge of EC effects and relaxor ferroelectrics. Z.J., Sergei P. and L.B. thank the DARPA grant HR0011-15-2-0038 (MATRIX program). Z.J. also acknowledges support from the National Natural Science Foundation of China (NSFC), Grant No. 51390472, 11574246, U1537210, National Basic Research Program of China, Grant No. 2015CB654903, and China Scholarship Council. Sergey P. thanks ONR Grant N00014-12-1-1034, the grants 3.1649.2017/4.6 from RMES (Russian Ministry of Education and Science) and 16-52-0072 Bel_a from RFBR (Russian Foundation for Basic Research). Y.N. thanks ARO grant W911NF-16-1-0227. We also acknowledge funding from the Luxembourg National Research Fund through the inter-mobility (Grant 15/9890527 Greenox, J.I and L.B.) and Pearl (Grant P12/4853155 Cofermat, J.I. and E.D.) programs. Some computations were also made possible thanks to the MRI grant 0722625 from NSF, the ONR grant N00014-15-1-2881 (DURIP) and a Challenge grant from the Department of Defense.^* The first two authors contributed equally to this work.10 Lines1997M. E. Lines and A. M. 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Commun. 151, 272 (2003).footnoterenrmalizationMD Note that the molecular dynamics simulations conducted here were done without a full account of strain dynamics following the approach of Ref. <cit.>. As a result, the results for the EC coefficient for the MD method were renormalized (multiplied by a factor of C_ E^'/C_ E, where C_ E^' corresponds to specific heat computed from MD simulations using fluctuation-dissipation theorem), in order to account for the concomitant change in the specific heat. Note, however, that we also performed MD simulations that include strain dynamics for some temperatures and found that the corresponding α coefficient is very similar (namely, within 2%) to this renormalized one for investigated electric field values.Lu2010S. G. Lu, B. Roič, Z. Kutnjak, and Q. M. Zhang, Electrocaloric Effect (ECE) in Ferroelectric Polymer Films, I. 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Commun. 8, 15682 (2017). | http://arxiv.org/abs/1706.08963v1 | {
"authors": [
"Zhijun Jiang",
"Sergei Prokhorenko",
"Sergey Prosandeev",
"Y. Nahas",
"D. Wang",
"J. Íñiguez",
"E. Defay",
"L. Bellaiche"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170627175143",
"title": "Electrocaloric effects in the lead-free Ba(Zr,Ti)O$_{3}$ relaxor ferroelectric from atomistic simulations"
} |
-1 s.t. ℤ ℝstyle green/.style= set fill color=green!50!lime!60, set border color=white, , style cyan/.style= set fill color=cyan!90!blue!60, set border color=white, , style orange/.style= set fill color=orange!80!red!60, set border color=white, , hor/.style= above left offset=-0.15,0.31, below right offset=0.15,-0.125, #1 , ver/.style= above left offset=-0.1,0.3, below right offset=0.15,-0.15, #1 *§ * §.§ *§.§.§=7500fancy[LE] [LO][R]firststyle [L]2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license <http://creativecommons.org/licenses/by-nc-nd/4.0/>.The formal publication of this article is available at <https://link.springer.com/article/10.1007/s10601-016-9267-5>. optimize[ 1]Y. Puranik1]Nikolaos V. Sahinidis[1]Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA [Domain reduction techniques for global NLP and MINLP optimization [================================================================= firststyle plainnatOptimization solvers routinely utilize presolve techniques, including model simplification, reformulation and domain reduction techniques. Domain reduction techniques are especially important in speeding up convergence to the global optimum for challenging nonconvex nonlinear programming (NLP) and mixed-integer nonlinear programming (MINLP) optimization problems. In this work, we survey the various techniques used for domain reduction of NLP and MINLP optimization problems. We also present a computational analysis of the impact of these techniques on the performance of various widely available global solvers on a collection of 1740 test problems.Keywords: Constraint propagation; feasibility-based bounds tightening; optimality-based bounds tightening; domain reduction ]§ INTRODUCTION We consider the following mixed-integer nonlinear programming problem (MINLP): . [ min f(x⃗); g⃗(⃗x⃗⃗⃗)⃗≤ 0;x⃗_l≤x⃗≤x⃗_u;x⃗∈ℝ^n-m×ℤ^m ]}MINLP is a very general representation for optimization problems and includes linear programming (LP), mixed-integer linear programming (MIP) and nonlinear programming (NLP) in its subclasses. A variety of applications in diverse fields are routinely formulated using this framework including water network design <cit.>, hydro energy systems management <cit.>, protein folding <cit.>, robust control <cit.>, trim loss <cit.>, heat exchanger network synthesis <cit.>, gas networks <cit.>, transportation <cit.>, chemical process operations <cit.>, chemical process synthesis <cit.>, crystallographic imaging <cit.>, and seizure predictions <cit.>. Modelling via nonconvex objective functions or constraints is necessitated for many of these practical applications. Aside from the combinatorial complexity introduced by the integer variables, nonconvexities in objective function or the feasible region lead to multiple local minima and provide a challenge to the optimization of such problems.Branch-and-bound <cit.> based methods can be exploited to solve Problem <ref> to global optimality. Inspired by branch-and-bound for discrete programming problems <cit.>, branch-and-bound was adapted for continuous problems by Falk and Soland <cit.>. The algorithm proceeds by bounding the global optimum by a valid lower and upper bound throughout the search process. Whenever these bounds are within an acceptable tolerance, the algorithm terminates with the upper bound as a (near-)global optimum. The algorithm exhaustively searches the space by branching on variables to divide the space into subdomains. The lower and upper bounding procedures are recursively applied to the new subdomains in a tree search until the bounds converge. Branch-and-bound algorithms are known to terminate within ϵ-accuracy (ϵ>0) to a global optimum as long as branching, node selection and lower bounding are designed to satisfy certain conditions <cit.>. For special cases, the true global optimum can be finitely achieved with branch-and-bound <cit.>. The success of branch-and-bound methods for global optimization is evident from the numerous software implementations available, including ANTIGONE <cit.>, BARON <cit.>, Couenne <cit.>,LindoGlobal <cit.> and SCIP <cit.>.In this work, we survey the various domain reduction techniques that are employed within branch-and-bound algorithms. While these techniques are not necessary to ensure convergence to the global optimum, they typically speed up convergence. These techniques often exploit feasibility analysis to eliminate infeasible parts of the search space. Alternatively, the methods can also utilize optimality arguments to shrink the search space while ensuring at least one optimal solution is retained. Domain reduction techniques constitute the major component of the solution methods for satisfiability problems through unit propagation <cit.> and for constraint programming (CP) through various filtering algorithms that achieve differing levels of consistencies <cit.>. They are also exploited in artificial intelligence (AI) <cit.> and interval analysis <cit.>. Some of the other names used in the literature for these methods include bound propagation, bounds tightening, bound strengthening, domain filtering, bound reduction and range reduction.Mathematical-programming-based methods for solving nonconvex MINLPs often rely on the solution of a relaxation problem for finding a valid lower bound. The strength of the relaxations employed for lower bounding depends on the diameter of the feasible region. Smaller domains lead to tighter relaxations. For example, Figure <ref> shows the convex relaxation for a simple univariate concave function. A convex relaxation for a given function defined on a nonempty convex set is a convex function that underestimates the given function on its domain. The convex relaxation over the reduced domain provides a better approximation for the univariate concave function thereby providing better lower bounds. Domain reduction techniques not only reduce the diameter of the search space, but they also improve the tightness of convex relaxations.Domain reduction techniques have been extensively studied in various communities including AI, CP, mathematical programming, interval analysis and computer science where they can be viewed as chaotic iterations <cit.>. The primary objective of this paper is to review key domain reduction techniques applicable to (nonconvex) NLPs and MINLPs and point to connections between methods from constraint programming and interval arithmetic communities wherever applicable. We also present results on standard test libraries with different global solvers to demonstrate the impact of domain reduction strategies on their performance.The remainder of the paper is organized as follows. Representation of general MINLPs through factorable reformulations and directed acyclic graphs is described in Section <ref>. Methods for constraint propagation and bounds tightening in global optimization of MINLPs often rely on interval arithmetic. A brief introduction to this topic is provided in Section <ref>. We introduce presolving for optimization in Section <ref>. Domain reduction techniques that rely on eliminating infeasible regions for the problem are described in Section <ref>. Techniques that utilize optimality arguments for carrying out domain reduction are described in Section <ref>. Computational tradeoffs in the implementation of some of the advanced techniques are discussed in Section <ref>. The computational impact of many of these techniques on the performance of widely available solvers is investigated in Section <ref>. Finally, we conclude in Section <ref>.§ REPRESENTATIONA crucial step in the branch-and-bound algorithm is the construction of relaxations. One of the most widely used methods for this purpose is the idea of factorable reformulations. It involves splitting a problem into basic atomic functions that are utilized for computing the function, an idea exploited by McCormick <cit.>, who developed a technique that constructed non-differentiable relaxations of optimization problems. Ryoo and Sahinidis <cit.> gain differentiability by introducing new variables and equations for each of the intermediate functional forms. These functional forms are simple in nature like the bilinear term. A similar idea was proposed by Kearfott <cit.> where new variables and equations are introduced to decompose nonlinearities to allow for more accurate computations of interval Jacobian matrices.Consider the following example: . [ min 3x + 4y; x - y ≤ 4;xy ≤ 3; x^2 + y^2 ≥ 1; 1 ≤ x ≤ 5; 1 ≤ y ≤ 5; ]}A factorable reformulation can proceed by introducing a new variable for every nonlinearity occurring in the model. We replace z_1 for x^2, z_2 for y^2 and z_3 for xy.A factorable reformulation of the model is thus given by:min3x + 4y x - y ≤ 4z_3 ≤ 3z_1 + z_2 ≥ 1z_1 = x^2z_2 = y^2z_3 = xy1 ≤ x ≤ 51 ≤ y ≤ 5It is trivial to outer approximate the univariate terms of this model (cf. Figure <ref>), while the bilinear term in <ref> may be outer approximated by its convex and concave envelopes <cit.>.The combination of these outer approximators provides a convex relaxation of the original problem.In general, factorable reformulations decompose the problem into simpler functional forms for which convex relaxations are known. Thus, a convex relaxation can be obtained by reformulating a problem into its factorable form and relaxing each of the simple functional forms present in the model. For example, see Algorithm Relax f(x) in Tawarmalani and Sahinidis <cit.>.Factorable reformulations can be conveniently represented through a directed acyclic graph <cit.>. In this graph representation, variables (x,y,z) and constants are leaf nodes, vertices are elementary operations (+, -, *, /, log, exp, etc.) and the functions to be represented are the root nodes. Variable and constraint bounds are represented through suitable intervals for the root and leaf nodes. Common subexpressions are combined to reduce the size of the graph as doing so is known to tighten the resulting relaxations <cit.>. Different mathematical formulations can be generated from the same DAG depending on the needs of the solver. Expressions are evaluated by propagating values from the leaves to the root node through the edges in a forward mode. Backward propagation is utilized for the computation of slopes and derivatives. Slopes can also be utilized to construct linear relaxations for the problem. Figure <ref> represents the DAG for Problem <ref>.§ INTERVAL ARITHMETICInterval arithmetic is a system of arithmetic based on intervals of real numbers <cit.>. An interval variable is defined using a variable's lower and upper bounds; the variable itself is restricted to lie between the bounds. Consider the interval variables x⃗ = [x⃗^l, x⃗^u] and y⃗ = [y⃗^l, y⃗^u]. Addition on two intervals can be defined as:x⃗ + y⃗ = [x⃗^l + y⃗^l, x⃗^u + y⃗^u]The addition operator defined by equation <ref> has the following property:x⃗ + y⃗ = {x + y|x ∈x⃗and y ∈y⃗}Extensions to the definitions of other classic operators can be similarly defined:x⃗ - y⃗= [x⃗^l - y⃗^u, x⃗^u - y⃗^l] x⃗×y⃗= [min (x⃗^l ×y⃗^l, x⃗^l ×y⃗^u, x⃗^u ×y⃗^l, x⃗^u ×y⃗^u)] 1/x⃗= [1/x⃗^u, 1/x⃗^l],0 ∉ [x⃗^l, x⃗^u] x⃗/y⃗ = x⃗× 1/y⃗,0 ∉ [y⃗^l, y⃗^u]These operators can be suitably defined to account for infinities within the intervals and for the case when 0 lies in the interval of the denominator in a division operation <cit.>. Natural extensions of factorable functions can be computed by replacing all the elementary operations involved in the computation of a factorable function with their interval counterparts. A natural extension provides valid lower and upper bounds for the value of a function.Floating point arithmetic is susceptible to rounding errors, which can lead to solutions being lost even for simple problems. Neumaier and Shcherbina <cit.> provide an example of a simple MIP problem that leads to incorrect solutions by major commercial solvers due to roundoff errors. Interval arithmetic methods utilize outward rounding to ensure no points are lost due to roundoff errors. These methods are utilized to design rigorous branch-and-bound-based optimization and root finding methods that ensure no solutions are lost due to roundoff errors <cit.>. The methods bound function values through interval arithmetic and carry out domain reduction through branching and fathoming tests based on monotonicity, convexity, infeasibility as well as interval Newton type methods which provide conditions for existence and uniqueness of solutions within a box <cit.>.Interval-based methods have been used for practical applications like solvent blend design <cit.>. Interval-based solvers include ICOS <cit.>, Globsol <cit.> and Numerica <cit.>. Interval-based solvers are often slower than their nonrigorous counterparts. More recently, aspects of safe computations are being introduced with different parts of nonrigorous optimization algorithms.For example, Neumaier and Shcherbina <cit.> describe methods to guarantee safety in the solution of MIPs through suitable preprocessing of the LP relaxations and postprocessing of their solutions. Their methods lead to valid results even when the LP solver itself does not use rigorous methods for obtaining a solution. Borradaile and van Hentenryck <cit.> provide ways of constructing numerically safe linear underestimators for univariate and multivariate functions. Numerically safe methods for computation have also been developed by the CP community, and are referenced in the remainder of the paper including in Sections <ref> and <ref>.§ PRESOLVING OPTIMIZATION MODELSThe idea of analyzing and converting an optimization model into a form more amenable to fast solution is old. Starting with the work of Brearley et al. <cit.>, a number of techniques have been used for analyzing models in the operations research community. We use the term presolve to denote all the techniques used for simplification of optimization models. These techniques have been developed extensively for linear programming models <cit.> and for mixed-integer linear programming models <cit.>. Bixby and Rothberg <cit.> observe that turning off root node presolve at the start of a branch-and-bound search degrades performance of CPLEX 8.0 by a factor of 10.8, while turning off presolve at every node other than the root node degrades the performance by a factor of 1.3 on certain MIP models, demonstrating the importance of presolve. See Achterberg and Wunderling <cit.> for an extensive computational analysis of the impact of various components of presolve algorithms for MIPs. Some of the ideas for simplification of models include:*Elimination of redundant constraints*Identification and elimination of dominated constraints (dominated constraints are constraints with a feasible region that is a superset of the feasible region of other constraints in the model)*Elimination of redundant variables*Assimilating singleton rows into bounds on variables*Tightening bounds on dual variables*Fixing variables at their bounds*Increasing sparsity in the model*Rearrangement of variables and constraints to induce structureSimilar preprocessing operations can be derived for nonlinear problems <cit.>. Some of the general guidelines include:*Avoid potentially undefined functions*Reduce nonlinearity in the model*Improve scaling of model*Increase convexity in the model through reformulations Amarger et al. <cit.> provide a software implementation REFORM to carry out many of these reformulations. Presolve techniques are usually implemented at the solver level by developers of various software for optimization. The modelling systems AIMMS <cit.> and AMPL <cit.> also provide dedicated presolve systems that are applied to all optimization models <cit.>. The success of presolve in mathematical programming has led to efforts for its extension to general constraint programming such as automated reformulation of CP models <cit.>. These methods can lead to considerable simplifications in the model, and can also lead to a reduction in the memory required to solve the problem. Another advantage of these presolve methods is that they are often able to detect infeasibility in optimization models. If a presolved model is infeasible, then the original model is also infeasible. Presolve methods can also be used to detect and correct the causes of infeasibilities for infeasible optimization models. Chinneck <cit.> provides examples of simple cases where infeasibilities can be correctly diagnosed by analyzing the sequence of reductions obtained with presolve. More recently, Puranik and Sahinidis <cit.> provide an automated infeasibility diagnosis methodology through the isolation of irreducible inconsistent sets (IISs). An IIS is defined as an infeasible set of constraints that has every subset feasible. Isolating an IIS can help accelerate the process of model correction by allowing the model expert to focus onto a smaller problem area within the model. The authors of <cit.> propose a deletion presolve procedure that exploits feasibility-based bounds tightening techniques in order to accelerate the isolation of an IIS.While simplified models are often easier to solve than their original counterparts, once an optimal solution has been obtained, the modelling system must return a solution that can be interpreted by the user with respect to the original model form that was specified. Restoration procedures to obtain primal and dual solutions to the original problem are described by Andersen and Andersen <cit.> and Fourer and Gay <cit.>. Such procedures are not discussed here.In the subsequent sections, we review domain reduction techniques that are a major component of all presolve algorithms.§ REDUCTION OF INFEASIBLE DOMAINS The methods described in this section eliminate regions from the search space where no feasible points of Problem <ref> can exist. Global optimization algorithms maintain continuous and discrete variable domains through upper and lower bounds. Domain reduction is achieved by making these bounds tighter. Therefore, domain reduction is also commonly referred to as bounds tightening.The tightest possible bounds based on feasibility of the constraints of Problem <ref> can be obtained by solving the following problems for each of the n variables (k = 1,…,n): . [ min ± x_k; g⃗(⃗x⃗⃗⃗)⃗≤ 0;x⃗^l≤x⃗≤x⃗^u;x⃗∈ℝ^n-m×ℤ^m ]}± x_k in problem <ref> denotes two optimization problems, where x_k and -x_k are individually minimized. Solution of these optimization problems returns the tightest possible bounds on the feasible region. If these bounds are tighter than the user-specified bounds in the model, we can achieve domain reduction by using them. However, since the constraints of Problem <ref> are potentially nonconvex and due to the presence of integer variables, these are expensive global optimization problems, which might be as hard to solve as Problem <ref>. A computationally inexpensive way as compared to problem <ref> to obtain valid bounds for each of the variables is by solving the following problems for each of the n variables (k = 1,…,n): . [min± x_k;g⃗_ conv(x⃗) ≤ 0;x⃗^l≤x⃗≤x⃗^u;x⃗∈ℝ^n ]}where g⃗_ conv(x⃗) ≤ 0 refers to a convex relaxation of the constraints. The convex relaxation can be nonlinear. The integrality restrictions on the variables are relaxed as well. While bounds obtained from Problem <ref> in general are weaker than the bounds obtained from Problem <ref>, they require the solution of convex optimization problems and are therefore obtained more efficiently. Convex relaxations are utilized in branch-and-bound methods for obtaining valid lower bounds to the optimum, and are thus already available for bounds tightening. To exploit the efficiency and robustness of LP solvers, these convex relaxations are linearized through outer approximation to obtain linear relaxations <cit.>. Linear relaxations may provide weaker bounds than nonlinear ones, but can be solved very efficiently. This linearization can also be utilized for obtaining tighter bounds on variables through the solution of the following problems for each of the n variables (k = 1,…,n): . [ min ± x_k; g⃗_ lin(x⃗) ≤ 0;x⃗^l≤x⃗≤x⃗^u;x⃗∈ℝ^n ]}where g⃗_ lin(x⃗) ≤ 0 refers to a linearized outer approximation of the feasible region. The bounds obtained from Problem <ref> are in general weaker than the bounds obtained from Problem <ref>, but require the solution of linear instead of nonlinear programming problems and are therefore solved more efficiently.Feasibility-based arguments for reduction can also be utilized for tightening constraints. Consider a linear set of constraints A⃗x⃗≤b⃗. Tight bounds on constraint i can be obtained with the solution of following optimization problems <cit.>: . [ min ±a⃗_i^Tx⃗; a⃗_j^Tx⃗≤ b_jj = 1,…,i-1, i+1,…,n;x⃗^l≤x⃗≤x⃗^u;x⃗∈ℝ^n ]}Problem <ref> can help identify redundancy if the maximum value of a⃗_i^Tx⃗ is strictly less than b_i. Conversely, if the minimum value of a⃗_i^Tx⃗ is strictly greater than b_i, this also identifies infeasibility.In general, full solution of LP or NLP problems for bounds tightening can be expensive. To balance the computational effort involved with the reduction in bounds obtained, these techniques are usually carried out only at the root node and/or for a subset of variables. They are utilized only sparingly through the rest of the search or not at all. In some cases, however, solving optimization problems for tightening methods throughout the search has been shown to provide significant computational benefits <cit.>.Note that reduction methods exploiting Problems <ref>, <ref> or <ref> are often referred to in the literature as optimality-based bounds tightening. While these methods utilize the solution of optimization problems, they only carry out reduction of infeasible regions from the search space. For this reason, we prefer to classify them under feasibility-based reduction techniques. §.§ Bounds propagation techniques Propagation-based bounds tightening techniques will be simply referred to as propagation in the remainder of the paper.These techniques find their roots in several works in the literatures of mathematical programming <cit.>, constraint logic programming <cit.>, interval arithmetic <cit.>, and AI <cit.>. These methods are often referred to as bounds propagation techniques in CP and are a specialization of constraint propagation. Davis <cit.> refers to a constraint network with nodes (variables) which can take possible labels (domains) and are connected by constraints. He further refers to six different categories of constraint propagation based on the type of information which is updated:*Constraint inference: New constraints are inferred and added.*Label inference: Constraints are utilized to restrict the sets of possible values for nodes.*Value inference: Nodes are partially initialized and constraints are utilized to complete assignments for all nodes.*Expression inference: Nodes are labelled with values expressed over other nodes.*Relaxation: Nodes are assigned exact values which may violate certain constraints.*Relaxation labelling: Nodes are assigned labels using probabilities. Updates in the network involve update of the probabilities.Propagation is equivalent to label inference in the AI community. Davis utilizes the Waltz algorithm <cit.> to describe bounds propagation in a constraint network. In the CP community, these methods are often utilized for solving discrete constraint satisfaction problems (CSP) <cit.> with backtracking-based search methods <cit.>. However, they are also utilized for continuous CSPs <cit.>.Hager <cit.> discusses a reduce operator to eliminate regions outside the solution set for solving systems of constraints. These methods have also been developed extensively in the mathematical programming community, first for LPs, and later for nonlinear programming problems <cit.>. Mclinden and Mangasarian <cit.> demonstrate inference of bounds for simple monotonic complementarity problems and convex problems. Lodwick <cit.> analyzed the relationship between the constraint propagation methods from AI and bound tightening methods from mathematical programming. These methods typically operate by systematically analyzing one constraint at a time to infer valid bounds on variables.Propagation techniques are computationally inexpensive. For example, consider a set of linear constraints:∑_j=1^na_ijx_j≤b_i i = 1,...,k x⃗^l ≤x⃗≤ x⃗^uThe following inequalities are implied by every linear constraint: x_h ≤1/a_ih(b_i - ∑_j ≠ hmin(a_ijx_j^U,a_ijx_j^L )),a_ih > 0x_h ≥1/a_ih(b_i - ∑_j ≠ hmin(a_ijx_j^U,a_ijx_j^L) ),a_ih < 0If these inequalities imply tighter bounds on x_h than the ones specified by the model, the bounds can be updated. For example, consider the set of inequalities:x_1 + x_2≥ 4x_2 + x_3≤ 1x_1 ∈[-2, 4] x_2 ∈ [0, 4] x_3 ∈[-1, 1] From inequality <ref>, we can infer x_1 ≥ 4 - max(0, 4)= 0. Thus, the lower bound of x_1 is updated to 0. From inequality <ref>, we can infer that x_2 ≤ 1 - min(-1, 1) = 2. Thus, the upper bound of x_2 is updated to 2. By analyzing inequality <ref> again, we can update the lower bound of x_1 to 2. The domains of the variables after these bounds tightening steps are x_1 ∈ [2, 4], x_2 ∈ [0, 2]andx_3 ∈ [-1, 1]. Harvey and Schimpf <cit.> describe how bounds can be iteratively tightened in sublinear time for long linear constraints with many variables.Inequalities <ref> and <ref> indicate that only one of the bounds for a variable can be updated from an inequality constraint based on the sign of its coefficient. Achterberg <cit.> formalizes this through the concept of variable locks. Thus, inequality <ref> down locks variables x_1 and x_2, since the lower bounds for x_1 and x_2 cannot be moved arbitrarily without violating inequality <ref>. The concept of variable locks allows for efficient duality fixing of variables. Note that there is no up lock for variable x_1 and no down lock for variable x_3 based on the inequalities <ref> and <ref>. Thus, if the coefficient of x_1 in the objective function is negative, x_1 can be set to its upper bound. Similarly, if the coefficient of x_3 in the objective function is positive, x_3 can be set to its lower bound. The concept of variables locks was extended for CP to develop presolve procedures for cumulative constraints by Heinz et al. <cit.>. Duality fixing is extended by Gamrath et al. <cit.> to allow for fixing of a singleton column. The authors also define dominance between two variables for a MIP and show how dominance information can be used for fixing variables at their bounds.Figure <ref> considers three cases of propagation. In Case 1, bounds for x and y are successfully tightened through iterative application of propagation, with no further tightening possible via Problem <ref>. In Case 2, the bound obtained for x by propagation is the tightest, however the bound for y can be tightened further by Problem <ref>. In Case 3, bounds for neither x nor y can be tightened by propagation, whereas significant reduction is possible via Problem <ref>.In general, reduction in domain through propagation is not guaranteed. Consider the following example:x_1 + x_2≥ 0x_1 + x_2≤ 4-x_1 + x_2≥ -2-x_1 + x_2≤ 2x_1 ∈[-3, 5] x_2 ∈ [-3, 5] The tightest possible domain for x_1 and x_2 based on feasibility arguments is [-1, 3] and can be obtained by solving Problem <ref>. However, analyzing constraints one-at-a-time leads to no reduction of bounds for this problem. The reason for this drawback is that propagation only analyzes one constraint at a time, whereas Problem <ref> utilizes information from all the constraints in the model simultaneously.Belotti <cit.> considers bound reduction by generating a convex combination of two linear inequalities and utilizing this combination for bound reduction. The author proposes a univariate optimization problem to determine the optimal value for the convex multipliers. However, the computational complexity of considering all possible combinations of m constraints in n variables is O(m^2n^3). The author proposes a heuristic scheme which, when implemented only on a subset of the nodes of the branch-and-bound tree, shows computational benefits of using this method. For other presolve-based methods for mixed-integer programs analyzing more than one constraint at a time, see Achterberg et al. <cit.>. Domes and Neumaier <cit.> propose the generation of a new constraint by taking the linear combination or aggregation of all constraints for quadratic problems. This new constraint can uncover new relationships between the variables and can lead to domain reduction. The authors propose the use of the dual multipliers of a local solution in the case of a feasible subproblem and the use of a constraint violation measure in the case of an infeasible subproblem to aggregate constraints. Their method shows computational benefits in reducing the cluster effect (Section <ref>). The notion of global constraints in CP <cit.> is related. A global constraint provides a concise representation for a set of individual constraints. Filtering algorithms on a global constraint effectively carry out domain reduction by considering multiple individual constraints simultaneously. Lebbah et al. <cit.> present a global constraint for a set of quadratic constraints and describe filtering algorithms. Similar ideas have been extended to polynomial constraints <cit.>.Bounds can also be inferred for nonlinear constraints via propagation. Consider a bilinear term of the form x_i = x_jx_k. Then, x_i ≤max{ x_j^Lx_k^L, x_j^Lx_k^U,x_j^Ux_k^L,x_j^Ux_k^U}and x_i ≥min{ x_j^Lx_k^L, x_j^Lx_k^U,x_j^Ux_k^L,x_j^Ux_k^U}represent valid bounds for variable x_i and can be used to potentially tighten any user-specified bounds for this variable.In the context of factorable reformulations, an optimization problem is already decomposed into simple functional forms, which can be readily utilized for domain propagation.Consider, for instance, the factorable reformulation for Problem <ref>. From constraint <ref>, we can infer that 0≤ z_3 ≤ 3. From constraint <ref>, we can infer that 1≤ y ≤ 3.The iterative application of simple constraints such as <ref>–<ref> and <ref>–<ref> is referred to as poor man's LPs and poor man's NLPs <cit.> because this iterative application is an inexpensive way to approximate the solution of linear and nonlinear optimization problems that aim to reduce domains of variables.Propagation based on these techniques can be applied not only to the original problem constraints but also to any cutting planes and other valid inequalities that might be derived by the branch-and-bound solution algorithm.Moreover, all these techniques can be implemented efficiently via the DAG representation. If a bound on a variable x has changed, it can be propagated to other variables that depend on x through a simple forward propagation on the DAG based on interval arithmetic. Similarly, the variables on which x depends can be updated through a backward propagation. §.§ Convergence of propagationPropagation methods can be carried out iteratively as long as there is an improvement in variable bounds. However, these methods can fail to reach a fixed point finitely. Consider as an example <cit.>: x_1 + x_2 = 0, x_1 - qx_2 = 0 with q ∈ (0,1). Propagation will converge to (0, 0) at the limit, i.e., as the number of iterations approaches infinity. On the contrary, any linear programming based method will terminate at (0,0) quickly as the only feasible point. A necessary condition for nonconvergence of propagation iterations is the presence of cycles in expression graphs <cit.>. The problem of achieving a fixed point with propagation iterations in the presence of only integer variables has been shown to be NP-complete <cit.>. The fixed point obtained is independent of the order in which the variables are considered <cit.>. In practice, the iterations are usually terminated when the improvement in variable domains is insignificant or when an upper limit on the number of iterations is reached.For linear inequalities in the presence of continuous variables alone, the fixed point can be obtained in polynomial time by the solution of a large linear program <cit.>. For nonlinear problems and for problems with integer variables, a linear relaxation can be solved to obtain reduced bounds <cit.>. §.§ ConsistencyConstraint programming algorithms for constraint satisfaction problems rely on the formalized notion of consistency to propagate domains along constraint networks and remove values from variable domains that cannot be a part of any solution. Differing levels of consistency exist, including arc consistency and k-consistency, while various algorithms have been proposed to achieve these consistencies <cit.>. These concepts were originally proposed for discrete problems but were also extended to continuous constraint satisfaction problems <cit.>.The domain reduction algorithms typically employed for global optimization of MINLPs are usually not iterated until they reach a certain level of consistency because attempting to establish consistency can lead to a prohibitively large computational effort. However, domain reduction techniques are not even necessary to prove convergence of branch-and-bound based methods for global optimization. In contrast to CP methods, the availability of relaxation methods for generating bounds allows the mathematical-programming-based branch-and-bound algorithms to converge without establishing any levels of consistency. §.§ Techniques for mixed-integer linear programsMany techniques for presolving LPs can also be directlyutilized for MIPs. However, specialized reduction methods can be developed for MIPs. For example, a number of methods have been developed for the analysis of 0-1 binary programs <cit.>. Probing and shaving techniques are equivalent to the singleton consistency techniques from CP. Singleton consistency techniques <cit.> proceed by assigning a fixed value to a variable and carrying out propagation. If this assignment leads to infeasibility of the constraint program, the value assigned to the variable cannot be a part of any solution and can be eliminated. Probing techniques, that can be used for binary programs, involve the fixing of a binary variable x_i to say 0. If basic preprocessing and other domain reduction methods are able to prove infeasibility for this subproblem, then the variable x_i can be fixed to 1. If fixing the binary variable x_i to both 0 and 1 lead to infeasibility, the problem can be declared as infeasible. For binary programs, if probing is carried on all binary variables, it leads to singleton consistency. Probing has also been utilized for satisfiability problems <cit.>. A similar technique for continuous problems is called shaving <cit.>. The method proceeds by removing a fraction of the domain of a variable [x_i^l , x_i^u] as either [x_i^l+ ϵ, x_i^u] or [x_i^l, x_i^u - ϵ] and testing whether the reduced domain leads to provable infeasibility of the model. If infeasibility is indeed proved, the domain of the variable can be reduced to the complement of the box used for testing. The author suggests using ϵ = 0.1× (x_i^u - x_i^l) for this method. Shaving is widely used in the sub-field of CP based scheduling <cit.>. Belotti et al. <cit.> call this technique aggressive bounds tightening (ABT) and implement it in the solver COUENNE. Once a fraction of the domain of a variable has been eliminated, propagation is invoked in the hopes of proving infeasibility in the subproblem. However, ABT and shaving techniques in general are expensive and reduction in the domain is not guaranteed. Faria and Bagajewicz <cit.> propose several variants of this shaving strategy for bilinear terms and utilize it in a branch-and-bound algorithm for water management and pooling problems <cit.>. Nannicini et al. <cit.> propose a similar strategy which they refer to as aggressive probing. In contrast to ABT technique, the nonconvex restrictions created by restricting a variable domain in their method are solved to global optimality with branch-and-bound in order to carry out the maximum possible domain reduction.Implication-based reductions utilize relations between the values of different variables that must be satisfied at an optimal solution. For instance, if fixing two binary variables x_i and x_j to 0 leads to infeasibility, then we have an implication requiring that one of the variables must be nonzero, which can be represented by the inequality x_i + x_j ≥ 1. Such relations can be efficiently represented through the use of conflict graphs <cit.>. Conflict graphs are utilized to generate valid inequalities that strengthen the MIP formulation. In general, if fixing a binary variable x_i to 0 implies that variable x_j must take value v, then the following inequalities are valid for the problem <cit.>: x_j ≤ v + (x_j^u - v)x_ix_j ≥ v - (v - x_j^l )x_iImplications can be derived by carrying out probing by fixing more than one variable at a time and/or through the analysis of problem structure. Implications can also be utilized for identifying and eliminating redundant constraints. Inequalities derived from implications lead to automatic disaggregation for some constraints <cit.>. Disaggregated constraints, while redundant for the MIP formulation, lead to tighter LP relaxations for the problem. Achterberg et al. <cit.> show how implications can be derived from conflicts (See Section <ref>)and from knapsack covers.Tighter formulations can also be obtained for a problem through coefficient reduction. Consider the example of a linear constraint on binary variables from <cit.>: -230x_10-200x_16-400x_17≤ -5The coefficients of this constraint can be reduced to: -x_10-x_16-x_17≤ -1While the set of binary values satisfying both of the above constraints are the same, the reduced constraint has a tighter LP relaxation. Approaches for coefficient reduction are provided by <cit.>. Andersen and Pochet <cit.> prove that, if no coefficients for an MIP system can be strengthened, then there does not exist a dominating constraint that can be used to replace an existing constraint in the MIP system to tighten its relaxation. They also describe an optimization formulation and an algorithmic solution to the problem of strengthening a coefficient in a constraint as much as possible. §.§ Conflict analysisBranch-and-bound based algorithms often encounter infeasibility in subproblems during the search for global optimum. Solvers for the satisfiability problem (SAT) also utilize a backtracking-based branching scheme for their solution <cit.>. The SAT problem consists of binary variables constrained by a set of logical conditions. The SAT problem has a solution if there exists an instantiation of the binary variables satisfying all the logical conditions. SAT-based solvers learn and add conflict clauses from infeasible subproblems <cit.>. These clauses are created by identifying the corresponding instantiations of a subset of variables leading to infeasibility and prohibiting them. This often leads to a reduction in the search tree. The idea of conflict analysis is similar to the idea of no-good learning from the CP community <cit.>. No-good is a generalization of a conflict clause from SAT to CP.The ideas of conflict analysis have been extended for MIP <cit.>. However, generation of conflict clauses is more complicated for MIP due to the presence of both continuous and general integer variables. Infeasibility in SAT problems and CP problems is detected through a chain of logical deductions caused by fixing some variables. However, in MIP, infeasibility can be identified through either such deductions or an infeasible LP relaxation. Achterberg <cit.> proposes a generalized conflict graph (termed implication graph in <cit.>) for representation of bound propagations that can be used to identify cause of infeasibility. Note that the conflict graph and the implication graph are defined differently than in Section <ref>. In the case of an infeasible LP relaxation, Achterberg proposes identifying a minimum bound-cardinality IIS. Representation of conflict causes for MIP requires use of disjunctive constraints which must be reformulated with additional binary variables and inequalities. Limited computational analysis demonstrates a reduction in the number of nodes and time with conflict analysis.§ REDUCTION OF SUBOPTIMAL DOMAINS In contrast to the methods of the previous section, the methods described here can lead to the elimination of feasible points from the domain under the condition that at least one globally optimal solution remains within the search space.Consider the following convex relaxation of Problem <ref>: . [min f_conv(x⃗);g⃗_ conv(x⃗) ≤ 0;x⃗^l≤x⃗≤x⃗^u;x⃗∈ℝ^n ]}As discussed before, convex relaxations are often constructed and linearized in a global branch-and-bound search for obtaining valid lower bounds.Assume that the optimal objective for Problem <ref> has value L and, at the optimal solution, a bound x_j ≤ x_j^u is active with a Lagrange multiplier λ_j > 0. Let U be a known valid upper bound for the optimal objective function value of Problem <ref>. Then, the following constraint does not exclude any optimal solutions better than U (Theorem 2 in <cit.>): x_j^l ≥ x_j^u - U-L/λ_jA geometric interpretation of this cut can be observed in Figure <ref>, where x_j^* denotes the right-hand-side of equation <ref>. The constraint excludes values of x_j for which the convex relaxation is guaranteed to have its value function to be greater than or equal to U. Consequently, the nonconvex problem also has its value function guaranteed to be greater than or equal to U in this domain. A corresponding cut can also be derived if, at the optimal solution L of the convex relaxation, a variable is at its lower bound, i.e., x_j = x_j^l, with the corresponding Lagrange multiplier λ^j > 0. The upper bound can then be potentially tightened without losing optimal solutions by the following cut: x_j^u ≤ x_j^l + U-L/λ_j For variables that are not at their bounds at the relaxation solution, probing tests can be carried out by temporarily fixing variables at their bounds and solving the restricted problem. For example, Tests 3 and 4 in <cit.> work as follows.Set x_j = x_j^u and solve Problem <ref>. If the corresponding multiplierλ^j is positive, then the following constraint is valid: x_j^l ≥ x_j^u - U-L/λ_jA similar probing test can be developed by fixing a variable at its lower bound <cit.>. Reduction by <ref> and <ref> only requires the solution of the relaxation problem, and thus can be implemented at every node of the branch-and-bound tree without much computational overhead. On the other hand, probing with fixing variables at their bounds requires the solution of 2n convex problems. Probing is usually carried out only at the root node and then only at some nodes of the tree or only for a subset of the variables. The probing tests are a generalization of probing for the integer programming case. However, they differ from the integer programming case in the sense that probing leads to variable fixing in the case of binary variables, whereas in the continuous case it leads to reduction in the variable domains.Similarly, if at the optimal solution of the relaxation Problem <ref>, a constraint g^j_ conv(x⃗) ≤ 0 is active with the corresponding multiplier μ_i ≥ 0, then the following constraint does not violate any points with objective values better than U <cit.>: g^j_ conv(x⃗) ≥- U-L/μ_jThese inequalities can be appended to the original formulation.However, this process can lead to the accumulation of a large number of constraints. Alternatively, these are often utilized for propagation-based tests locally at the current node and then discarded. Lebbah et al. <cit.> provide a safe way of implementing duality-based reduction techniques in the context of floating point arithmetic. It must be noted that suboptimal Lagrangian multipliers often provide greater reduction through constraints <ref>, <ref>, <ref> and <ref>. A more detailed discussion on the use of other than optimal multipliers is provided in Tawarmalani and Sahinidis <cit.> and Sellmann <cit.>.If a valid upper bound U is available, it can also be exploited by reformulating Problem <ref> as follows: . [min± x_k; g⃗(⃗x⃗⃗⃗)⃗≤ 0;f_conv(x⃗) ≤ U;x⃗^l≤x⃗≤x⃗^u;x⃗∈ℝ^n ]}Problem <ref> differs from Problem <ref> by the addition of a single constraint. In this constraint, f_conv(x⃗) refers to the convex relaxation of the objective function. It can be replaced by a suitable linearizationf_lin(x⃗) for use in formulation <ref>. . [ min ± x_k; g⃗_ lin(x⃗) ≤ 0; f_lin(x⃗) ≤ U;x⃗^l≤x⃗≤x⃗^u;x⃗∈ℝ^n ]}Zamora and Grossmann <cit.> refer to Problem <ref> as the contraction subproblem. They define a contraction operation that uses the solution of Problem <ref> along with multipliers-based reduction steps described in this section to carry out domain reduction. Their algorithm is therefore referred to as branch-and-contract. Optimality-based arguments can also be used for filtering in CP, see for example <cit.>.Note that problem <ref> can be solved iteratively to obtain further tightening. However, convergence to a fixed point can be slow. Caprara and Locatelli <cit.> propose a different approach to carry out bounds tightening called nonlinearities removal domain reduction (NRDR). NRDR relies on the solution of parametric univariate optimization problems to find tight bounds. The method reduces nonlinearities due to a single variable at every iteration. The authors further demonstrate that, under certain assumptions, their domain reduction method is equivalent to carrying out optimization-based bounds tightening iteratively until it reaches a fixed point. Caprara et al. <cit.> show that the NRDR method leads to bounds consistency for a special case of linear multiplicative programming problems with only two variables in the objective function. §.§ A unified theory for feasibility- and optimality-based tightening Tawarmalani and Sahinidis <cit.> provide a unified theory of feasibility- and optimality-based bounds tightening techniques.This theory evolves around Lagrangian subproblems created by the dualization of constraints.Consider the problem: . [ min f(x⃗); g⃗(⃗x⃗⃗⃗)⃗≤ 0;x⃗_l≤x⃗≤x⃗_u;x⃗∈ℝ^n ]}Define the Lagrangian subproblem as follows: inf_x⃗_l≤x⃗≤x⃗_u{-y_0 f(x⃗) - y⃗g⃗(⃗x⃗)⃗}The dual variables y_0 and y are nonpositive. Suppose an upper bound U is available for the optimal solution of Problem <ref>. A domain reduction master problem can be constructed as follows: . [inf h(μ_0, μ⃗);f(x⃗) ≤μ_0 ≤ U; g⃗(x⃗) ≤μ⃗≤ b;x⃗_l≤x⃗≤x⃗_u ]}Using the linear objective function h(μ_0, μ⃗) = a_0μ_0 + a⃗^Tμ⃗, Problem <ref> can be equivalently stated as: . [ inf a_0μ_0 + a⃗^Tμ⃗;-y_0(f(x⃗) - μ_0 ); -y⃗^T(g⃗(x⃗)-μ) ≤ 0 ∀(y_0,y⃗) ≤ 0;(μ_0,μ⃗) ≤ (U,0);x⃗_l≤x⃗≤x⃗_u ]}Solving Problem <ref> can be as hard as solving Problem <ref>. Instead, the lower bound for Problem <ref> can be obtained from the solution of the following relaxed master problem: . [infa_0μ_0 + a⃗^Tμ⃗; y_0μ_0 + y⃗^Tu⃗;+ inf_x⃗_l≤x⃗≤x⃗_u{-y_0f(x⃗) - y⃗^Tg⃗(⃗x⃗)⃗}; ≤ 0∀ (y_0,y⃗) ≤ 0;(μ_0,μ⃗) ≤ (U,0) ]}Many domain reduction operations can be obtained by suitable choice of the coefficients (a_0, a⃗) in the objective function for Problem <ref>. For example, equation <ref> can be derived by setting (a⃗,a_0) to e⃗_j in Problem <ref>, where e⃗_j is the j^ th column of the identity matrix.Similarly, propagation for linear constraints described by equations <ref> and <ref> can derived by application of duality theory to the relaxed problem formed by relaxing all but the i^ th linear constraint under consideration: minx_h a⃗_i^T x⃗≤ b_ix⃗≤x⃗^ux⃗≥x⃗^lAssuming a_ih < 0 and x⃗_h is not at its lower bound, the optimal dual solution for the linear program can be obtained as: μ =1/a_ih λ⃗_j =-max{a_ij/a_ih, 0} for allj ≠ h σ⃗_j =min{a_ij/a_ih, 0} for allj ≠ h λ⃗_h =σ⃗_h = 0Here, μ is the dual multiplier corresponding to a⃗_i^T x⃗≤ b_i, λ⃗_j is the dual multiplier corresponding to x⃗_j ≤x⃗_j^u, and σ⃗_j is the dual multiplier corresponding to x⃗_j ≥x⃗_j^l. The domain reduction master problem can be constructed as: . [minx_h;- x_h + μ u + λ⃗^T v⃗ - σ⃗^T w⃗≤ 0; u ≤ b_i; v⃗≤x⃗^u; w⃗≤x⃗^l ]}Equation <ref> follows from Problem <ref>. Equation <ref> can be similarly derived.Tawarmalani and Sahinidis <cit.> also present a duality-based reduction scheme that utilizes dual feasible solutions and a learning reduction heuristic. In branch-and-bound algorithms, branching is typically carried out by various heuristics for the selection of the branching variable and the branching point. If one of the nodes created due to a branching decision is proven to be inferior, the learning reduction heuristic attempts to expand on the region defined by this node that is proven to be inferior by the construction of dual solutions for the other node. The authors remark that all dual feasible solutions can be utilized to carry out reductions. This idea is instantiated in the Lagrangian variable bounds propagation by Gleixner and coworkers <cit.>. To avoid solving 2n optimization problems at every node with Problem <ref>, valid Lagrangian variable inequalities can be generated by aggregating the linear relaxation constraints with the dual solution of Problem <ref>. While redundant for the linear relaxation, these inequalities approximate the effect of solving Problem <ref> locally at every node and can be used to infer stronger bounds on variables when variable bounds are updated through branching or otherwise if a better upper bound is obtained. Gleixner et al. <cit.> provides heuristics for ordering the generated Lagrangian variable inequalities to achieve maximum tightening. An aggressive filtering strategy is proposed that involves the solution of LPs to determine variables for which bounds cannot be tightened. The optimization steps for these variables in Problem <ref> can therefore be skipped, leading to computational savings. The authors also provide heuristics for ordering the LP solves in Problem <ref> to allow for more efficient warm starting and reduce the number of simplex iterations.§.§ PruningIf the lower bound obtained at a node of the branch-and-bound search is worse than the best known upper bound U, the current node can be fathomed. We are guaranteed that the globally optimal solution cannot lie at this node, since all feasible solutions at this node have an objective value greater than U. This process is also referred to as pruning and is a special case of a general concept called dominance; node n_1 dominates node n_2 if, for every feasible solution s in n_2, there is a complete solution in n_1 that is as good or better than s. The idea of dominance is old, first introduced by Kohler and Steiglitz <cit.> and developed in more detail by Ibaraki <cit.>. Dominance relations can be utilized to speed up the search. For MIP problems, Fischetti and Toth <cit.> propose solution of an auxiliary MIP involving only fixed variables at a node to determine whether the current node is dominated by another node which may or may not have been explored yet. However, the overhead of solving the auxiliary MIP is fairly large. Fischetti and Salvagnin <cit.> propose improvements to this scheme and demonstrate computational benefits for network loading problems arising in telecommunication. Sewell et al. <cit.> propose a memory-based dominance rule for a scheduling application. Memory-based dominance rules require storage of the entire search tree, and their performance is dependent on the memory available. However, since information from the entire tree is available, considerably stronger pruning rules can be determined. Memory-based search algorithms are related to the heuristic search algorithms from AI <cit.>.Problems involving integer variables often have a high degree of symmetry. Symmetry is detrimental for branch-and-bound algorithms as it can lead to repetitive work for the solver. Most symmetry breaking approaches rely on exploiting information about the specific problem being considered. A more general technique called isomorphic pruning has been proposed by Margot <cit.>. Isomorphic pruning relies on lexicographic tests to determine if a node can be pruned. Orbital branching <cit.> is another method that can be utilized to tackle symmetry. The branching method identifies orbits of equivalent variables. Orbital branching proceeds by fixing a variable in the orbit to one at a node, and fixes all the variables in the node to zero in another node. Thus, orbital branching implicitly prunes all the other nodes which involve each of the other variables in the orbit fixed to one. §.§ Exploiting optimality conditionsThe Karush-Kuhn-Tucker <cit.> conditions are necessary for a point to be locally optimal for a nonlinear programming problem under certain constraint qualifications. See Schichl and Neumaier <cit.> for a derivation and a general discussion of these conditions. The conditions can be used to reduce the search space for a nonconvex NLP, since globally optimal points also satisfy them. Vandenbussche and Nemhauser generate valid inequalities for quadratic programs with box constraints through the analysis of optimality conditions <cit.> and also utilize them in a branch-and-cut scheme <cit.>. Optimality conditions have been extensively utilized in branch-and-bound algorithms for quadratic programs <cit.>. Optimality conditions can also be utilized for pruning of nodes. For example, for an unconstrained optimization problem, if 0 is not contained within the interval inclusion function for the partial derivatives of the objective function at a node, the corresponding node can be pruned <cit.>. This test is often referred to as the monotonicity test.Sahinidis and Tawarmalani <cit.> have added a modelling language construct for BARON which allows for the specification of certain constraints as relaxation-only. Relaxation-only constraints are utilized for the construction of convex relaxations and for domain reduction, but are not utilized in local search for obtaining upper bounds. The authors use first-order optimality conditions explicitly as relaxation-only constraints and observe improved convergence for some univariate optimization problems. Amaran and Sahinidis <cit.> use a similar strategy and show significant computational benefits for parameter estimation problems. They analyze their results and demonstrate that explicit use of optimality conditions aids in domain reduction steps of BARON leading to computational speedups. Puranik and Sahinidis <cit.> propose a strategy for carrying out implicit bounds tightening on optimality conditions for bound-constrained optimization problems. Their method does not require the generation of optimality conditions which can be time consuming and lead to increase in memory requirements. For a large collection of test problems, this strategy leads to computational speedups and reduction in the number of nodes. §.§ Cluster effectBranch-and-bounds methods often undergo repeated branching in the neighbourhood of the global solution before converging. This problem is referred to as the cluster effect and was first studied by Du and Kearfott <cit.> in the context of interval-based branch-and-bound methods. Subsequent analysis <cit.> also demonstrates the importance of convergence order of the bounding operation (see Definition 1 in <cit.>). These results indicate that at least second-order convergence is required to overcome the cluster effect.Thus, tighter relaxations can indeed help mitigate the cluster effect and their development is the subject of extensive research in the area <cit.>. Neumaier and coworkers provide methods to construct exclusion regions for the solution of systems of equations <cit.> and for the solution of optimization problems <cit.>. These exclusion regions guarantee that no other solution can lie within the exclusion box around a local minimizer or a solution for systems of equations. These boxes can then be eliminated from the search space.These boxes are constructed through existence and uniqueness tests based on Krawczyk operator or the Kantorovich Theorem (see Chapter 1, <cit.>). Other methods to construct exclusion regions include back boxing <cit.> and ϵ-inflation <cit.>.§ IMPLEMENTATION OF DOMAIN REDUCTION TECHNIQUESDomain reduction strategies, if successful, typically lead to a reduction in the number of nodes searched in a branch-and-bound tree. Techniques like propagation are computationally inexpensive and can be applied at every node without much overhead. However, they are not as efficient in carrying out domain reduction as the more computationally intensive strategies like Problem <ref> or probing. Thus, there exists a need to balance the effort involved in domain reduction in order to reduce the average effort per node. Multiple heuristic or learning strategies are employed for this purpose. These ideas are important in constraint satisfaction problems where multiple filtering algorithms are available achieving varying degrees of consistency. Stergiou <cit.> experimentally analysed the domain reduction events through filtering techniques. Drawing insights from the clustering of this experimental data, various heuristics are proposed for choosing the propagation algorithm on the fly. Based on insights from Stergiou <cit.> that propagation events often occur in close clusters, Araya et al. <cit.> propose an adaptive strategy. Thus, if a constraint propagation mechanism succeeds in carrying out domain reduction at a given node, it should be exploited repetitively in other nodes that are geometrically close to the node until the method fails. Similar heuristics are also utilized for solution of MINLP problems. For example, the aggressive probing strategy of Nannicini et al. <cit.> solves probing problems to global optimality and thus has a huge computational overhead. To avoid excessive work, Nannicini et al. propose a strategy based on support vector machines <cit.> to predict when this aggressive probing strategy is likely to succeed based on the success of propagation operations. Aggressive probing is only carried out when its chances of success are high. Couenne solves linear versions of Problem <ref> in the branch-and-bound tree in all nodes up to a depth specified by a parameter L. For nodes at a depth d>L, the strategy is applied with a probability 2^L-d. A similar strategy is employed by ANTIGONE. The idea of propagation of Lagrangian variable bounds <cit.> can also be thought of as a means of balancing the computational effort for tightening based on solving variants of Problem <ref>. Vu et al. <cit.> show significant computational benefitsby carrying out propagation on a subset of the nodes and a partial subgraph of the DAG rather than the entire graph. Vu et al. <cit.> present multiple strategies for combining the various reduction schemes from constraint programming and mathematical programming for constraint satisfaction problems.§ COMPUTATIONAL IMPACT OF DOMAIN REDUCTIONWe demonstrate the computational impact of domain reduction techniques on three widely available solvers: BARON <cit.>, Couenne <cit.> and SCIP <cit.>. BARON is commercial <cit.> and also free through the NEOS server <cit.>.SCIP is free for academics and commercial for all others.Couenne is open-source and free software.All three solvers are available under the GAMS modeling system <cit.> and provide options that allow us to turn off their domain reduction algorithms.The global MINLP solvers Antigone <cit.> and LindoGlobal <cit.> are also available under GAMS but were not used in these experiments since they do not offer facilities that turn off their presolve routines.The test libraries used in our tests are the Global library <cit.>, Princeton library <cit.>, MINLP library <cit.> and the CMU-IBM library <cit.>. Global and Princeton libraries consist of NLP problems, while the MINLP and CMU-IBM libraries consist of MINLP problems. While over 25% of the Princeton library are convex and the NLP relaxations of the CMU-IBM library problems are all convex, the Global and MINLP libraries contain mostly nonconvex problems. In the sequel we present results for each library separately so as to demonstrate that the observed trends are not dominated by any particular library, convexity, or integrality properties.Key statistics on the test sets are summarized in Table <ref>.All computational tests were run on a 64-bit Intel Xeon X5650 2.66 Ghz processor running CentOS release 7. The tests were carried out with a time limit of 500 seconds and absolute and relative optimality tolerances set to 10^-6.All solvers were run under two different settings: (1) default options and (2) default options with domain reduction turned off.Our main objective is to compare these two settings. Comparing the relative impact of the individual reduction techniques on the performance of global solvers is beyond the scope of this work. Readers can refer to Kılınç and Sahinidis <cit.> for experiments showing the effect of different bounds reduction strategies in BARON. In the results presented below, the suffix “nr" is used to denote a solver applied with domain reduction techniques turned off.Comparisons are presented in terms of performance profiles generated through PAVER <cit.>.In these profiles, a solver is considered to have solved a problem if it obtains the best solution amongst the solvers compared in the profile within a given multiple of the time taken by the fastest solver that solves a problem. CAN_SOLVE denotes the number of problems in the library that can be solved with all the solvers compared in the profile. §.§ BARONThe solver options and their values utilized for tests with BARON are summarized in Table <ref>. These turn off the various domain reduction techniques utilized in BARON. The remaining options are utilized at their default settings. Figures <ref>, <ref>, <ref> and <ref> demonstrate the performance of BARON with and without domain reduction techniques employed on the Global, Princeton, CMU-IBM and MINLP libraries. The profiles indicate a huge deterioration in performance across all test libraries when reduction is turned off.Interestingly, for the continuous test libraries, the performance profile of the no-reduction version of BARON “catches up” with that of the reduction-based version at the end of the profiles.This simply means that, without reduction, BARON's heuristics are still able to come up with (near-)global solutions but the branch-and-bound algorithm is not able to provide sufficiently strong lower bounds in order to prove global optimality.For the MINLP case, the no-reduction-based algorithm is not even able to find good feasible solutions.Results in Table <ref> show that turning off domain reduction techniques leads to a huge increase in the number of nodes required by BARON. The increase in nodes is also accompanied by a significant increase in computational time across all test libraries. §.§ CouenneThe solver options and their values utilized for tests with Couenne are summarized in Table <ref>. These turn off the various domain reduction techniques utilized in Couenne. The remaining options are utilized at their default settings. Figures <ref>, <ref>, <ref> and <ref> demonstrate the performance of Couenne with and without domain reduction techniques employed on the Global, Princeton, CMU-IBM and MINLP libraries. Similar to BARON, turning off domain reduction techniques has a huge impact on the performance of Couenne. Contrary to BARON, without reduction, this solver is unable to find good solutions for the NLP test libraries; as a result, the performance profiles for the no-reduction-based algorithm do not catch up with the reduction-based algorithm.Table <ref> indicates a significant increase in computational time and the number of nodes explored in the branch-and-bound search when reduction is turned off. It should be pointed out that no comparisons are possible between BARON and Couenne by looking at their respective performance profiles since these profiles depend solely on the solvers included in each figure. §.§ SCIPThe solver options and their values utilized for tests with SCIP are summarized in Table <ref>. As before, the remaining options are used at their default settings.Figures <ref>, <ref>, <ref> and <ref> demonstrate the huge impact of turning off domain reductions with SCIP on the Global, Princeton, CMU-IBM and MINLP libraries. Similar to Couenne, this solver is also not able of finding good solutions for continuous problems when reduction is turned off.Results in Table <ref> indicate a significant deterioration in performance for SCIP as other solvers. §.§ Relative solver performance Figure <ref> describes the performance of BARON, Couenne and SCIP on all test libraries aggregated together. Even without reduction, BARON dominates over the other two solvers, suggesting that this solver has an edge over the other two solvers in terms of its technology for relaxation construction, branching schemes, and primal feasibility heuristics.However, the impact of domain reduction techniques is clear and rather substantial on all three solvers.Interestingly, the relative order of SCIP and Couenne changes when reduction is turned off, suggesting that SCIP is relying much more on domain reduction than Couenne does.Equivalently, SCIP enjoys a substantial advantage over Couenne thanks to its implementation of a more extensive set of reduction techniques.§ CONCLUSIONS We have presented a review of the various domain reduction techniques proposed in literature for the purpose of global NLP and MINLP optimization. These techniques vary in complexity including simple ones like propagation to more computationally intensive ones that involve full solution of optimization subproblems. Application of some of the more complex techniques requires the use of smart heuristics to ensure that they are utilized only when domain reduction is likely. We have also presented computational results with BARON, SCIP and Couenne on publicly available test libraries. The results show that domain reduction techniques have a significant impact on the performance of these solvers. Incorporation of domain reduction within branch-and-bound leads to huge reductions in computational time and number of nodes required for solution. 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Winston (ed.) The Pyschology of Computer Vision. McGraw-Hill, New York (1975)wsb14 Wechsung, A., Schaber, S.D., Barton, P.I.: The cluster problem revisited.Journal of Global Optimization 58, 429–438 (2014)zg99 Zamora, J.M., Grossmann, I.E.: A branch and contract algorithm for problems with concave univariate, bilinear and linear fractional terms.Journal of Global Optimization 14, 217–249 (1999)zs:13:oms Zorn, K., Sahinidis, N.V.: Global optimization of general nonconvex problems with intermediate bilinear substructures.Optimization Methods and Software 29, 442––462 (2013) ] | http://arxiv.org/abs/1706.08601v1 | {
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Near-Earth Object Orbit Linkingwith the Large Synoptic Survey Telescope Steven R. Chesley December 30, 2023 ========================================================================= We further study sets of labeled dice in which the relation “is a better die than” is non-transitive.Focusing on sets with an additional symmetry we call “balance,” we prove that sets of n such m-sided dice exist for all n,m ≥ 3. We then show how to construct a set of n dice such that the relation behaves according to the direction of the arrows of any tournament (complete directed graph) on n vertices. IntroductionConsider the following game: choose a die in Figure <ref>, and then I choose a different die (based on your choice).We roll our dice, and the player whose die shows a higher number wins. In the long run, I will have an advantage in this game:Whichever die you choose, I will choose the one immediately to its left (and I will choose die C if you choose die A).In any case, the probability of my die beating yours is 19/36 > 1/2. This is a case of the phenomenon of non-transitive dice, first introduced by Martin Gardner in <cit.>, and further explored in <cit.>, <cit.>, and <cit.>. Fix integers n, m ≥ 3.For our purposes, a set of n m-sided dice is a collection of pairwise-disjoint sets A_1, A_2,…, A_n with |A_i| = m and ∪ A_i = [n· m] (here and throughout, [k] = {1, 2, …, k}).We think of die A_i as being labeled with the elements of A_i.Each die is fair, in that the probability of rolling any one of its numbers is 1/m.We also write P(A≻ B) for the probability that, upon rolling both A and B, the number rolled on A exceeds that on B, and A≻ B if this probability exceeds 1/2. A set of dice is non-transitive if A_i≻ A_i+1 for all i (and A_n≻ A_1).That is, the relation “is a better die than” is non-transitive. In this paper we (mostly) examine non-transitive sets of dice, but we introduce a new property as well. A set of dice is balanced if P(A_i≻ A_i+1) = P(A_j≻ A_j+1) = P(A_n≻ A_1) for all i and j. This value is called the victorious probability of the set. Note that the set of dice in Figure <ref> is balanced, as P(A ≻ B) = P(B ≻ C) = P(C ≻ A) = 19/36.A graph is an ordered pair (V,E) where V is a set of vertices and E is a set of unordered pairs of vertices called edges. A directed graph is a graph where the edges are ordered pairs. We can then also think of a set of dice as the set of vertices of a graph, and having P(A≻ B)>1/2 may correspond to a directed edge from A to B.In <cit.>, Schaefer and Schweig showed that non-transitive balanced sets of n m-sided dice exist for n=3,4 and all m ≥ 3. A directed graph G is realizble by a set of dice D if there is a one-to-one map f:D→ V(G) such that P(A≻ B)>1/2 implies that (f(A),f(B)) is a directed edge of G. The main results of <cit.> could then be restated as follows. Directed 3- and 4-cycles are realizable by sets of balanced dice with any number m≥3 of sides. Our first goal here is to generalize this statement so that the cycle may also be of any length. Then, we will generalize again from directed C_n (cycles) to directed K_n (complete graphs).Realizing cycles as diceThe main goal in this section is to prove the following. For any n,m ≥ 3, there exists a non-transitive set of n balanced m-sided dice. An example to illustrate our procedure will be useful.We start with a set of balanced non-transitive dice, and would like to add another one.[ A:951; B:843; C:762 ]↦[ A: 9̂ 5̂ 1̂; B: 8̂ 4̂ 3̂; C: 7̂ 6̂ 2̂; D:??? ]We require that C≻ D and D≻ A. But We already have C≻ A, and so these three dice are totally ordered: C>D>A. As such, we can move outside of ℕ by copying C to D and lowering all values by, say, 1/10.[A: 9 5 1;B: 8 4 3;C: 7 6 2;D: 6.9 5.9 1.9 ]This new set of dice does indeed have C≻ D≻ A, and so we are done if we only seek non-transitivity. However, the original set was balanced, and this will likely not be (we have P(C≻ D)=m+12/m^2>P(D≻ A)). So, if we count the number of “victories” of our original set (the numerator of our probability), we can raise values on D by 1/10 instead of lowering them to lower the number of victories of C over D. We can alter this number by any amount we desire, from 1 to m2 (by raising every value on D), and so can match the desired victorious probability. The last step is to return to ℕ by relabeling linearly.[A: 9 5 1;B: 8 4 3;C: 7 6 2;D: 6.9 5.9 1.9 ]↦[A: 9 5 1;B: 8 4 3;C: 7 6 2;D: 6.9 5.9 2.1 ]↦[ A: 1261; B: 1154; C: 1082; D:973 ]. This procedure is general. We proceed by induction. Our base case, with three dice (n=3), is done (for arbitrary number of sides m≥3, see <cit.>). So assume we have a set of k balanced non-transitive m-sided dice, A_1,…,A_k. Create a new die A_k+1 whose entries are those of A_k, each shifted down by some ϵ<1. The set of dice A_1,…,A_k+1 could be relabeled linearly from [(k+1)m], which would complete the proof if the condition of balance were omitted from the theorem (we could then also omit it from the proof). But, by shifting the entries of A_k+1 up by ϵ rather than down, we alter P(A_k≻ A_k+1) while keeping P(A_k+1≻ A_1) the same (the victorious probability we started with). This recovers the condition of balance. TournamentsA tournament is a directed complete graph. Two vertices x,y in a directed graph are strongly connected if there is a directed path from x to y and also one from y to x. Under this equivalence relation, the vertices of a directed graph are sorted into strongly connected components (or strong components). A strongly connected directed graph is one with only one strong component. We know from Moon <cit.> that a tournament is strong if and only if it contains a directed cycle of every length. He also shows, in particular, a tournament is strong if and only if it contains a directed Hamilton cycle.Given a directed graph, we may form a new directed graph from it by contracting each connected component down to a single vertex. The result will likely have parallel edges, but all edges between any two vertices point in the same direction; delete all but one of each parallel edge group. The result, called the condensation, is always acyclic. For a strong directed graph, the condensation is a single vertex.Because each vertex in the condensation of a directed graph contains a directed cycle on all its vertices of the original directed graph, we can give a set of non-transitive dice (one for each vertex of the condensation) that realizes the cycle. The question, then, is about any edges between vertices not adjacent (with an edge in either direction) in the cycle. Namely, can we choose or manipulate our dice to obey these edges as well? We will answer this question by constructing dice that realize any tournament.If the tournament is not strong, the condition of balance will be impossible. However, strong tournaments can be realized by balanced dice, and so we can create a set of balanced dice for each vertex in the condensation (strong component), and then shift the labels to obey the total order given by the condensation. The problem of realizing tournaments then reduces to the problem of strong tournaments, which we will use to our advantage.Given a strong tournament, locate a directed Hamilton cycle as a subgraph. This cycle alone can be realized by balanced non-transitive dice by Theorem <ref>. We will then augment our dice to account for the other edges.Again, an example will be helpful. Start with a directed 5-cycle, and a set of 5 balanced non-transitive dice with 3 sides (constructed by repeating the procedure of the previous example).3 [scale=.4] [draw, shape=circle] (1) at (90:4) A; [draw, shape=circle] (2) at (18:4) B; [draw, shape=circle] (3) at (306:4) C; [draw, shape=circle] (4) at (234:4) D; [draw, shape=circle] (5) at (162:4) E;[->, line width=2pt] (1)–(2); [->, line width=2pt] (2)–(3); [->, line width=2pt] (3)–(4); [->, line width=2pt] (4)–(5); [->, line width=2pt] (5)–(1);[ ; A: 1571; B: 1465; C: 13 102; D: 1293; E: 1184 ] For every edge we add, we will need to add sides to our dice: one above and one below, for a total of 2(n2-n)=n^2-3n extra entries. Because half of them are below, we shift all our labels up by (n2-n)=n^2-3n/2.3 [scale=.4] [draw, shape=circle] (1) at (90:4) A; [draw, shape=circle] (2) at (18:4) B; [draw, shape=circle] (3) at (306:4) C; [draw, shape=circle] (4) at (234:4) D; [draw, shape=circle] (5) at (162:4) E;[->, line width=2pt] (1)–(2); [->, line width=2pt] (2)–(3); [->, line width=2pt] (3)–(4); [->, line width=2pt] (4)–(5); [->, line width=2pt] (5)–(1);[ ; A: 25 17 11; B: 24 16 15; C: 23 20 12; D: 22 19 13; E: 21 18 14 ] We will add the missing edges (which can be done in any order) by choosing the two numbers above and the two numbers below our existing labels. Count the number of victories that existed to begin with on the missing edge. For three-sided non-transitive dice, it will be either 4 or 5. If the die we want to be victorious had 5, it gets the smaller number of the two below (it doesn't need one more). Otherwise it gets the larger. The other die gets the opposite. Of the two numbers above, the larger goes on the die we want to be victorious (the smaller on the other). So to add the edge (A,C), A will get the larger of {9,10} (as currently P(A≻ C)=4/9), and it also gets the larger of {26,27} (so that A will beat C).3 [scale=.4] [draw, shape=circle] (1) at (90:4) A; [draw, shape=circle] (2) at (18:4) B; [draw, shape=circle] (3) at (306:4) C; [draw, shape=circle] (4) at (234:4) D; [draw, shape=circle] (5) at (162:4) E;[->, line width=2pt] (1)–(2); [->, line width=2pt] (2)–(3); [->, line width=2pt] (3)–(4); [->, line width=2pt] (4)–(5); [->, line width=2pt] (5)–(1); [->, line width=2pt] (1)–(3);[ A: 27 25 17 11 10; B:24 16 15 ; C: 26 23 20 129; D:22 19 13 ; E:21 18 14] Now add, say, (B,D). That means B gets 29 and D gets 28. In the original, P(B≻ D)=5/9, so B gets 7 and D gets 8.3 [scale=.4] [draw, shape=circle] (1) at (90:4) A; [draw, shape=circle] (2) at (18:4) B; [draw, shape=circle] (3) at (306:4) C; [draw, shape=circle] (4) at (234:4) D; [draw, shape=circle] (5) at (162:4) E;[->, line width=2pt] (1)–(2); [->, line width=2pt] (2)–(3); [->, line width=2pt] (3)–(4); [->, line width=2pt] (4)–(5); [->, line width=2pt] (5)–(1); [->, line width=2pt] (1)–(3); [->, line width=2pt] (2)–(4);[ A: 27 25 17 11 10; B: 29 24 16 157; C: 26 23 20 129; D: 28 22 19 138; E:21 18 14] Note that (A,B) (and all others) remain correct! When we add (B,D), a value larger than any yet on A appears on B, but so does a value smaller than any yet on A, so the net change on the number of victories of A over B is zero.We proceed in this fashion. The order the edges were added here was: (B,E), (C,E), (A,D). This gives the set of dice below. If edges were added in a different order, a different set of dice, also with the features of this one, would be produced.3 [scale=.4] [draw, shape=circle] (1) at (90:4) A; [draw, shape=circle] (2) at (18:4) B; [draw, shape=circle] (3) at (306:4) C; [draw, shape=circle] (4) at (234:4) D; [draw, shape=circle] (5) at (162:4) E;[->, line width=2pt] (1)–(2); [->, line width=2pt] (2)–(3); [->, line width=2pt] (3)–(4); [->, line width=2pt] (4)–(5); [->, line width=2pt] (5)–(1); [->, line width=2pt] (1)–(3); [->, line width=2pt] (2)–(4); [->, line width=2pt] (2)–(5); [->, line width=2pt] (3)–(5); [->, line width=2pt] (1)–(4);[ A: 35 27 25 17 11 102; B: 31 29 24 16 1575; C: 33 26 23 20 1293; D: 34 28 22 19 1381; E: 32 30 21 18 1464 ]Let G be a strong tournament. There is a set of balanced non-transitive dice realizing G.Let G be a strong tournament on n≥3 vertices; we construct a set of dice which will have 3+2(n-3)=2n-3 sides (to make the condition of balance easy to recover). Because G is strong, it contains a directed Hamilton cycle. By Theorem <ref>, there is a set {A_1,…,A_n} of balanced non-transitive dice realizing this subgraph of G with 3 sides. The only possible victorious probability of such a set of dice is 5/9. Further, the number of victories of any die over any other (i.e. those not adjacent in the cycle) is either 4 or 5. We will add sides to our dice to account for the edges other than those in the cycle. Add n^2-3n/2 (an integer) to all the labels for the A_i, which will now be labeled by {n^2-3n/2,n^2-3n/2+1,…,3n+n^2-3n/2}. Choose a directed edge (A_i,A_j) other than one in the cycle, and assume (without loss) A_i≻ A_j. Calculate P(A_i≻ A_j) as it stands in the original set, which is either 4/9 or 5/9. Append the label 3n+n^2-3n/2+2 to A_i and 3n+n^2-3n/2+1 to A_j. Append the smaller two labels as follows: A_i gets 3n+n^2-3n/2-2, A_j gets 3n+n^2-3n/2-1 if P(A_i≻ A_j)=5/9, the opposite If P(A_i≻ A_j)=4/9.The placement of the larger number on A_i will add 5 additional victories, and the placement of a smaller number on A_j will add 3 victories to A_i (one for each original element of A_i and either 0 or 1 more based on what P(A_i≻ A_j) was to begin with. The new victorious probability is 13/25. It is clear that the repetition of this process allows the new set of dice to obey the edges outside the cycle. We also remark that we do not negatively impact the edges in the cycle: if, at the j^th stage we append two numbers to A_i, and in step k>j we append two numbers to A_i+1, A_i+1 gains k extra victories over A_i from the large label, but k extra losses from the small label. We can iterate through all the “chords” of the Hamilton cycle (in any order), and we finish with a set of dice labeled by [3n+2(n2-n)]=[n^2] which is balanced, non-transitive, and obeys all arrows in the tournament. Further QuestionsIf a directed graph is acyclic, it is trivially realizable even by one sided dice (acyclic graphs correspond to total orderings). By this and previous observations, along with Theorem <ref>, we have the following, which puts everything together nicely. Let G be a directed graph. There is a set of dice realizing G. Moreover, the dice may be made balanced if and only if G is a subgraph of a strong tournament. The condition on balance suggests the following. A directed graph is strongly connectable if the missing edges can be added and directed in a way such that the resulting tournament is strong.Is there a necessary and sufficient condition for strong connectability? This question was answered (in this form) by Joyce, Schaefer, West, and Zaslavsky, who showed in <cit.> that the obvious necessary condition on G of containing no complete directed cut (a complete collection of edges between a partition of the vertices into two nonempty sets, all pointing one way) is also sufficient. The method of proof of Theorem <ref> suggests: What is the minimum number of sides required for a set of n dice to realize a strong tournament? Any tournament? A natural generalization is to the realm of weighted directed graphs. Let G be a weighted directed graph. Can G be realized by a set of dice such that the probability of one die beating another depends (in some way) on the weight of the edge between the corresponding vertices? 1FFF Edward J. Barbeau. Mathematical Fallacies, Flaws, and Flimflam. The Mathematical Association of America, 2000.MGntD M. Gardner. The paradox of the nontransitive dice and the elusive principle of indifference. Scientific American, 223:110–114, 1970.MGnt M. Gardner. On the paradoxical situations that arise from nontransitive relations. Scientific American, 231:120–125, 1974.JSWZ S. Joyce, A. Schaefer, D. West, and T. Zaslavsky. Strongly connectable digraphs and non-transitive dice. In submission.moon John W. Moon, Topics on Tournaments. Holt, Rinehart and Winston, New York, 1968.ntD1 Richard P. Savage Jr. The paradox of nontransitive dice. The American Mathematical Monthly, 101(5):429–436, May 1994.bntd1 A. Schaefer and J. Schweig. Balanced Non-Transitive Dice. College Math. J., 48 (2017), no. 1, 10–16. | http://arxiv.org/abs/1706.08986v1 | {
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1Classical Spin Nematic Transition in LiGa_0.95In_0.05Cr_4O_8 G. J. Nilsen December 30, 2023 ============================================================ Continuing the study of <cit.> on the critical probability of the bootstrap percolation on Galton-Watson trees, we analyze the metastable states near criticality. We find that, depending on the exact choice of the offspring distribution, it is possible to have several distinct metastable states, with varying scaling of their duration while approaching criticality. § INTRODUCTION Bootstrap percolation is a deterministic dynamics in discrete time first introduced in <cit.> in order to model disordered magnetic systems, and broadly studied since in many different contexts. Fix a graph G and a parameter r∈ℕ. Each vertex of the graph can be in one of two states – infected or healthy, which are initially distributed independently with probabilities p and q=1-p. At each time step we update these states, such that the infected vertices remain infected, and a healthy vertex becomes infected if it has at least r infected neighbors. One may also consider more general infection conditions, such as the oriented bootstrap percolation – when the graph G is oriented, and we require at least r edges to point at infected vertices. Bootstrap percolation on various deterministic graphs has been the subject of extensive research. On the grid [n]^d, the probability that all vertices are eventually infected, as a function of p (or equivalently q), has been profoundly studied in <cit.>. For (d+1)-regular infinite trees, with 2≤ r ≤ d, it is shown in <cit.> that a phase transition occurs. Defining q_c to be the supremum over all q such that starting with probability q to be healthy all vertices end up being infected with probability 1, an explicit expression for q_c is found, and it is furthermore proven that q_c lies in the open interval (0,1). In addition, it is determined, depending on d and on r, when the transition is continuous and when it is discontinuous. In <cit.> the details of the metastability properties are studied, describing the time evolution of the probability that the root stays healthy near criticality.Random environments have also been of interest in this field, e.g., the bootstrap percolation on a polluted grid <cit.>, the random graph G_n,p <cit.>, the random regular graph <cit.>, and the Galton-Watson tree <cit.>.In this paper, we will analyze the metastability of the bootstrap percolation on a directed Galton-Watson tree, i.e., the time behavior near criticality of the probability that the root is infected. In Prevalence we present an interpretation of this probability as the almost sure prevalence – the limiting ratio of infected vertices. In Critical-Behavior we will study the zoology of the metastabilities for different offspring distributions, showing that this model introduces a vast variety of possible behaviors. Finally, in othertransitions we comment on other phase transitions that may occur. § MODEL AND NOTATIONS Fix an infection threshold r≥2, and consider a Galton-Watson tree G whose offspring distribution is supported on r,r+1,… That is, defining ξ_k to be the probability that a vertex has k children, we require ξ_k=0 for k<r.In the beginning, we decide for each vertex of G whether it is infected or healthy, independently with probabilities p and q=1-p respectively. Then, at each time step t, a healthy vertex will get infected if it has at least r infected children. Let us denote by ϕ_t^G the (random) probability that the root is healthy at time t, so in particular ϕ_0^G=q. Note also that ϕ_t^G is decreasing in t. The expected value of ϕ_t^G over all graphs G, generated with offspring distribution ξ, will be denoted ϕ_t^ξ.One particular case, that has been studied in <cit.>, is the case of a rooted (d+1)-regular tree, i.e., ξ_k=_k=d. Here, one can find ϕ_t^d recursively using the relationϕ_t+1^d= h_d(ϕ_t^d);h_d(x) = qℙ[(d,1-x)≤ r-1].For the GW tree, such a recursion still holds for the expected value ϕ_t^ξ:ϕ_t+1^ξ= h_ξ(ϕ_t^ξ);h_ξ(x) =∑_k=r^∞ξ_k h_k(x). § RESULTS §.§ Prevalence and ϕ_t The relation in (<ref>) allows us to find the expected value of ϕ_t^G, but for a specific realization of G, ϕ_t^G may differ from that value. For example, fixing t, there is a nonzero probability that a finite neighborhood of the root will have many vertices of high degree, which will result in a smaller ϕ_t^G. However, we will see that ϕ_t^ξ describes almost surely another observable – the prevalence, i.e., the limiting fraction of infected vertices.First, denote by B(R) the ball of radius R around the root. We can then define the R-prevalence at time t asρ_R(t)=|{B(R)t}|/|B(R)|.It is natural to expect ρ_R(t) to be close to 1-ϕ_t^ξ, and this is indeed the case, as shown in the following proposition:Fix t. Then lim_R→∞ρ_R(t)=1-ϕ_t^ξ almost surely (in both the graph and the initial state measures).§.§ Critical Behavior Following <cit.>, we define the critical probabilityq_c=sup_[0,1]{ q: ϕ_∞^ξ=0} .In order to analyze this criticality, defineg_k(x) =h_k(x)/qx,g_ξ(x) =h_ξ(x)/qx.In <cit.>, the following fact is shown: Fix ξ. Then: * For a given q, ϕ_∞^ξ is the maximal solution in [0,1] of g_ξ(x)=1/q, and 0 if no such solution exists.* q_c=1/max_[0,1]g_ξ(x). We will consider here the behavior near criticality, at q slightly smaller than q_c.For 0<x<1 and some positive δ, the δ-entrance time of x isτ^-_x,δ(q) = min{t: ϕ^ξ_t<x+δ},and the δ-exit time is defined asτ^+_x,δ(q) = min{t: ϕ^ξ_t<x-δ}. Fix δ>0. We say that the critical point is δ-(ν_1,…,ν_n)-metastable at x_1>…>x_n>0 if, for q↗ q_c, the following hold: * τ_x_1,δ^-=O(1).* log(τ_x_i,δ^+-τ_x_i,δ^-)/log(q_c-q)-1+1/2ν_i for i=1,…,n.* τ_x_i+1,δ^--τ_x_i,δ^+=O(1) for i=1,…,n and x_n+1=0.We say that the critical point is (ν_1,…,ν_n)-metastable at x_1>…>x_n if it is δ-(ν_1,…,ν_n)-metastable at x_1>…>x_n for small enough δ. See multiplemetastabilities.The following theorem gives a full classification of the metastability properties:Fix ξ. Then the metastable behavior is determined by one of the following cases: * g_ξ attains its maximum at 1. In this case the critical probability is 1.* g_ξ has a unique maximum at 0. In this case the phase transition is continuous. At the critical pointlog(ϕ_t^ξ)/log t -1/ν,where ν is determined by the asymptotic expansion g_ξ(x)=1/q_c-Cx^ν+o(x^ν). * The maximum of g_ξ is attained at the points x_1,…,x_n for 1>x_1>…>x_n>0, and possibly also at 0. In this case the phase transition is discontinuous. For i=1,…,n we may write around x_ig_ξ(x)=1/q_c-C_i(x-x_i)^2ν_i+o((x-x_i)^2ν_i),with some C_i>0.Then the critical point is (ν_1,…,ν_n)-metastable at x_1>…>x_n. In the first case, where the critical probability is 1, it is not clear whether or not an asymptotic expansion exists, since g_ξ is not guaranteed to be analytic. When it does exist, one can recover a result similar to Case 3.Finally, we show the main result – that the different metastability behaviors described above are possible: * Let ν∈ℕ. Then there exists ξ such that the phase transition is continuous, and satisfies (<ref>) at criticality.* Let (ν_1,…,ν_n)∈ℕ^n. Then there exist ξ and x_1>…>x_n such that the critical point is (ν_1,…,ν_n)-metastable at x_1>…>x_n. § PROOFSThe idea of the proof is to notice that the main contribution to the prevalence comes from the sites close to the boundary, and then use their independence. Thus, we fix a width w, and considerρ_R,w(t)=|{B(R)∖ B(R-w)t}|/|B(R)∖ B(R-w)|.First, we claim that ρ_R(t) is approximated by ρ_R,w(t) for large w. More accurately, we have |B(R-w)|≤2^-w|B(R)|, which also implies that the number of infected vertices in B(R)∖ B(R-w) is the same as the number of infected vertices in B(R), up to a correction of order 2^-w|B(R)|. Thenρ_R(t)=ρ_R,w(t)+O(2^-w).We would now like to bound the distance between ρ_R,w(t) and 1-ϕ_t^ξ. Let ε>0, and, by (<ref>), take w big enough such that |ρ_R(t)-ρ_R,w(t)|<ε/2 uniformly in R. Note that ρ_R,w(t) is a weighted average of the w random variables ρ_R,1(t),ρ_R-1,1(t),…,ρ_R-w+1,1(t), and consider one of these variables, ρ_r,1(t). This variable is the average of the random variables _v for all vertices v of distance r from the root, and since these are independent Bernoulli random variables with mean 1-ϕ_t^ξ, and since there are at least 2^R-w+1 such variables, we can use a large deviation estimate, yielding ℙ[|ρ_r,1(t)-(1-ϕ_t^ξ)|>ε/2]≤ e^-c 2^R-w+1for a positive c that only depends on ε and on ϕ_t^ξ. Since for 1-ϕ_t^ξ to be far from ρ_R,w(t) it must be far from at least one of the variables ρ_R,1(t),ρ_R-1,1(t),…,ρ_R-w+1,1(t), we have ℙ[|ρ_R,w(t)-(1-ϕ_t^ξ)|>ε/2]≤ we^-c 2^R-w+1.Hence, ρ_R(t) is ε-close to 1-ϕ_t^ξ with probability larger than 1-we^-c 2^R-w+1, which concludes the proof by the Borel-Cantelli lemma.Before proving Theorems <ref> and <ref>, we will need a couple of small results.g_k is a polynomial of degree k-1, whose lowest degree monomial is of degree k-r.By equations <ref> and <ref>g_k(x) =ℙ[(k,1-x)≤ r-1]/x =∑_i=0^r-1ki(1-x)^ix^k-i-1;therefore all monomials are of degree between k-r and k-1. The coefficient of x^k-r is kr-1≠0, and the coefficient of x^k-1 is ∑_i=0^r-1ki(-1)^i, which is also nonzero since 0<r-1<k. This concludes the proof.g_r(x),…,g_m(x),x^m-r+1,…,x^m-1 is a basis of the linear space of polynomials of degree smaller or equal to m-1.Denote v_1(x)=g_r(x),…,v_m-r+1(x)=g_m(x),v_m-r+2(x)=x^m-r+1,v_m(x)=x^m-1. By gkispolynomial, all v's are of degree smaller or equal to m-1. Moreover, the matrix whose (i,j) entry is the coefficient of x^j in the polynomial v_i is upper triangular, with nonzero diagonal. This shows that { v_i} _i=1^m is indeed a basis. We will also use the following result from <cit.>:[Claim 3.9 of <cit.>]For ξ_k=r-1/k(k-1), g_ξ(x)=1.We are now ready to prove Theorems <ref> and <ref>. First, we note that g_k(1)=1 for all k, so in particular the series ∑_k=r^∞ξ_kg_k(x) converges at x=1.By gkispolynomial, the monomials of degree up to n of the partial sum ∑_k=r^Nξ_kg_k(x) are fixed once N>n+r. From these two facts we conclude that g_ξ(x) is analytic in (-1,1) and continuous at 1. Thus, cases 1, 2 and 3 exhaust all possibilities.The result will then follow from general arguments of dynamical systems near a bifurcation point. Since the exact calculations are a bit tedious, we only give here a short sketch of the argument, referring to the appendix for the complete proof.For case 2, the expressionϕ_t+1 = ϕ_t - Cq_c ϕ_t^ν+1 + o(ϕ_t^ν+1)could be estimated by comparing to the differential equation dϕ/dt = - Cq_c ϕ_t^ν+1.This equation could be solved explicitly, yielding the asymptotics of (<ref>).For case 3, the approximate differential equation will bedϕ/dt = - x_i/q_c(q_c-q) -C_i q_c x_i (ϕ - x_i)^2ν_i.The solution of this equation has a plateau around x_i, whose length diverges as (q_c-q)^-1+1/2ν_i. For the first part, it will be enough to show that there exist an offspring distribution ξ and a polynomial Q(x)=b_0+…+b_r-2x^r-2 such that * g_ξ(x)=Const-x^νQ(x).* Q(x)>0 for all x∈[0,1]. This ξ, according to zeologyofmetastabilities and the fact that b_0>0, will indeed satisfy (<ref>). Rather than ξ, it will be easier to find a sequence {χ_k} _k=r^∞ with a finite sum together with a polynomial P(x)=a_0+…+a_r-2x^r-2, such that * g_χ(x)=∑_kχ_kg_k(x)=1-x^νP(x).* χ_k≥0.* P(x)>0 for all x∈[0,1]. Taking ξ=1/∑χ_kχ_k will then conclude the proof. Letχ_k=r-1/k(k-1)r≤ k≤ν+r-10 k≥ν+r . Using constantg, we may writeg_χ(x)=1-∑_k=ν+r^∞r-1/k(k-1)g_k(x). By gkispolynomial g_χ is a polynomial of degree ν+r-2, therefore ∑_k=ν+r^∞r-1/k(k-1)g_k(x) equals a polynomial of degree ν+r-2. Using again gkispolynomial, we can define the polynomialP(x)=∑_k=ν+r^∞r-1/k(k-1)g_k(x)/x^ν. It is left to show that P(x)>0 for all x∈[0,1]. By equations <ref> and <ref>, P(x) is non-negative and could only vanish at x=0. But by gkispolynomial, P(0)=r-1/(ν+r)(ν+r-1)(g_ν+r(x)/x^ν)_x=0≠0. This concludes the first part.Note that, by basisofpolynomials, we can define the projection Pr from the space of polynomials of degree at most r+ν-2 to its subspace spanned by x^ν,…,x^ν+r-2 with kernel spanned by g_r(x),…,g_ν+r-1(x). Define also M_0 to be the map from the space of polynomials of degree at most r-2 to the space of polynomials of degree at most r+ν-2 given by the multiplication by x^ν. Then the first of the conditions above can be written asM_0 P=1.Since ∘ M_0 is bijective, this equation has a unique solution; and what we have shown in the proof is that this solution satisfies the necessary positivity conditions. We will now prove the second part of the theorem. In analogy with the first one, we will find ξ, Q(x)=b_0+…+b_r-2x^r-2 and x_1>…>x_n such that: * g_ξ(x)=Const-(x-x_1)^2ν_1… (x-x_n)^2ν_nQ(x).* Q(x)>0 for all x∈[0,1]. Similarly to the previous part, we will look for {χ_k} _k=r^ν+r-1 and P(x)=a_0+…+a_r-2x^r-2 satisfying: * g_χ(x)=∑_kχ_kg_k(x)=1-(x-x_1)^2ν_1… (x-x_n)^2ν_nP(x).* χ_k>0.* P(x)>0 for all x∈[0,1]. Note that choosing ν=2ν_1+…+2ν_n, χ_k (defined in (<ref>)) is strictly positive for r≤ k≤ν+r-1. Since P was required to be strictly positive, we may hope that also after adding a small perturbation (x_1,…,x_n) around 0 there still exists a positive solution P. More precisely, let us denote by M_x_1,…,x_n the multiplication by (x-x_1)^2ν_1… (x-x_n)^2ν_n, acting on the polynomials of degree at most r-2. In particular, for x_1,…,x_n=0 this is M_0 defined in uniquesolforP. Then, we want to show that the solution ofM_x_1,…,x_n P=1satisfies the positivity conditions 2 and 3. By continuity of the determinant, when (x_1,…,x_n) is in a small neighborhood of 0 the operator M_x_1,…,x_n is invertible. Moreover, in an even smaller neighborhood of 0 the polynomial (M_x_1,…,x_n)^-11 will satisfy the positivity condition 3 – matrix inversion is continuous, and the set of polynomials satisfying this condition is open and contains (M_0)^-11 by the first part of the proof. Finally, since coordinate projections of 1 - (x-x_1)^2ν_1… (x-x_n)^2ν_n(M_x_1,…,x_n)^-11 with respect to the basis defined in basisofpolynomials are continuous in (x_1,…,x_n), and since for (x_1,…,x_n)=0 condition 2 is satisfied, by taking (x_1,…,x_n) in a further smaller neighborhood of 0 we are guaranteed to find a polynomial P satisfying the required conditions.§ REMARKS ON TWO OTHER PHASE TRANSITIONS §.§ More Discontinuities of ϕ_t Consider, for example, r=2 and ξ_k=3/5_k=2+2/5_k=5. The function g_ξ(x) is maximal at g_ξ(0)=6/5, then it has a local minimum, followed by a local maximum (see secondphasetransition). In this case, recalling Fact <ref>, ϕ_t^ξ will have a discontinuity at this local maximum, that is, a second phase transition occurs. We may then expect that one can find ξ giving rise to as many (decreasing) local maxima of g_ξ as we wish:Let ν_1^(1),…,ν_n_1^(1),ν_1^(2),…,ν_n_2^(2),…,ν_n_m^(m). Then there exists g_ξ, { q_i} _i=1^m,{ x_j^(i)} _1≤ i≤ m, 1≤ j≤ n_i such that q_i is a critical point which is (ν_1^(i),…,ν_n_i^(i))-metastable at x_1^(i),…,x_n_i^(i).§.§ Percolation of Infection Another possible phase transition, studied in <cit.> for the case of regular trees, is when infinite infected clusters start to appear, but the prevalence is still smaller than 1. Following the proof of Proposition 3.9 in<cit.>, one sees that it applies also for the bootstrap percolation on GW trees, showing that the critical probability q_c^(∞) above which infinite clusters no longer appear is strictly bigger than q_c defined in (<ref>), unless ξ_k=_r. § MORE QUESTIONS The problem of bootstrap percolation in disordered systems raises many questions. Related to the work presented here, one may be interested in the metastable regime for other systems, such as G_n,p or the random regular graph. Another natural problem is the analysis of the bootstrap percolation on the random graph with a given degree sequence, that has a GW local structure, with analogy to the regular tree structure of the random regular graph.§ ACKNOWLEDGMENTS I would like to thank Cristina Toninelli for the introduction of the subject and for useful discussion, and to Lucas Benigni for his help throughout the writing of this paper.§ APPENDIXThis paper concerns with the analysis of a phase transition originating in the appearance of a new fixed point for a certain recurrence relation, i.e., a bifurcation. In this appendix, we will try to understand in a more general context the time scaling in systems of that type. Let us then consider a sequence of reals { x_n} _n=0^∞ , defined by the value x_0 and a recursion formula for n>0:x_n=f(x_n-1). We will also fix now some positive δ<1, that will be used throughout this appendix as the window around the new fixed point in which we are interested.First, we will study the time scaling at the bifurcation point, when the new fixed point is first created. In this case, we may expect f to be tangent to the identity function at the fixed point, so we will start our discussion with the following assumptions:f has a fixed point y_0, such that for y∈(y_0,y_0+δ): y-c(y-y_0)^α≤ f(y)≤ y-c(y-y_0)^α,for some α>1, 0<c≤c<δ^-(α-1).x_0∈(y_0,y_0+δ).We first mention the following fact:The sequence is decreasing and bounded from below by y_0.By Assumption <ref>, x_n+1<x_n whenever x_n∈(y_0,y_0+δ). Moreover,x_n+1-y_0 ≥x_n-y_0-c(x_n-y_0)^α =(x_n-y_0)(1-c(x_n-y_0)^α-1)≥ (x_n-y_0)(1-cδ^α-1) > 0.Therefore, since x_0∈(y_0,y_0+δ) by assumption <ref>, the entire sequence is in the interval (y_0,y_0+δ), and it is decreasing.The following theorem will describe the asymptotic of the sequence: Let {x_n}_n=0^∞ be the sequence defined in (<ref>), satisfying Assumptions <ref> and <ref>. Theny_0+a(n+n_0)^-1/α-1≤ x_n≤ y_0+an^-1/α-1,where a=[(α-1)(1-δ)^-αc]^-1/α-1, a=[(α-1)c]^-1/α-1, and n_0=(x_0-y_0)^1-α/(α-1)(1-δ)^-αc are all positive constants.Let us first define a sequence t_n=(x_n-y_0)^1-α, and note that t_n is positive for all n. Then using Fact <ref> and Assumption <ref>, fixing c^'=(α-1)(1-δ)^-αc and c^'=(α-1)c, we can estimate:t_n=(f(x_n-1)-y_0)^1-α t_n=(f(x_n-1)-y_0)^1-α≤(x_n-1-c(x_n-1-y_0)^α-y_0)^1-α ≥(x_n-1-c(x_n-1-y_0)^α-y_0)^1-α=(t_n-1^1/1-α-ct_n-1^α/1-α)^1-α =(t_n-1^1/1-α-ct_n-1^α/1-α) =t_n-1(1-ct_n-1^-1)^1-α =t_n-1(1-ct_n-1^-1)^1-α ≤ t_n-1(1+c^'t_n-1^-1) ≥ t_n-1(1+c^'t_n-1^-1) =t_n-1+c^'; =t_n-1+c^'. We have used here the fact that, for any 0<z<δ<1, we can approximate (1-z)^1-α using its derivatives at 0 and at δ:-(1-α) ≤(1-z)^1-α-1/z≤ -(1-α) (1-δ)^-α.We then also use ct_n-1^-1=(x_n-y_0)^α-1<δ^α-1<δ.Finally, x_n=y_0+t_n^-1/α-1x_n≥ y_0+((x_0-y_0)^1-α+c^'n)^-1/α-1 ≤ y_0+((x_0-y_0)^1-α+c^'n)^-1/α-1 = y_0+(c^'(n+(x_0-y_0)^1-α/c^'))^-1/α-1 ≤ y_0+an^-1/α-1;=y_0+a(n+n_0)^-1/α-1.Next, we will be interested in the behavior near the bifurcation point, just before the new fixed point appears. For this purpose we will consider a family { x_n^ε} _n=0^∞ of sequences, each defined by the value x_0^ε and a recursion formula for n>0:x^ε_n=f_ε(x^ε_n-1),and assume:There is a point y_0 such that for |y-y_0|<δ and ε<ε_0 y-c(y-y_0)^2α-ε≤ f_ε(y)≤ y-c(y-y_0)^2α-ε,for an integer α>1 and positive constants c and c.0<x_0-y_0<δ.In order to study the asymptotic behavior of x^ε_n for small values of ε, we will need the following definition:The exit time N_δ(ε) is the minimal n such that x^ε_n<y_0-δ. Replacing Fact <ref> will be the following:For all ε<ε_0, N_δ(ε) is finite, and for n<N_δ(ε) the sequence x^ε_n is decreasing.By Assumption <ref>, for n<N_δ(ε), if x^ε_n<y_0+δ then x^ε_n+1<x^ε_n<y_0+δ. Hence, the sequence remains in the interval (y_0-δ,y_0+δ) an long as n<N_δ(ε). Since in this interval the sequence is decreasing, the result follows by Assumption <ref>.For our analysis, we will compare this sequence to the solution of the following differential equations, that will approximate x^ε_n-y_0: /s=-c ^2α-ε, ζ/s=-c ζ^2α-ε,(0)=z^ε_0=x^ε_0-y_0; ζ(0)=z^ε_0=x^ε_0-y_0.The solution ζ is strictly decreasing, and in particular one can define its inverse t: [-∞,z^ε_0]→[0,∞], and τ_n=t(x^ε_n-y_0). t and τ_n will be defined in the same manner. Note that these all depend on ε, even though this dependence is omitted from the notation. The next lemma will show that the continuous crossing times τ_n and τ_n are close to the discrete one, namely n.For all n≤ N_δ(ε),(1-κ_c,δ,ε)n≤τ_n≤τ_n≤(1+κ_c,δ,ε)n,where for all c>0, κ_c,δ,ε_0 = max(C_4 ε^2α-1,2αδ ^2α-1). C_4 is a positive constant depending on δ,c and ε_0 given explicitly in the proof, and bounded when δ and ε_0 are not too big. For example, if ε_0<1 and cδ^2α-1<1/2, C_4 < (3+ 4^α c)^4α. Let z_n=x_n-y_0. Thenτ_n=t(f_ε(x_n-1)-y_0)≤ t(z_n-1-cz_n-1^2α-ε) =∫ _z_0^z_n-1-cz_n-1^2α-εz/-cz^2α-ε =t(z_n-1)-∫ _z_n-1^z_n-1-cz_n-1^2α-εz/cz_n-1^2α+ε-∫ _z_n-1^z_n-1-cz_n-1^2α-ε(z/cz^2α+ε-z/cz_n-1^2α+ε) =τ_n-1+1-∫ _z_n-1^z_n-1-cz_n-1^2α-ε(z/cz^2α+ε-z/cz_n-1^2α+ε).In order to study the error term, we will use the following estimation: Fix w_0∈(-δ,δ), and c>0. LetI=∫ _w_0-cw_0^2α-ε^w_0(1/cw^2α+ε-1/cw_0^2α+ε) w.Then|I|≤κ_c,δ,ε_0.We will first consider the case in which the integration interval passes through 0, that is 0<w_0<cw_0^2α+ε. In this case, w_0 ≤w_0(1-cw_0^2α-1)(1-cδ^2α-1)^-1≤ C_1ε,cw_0^2α+ε ≤ [1+C_2ε^2α-1]ε,for C_1=(1-cδ^2α-1)^-1 and C_2=c(1-cδ^2α-1)^-2α.We may then bound the nominator of the integrand for all w∈[w_0-cw_0^2α-ε,w_0] by |cw_0^2α+ε-cw^2α-ε|≤ cw_0^2α+cw^2α≤ C_3ε^2α,where C_3=(1+C_2ε_0^2α-1)^2α+C_1^2α.For the denominator, (cw^2α+ε)(cw_0^2α+ε)≥ε^2.Putting everything together |I|≤ ∫ _w_0-cw_0^2α-ε^w_0|cw_0^2α+ε-cw^2α-ε/(cw^2α+ε)(cw_0^2α+ε)|≤ (cw_0^2α+ε)C_3ε^2α-2≤C_4ε^2α-1,for C_4=[1+C_2ε_0^2α-1]C_3.Next, we consider the case where the integral is over a positive interval, i.e., w_0≥ cw_0^2α+ε. We can bound the integrand using convexity – for all w∈(w_0-cw_0^2α-ε,w_0) 1/cw^2α+ε-1/cw_0^2α+ε/w-w_0≥-2α c w^2α-1/(cw^2α+ε)^2.This implies that |I|≤ (cw_0^2α+ε)2α c w^2α-1/(cw^2α+ε)^2(w_0-w)≤2α c w^2α-1≤ 2α c δ^2α-1.We are left with the case w_0≤-cw_0^2α-ε, which could be analyzed using the exact same argument as the previous one to obtain the result.Using the claim we can continue with our estimation, obtaining_n≤_n-1+1+κ_c,δ,ε_0,and proving the upper bound. The lower bound could be found using the exact same calculation replacing c by c. The result follows since c≤c, and thus τ_n≤τ_n by monotonicity of the integral.We are now ready to formulate the final result:Fix a family of sequences (indexed by ε) defined in (<ref>) satisfying Assumptions <ref> and <ref>, and consider their exit times N_δ(ε) (see Definition <ref>). Let I=∫ _-∞^∞u/cu^2α+1, I=∫ _-∞^∞u/cu^2α+1, and κ_δ,0=max(κ_c,δ,0,κ_c,δ,0), where κ_c,δ,0 and κ_c,δ,0 are the positive constants given in Lemma <ref>. Assume further that κ_δ,0<1. Then 0<1/2I/(1+κ_δ,0)≤lim inf_ε→0N_δ(ε)/ε^-1+1/2α≤lim sup_ε→0N_δ(ε)/ε^-1+1/2α≤I/(1-κ_δ,0)<∞.The factor 1/2 in front of I could be removed when ε^-1/2α(x^ε_0-y_0)^ε→∞ as ε→ 0 (e.g., when x^ε_0-y_0 is bounded away from 0 uniformly in ε). This theorem is a direct consequence of the fact that ζ shows an ε^-1+1/2α time scaling behavior. First, note thatτ_N_δ(ε)-1≤t(-δ)≤t(-δ)≤_N_δ(ε). We will then be interested in finding t(-δ),t(-δ):t(-δ) =∫ _z^ε_0^-δz/-cz^2α-ε =ε^-1+1/2α∫ _z_0^ε^-δε^-1/2αz/-c(zε^-1/2α)^2α-1 =ε^-1+1/2α∫ _-ε^-1/2αz^ε_0^ε^-1/2αδu/cu^2α+1,where for t one should take c=c, and c=c for t.All that is left is to use Lemma <ref>, finding t(-δ)/(1+κ_δ,ε)≤ N_δ(ε)≤1+t(-δ)/(1-κ_δ,ε),which, since the integrals defining I and I converge, concludes the proof.When f_ε satisfies not only Assumption <ref>, but alsof_ε(y)=y-c(y-y_0)^2α-ε+o((y-y_0)^2α)+o(ε),we can consider δ_ε that goes to 0 with ε, e.g. 1/|logε|, so that κ_δ,0 will converge to 0 as well. In this case, we may choose c_δ and c_δ that converge to c, and thus Theorem <ref> will give the limit of N_δ(ε)/ε^-1+1/2α, rather than just bounds on its limsup and liminf. Such a direct application of the theorem, however, forces us to choose an initial condition x_0^ε that converges to y_0 as ε goes to 0. To overcome this issue, we can use the estimation above with a fixed δ until x_n reaches δ_ε, which happens at n of order ∫ _z_0^δ_εz/-cz^2α-ε≪ε^-1+1/2α. Then restart the dynamics using the estimation with δ_ε until reaching -δ_ε, which takes an order ε^-1+1/2α of steps, and then using again the estimation for our fixed δ show that the number of steps required to reach -δ is much smaller than ε^-1+1/2α. This would yield lim_ε→0N_δ(ε)/ε^-1+1/2α=∫ _-∞^∞u/cu^2α+1. plain | http://arxiv.org/abs/1706.08390v2 | {
"authors": [
"Assaf Shapira"
],
"categories": [
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"primary_category": "math.PR",
"published": "20170626141353",
"title": "Metastable Behavior of Bootstrap Percolation on Galton-Watson Trees"
} |
1Physics Department, McGill University, 3600 Rue University, Montreal QC Canada, H3A 2T8.*[email protected] Fiber-based optical microcavities exhibit high quality factor and low mode volume resonances that make them attractive for coupling light to individual atoms or other microscopic systems. Moreover, their low mass should lead to excellent mechanical response up to high frequencies, opening the possibility for high bandwidth stabilization of the cavity length. Here, we demonstrate a locking bandwidth of 44 kHz achieved using a simple, compact design that exploits these properties. Owing to the simplicity of fiber feedthroughs and lack of free-space alignment, this design is inherently compatible with vacuum and cryogenic environments. We measurethe transfer function of the feedback circuit (closed-loop) and the cavity mount itself (open-loop), which, combined with simulations of the mechanical response of our device, provide insight into underlying limitations of the design as well as further improvements that can be made. (050.2230) Fabry-Perot; (140.3945) Microcavities; (140.3425) Laser stabilization. 10 berden2000cavity G. Berden, R. Peeters, and G. Meijer, Cavity ring-down spectroscopy: Experimental schemes and applications, Int. Rev. Phys. Chem. 19, 565–607 (2000).srinivasan2007linear K. Srinivasan and O. Painter, Linear and nonlinear optical spectroscopy of a strongly coupled microdisk–quantum dot system, 450, 862–865 (2007).udem2002optical T. Udem, R. Holzwarth, and T. W. Hänsch, Optical frequency metrology, 416, 233–237 (2002).hinkley2013atomic N. Hinkley, J. Sherman, N. Phillips, M. Schioppo, N. Lemke, K. Beloy, M. Pizzocaro, C. Oates, and A. 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Express 25, 1582–1597 (2017).rakhmanov2002dynamic M. Rakhmanov, R. Savage, D. Reitze, and D. Tanner, Dynamic resonance of light in Fabry–Perot cavities, Phys. Lett. A 305, 239–244 (2002).bechhoefer2005 J. Bechhoefer, Feedback for physicists: A tutorial essay on control, 77, 783 (2005).ansys ANSYS® Workbench, 17th ed.§ INTRODUCTION High finesse Fabry-Perot cavities have extensive applications in spectroscopy <cit.> and precision measurement <cit.>, as well as fundamental research in quantum optics <cit.>. In many situations,the cavity must be locked to a frequency reference (for example, an atomic transition) to compensate for external disturbances and maintain a specific resonant frequency <cit.>. Such locking is typically accomplished by monitoring the transmission or reflection of light at the reference frequency, and feeding the signal back to a mechanical transducer that adjusts a mirror to stabilize the cavity length. The bandwidth of the lock (here defined as the frequency below which noise is suppressed) represents a key figure of merit, and determines the maximal noise suppression that can be achieved at low frequencies. The mechanical response of the mirrors and mirror mount typically limit the bandwidth, requiring careful engineering to suppress low-frequency vibrational modes.Fiber-based micro-mirrors<cit.> offer a promising technology for creating tunable, high-finesse micro-cavities. These cavities can achieve very small mode waists, which are advantageous for cavity quantum electrodynamics applications; so far, they have been coupled to atoms<cit.>, ions<cit.>, optomechanical systems <cit.>, molecules<cit.>, and crystalline defect centers<cit.>. Moreover, the light weight of the fiber mirror suggests that it should be possible to achieve a high bandwidth feedback loop for length stabilization of such a cavity.Previous work on locking fiber cavities has demonstrated mechanical feedback with bandwidths of only 1-3 kHz<cit.> (we note that for the 1 kHz case the first limiting mechanical resonance occurred at 25 kHz). Adding photothermal stabilization can further improve noise suppression for frequencies up to 500 kHz<cit.>. Such “self-stable" operation is achieved via intra-cavity heating of the mirror coatings by an incident laser; disturbances that change the length of the cavity affect the intra-cavity power, which in turn induces thermal expansion that stabilizes the effective length. This method of thermal stabilization comes at the cost of high intra-cavity power on the order of 1-10 Watts<cit.>, presenting a challenge for cryogenic operation or for coupling cavities to solid-state systems where non-resonant absorption of the locking light cannot be neglected. In contrast, higher locking bandwidths can also be achieved using careful mechanical and electrical engineering of the cavity mount and feedback circuit. An optimized design for a macroscopic Fabry-Perot cavity composed of two small free-space mirrors achieved a bandwidth of up to 180 kHz<cit.>. However, this result relied on the damping properties of lead inside the mirror mount to reduce the impact of low-frequency mechanical resonances on the feedback circuit, which limits its function to non-cryogenic applications.Here, we investigate the bandwidth attainable when electronically feeding back to a piezo-mounted fiber mirror. Our approach does not rely on specific material properties or intra-cavity heating; instead, we take advantage of the intrinsic high-frequency response available with a lightweight fiber mirror. We measure the full transfer function for the feedback circuit, and find that a locking bandwidth of 44 kHz is readily obtained.With a combination of direct measurements of the system's transfer function and finite-element simulations, we identify limiting features in the mechanical response associated with resonances in the mount, fiber, and epoxy, and provide an additional set of design considerations. § EXPERIMENTAL SETUP §.§ Device Design and ConstructionOur cavity is formed by a macroscopic flat mirror and a microscopic spherical mirror fabricated on the tip of a single mode optical fiber (Fig. <ref>a), using a CO_2 laser ablation process<cit.>. Both mirror substrates are coated with a high reflectivity dielectric mirror (LASEROPTIK) with the reflection band centered at 1550 nm. This coating exhibits a finesse of ℱ=21000±2000, measured after annealing for 5 hours at 300^∘C under atmospheric conditions<cit.>.The device is composed of two aluminum pieces onto which each mirror is mounted. The assembled device has dimensions of 28 mm in all directions. This aspect ratio was chosen heuristically to maximize the frequencies of the structure's normal modes, thereby minimizing the coupling between ambient (or driven) vibrations and the cavity length. The upright aluminum piece serves as a mount for the flat mirror, and (in this case) the mirror is glued in place using Stycast 2850.On a second aluminum piece, a shear piezoelectric actuator (Noliac CSAP03) supports the fiber mirror and controls the length of the cavity. This piezo was chosen for its high unloaded resonance frequency of 1.75 MHz, which ultimately limits the theoretically achievable locking bandwidth. The bottom electrode is connected to a large DC voltage to coarsely tune the cavity length, while the top electrode is connected to a fast signal for feedback (maximum ±10 V). The combined voltages we can achieve limit our travel range to ≈600 nm at room temperature, less than a free spectral range of our cavity (λ/2=775 nm). We therefore employ a heating pad and thermocouple to control the temperature of the aluminum, and tune a cavity resonance within range of the piezo. For the case of a cryogenic environment, piezo travel should decrease by a factor of ∼5, requiring a combination of improved fabrication tolerances, higher drive voltages, and/or longer-travel piezo elements.To mount the actuator, an alumina plate is first glued (Stycast 2850) to the second aluminum piece for electrical insulation. A piece of copper slightly longer than the piezo is then glued to the alumina, where the exposed copper is used for electrical contact to the bottom electrode. The piezo is attached to the copper sheet using silver epoxy (Epotek H20E), and kapton-coated copper wires are similarly glued to the copper sheet and top electrode. The two aluminum pieces are then screwed together.Finally, the fiber mirror is pre-aligned above the actuator and glued in place with Stycast 2850 (the fiber is further aligned while submerged in the epoxy to maximize the contrast of the cavity reflection dip for a fundamental mode). The system is left to cure for 24 hours under ambient conditions. The assembled device is then clamped to an optical table between two viton O-rings (visible above and below the device in Fig. <ref>a) to help isolate the system from high-frequency noise transmitted through the table. §.§ Pound-Drever-Hall Locking CircuitTo lock the cavity length to our laser frequency, we employ active feedback via the Pound-Drever-Hall locking scheme <cit.>. Briefly, the incident laser frequency is modulated and the reflected light is demodulated to produce an error signal that dependslinearly on detuning from the cavity resonance. This signal is filtered, amplified, and fed back to the shear piezo under the fiber mirror.The circuit used for generating an error signal is shown in Fig. <ref>b, with optical (electrical) signal paths indicated in red (blue). The laser (Koheras Adjustik E15) is frequency modulated using an electro-optic phase modulator (EOM), driven with the output of a voltage controlled oscillator (VCO). The VCO is capable of generating output frequencies f_m=76-80 MHz, which corresponds to the so-called “low-modulation regime"<cit.> for our cavity, which has a (full) linewidth of 189±7 MHz. We find it is important to use only APC terminated fibers as well as isolators in our optical circuit to prevent back-reflections and standing waves that lead to residual amplitude modulation <cit.>. The modulated beam passes through a circulator (C) and couples to the cavity through the fiber mirror. The degeneracy of the two linear polarization states of the fundamental cavity mode is lifted by ∼100 MHz, likely due to a combination of fiber mirror ellipticity and birefringence<cit.>. The polarization is adjusted to select only one of these modes using a fiber polarizer (P), which we correct on an hourly basis as the strain in the fibers changes with lab temperature (typical temperature excursions <1 ^∘C).The reflected power is measured on a high bandwidth photodetector (bandwidth 150 MHz), and the output is demodulated by mixing with the VCO signal. The mixer output is sent through a 32 MHz low pass filter to remove the f_m and 2f_m oscillating terms, and the relative phase between the photodiode and mixing signal (ϕ) is adjusted by tuning the VCO frequency to obtain the desired error signal <cit.>, similar to what is shown in the inset of Fig. <ref>b. The residual signal from the incompletely suppressed orthogonal polarization mode is also visible to the right of the main features (indicated by the red arrow). This error signal is then sent through a combination of filters and a servo controller consisting of a “proportional-integral" (PI) amplifier (Newfocus LB1005) before finally being applied to the shear piezo. As discussed below, the filters andcontroller allow us to maximize the feedback gain at low frequencies for a given lock bandwidth.§ CLOSED-LOOP MEASUREMENTS §.§ Measurement Block DiagramWe can quickly gain insight into the limitations of this feedback system and the mechanical response of the mount by injecting a disturbance into the locked circuit and measuring its response, as described in Ref. <cit.>. As discussed below, this provides an estimate of the closed-loop transfer function.A block diagram representing our feedback system is shown in Fig. <ref>a, where all circuit element transfer functions and signals are complex functions of frequency. The error signal (a) is connected to the positive input of our servo controller, which also has an inverting input (b). The controller has an error monitor port that produces a voltage e = T(a-b), where T is the associated transfer function. The controller itself has a PI transfer function P, such that its output voltage is u = P(a-b). The output u then drives the system (G), which represents transduction between applied voltage and change in cavity length. Practically, this comprises a 200 Ω resistor in series with the piezo actuator supporting the fiber mirror. These two elements form a low-pass filter with a cut-off frequency ≈240 kHz. We found that it was not possible to lock the system without a resistor, as the low-pass filter behaviour is essential to suppress high-frequency perturbations. Consequently, the resistor is included in the system transfer function G for all closed-loop measurements. The piezo converts the output voltage into shear displacement (with a harmonic-oscillator-like transfer function), and all mechanical noise experienced by the cavity is modeled by the addition of a driving term d. The deviation of the length of the cavity from resonance is measured by the cavity mirrors, photodiode, and mixer circuit, which together have transfer function -M. M represents the cavity light's dynamical response to mechanical motion (essentially a low-pass filter with time constant equal to the cavity's ringdown time<cit.>), along with the transfer functions of the fiber components, photodiode, cables, mixer, and 32 MHz low-pass filter. The sign of M is determined by the phase of the local oscillator, and is negative in our case. As discussed below, we also have the freedom to introduce an additional filter F prior to the PI controller, where the output is the error signal a that is fed back to the servo controller.To probe the frequency response of different elements in the loop, we can apply a known perturbation b on the inverting input of the servo controller and observe the closed-loop response. Solving for the measurable quantity e in terms of the inputs b and d yields:e= -T/1+PGMFb--TMF/1+PGMFd .All of the perturbations acting on the locking system are scaled by a term proportional to 1/(1+PGMF). Thus, when the magnitude of PGMF is large, the effect of these disturbances is minimized and the error signal tends to 0. Conversely, if PGMF approaches the value -1 at some frequency, we enter a situation of positive feedback where the signal on the inputs is amplified <cit.>. Using a dual-phase lock-in amplifier (Zurich Instruments HF2LI) to supply the perturbation b and to measure e, the noise term d (which is uncorrelated with the lock-in's output b) can be eliminated, and it is straightforward to extract the circuit transfer function:PGMF =-Tb/e-1 .Here the only unknown is the error monitor port transfer function T, which can be measured independently (i.e. while unlocked). We hereafter refer to PGMF as the transfer function of the circuit.§.§ Closed-loop Measurements The extracted circuit transfer functions can be seen in Fig. <ref>b forthe “no filter" (F=1) case, and for the case where F is an analog electronic filter (schematic shown in Fig. <ref>c). The filter increases the noise suppression of the lock at low-frequencies by adding a second pole of roll-off to PGMF from ≈ 300 Hz - 5 kHz, with the transfer function:F=R_2(C_1R_1 ω-i)/(C_1+C_2)R_1 R_2 ω-i(R_1+R_2) where C_1=330 pF, C_2=4.7 nF, R_1=100 kΩ, and R_2=1 MΩ.For the “no filter" case, PGMF exhibits one pole of roll-off at low frequencies from the PI controller, which has transfer function P=K(1-iω_PI/ω), where K is an overall scaling factor, and ω_PI is the “PI corner" frequency of ω_PI/2π=100 kHz. The inclusion of the additional electronic filter allows us to attain a higher low frequency gain (PGMF=272± 40 at 1 kHz), but does not affect the transfer function considerably at higher frequencies. Between 20-68 kHz some small resonances appear that are mainly related to motion of the macroscopic mirror, deformation of the aluminum jig, and bending resonances of the fiber tip, as discussed in detail in Section <ref>. We refer to these as “indirect" resonances since they correspond to small phase excursions of <π, and could in principle be compensated for with electronics. Starting at 68 kHz, an increasingly dense set of resonances appears, stemming from the mechanical modes of the fiber mount and the clamped fiber. We refer to these resonances as “direct", since they behave similarly to a directly driven harmonic oscillator, exhibiting a phase decrease on the order of π when passing through resonance.§.§.§ Circuit BandwidthWe define the bandwidth of our circuit as the frequency range over which |PGMF|>1 and arg(PGMF)>-π; if the limiting conditions occur at the same frequency, it corresponds to the ringing condition of our system (PGMF=-1). One can see that for both plots in Fig. <ref>b, the first instance of |PGMF|=1 occurs at 44 kHz, while arg(PGMF)>-π for frequencies up to 68 kHz (the first direct resonance). We therefore conservatively define the bandwidth of our system as 44 kHz for both measurements (the apparent -π-phase crossing point at 1.5 kHz in the “filter" plot is a result of measurement noise). §.§.§ Ringing frequencyFor many systems, one would expect the amplitude and phase response to decrease monotonically with frequency after the first direct resonance (as, for example, in the case of a simple harmonic oscillator). In this case, the proportional gain provided by the PI controller could be increased until |PGMF|=1 at the lowest frequency where arg(PGMF)=-π, and the system would ring at this frequency. The electro-mechanical modes of our system are not so simply modeled, and result in many high frequency resonances that exhibit an increase in magnitude and phase. Consequently, the bandwidth, first zero-phase crossing frequency, and first ringing frequency are all different. Fig. <ref>d shows a time trace of the the locked error signal where the proportional gain has been increased to hit the first ringing pointat 179 kHz. § SYSTEM TRANSFER FUNCTIONSAs discussed in the previous section, the mechanical response of our cavity has a complicated structure. Itis therefore of interest to isolate the system transfer function G to determine limiting factors in the design with regard to the locking bandwidth. We measure the response of the cavity mount (G) by coupling a 1310 nm laser through the fiber mirror where the dielectric mirror coatings are low finesse (with power reflection coefficient R≈45%). A high voltage DC signal is first applied to the bottom electrode of the piezo to select a cavity length offset corresponding to high measurement sensitivity (approximately the point of highest slope on the reflection fringe). An AC drive voltage from a lock-in amplifier is then applied to the top electrode while recording the modulation in reflected light, thereby probing the mechanical response of the system at different frequencies. The resulting system transfer function is shown in Fig. <ref>a (normalized to the low frequency amplitude). We simulated the mechanical modes of the assembled cavity device using the ANSYS <cit.> finite element analysis program to understand the origin of different resonances. The results are overlaid with the measured amplitude response between 10-100 kHz in Fig. <ref>b, which yield a decent correspondence given the difficulties in modeling the exact system. Interestingly, the simulation suggests that the mechanical resonances can be divided into three distinct frequency ranges:* The low frequency region (20-50 kHz) contains only a few modes and is mainly characterized by the deformation of the jig and the slip/rotation of the macroscopic flat mirror. We attribute the measured resonance near 20 kHz to the movement of the flat mirror in its housing (this mode is calculated to occur at 24.7 kHz in our simulations). These resonant frequencies could be improved (increased) by optimizing the geometry of the mount, reducing the mass of the macroscopic mirror, and changing how the mirror is fixed to the mount (for example, using a flexural clamp similar to a shaft collar).* The mid-frequency region (50-70 kHz) is characterized by the low frequency bending resonances of the overhanging fiber tip, as well as the bending/folding of the jig under the piezo. Our simulations suggest that further reducing the length of overhanging fiber (from L≈500 μm) would increase the frequencies of the clamped fiber modes, where the resonant frequencies scale approximately with 1/L^2 (up to frequencies where the mechanical modes of the glue structure are excited). * The high-frequency region (>70 kHz) is the most interesting with regard to fiber mounting considerations. In this frequency range, the fiber no longer acts as a simple clamped beam and the mechanical modes begin to incorporate motion of both the fiber and the glue bonding it to the piezo. The glue will deform along the fiber axis when driven with shear motion, impacting the cavity length directly. This could be improved by ensuring the fiber is in contact with the piezo while the epoxy cures, and by using less epoxy.At these high frequencies it is also necessary to consider vibration of the clamping screws, and “flapping" at unbonded corners for the different layers in the piezo stack. §.§ Cryogenic Operation Many experiments in quantum optics require moving to cryogenic temperatures where mechanisms for thermal decoherence are suppressed. Although our device has not been expressly designed for cryogenic operation, its materials are in principle cryogenically compatible, and thus we explore the impact of thermal cycling on the system. We clamp our device using a stainless steel strap to the base plate of a Montana Instruments Nanoscale Workstation cryostat. The fiber mirror used for the previous measurements was replaced to facilitate a longer cavity length of ≈60 μm that would prevent the fiber from crashing into the flat mirror due to thermal contraction of the mount. We estimate that this increase in cavity length corresponds to reducing the overhanging fiber length by ∼10%. The piezo stack is unchanged to allow for comparison with previous measurements. The device survived two thermal cycles from room temperature to 6K. Due to the limited range of our piezo and voltage supplies, we were not able to make measurements at the base temperature. Nevertheless, after thermal cycling, we were able to maintain a cavity lock at 270K with the cryo-cooler running, but the system transfer function G changed considerably (see Fig. <ref>), likely due to a loosening of the aluminum screws holding the device together or delamination of the piezo stack. The new strong resonance at 37 kHz set a new ringing point for the locking circuit, and limited the achievable lock bandwidth to 3 kHz. The device broke at the epoxy connection between the alumina plate and aluminum mount as we tried to add Belleville washers to the screws, indicating that mount materials with a closer thermal expansion match to alumina (stainless steel, titanium) may prove advantageous for long-term cryogenic operation.§ CONCLUSIONFiber-based micro-mirrors offer a new platform for applications of high finesse Fabry-Perot cavities. We showed that a simple electronic feedback circuit can take advantage of the intrinsic high mechanical resonance frequency of a fiber mirror, achieving a lock bandwidth of 44 kHz. We also modeled the system, and our results suggest that the lock bandwidth may be further improved by minimizing the length of overhanging fiber from the piezo, and reducing the thickness of the epoxy holding the fiber. The resonance frequencies of the mount can be further increased by optimizing the geometry of the mount and, in particular, reducing the size (mass) of the flat mirror. Since our locking approach does not require strong intra-cavity laser power or specialized material damping properties, it should be extendable to cryogenic applications, or to systems requiring minimal perturbations to the cavity's light field and/or weak probe beams. Notably, even though our device is not expressly designed for cryogenic operation, we were still able to lock the cavity, clamped to the cold plate, with our closed-cycle cryostat running. This illustrates the potential for fiber-cavity systems to operate in noisy environments, extending the range of applications for high finesse cavity measurements.§ FUNDINGNSERC Discovery(435554-2013 and 418459-12); CRC (950-229003 and 235060); CFI (229003, 228130 and 33488); FRQNT (NC-172619); Alfred P. Sloan Foundation (BR2013-088); Centre for the Physics of Materials at McGill; Institute Transciplinaire d'Information Quantique; Y.F. acknowledges support by a Swiss National Foundation Early Postdoc Mobility Fellowship.§ ACKNOWLEDGMENTSWe thank Christoph Reinhardt, Tina Muller, and Yoichi Miyahara for helpful discussions. | http://arxiv.org/abs/1706.09843v1 | {
"authors": [
"E. Janitz",
"M. Ruf",
"Y. Fontana",
"J. Sankey",
"L. Childress"
],
"categories": [
"physics.ins-det",
"physics.optics"
],
"primary_category": "physics.ins-det",
"published": "20170627224716",
"title": "A High-Mechanical Bandwidth Fabry-Perot Fiber Cavity"
} |
We analyze the finite-temperature phase diagram of the boson-fermion-Hubbard model with Feshbach converting interaction, using the coherent-state path-integral method. We show that depending on the position of the bosonic band, this type of interaction, even if weak, can drive the system into the resonant superfluid phase in the strong bosonic interaction limit. It turns out that this phase can exist for an arbitrary number of fermions (i.e., fermionic concentration between 0 and 2) but with the bosonic particle number very close to an integer value. We point out that the standard time-of-flight method in optical lattice experiments can be an adequate technique to confirm the existence of this resonant phase. Moreover, in the non-resonant regime, the enhancement of the critical temperature of the superfluid phase due to Feshbach interaction is also observed. We account for this interesting phenomena for a hole- or particlelike pairing mechanism depending on the system density and mutual location of the fermionic and bosonic bands.67.85.Hj, 67.85.Bc, 64.70.Tg, 74.20.-z Solid State Theory Division, Faculty of Physics, Adam Mickiewicz University, ulica Umultowska 85, 61-614 Poznań, PolandEffect of boson on-site repulsion on the superfluidityin the boson-fermion-Hubbard model A. S. Sajna and R. Micnas December 30, 2023 ==========================================================================================§ INTRODUCTION The boson-fermion-Hubbard model (BFHM) with resonant pairing mechanism has a very long history in the context of high temperature superconductivity (see, e.g. <cit.> and references therein). Recently, the interest in this model has been also extended to the ultracold atomic systems because they are a versatile tool for simulating many-body physics <cit.> and BFHM can be studied by using Feshbach resonance experiments in which the BCS-BEC crossover is realized <cit.>.The impact of strong bosonic interaction on the superfluid (SF) phase in the lattice bosons system has been widely investigated in literature in the terms of Bose-Hubbard model (BHM) (e.g. see <cit.> and reference therein). However, the superfluidity in the regime of strong bosonic repulsion in which Feshbach interaction is included is much less understood. So far only hard-core limit <cit.> and some qualitative studies have been performed <cit.>. Therefore in this paper, quantitative investigation of the non-zero temperature BFHM phase diagram with finite bosonic repulsion interaction is carried out, which is relevant for working out realistic experimental conditions. The effective field theory description of the BFHM is constructed by using the coherent state path integral formalism. This analytical method seems to be a good starting point for analysis of BFHM because it provides a reasonable description of the standard Fermi Hubbard model at weak inter-particle interaction (i.e. in the BCS regime) <cit.> and it also gives a correct description of BHM <cit.>. In this paper, we show that besides the standard superfluid phase which is governed by the pure bosonic correlation mechanism present in BHM, there appears also a resonant superfluid (RSF) phase due to Feshbach resonance phenomena. Moreover, we explain that the standard superfluid phase (not RSF) is enhanced by the hole or particle pairing mechanism of fermions. The results allow us to discuss experimental proposal for possible investigation of RSF phase in BFHM.In the following sections, we first describe the model and the coherent state path integral method applied (Sec. II). Then, in Sec. III, we use this method in analysis of the finite temperature phase diagram of BFHM and its thermodynamic quantities. At the end of Sec. III we also discuss experimental setups that could be used to prove some results of our theory. Finally in Sec. IV we give a summary of our work. Moreover, Appendix A and B contains additional investigation of BFHM model within the operator approach.§ MODEL AND METHOD§.§ Model We consider the boson-fermion Hubbard model (BFHM) with converting interaction energy I whose Hamiltonian is given by <cit.> H=-∑_ijσ(t_ij+μδ_ij)c_iσ^†c_jσ-V∑_ic_i↑^†c_i↓^†c_i↓c_i↑-∑_ij(J_ij+μ^*δ_ij)b_i^†b_j+U/2∑_ib_i^†b_i^†b_ib_i+I∑_i[c_i↑^†c_i↓^†b_i+b_i^†c_i↓c_i↑],where μ is the chemical potential, μ^*=2μ-2Δ_B and σ is a spin-1/2 index (σ∈{↑,↓}). c_iσ (c_iσ^†) is fermionic annihilation (creation) operator at site i with spin σ and b_i (b_i^†) is bosonic annihilation (creation) operator at site i. The hopping energies for fermions and bosons are t_ij and J_ij, respectively. Throughout this work we restrict hopping parameters to the nearest-neighbour sites. Moreover, U denotes the on-site interaction energy of bosons which will be treated exactly during calculations and V is the efermionic on-site interaction strength. The bottom of bosonic band is shifted by 2Δ_B parameter which could be tuned in ultracold atoms experiments with the Feshbach resonance <cit.>.Interestingly, if we assume I=0 and independent chemical potentials, the BFHM Hamiltonian (Eq. (<ref>)) describes two independent modelsi.e. the fermionic and bosonic Hubbard models. However, in the presence of finite resonant interaction (I≠ 0), there is only one phase transition from the superfluid phase which we will show shortly.Further, in the case of U=V=0 the model described by the Hamiltonian in Eq. (<ref>) has been investigated earlier in the continuum and lattice systems <cit.>. Moreover, when U→∞ the hard-core bosonic limit is obtained for which bosonic operators satisfy the Pauli spin 1/2 commutations relations <cit.>.In the coherent state path integral representation, the partition function of BFHM reads Z=∫𝒟[c̅,c,b̅,̅b]e^-1/ħS[c̅,̅c,b̅,̅b],where the action is given by S[c̅,̅c,b̅,̅b]=S_0^F[c̅,̅c]+S_0^B[b̅,̅b]+S_0^FB[b̅,̅b,c̅,̅c]+S_1^B[b̅,̅b].The denotation is related with perturbed and unperturbed parts of the action which we exploit further, i.e. unperturbed parts areS_0^F[c̅,̅c]=∫_0^ħβdτ{∑_iσc̅_iσ(τ)ħ∂/∂τc_iσ(τ).+∑_ijσ(-t_ij-μδ_ij)c̅_iσ(τ)c_jσ(τ) .-V∑_ic̅_i↑(τ)c̅_i↓(τ)c_i↓(τ)c_i↑(τ)} , S_0^B[b̅,̅b]=∑_i∫_0^ħβdτ{b̅_i(τ)ħ∂/∂τb_i(τ). . -μ^*b̅_i(τ)b_i(τ)+U/2b̅_i(τ)b̅_i(τ)b_i(τ)b_i(τ)}, S_0^FB[b̅,̅b,c̅,̅c] = I∑_i∫_0^ħβdτ[c̅_i↑(τ)c̅_i↓(τ)b_i(τ)+c.c.],and the part of the action which we will be treated approximately is S_1^B[b̅,̅b] = -∑_ij∫_0^ħβdτ J_ijb̅_i(τ)b_j(τ) .The fields c_iσ(τ), c̅_iσ(τ) are Grassman variables, the b_i(τ), b̅_i(τ) are complex variables, ħ is reduced Planck constant, β=1/k_B T where k_B and T denote Boltzmann constant and temperature, respectively. Throughout this work we denote the complex conjugation of arbitrary x variable by x̅.§.§ Effective actionWe are interested in the influence of the fermionic degrees of freedom on the bosonic part in the BFHM model within the J≪ U limit. In the first step, the therm describing the interaction between fermionic particles is decoupled by the Hubbard-Stratonovich (HS) transformation in the pairing channel which introduces Δ_i(τ), Δ̅_i(τ) fields <cit.>. Then S_0^F[c̅,̅c,]→S̃_0^F[c̅,̅c,Δ̅,Δ] where S̃_0^F[c̅,̅c,Δ̅,Δ]=∫_0^ħβdτ{∑_iσc̅_iσ(τ)ħ∂/∂τc_iσ(τ).-∑_ic̅_i↑(τ)c̅_i↓(τ)Δ_i(τ)-∑_iΔ̅_i(τ)c_i↓(τ)c_i↑(τ) .+∑_ijσ(-t_ij-μδ_ij)c̅_iσ(τ)c_jσ(τ)+1/V∑_i|Δ_i(τ)|^2} .and for which the HS measure 𝒟[Δ̅,Δ] contains the determinant [V^-1]. Then, in the J≪ U limit, we decouple the term in the action from Eq. (<ref>) which is proportional to J. It is performed by introducing the HS transformation∑_ij∫_0^ħβdτ J_ijb̅_i(τ)b_j(τ)→-∑_ij∫_0^ħβdτ J_ij^-1ψ̅_i(τ)ψ_j(τ)+∑_i∫_0^ħβdτψ̅_i(τ)b_i(τ)+∑_i∫_0^ħβdτb̅_i(τ)ψ_i(τ) .Going further, integrating out of bosonic fields b̅_i(τ), b_i(τ) is desirable. Before, we do that, we have to apply some approximation of these fields since in the present form, the action considered above, is non-integrable in b̅_i(τ), b_i(τ) because of the interaction term proportional to U. Therefore we rewrite the partition function from Eq. (<ref>) to the following formZ=Z_0^B[𝐉^-1]∫𝒟[c̅,c,ψ̅,ψ,Δ̅,Δ] × e^-1/ħ∑_ij∫_0^ħβdτ J_ij^-1ψ̅_i(τ)ψ_j(τ)-1/ħS̃_0^F[c̅,̅c,Δ̅,Δ] ×⟨ e^-1/ħ∑_i∫_0^ħβdτ([-ψ̅_i(τ)+Ic̅_i↑(τ)c̅_i↓(τ)]b_i(τ)+c.c.)⟩ _0^Bwhere 𝐉 is the hopping matrix J_ij which results from the HS transformation in Eq. (<ref>) and the statistical average ⟨ ...⟩ _0^B is defined as (Z_0^B)^-1∫𝒟[b̅,b]...e^-S_0^B[b̅,̅b]/ħ with Z_0^B=∫𝒟[b̅,b]e^-S_0^B[b̅,̅b]/ħ.Because ψ_i(τ), ψ̅_i(τ) fields have quadratic form with linear terms we can make the shift ψ_i(τ)→ψ_i(τ)+Ic_i↓(τ)c_i↑(τ) and ψ̅_i(τ)→ψ̅_i(τ)+Ic̅_i↑(τ)c̅_i↓(τ) and obtainZ=Z_0^B[𝐉^-1]∫𝒟[c̅,c,ψ̅,ψ,Δ̅,Δ] × e^-1/ħ∑_ij∫_0^ħβdτ J_ij^-1[ψ̅_i(τ)+Ic̅_i↑(τ)c̅_i↓(τ)][ψ_j(τ)+Ic_j↓(τ)c_j↑(τ)] × e^-1/ħS̃_0^F[c̅,̅c,Δ̅,Δ]-1/ħW_1[ψ̅,̅ψ],where we defineW_1[ψ̅,̅ψ]=-ħln⟨ e^-1/ħ∑_i∫_0^ħβdτ(-ψ̅_i(τ)b_i(τ)+c.c.)⟩ _0^B.Within the strong-coupling approach (J≪ U) it is convenient to expand W_1[ψ̅,̅ψ] in terms of ψ_i(τ), ψ̅_i(τ) fields, namely W_1[ψ̅,̅ψ]=∑_p=1^∞(-1)^p/(p!)^2∫_0^ħβdτ_1...dτ_pdτ_1'...dτ_p' ×∑_iG_i^p,c(τ_1', ..., τ'_p, τ_1, ..., τ_p) ×ψ̅_i(τ_1')...ψ̅_i(τ_p')ψ_i(τ_1)...ψ_i(τ_p),where G_i^p,c(τ_1', ..., τ'_p, τ_1, ..., τ_p) are connected local Green functions G_i^p,c(τ_1', ..., τ'_p, τ_1, ..., τ_p) .=(-1)^pδ^(2p)W_1[ψ̅,̅ψ]/δψ̅_i(τ_1')...δψ̅_i(τ_p')δψ_i(τ_1)...δψ_i(τ_p)|_ψ̅=ψ=0.Then, truncating W_1[ψ̅,̅ψ] to quartic order and inserting the results to Eq. (<ref>), one gets the following effective actionS^eff[c̅,c,ψ̅,ψ,Δ̅,Δ]=S̃_0^B[ψ̅,̅ψ]+∑_ij∫_0^ħβdτ[ψ̅_i(τ)+Ic̅_i↑(τ)c̅_i↓(τ)] × J_ij^-1[ψ_j(τ)+Ic_j↓(τ)c_j↑(τ)]+S̃_0^F[c̅,̅c,Δ̅,Δ] -1/4∑_i∫_0^ħβdτ dτ'dτ”dτ”'G_i^2,c(τ,τ',τ”,τ”') ×ψ̅_i(τ”')ψ̅_i(τ”)ψ_i(τ')ψ_i(τ) .,withS̃_0^B[ψ̅,̅ψ] =∑_i∫_0^ħβdτ dτ'G_i^1,c(τ,τ')ψ̅_i(τ')ψ_i(τ)It is interesting to point out here that the pair hopping term naturally emerges in the effective action from Eq. (<ref>), i.e. the term I^2∑_ij∫_0^ħβdτ J_ij^-1c̅_i↑(τ)c̅_i↓(τ)c_j↓(τ)c_j↑(τ) and is induced by the resonant interaction I. Further, we perform the second HS transformation in terms of J_ij^-1 , i.e.-∑_ij∫_0^ħβdτ[ψ̅_i(τ)+Ic̅_i↑(τ)c̅_i↓(τ)] × J_ij^-1[ψ_j(τ)+Ic_j↓(τ)c_j↑(τ)] →∑_ij∫_0^ħβdτ J_ijϕ̅_i(τ)ϕ_j(τ)-{∑_i∫_0^ħβdτϕ̅_i(τ)[ψ_i(τ)+Ic_i↓(τ)c_i↑(τ)]+c.c.} ,where the new HS fields are ϕ_i(τ), ϕ̅_i(τ). In comparison to the fields from the first HS (Eq. (<ref>)), the ϕ_i(τ), ϕ̅_i(τ) fields have the same generating functional as the original b_i(τ), b̅_i(τ) fields. Therefore using the ϕ_i(τ), ϕ̅_i(τ) fields is more suitable in the physical analysis because their correlation functions havethe same interpretation as the correlation functions for the original b_i(τ), b̅_i(τ) fields. To clarify this, in Appendix <ref>, we add the proof that both fields have the same generating functional. Moreover, beyond this useful fact about ϕ_i(τ), ϕ̅_i(τ), it is worth mentioning here that these fields, in the limit of BHM (when I=0), yield properly normalized density of states in the BHM superfluid phase <cit.> (properties of the SF spectrum in the full BFHM need further studies).After applying second HS (Eq. (<ref>)) to the Eq. (<ref>), corresponding effective action is S^eff[c̅,c,ϕ̅,̅ϕ,Δ̅,Δ]=-∑_ij∫_0^ħβdτ J_ijϕ̅_i(τ)ϕ_j(τ)+{ I∑_i∫_0^ħβdτϕ̅_i(τ)c_i↓(τ)c_i↑(τ)+c.c.}+S̃_0^F[c̅,̅c,Δ̅,Δ]+W_2[ϕ̅,̅ϕ] ,with denotationW_2[ϕ̅,̅ϕ]=-ħln⟨ e^-1/ħ∑_i∫_0^ħβdτ(ϕ̅_i(τ)ψ_i(τ)+c.c.)+1/4∑_i∫_0^ħβdτ dτ'dτ”dτ”'G_i^2,c(τ,τ',τ”,τ”')ψ̅_i(τ”')ψ̅_i(τ”)ψ_i(τ')ψ_i(τ)⟩ _0^B,eff,and where the statistical average ⟨ ...⟩ _0^B,eff is defined as (Z̃_0^B)^-1∫𝒟[ψ̅,ψ]...e^-S̃_0^B/ħ with Z̃_0^B=∫𝒟[ψ̅,ψ]e^-S̃_0^B/ħ. And once again truncating W_2[ϕ̅,̅ϕ] to the quartic order and retaining only the terms which are not “anomalous” <cit.>, we obtained the final form of statistical sum Z̃^eff with effective action S̃^eff (in which the fermionic degrees of freedom were integrated out), i.e.Z̃^eff=∫𝒟[ϕ̅,ϕ,Δ̅,Δ]e^-1/ħS̃^eff[ϕ̅,ϕ,Δ̅,Δ], S̃^eff[ϕ̅,ϕ,Δ̅,Δ]=-Trln(-G_F^-1(i,j, τ))+1/V∑_i|Δ_i(τ)|^2-∑_ij∫_0^ħβdτ J_ijϕ̅_i(τ)ϕ_j(τ)-∑_i∫_0^ħβdτ dτ'[G_i^1,c(τ,τ')]^-1ϕ̅_i(τ')ϕ_i(τ)+1/4∑_i∫_0^ħβdτ dτ'dτ”dτ”'Γ_i^2,c(τ,τ',τ”,τ”')ϕ̅_i(τ”')ϕ̅_i(τ”)ϕ_i(τ')ϕ_i(τ),where we introduced the matrix fermionic Green functionG_F^-1(i,j, τ)==[[ (-ħ∂/∂τ+μ)δ_ij+t_ijΔ_i(τ)-Iϕ_i(τ);Δ̅_i(τ)-Iϕ̅_i(τ) (-ħ∂/∂τ-μ)δ_ij-t_ij ]],and effective interaction between bosonsΓ_i^2,c(τ,τ',τ”,τ”') .=δ^(4)W_2[ϕ̅,̅ϕ]/δϕ̅_i(τ_1')δϕ̅_i(τ_2')δϕ_i(τ_1)δϕ_i(τ_2)|_ϕ̅=ϕ=0.In the following, to analyze the phase diagrams of BFHM, we focus on the saddle point approximation for the effective action from Eq. (<ref>). Moreover, we point out that this effective action could be also used as a starting point for more general considerations which include the fluctuations around saddle point approximation. Formally, it can be performed by expanding G_F^-1(i,j, τ) in terms of Δ_i(τ)-Iϕ_i(τ) fields.§.§ Saddle point approximation of the effective actionTo investigate the phase diagram which is described by the BFHM effective action from Eq. (<ref>), we apply the mean-field type approximations.At first, we rewrite Eq. (<ref>) in the Matsubara frequencies (ω_m, ν_n) and wave vector (𝐤, 𝐪, 𝐩) representation, which results in c_i(τ)→ c_𝐤m, ϕ_i(τ)→ϕ_𝐪n, Δ_i(τ)→Δ_𝐪n. The Matsubara frequencies are defined as ω_m=(2m+1)π/β and ν_n=2nπ/β where m,n∈ℤ. Then, applying the Bogoliubov like substitution to the ϕ_00 and Δ_00 components, i.e. ϕ_00→√(Nħβ)ϕ_0 and Δ_00→√(Nħβ)Δ_0 and omitting the fluctuating bosonic parts Δ_𝐪n and ϕ_𝐪n, the mean-field effective action is obtained, i.e.S_MF^eff={ϵ_0-ħ[G^1,c(iν_n=0)]^-1} Nħβ|ϕ_0|^2+g/2(Nħβ)^2|ϕ_0|^4+Nħβ/V|Δ_0|^2-Trln(-Nβ G_F^-1(iω_m, 𝐤)) ,whereG_F^-1(𝐤,iν_m)=[[iħω_m-ξ_𝐤 Δ_0-Iϕ_0; Δ̅_0-Iϕ̅_0iħω_m+ξ_𝐤 ]] ,with ϵ_𝐪=-2J∑_αcos q_α, ξ_𝐤=t_𝐤-μ, t_𝐤=-2t∑_αcos k_α (symbol α∈{ x,y,z} denotes Cartesian coordinates). Moreover, in further calculations we also define coordinate number z=6 which is related to the ϵ_𝐪 by expression ϵ_0=-Jz. Here, we restrict our consideration to the simple cubic lattices.The explicit form of G^1,c(iν_n) is given in Appendix <ref>. Moreover, in Eq. (<ref>) we use static approximation to the Γ_i^2,c function and denote this limit by 2g (here we do not use the explicit form of g but it could be found in Ref. <cit.>). To describe the ordered phase in terms of ϕ_0 and Δ_0 we calculate the saddle point of the above effective action ∂/∂b̅_0S_MF^eff=0 , ∂/∂Δ̅_0S_MF^eff=0 .This results in the following coupled equations{[ {ϵ_0-ħ[G^1,c(iν_n=0)]^-1}ϕ_0+gNħβ|ϕ_0|^2ϕ_0=-I/Nħβ∑_m𝐤G_F^12(𝐤,iħω_m)=-I/N∑_𝐤(Vx_0-Iϕ_0)/2E_𝐤tanh(β/2E_𝐤),; x_0=1/Nħβ∑_m𝐤G_F^12(𝐤,iħω_m)=1/N∑_𝐤(Vx_0-Iϕ_0)/2E_𝐤tanh(β/2E_𝐤), ].whereVx_0=Δ_0 andE_𝐤=√(ξ_𝐤^2+|Iϕ_0-Vx_0|^2) .From Eqs. (<ref>) one immediately sees that x_0 and ϕ_0 are non-linearly coupled to each other, i.e. {ϵ_0-ħ[G^1,c(iν_n=0)]^-1}ϕ_0+gNħβ|ϕ_0|^2ϕ_0=-Ix_0.which suggests that there is only one phase transition from the superfluid phase to normal phase.Moreover, it is interesting to point out here, that above equation correctly recovers the limiting cases of non-interacting (U=0) and hard core (U→∞) bosons (in which fermionic interaction can be finite i.e. V≠0). For U=0 the term with g disappears and one has ħ[G^1,c(iν_n=0)]^-1=μ^*, therefore ϕ_0=-I/ϵ_0-(2μ-2Δ_B)x_0 ,which corresponds to the well-known result without a lattice <cit.>. For U→∞, two Fock states are taken in Eq. (<ref>), i.e. n_0=0, 1, which gives ħ[G^1,c(iν_n=0)]^-1=μ^*/(1-2n_B,0) with n_B,0=e^βμ^*/(1+e^βμ^*). Therefore, for the hard-core bosons case one getsϕ_0=(Ix_0+ϵ_0ϕ_0)1-2n_B,0/2μ-2Δ_B ,where we neglect the contribution from g term by assuming a limit of small order parameter ϕ_0. This result (Eq. (<ref>)) recovers the previous one from Ref. <cit.>.We have also confirmed that Eqs. (<ref>), in the limit of small amplitude of ϕ_0 (in which the term proportional to g could be neglected), can be recovered from the mean-field and linear response considerations, see Appendix <ref>. Therefore, these both approaches lead to the same equation for critical line considered in the rest of the paper.At the end of this subsection, it is worth pointing out that the results, obtained in Secs. <ref> and <ref>, are quite general and can be used for further analytical and numerical considerations in which I, U and V interactions are finite quantities. These results are interested on its own right and can be applied to study of e.g. superfluidity or critical phenomena.In our further analysis, we focus on the specific physical regime of derived theory in which BHM is set as our reference point. §.§ Phase diagram In this work, we are interested in the phase diagram of strongly correlated bosonic regime (J≪ U). Therefore, at the phase boundary where x_0→0, ϕ_0→0 in Eqs. (<ref>), critical line is obtained fromϵ_0-ħ [G^1,c(iν_n=0)]^-1=I^2Π(T_c)/1-VΠ(T_c),whereΠ(T_c)=1/N∑_𝐤1/2ξ_𝐤tanh(ξ_𝐤/2k_B T_c).It is interesting to notice here that in the case of I=0, the Eq. (<ref>) and the equation in the second line of (<ref>), get the forms which are known in the phase diagram analysis of BHM and BCS systems, respectively. However, in our furhter analysis, we limit considerations to the case of V=0 for simplicity. Therefore we focus on the paring mechanism of fermions which comes from the converting interactions I. Then, by direct substitution of V=0 tothe Eq. (<ref>), the phase boundary in BFHM is obtained from the equationϵ_0-ħ [G^1,c(iν_n=0)]^-1=I^2Π(T_c).In further discussion we set ħ=1 and k_B=1 for simplicity.§.§ Average particle numberDuring the analysis of the boson-fermion mixture phase diagram in the following sections, the additional considerations of the average particle number per site n are made; n is calculated within the unperturbed part of the action from Eq. (<ref>) at the phase boundary (it is consistent with the mean-field calculation of average particle number per site at phase boundary within the operator approach method, see Appendix <ref>). This means that the 0-th order partition function has the form Z_0=Z_0^FZ_0^B where Z_0^F=∫𝒟[c̅,c]e^-S_0^F[c̅,̅c]/ħ and Z_0^B is defined in Eq. (<ref>). Therefore n is calculated by using n=-∂ln Z_0/∂μ and we getn=n_F+2n_B ,where n_F is the average particle number of fermions for both spin componentsn_F=2∑_𝐤1/e^β(t_𝐤-μ)+1 ,and n_B is an average particle number of bosons n_B=∑_n_0=0^∞n_0e^-β E_n_0/∑_n_0=0^∞e^-β E_n_0 ,where on-site bosonic energy E_n_0 is defined in Eq. (<ref>). There is also a possibility to obtain Eqs. (<ref>-<ref>) directly by taking into account Gaussian fluctuations over a saddle point action S_MF^eff from Eq. (<ref>) at the phase boundary.It is also worth adding here that improved approach which includes the effect of resonant interaction I, bosonic hopping J and fermionic interaction V, in the normal phase, can be achieved by using the self-consistent T-matrix theory <cit.>. § RESULTS AND DISCUSSION§.§ Phase diagram of the BHM In order to clarify further discussion, we shortly review the finite temperature phase diagram of the standard BHM in terms of reduced critical temperature T_c/zJ versus average concentration of bosons per site n_B. Using previously defined bosonic annihilation and creation operators b_i and b_i^†, BHM Hamitonian has the form H_BHM=-∑_ij(J_ij+μδ_ij)b_i^†b_j+U∑_ib_i^†b_i^†b_ib_i. The phase diagram comprising SF, bosonic Mott insulator (BMI) and normal (N) phases is well-known <cit.> and in the mean-field approximation the critical line is given by ϵ_0-[G^1,c(iν_n=0)]^-1=0. In Fig. <ref>, we plot critical temperature T_c/zJ dependence on the average density of bosons per site n_B for the critical boundary in BHM. BMI for different integer values of n_B are located only between lobes at zero temperatures which are indicated in Fig. <ref> by black arrows (at finite temperatures there is no true insulating state <cit.>). Here and in the following subsection we choose U/Jz=20 to analyze strong interaction limit of bosonic particles.§.§ Density phase diagram of BFHM modelWe are interested in the density phase diagram of BFHM in the limit J≪ U and V=0 (as was mentioned in Sec. <ref>). The critical boundary line at finite temperatures is obtained from Eq. (<ref>). In the following subsections <ref>, <ref>, <ref>, the phase diagram of BFHM is analyzed in three different regimes of parameter Δ_B which controls mutual position of fermionic and bosonic band, namely: (a) Δ_B/zt=0, (b) Δ_B/zt>0, (c) Δ_B/zt<0. In particular, the value of parameter Δ_B is directly related to the position of the bottom of bosonic band with respect to that of fermionic one. It is clear from considering BFHM Hamiltonian from Eq. (<ref>) and from relation J=t/2 which corresponds to assumption that one molecule is made of two fermionic particles. The bottom of the boson band is located at the center of the fermion band at Δ_B/zt=0.25 and it starts to appear below the fermionic band for Δ_B/zt<-0.75 and above for Δ_B/zt>1.25.§.§ Zero detuning (Δ_B=0)In Fig. <ref>, we show the finite temperature phase diagram for BFHM with zero detuning Δ_B/zt=0, finite bosonic interaction strength U/zJ=20 and converting interaction I/zt=1. These results explicitly show that if Δ_B/zt=0 parameter is close to Δ_B/zt=0.25 value (i.e. bottom of bosonic band is close to the middle of fermionic one), the critical line assumes a regular lobe structure similarly like in the phase diagram of standard BHM (see Fig. <ref>). However, in BFHM case the lowest lobe is relatively wider than the others (i.e. n∈(0,4) instead of width 2 in n units in comparison to pure BHM case, see. Fig. <ref>). This widening is related to the gradual filling up of the fermionic band with increasing value of total particles n (see, Fig. <ref> b). Indeed, chemical potential gradually crosses the fermionic band which is clearly visible in Fig. <ref> d and e, i.e. μ/zt appear at the bottom of fermionic band (μ=-zt) at n=0 and ending at the top of fermionic band (μ=zt) for n=4.It is interesting to notice here thatin comparison to the BHM case (Fig. <ref>), there is an enhancement of the superfluid critical temperature when I≠ 0. Starting from second lobe, this enhancement can be simple accounted forthe pairing mechanism of fermionic holes. This is confirmed by the slight deviations of fermionic density from a band insulator regime (n_F=2) for n>4 (see Fig. <ref> b and its corresponding enlargement in Fig. <ref>).The above picture is dramatically changed when detuning starts to deviate from zero value. It will be discussed below.§.§ Positive detuning (Δ_B>0)With increasing value of Δ_B/zt parameter, the bottom of bosonic band is above the fermionic one for Δ_B/zt>1.25. This should result in increasing fermionic density at the expense of bosonic one at low n which indeed is clearly visible in Fig. <ref> a-c. In particular, with increasing Δ_B/zt, the lower part of the first lobe gradually diminishes and the first lobe-like structure appears for n∈(2,4) (see, Fig. <ref> with Δ_B/zt=1.5). Such a situation is also confirmed by analysis of the chemical potential μ/zt (see, Fig. <ref> d and e) which shows that its value starts to appear only in region of fermionic band for n∈(0, 2) and for higher values of Δ_B/zt for which the bosonic density is very low (it should be compared to the situation with Δ_B/zt=0 in which μ∈⟨ -zt,zt⟩ for n∈(0, 4), Fig. <ref> d and e).§.§ Negative detuning (Δ_B<0)The situation is even more interesting for negative detuning for which the bottom of bosonic band is below the fermionic one for Δ_B/zt<-0.75. Intuitively, when the number of particles n is increased, at first the bosonic band should start to fill up. This intuition fully agrees with our simulation presented in Fig. <ref> for n_B and n_F versus n and is clearly observed in the regime of relatively high negative values of Δ_B/zt=-2.5. However, in comparison to the reference case at Δ_B/zt=0 the situation here is more complex, the critical line at Δ_B/zt=-2.5 for n∈⟨ 0, 4⟩ range decays into two lobes (see, Fig. <ref> a). The first lobe at n∈⟨ 0, 2⟩ contains the SF phase with gradually increasing average number of bosonic particles n_B (Fig. <ref> c) and the second lobe at n∈⟨ 2, 4⟩ is characterized by the almost integer bosonic density n_B (here close to one)i.e. it has the BMI character for bosonic particles (the bosonic density deviates from the integer number with order less than 10^-6) (Fig. <ref> c). Moreover, the fermionic part for n∈⟨ 2, 4⟩ gradually changes its density from n_F=0 to n_F=2 with increasing value of n. We also clearly see, that the phase is characterized by location of chemical potential inside the fermionic band, pointing out that the system is at the Feshbach resonance (see, Fig. <ref> d and e). Further, we argue that this superfluid phase withnumber of bosons close to integer value arises purely from the resonant mechanism and for simplicity we denote it as resonant superfluid (RSF) phase.To show the resonant character of RSF we check the sensitivity of this phase by tuning the amplitude of converting interaction in S_0^FB from Eq. (<ref>). Namely, in Fig. <ref>we plot the phase diagram for different values of I/zt. This phase diagram shows that the RSF phase is highly suppressed at finite temperatures and it almost disappear for I/zt=0.5. Therefore one can conclude that RSF phase originates from the Feshbach-like correlations. Moreover, it is worth adding here, that fermionic n_F and bosonic n_B densities are almost intact with respect to the change of I/zt in RSF phase (see Fig. <ref>b, c and <ref>). However, as expected we observe, that there is a slight change of these densities not visible in the presented density plots (the order of this change is less than 10^-6).We also checked the vicinity of the RSF region by analyzing normal phase above the critical temperature in terms of n_F and n_B densities at a constant n (see, Fig. <ref>). These densities correspond to the I/zt=0 regime at this level of approximation (see Sec. <ref> and Eqs. (<ref>-<ref>). From Fig. <ref>, we observe that in the T→0 limit, n_B is pinned to the integer value equal to one while n_F gradually increases for the corresponding total particle density n=2, 3, 4. This observation is consistent with the conclusion about RSF phase drawn in the previous paragraph.It is also worth adding here, that the above picture of BFHM phase diagram is also consistent with the work <cit.> which considered the hard-core limit of bosonic particles without bosonic hopping (J=0). It should not be surprising because our theory properly recover this limit at the mean-field level (see, Eq. <ref>). However, RSF phase with the number of bosons close to one is a novel behavior which appears beyond the hard-core limit.Moreover, when the system is beyond the Feshbach resonance for Δ_B/zt=-2.5 (i.e. chemical potential is below or above fermionic band) there is another interesting feature observed in Fig. <ref>. Namely, the SF phase is favored for n∈(0, 2) and n>4, but it is important to point out here that the mechanism behind it is quite different. In the n∈(0, 2) range SF is enhanced through paring of fermionic particles (BCS like character), but in the n>4 range paring mechanism is through fermionic holes. It is indicated by the corresponding low magnitude enhancement (for n∈(0, 2)) or reduction (for n>4) of the fermionic density part in numerical data.At the end of this section, we would like to also add that for higher values of negative detuning Δ_B/zt, the general behavior of phase boundary is similar to that discussed above. Namely, higher negative values of Δ_B/zt shift of chemical potential also to higher negative values causing that Feshbach resonance region around μ/zt∈⟨ -1, 1⟩ appears for higher densities. Then, depending on the Δ_B/zt value, a situation like that in the former cases appears, i.e. (1) the widening of one of the lobes like for Δ_B/zt=0 (see, Fig. <ref>) or (2) the emergence of RSF mixture like for Δ_B/zt=-2.5 (see, Fig. <ref>). In particular, up to Δ_B/zt=-10 with the same BFHM Hamiltonian parameters as before, we numerically check, that the first situation (1) appears for Δ_B/zt=-5 and Δ_B/zt=-10 and the second one (2) appears for Δ_B/zt=-7.5 (here RSF phase emerge for n∈(4, 6)).It would be also interesting in further investigations beyond mean-field approximation, to include the effects of pairing fluctuations into theory which should imply lowering of superfluid critical temperature. Then the temperature obtained in this work will correspond to the appearance of the pseudogap regime for fermionic particles <cit.>.§.§ RSF phase in the time of flight type experimentTime of flight (TOF) type spectroscopy is one of the most powerful methods of measurements in the state of art of current experimental setups in ultracold atoms. Within the optical lattice systems, it has been widely used for e.g. bosons <cit.>, fermions <cit.> or boson-fermion mixtures <cit.>. In particular, it is relatively simple to probe coherence via momentum distribution encoded in freely expanding cloud. As an example, it has been previously used to detect SF-BMI quantum phase transition in the bosonic Rb atoms <cit.> or resonant superfluidity in the fermionic Li atoms <cit.>. In a realistic experiment, the enhancement of coherence is observed as the appearance of peaks in the time of flight pattern <cit.>.We suggest that the footprint of the RSF phase can be tested by preparing ultracold fermionic gas at the Feshbach resonance with negative detuning of Δ_B parameter. The detuning should be about two and half times greater than the width of the fermionic band. Then repeating the experiment with increasing number of fermions which simulate BFHM (which is close to the ground state), one should observe a lowering of coherence at n∈⟨ 2, 4⟩ densities. It can be deduced from the phase diagram in Fig. <ref> where in the range n∈(0, 2) and n>4, SF phase has a higher critical temperature than in the n∈(2, 4) region. For instance, let's assume that the atomic gas is prepared at similar temperatures for different particle numbers which are represented by points A, B and C in Figs. <ref> a and <ref> a. Furthermore, let's assume that in each of these phases represented by points A, B and C, TOF experiment is performed. Then, it can be concluded that for the situation with positive detuning as in Fig. <ref> a, the coherence of bosonic particles should be an increasing function of n at corresponding points A, B and C, because of the deeper penetration of the system into SF phase for A, B and C, respectively. However, this situation should be quite different for negative detuning of Δ_B/zt. As shown in Fig. <ref> a, point B in comparison to point A and C is located beyond SF phase, which means that TOF pattern does not exhibit the behavior characteristic of SF phase <cit.>. Therefore, for negative detuning, one should observe non-monotonous behavior of coherence peaks which can be read off from TOF patterns for the corresponding points A, B and C. Moreover, increasing strength of Fershbach interaction I/zt should result in gradual disappearance of this non-monotonous behavior at point B (see Fig. <ref> a). Consequently, such coherence dependence which can be observed in experiment, could be accounted for by the appearance of RSF phase in the investigated system. § SUMMARY In this work, we investigated the limit of strongly correlated Feshbach molecules at finite temperatures in a three dimensional lattice. We show, that for negative detuning Δ_B/zt and at least for weak strength of converting interaction I/zt, a resonant superfluid phase (RSF) appears which is characterized by an arbitrary number of fermions per site (i.e. fermionic concentration between 0 and 2) and an integer number of bosonic atoms. This happens when fermions are in the Feshbach resonance. We show that this resonant character of RSF phase is unstable toward weakening converting interaction I/zt. In the situation when the fermions are beyond resonance the superfluid phase is strengthened. We explain that this enhancement is caused by hole pairing mechanism for higher densities, while for lower densities it is standard fermionic particle paring mechanism which corresponds to that known in the BCS theory.Moreover, we have also discussed the experimental protocol in which footprint of RSF phase can appear in TOF type experiment. Namely, the footprint of the RSF phase could be simply observed as a non-monotonous behavior of coherence peaks from time of flight pattern when the number of fermions is increased.In future investigation, it will be also interesting to study the system's behavior from the point of view of tuning the parameter Δ_B at fixed total n. Especially interesting analysis would be for the total density equal to two (n=2) in which two different peculiar regimes should appear depending on the Δ_B and U amplitude. Namely, tuning the system from positive Δ_B>0 to negative value Δ_B<0, should result in transition from fermionic band insulator (n_F=2, n_B=0) to SF phase and from SF to bosonic Mott insulator (n_F=0, n_B=1). We left this problem for future studies in which careful analysis of the BFHM ground state is also required.We would like to thank to Prof. T. K. Kopeć for useful discussions on the early stage of the presented work. We are also grateful to Dr T. P. Polak for careful reading of the manuscript. § APPENDIX §.§ Local Green functionOn-site single particle green function, defined as 1/ħG^1,c(τ-τ')=-⟨ψ̅_i(τ)ψ_i(τ')⟩ _0^B is given by 1/ħG^1,c(iν_n)=1/Z_0∑_n_0=0^∞(n_0+1)e^-β E_n_0+1-e^-β E_n_0/E_n_0+1-E_n_0-iħν_n ,where E_n_0=-μ^*n_0+Un_0(n_0-1)/2 , Z_0=∑_n_0=0^∞e^-β E_n_0 . §.§ Generating functional in the BFHM The generating function of statistical sum from Eq. (<ref>) has the formZ[γ̅,̅γ]=∫𝒟[c̅,c,b̅,̅b]e^∑_ij∫_0^ħβdτ J_ijb̅_i(τ)b_j(τ)-S_0^F[c̅,̅c]-S_0^B[b̅,̅b]-S_0^FB[b̅,̅b,c̅,̅c]+∑_i∫_0^βdτ(γ̅_i(τ)b_i(τ)+c.c.),where γ_i(τ), γ̅_i(τ) are external sources. It can be rewritten to the formZ[γ̅,̅γ]=∫𝒟[c̅,c,b̅,̅b]e^∑_ij∫_0^ħβdτ J_ijb̅_i(τ)b_j(τ)-S_0^F[c̅,̅c]-S_0^B[b̅,̅b]-∑_i∫_0^βdτ{[-ψ̅_i(τ)+Ic̅_i↑(τ)c̅_i↓(τ)-γ̅_i(τ)]b_i(τ)+c.c.}After first HS of bosonic fields b_i(τ), b̅_i(τ)(see also Eq. (<ref>)), one hasZ[γ̅,̅γ]=Z_0^B[𝐉^-1]∫𝒟[c̅,c,ψ̅,ψ] × e^-1/ħ∑_ij∫_0^ħβdτ J_ij^-1ψ̅_i(τ)ψ_j(τ)-1/ħ∑_i∫_0^ħβdτ([-ψ̅_i(τ)+Ic̅_i↑(τ)c̅_i↓(τ)-γ̅_i(τ)]b_i(τ)+c.c.) × e^-S_0^F[c̅,̅c]-S_0^B[b̅,̅b]-S_0^FB[b̅,̅b,c̅,̅c].Next, shifting ψ_i(τ)→ψ_i(τ)-γ_i(τ)+Ic_i↓(τ)c_i↑(τ), ψ̅_i(τ)→ψ̅_i(τ)-γ̅_i(τ)+Ic̅_i↑(τ)c̅_i↓(τ), we obtainZ=Z_0^B[𝐉^-1]∫𝒟[c̅,c,ψ̅,ψ] × e^-1/ħ∑_ij∫_0^ħβdτ J_ij^-1[ψ̅_i(τ)+Ic̅_i↑(τ)c̅_i↓(τ)-γ̅_i(τ)][ψ_j(τ)+Ic_j↓(τ)c_j↑(τ)-γ_i(τ)]-W_1[ψ̅,̅ψ] × e^-S_0^F[c̅,̅c].Finally, taking second HS (see also Eq. (<ref>))-∑_ij∫_0^ħβdτ[ψ̅_i(τ)+Ic̅_i↑(τ)c̅_i↓(τ)-γ̅_i(τ)] × J_ij^-1[ψ_j(τ)+Ic_j↓(τ)c_j↑(τ)-γ_i(τ)] →∑_ij∫_0^ħβdτ J_ijϕ̅_i(τ)ϕ_j(τ)-{∑_i∫_0^ħβdτϕ̅_i(τ)[ψ_i(τ)+Ic_i↓(τ)c_i↑(τ)-γ_i(τ)]+c.c.} ,we haveZ[γ̅,̅γ] = Z_0^B[𝐉^-1][-𝐉]∫𝒟[c̅,c,ψ̅,ψ,ϕ̅,ϕ]e^∑_ij∫_0^ħβdτ J_ijϕ̅_i(τ)ϕ_j(τ)+∑_i∫_0^ħβdτ{ϕ̅_i(τ)ψ_i(τ)+c.c.}, × e^-1/ħW_1[ψ̅,̅ψ]+S̃_0^F[c̅,̅c,Δ̅,Δ]+∑_i∫_0^ħβdτ{ϕ̅_i(τ)γ_i(τ)+c.c.}.From Eqs. (<ref>) and (<ref>), we see that the b_i(τ), b̅_i(τ) and ϕ_i(τ), ϕ̅_i(τ) fields have the same generating functional Z[γ̅,̅γ]. The above considerations about generating functional correspond to those in Appendix A of Ref. <cit.>. §.§ Mean-field equations for order parameters - the operator approachEqs. (<ref>) were derived by using coherent state path integral within double Hubbard-Stratonovich transformation within the bosonic part of action. Now, we show that these equations can be also recovered by using a standard operator approach, at least in the small ϕ_0 limit. In order to get the equations for order parameters ϕ_0 and x_0, we start from the mean-field approximation applied to the BFHM Hamiltonian defined in Eq. (<ref>), i.e.- for bosonic hopping term:-∑_ijJ_ijb_i^†b_j ≈NzJ|ϕ_0|^2-zJϕ_0∑_ib_i^†-zJϕ̅_0∑_ib_i ,- for fermionic interaction term (BCS type approximation in the pairing channel): V∑_ic_i↑^†c_i↓^†c_i↓c_i↑ ≈V/N∑_𝐤𝐤'c_𝐤'↑^†c_-𝐤'↓^†c_-𝐤↓c_𝐤↑ ≈-N/V|Δ_0|^2+∑_𝐤Δ̅_0c_-𝐤↓c_𝐤↑+∑_𝐤'c_𝐤'↑^†c_-𝐤'↓^†Δ_0 ,- for resonant interaction term:I∑_i(c_i↑^†c_i↓^†b_i+b_i^†c_i↓c_i↑) ≈ I∑_𝐤(c_𝐤↑^†c_-𝐤↓^†ϕ_0+ϕ̅_0c_-𝐤↓c_𝐤↑)+I1/V∑_i(Δ̅_0b_i+Δ_0b_i^†)-I1/V∑_i(Δ̅_0ϕ_0+Δ_0ϕ̅_0) .Then, the thermodynamic potential can be written in the form Ω=-1/βln Z,withZ=Tre^-β(H_eff^F+H_eff^B+H_eff^FB)and whereH_eff^F=∑_𝐤σξ_𝐤c_𝐤σ^†c_𝐤σ-∑_𝐤(Δ̅_0-Iϕ̅_0)c_-𝐤↓c_𝐤↑-∑_𝐤c_𝐤↑^†c_-𝐤↓^†(Δ_0-Iϕ_0)+N/V|Δ_0|^2, H_eff^B= NzJ|ϕ_0|^2+(I1/VΔ_0-zJϕ_0)∑_ib_i^†+(I1/VΔ̅_0-zJϕ̅_0)∑_ib_i-∑_iμ^*b_i^†b_i+U∑_ib_i^†b_i^†b_ib_i , H_eff^FB=-IN/V(Δ̅_0ϕ_0+Δ_0ϕ̅_0).Next, the ϕ_0 and Δ amplitudes can be obtained from the conditions∂Ω/∂Δ̅_0=0,∂Ω/∂ϕ̅_0=0,which give{[ 0=-N/VΔ_0+IN/Vϕ_0+∑_𝐤⟨ c_-𝐤↓c_𝐤↑⟩ -I/V∑_i⟨ b_i⟩ ,; 0=-I∑_𝐤⟨ c_-𝐤↓c_𝐤↑⟩ -NzJϕ_0+zJ∑_i⟨ b_i⟩ +IN/VΔ_0. ].This leads tox_0=1/N∑_𝐤⟨ c_-𝐤↓c_𝐤↑⟩ , ϕ_0=1/N∑_i⟨ b_i⟩ ,where in this section statistical average is defined as ⟨ ...⟩ =Tr ...e^-β(H_eff^fer+H_eff^bos+H_eff^fer-bos)/Z and we introduce x_0=Δ/V the same as in Sec. <ref>.Now we focus on the first equation, i.e. Eq. (<ref>). Expectation value ⟨ c_-𝐤↓c_𝐤↑⟩ for a given wave vector 𝐤 can be calculated by diagonalizing H_eff^fer Hamiltonian using the standard Bogoliubov transformationc_𝐤↑=u̅_𝐤γ_𝐤↑+v̅_𝐤γ_-𝐤↓^† , c_𝐤↓=u̅_𝐤γ_𝐤↓-v̅_𝐤γ_-𝐤↑^† ,with|u_𝐤|^2=1/2(1+ξ_𝐤/E_𝐤), |v_𝐤|^2=1/2(1-ξ_𝐤/E_𝐤),then we obtain⟨ c_-𝐤↓c_𝐤↑⟩ =Vx_0-Iϕ_0/2E_𝐤tanh(β/2E_𝐤) ,with a quasi-particle fermionic energy E_𝐤 defined as before in Eq. (<ref>). Next equation, i.e. Eq. (<ref>), we calculate by using the linear response theory. Assuming, that ϕ_0 and x_0 amplitudes are small one can expand ⟨ b_i⟩ in terms of these parameters which gives 1/N∑_i⟨ b_i⟩ ≈-1/ħzJϕ_0G^1,c(iν_n=0)+1/ħIx_0G^1,c(iν_n=0) ,Finally, combining Eqs. (<ref>, <ref>, <ref>, <ref>), one gets{[ (ϵ_0-ħ[G^1,c(iν_n=0)]^-1)ϕ_0=-I/N∑_𝐤Vx_0-Iϕ_0/2E_𝐤^Ftanh(β/2E_𝐤),; x_0=1/N∑_𝐤Vx_0-Iϕ_0/2E_𝐤tanh(β/2E_𝐤), ].which recovers the result from coherent state path integral, i.e. Eqs. (<ref>) in the limit of small ϕ_0, in which the term gNħβ|ϕ_0|^2ϕ_0 can be neglected (i.e. on the phase boundary). Moreover, it is also worth adding that the above derivation of equations for order parameters x_0 and ϕ_0 (i.e. Eq. (<ref>)), can be also handled by using an explicit form of thermodynamic potentialΩ=Ω_F+Ω_FB+Ω_B ,whereΩ_F/N=1/N∑_𝐤(ξ_𝐤-E_𝐤)+V|x_0|^2-2/β N∑_𝐤ln(1+e^-β E_𝐤), Ω_FB/N=-I(x̅_0ϕ_0+x_0ϕ̅_0), Ω_B/N= -1/βlnTre^-β(zJ|ϕ_0|^2+(Ix_0-zJϕ_0)b_i^†+(Ix̅_0-zJϕ̅_0)b_i-μ^*b_i^†b_i+Ub_i^†b_i^†b_ib_i).Then extremizing Ω in terms of x̅_0 and ϕ̅_0 yields general mean-field equations for order parametersx_0=1/N∑_𝐤Vx_0-Iϕ_0/2E_𝐤tanh(β/2E_𝐤) ϕ_0=1/N∑_i⟨ b_i⟩ _Bwhere ⟨ ...⟩ _B=Tr ...e^-β H_eff^bos/Z, Z=Tre^-β H_eff^bos and should be compared to Eqs (<ref>) or (<ref>) which was evolved close to the phase boundary. Moreover, from Eqs. (<ref>-<ref>) it is easy to notice that the thermodynamic potential Ω consists of standard BCS-like part Ω_fer, BHM-like part Ω_bos and part Ω_fer-bos which is proportional to Feshbach interaction energy I. Eqs. (<ref>-<ref>) make also a clear framework for further analysis of thermodynamic properties of BFHM. As an example the free energy F is now simply given by F/N=Ω/N+μ n in which n=-1/N∂Ω/∂μ=n_F+2n_B n_F=1/N∑_𝐤[1-ξ_𝐤/E_𝐤tanh(β/2E_𝐤)] n_B=1/N∑_i⟨ b_i^†b_i⟩ _BThese mean-field results should be also compared with Eqs. (<ref>-<ref>) in which the 0th order approximation was imposed on statistical sum. Interestingly, the form of Ω_B and ϕ_0 given in Eqs. (<ref>) and (<ref>) can be calculated exactly for limiting cases of hard-core bosonic interaction (U→∞) and for the case where U vanishes (U=0). For example within the hard-core limit on-site bosonic density basis is restricted to two occupation numbers (i.e. to 0 or 1 boson per site) and then one gets Ω_bos/N=zJ|ϕ_0|^2-μ^*-ln[2cosh(β E_g)]/β where E_g=√((μ^*)^2+|Ix_0-zJϕ_0|^2) and for order parameter ϕ_0 one finds ϕ_0=-(Ix_0-zJϕ_0)tanh(β E_g)/2E_g <cit.>.At the end of this section, we would like to also add that going beyond the critical line toward SF phase, it is worth mentioning that the functional integral approach presented in Sec. <ref> and the operator approach discussed here give different descriptions. Indeed, evaluation of the expansion in Eq. (<ref>) to the third order in the ϕ_0 and x_0 amplitudes, generates coefficients with four point local bosonic correlation function denoted by G_i^2,c(τ_1', τ'_2, τ_1, τ_p) (see Eq. (<ref>)), while the path integral method gives Γ_i^2,c(τ,τ',τ”,τ”') (see Eq. (<ref>)). This higher order term in the path integral formulation is denoted by g in Eq. (<ref>), which is proportional to Γ_i^2,c in the static limit. Therefore, on the grounds of the previous considerations within the BHM in Ref. <cit.> we would like to point out, that our path integral formulation, should be more relevant than the operator ones, because its gives better description of gaussian fluctuation in the BHM limit with SF phase.apsrev | http://arxiv.org/abs/1706.08375v2 | {
"authors": [
"A. S. Sajna",
"R. Micnas"
],
"categories": [
"cond-mat.quant-gas",
"cond-mat.str-el",
"cond-mat.supr-con"
],
"primary_category": "cond-mat.quant-gas",
"published": "20170626135528",
"title": "Effect of boson on-site repulsion on the superfluidity in the boson-fermion-Hubbard model"
} |
=1 =1 | http://arxiv.org/abs/1706.08985v3 | {
"authors": [
"Alessandro Davoli",
"Andrea De Simone",
"Thomas Jacques",
"Verónica Sanz"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170627180250",
"title": "Displaced Vertices from Pseudo-Dirac Dark Matter"
} |
An Efficient Algorithm for Matrix-Valued and Vector-Valued Optimal Mass Transport Yongxin Chen, Eldad Haber, Kaoru Yamamoto, Tryphon T. Georgiou, and Allen Tannenbaum Y. Chen is with the Department of Medical Physics, Memorial Sloan Kettering Cancer Center, NY; email: [email protected] E. Haber is with the Department of Mathematics, University British Columbia, Vancouver, Canada; email: [email protected] K. Yamamoto is with the Department of Electrical Engineering, Lund University, Sweden; email: [email protected] T. T. Georgiou is with the Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA; email: [email protected] A. Tannenbaum is with the Departments of Computer Science and Applied Mathematics & Statistics, Stony Brook University, NY; email: [email protected] December 30, 2023 ========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================= We present an efficient algorithm for recent generalizations of optimal mass transport theory to matrix-valued and vector-valued densities. These generalizations lead to several applications including diffusion tensor imaging, color images processing, and multi-modality imaging. The algorithm is based on sequential quadratic programming (SQP). By approximating the Hessian of the cost and solving each iteration in an inexact manner, we are able to solve each iteration with relatively low cost while still maintaining a fast convergent rate. The core of the algorithm is solving a weighted Poisson equation, where different efficient preconditioners may be employed. We utilize incomplete Cholesky factorization, which yields an efficient and straightforward solver for our problem. Several illustrative examples are presented for both the matrix and vector-valued cases. § INTRODUCTIONThe theory of optimal mass transport (OMT) <cit.> has proven its power and usefulness in both theory and applications. The theory part has been developed through a sequence of elegant papers, and the research is still going strong; see <cit.> and the references therein. On the other hand, during the past decade, the need for applications has engendered the fast development of efficient algorithms for OMT <cit.>. Recently, the OMT theory has been extended to study matrix <cit.> and vector-valued densities <cit.>.The mathematical approach to matrix optimal mass transport in <cit.> is based on the seminal work of Benamou-Brenier <cit.>, where optimal mass transport with quadratic cost is recast as the problem of minimizing kinetic energy (i.e., an action integral) subject to a continuity equation. In the matrix case, one needs to develop a non-commutative counterpart to scalar optimal transport where probability distributions are replaced by density matrices ρ (Hermitian positive-definite with unit trace) and where “transport” corresponds to a flow on the space of such matrices that minimizes a corresponding action integral. The work is motivated by a plethora of applications including spectral analysis of vector-valued time-series, which may encode different modalities (e.g., frequency, color, polarization) across a distributed array of sensors <cit.>. The associated power spectra are matrix-valued and hence there is a need for suitable metrics that quantify distances and provide tools to average and interpolate spectra.The generalization of the Benamou-Brenier theory is founded upon concepts from quantum mechanics, and allows us to formulate a continuity equation for matrix-flows, and then derive a Wasserstein distance between density matrices and matrix-valued distributions. Similar remarks apply to the vector-valued case in which one must also invoke some ideas from graph theory in formulating our generalization of scalar-valued densities. See <cit.> for all the details.In this paper, we focus on algorithms for the numerical solution of the optimal matrix-valued mass transport problems introduced in <cit.>, and the vector-valued case formulated in <cit.>. In <cit.>, both problems are reformulated as convex optimization problems.We adopt an inexact sequential quadratic programming (SQP) method <cit.> to tackle such convex optimization problems. Similar methods have been applied to scalar optimal mass transport <cit.>.The remainder ofthis paper is summarized as follows. Section <ref> is a brief introduction to the matrix-valued optimal transport theory. We develop the corresponding algorithm in Section <ref>, and then the algorithm for vector-valued optimal transport is described in Section <ref>. We conclude with several examples to demonstrate our algorithm in Section <ref>.§ MATRIX-VALUED OPTIMAL MASS TRANSPORTIn this section, we sketch the approach <cit.> for which the convex optimization algorithm given in the present note was formulated. As noted above, similar approaches to matrix-valued OMT were formulated independently in <cit.>. §.§ Gradient on space of Hermitian matricesDenote byandthe set ofn× n Hermitian and skew-Hermitian matrices, respectively. We will assume that all of our matrices are of fixed size n× n. Next, we denote the space of block-column vectors consisting of N elements inandas ^N and ^N, respectively. We also let _+ and _++ denote the cones of nonnegative and positive-definite matrices, respectively,and we use the standard notion of inner product, namely ⟨ X,Y⟩=(X^*Y),for bothand . For X, Y∈^N (^N), ⟨ X, Y⟩=∑_k=1^N (X_k^*Y_k).Given X=[X_1^*,⋯,X_N^*]^* ∈^N (^N), Y∈ (), set XY=[[ X_1; ⋮; X_N ]]Y := [[X_1Y; ⋮; X_N Y ]], YX=Y[[ X_1; ⋮; X_N ]] := [[ YX_1;⋮; YX_N ]],and X̅ = [[ X_1^*; ⋮; X_N^* ]].For a given L∈^N we define ∇_L: →^N, X ↦[ [L_1 X-XL_1; ⋮; L_N X-X L_N ]]to be the gradient operator. By analogy with the ordinary multivariable calculus, we refer to its dual with respect to the Hilbert-Schmidt inner product as the (negative) divergence operator, and this is ∇_L^*: ^N →, Y= [ [ Y_1; ⋮; Y_N ]] ↦∑_k^N L_k Y_k-Y_k L_k,i.e., ∇_L^* is defined by means of the identity ⟨∇_L X , Y⟩ =⟨ X , ∇_L^* Y⟩.A standing assumption throughout, is that the null space of ∇_L, denoted by ker(∇_L), contains only scalar multiples of the identity matrix. In this note, we use one such basis generated by the following N=2 components: L_1=[1 1⋯ 1 10⋯0 ⋮ ⋮ ⋱ ⋮ 10⋯0], L_2= ([1, 2, …, n-1, 0]). §.§ Matrix-valued Optimal mass transportWe next sketch the formulation for matrix-valued optimal mass transport proposed in <cit.>. Given a convex compact set E∈^m, denote = {ρ(·) ∈_+ | ∫_E(ρ(x)) dx =1},and _+ the interior of . Let ρ^0,ρ^1∈_+ be two matrix-valued densities defined on E with positive values. A dynamic formulation of matrix-valued optimal mass transport between these two given marginals is <cit.>, min_ρ∈_+, w∈^m, v∈^N∫_0^1∫_E{(ρ w^*w)+γ(ρ v^*v)}dxdt, ∂ρ/∂ t+1/2∇_x·(wρ+ρ w)-1/2∇_L^* (vρ+ρ v)=0,ρ(0,·)=ρ^0, ρ(1,·)=ρ^1 with ∇_x · being the standard divergence operator in ^m. By defining p=wρ, u=vρ, the above can be cast as a convex optimization problem min_ρ, p, u∫_0^1∫_E{(pρ^-1p^*)+γ(uρ^-1u^*)}dxdt ∂ρ/∂ t+1/2∇_x·(p+p̅)-1/2∇_L^* (u-u̅)=0,ρ(0,·)=ρ^0, ρ(1,·)=ρ^1. We remark that (p+p̅)/2∈^m and (u-u̅)/2∈^N, which is consist with the domain of ∇_L^*.For the sake of brevity, the set E is taken to be the unit cube [0,1]^m.§ DISCRETIZATION AND ALGORITHM: MATRIX-VALUED CASEWe follow closely the algorithm developed in <cit.> for scalar optimal mass transport problems. We restrict ourselves to the real-valued case, that is,anddenote symmetric and skew-symmetric matrices, respectively. In order to highlight the key parts of our methodology, we first consider the discretization in 1D case, i.e., m=1. In particular, we take E=[0,1]. The algorithm extends almost verbatim to the higher dimensional setting as we will see in Section <ref>.We discretize the space-time domain [0, 1]×[0, 1] into n_x× n_t rectangular cells. Denote Ω_ij, 1≤ i≤ n_x, 1≤ j≤ n_t as the (i,j) box. We use a staggered grid to discretize p and ρ. The variable u is, however, valued at the centers of the cells {Ω_ij}. More specifically, p=(p_i+1/2,j), 0≤ i≤ n_x, 1≤ j≤ n_t ρ=(ρ_i,j+1/2), 1≤ i≤ n_x, 0≤ j≤ n_t u=(u_i,j), 1≤ i ≤ n_x, 1≤ j≤ n_t.Note the boundary values are p_1/2,j=0, p_n_x+1/2,j=0, 1≤ j≤ n_tand ρ_i,1/2=ρ^0_i, ρ_i,n_t+1/2=ρ^1_i, 1≤ i≤ n_x.We exclude the boundary values from the variables and denote p=(p_i+1/2,j), 1≤ i≤ n_x-1, 1≤ j≤ n_t ρ=(ρ_i,j+1/2), 1≤ i≤ n_x, 1≤ j≤ n_t-1. §.§ Continuity equation We use the above discretizing scheme, together with the boundary conditions to rewrite the continuity equation (<ref>) as D_1 p+D_2 ρ+D_3 u=b.Here the linear operators D_1, D_2, D_3 are defined as (D_1 p)_i,j= 1/2(p_i+1/2,j+p_i+1/2,j^* -p_i-1/2,j-p_i-1/2,j^*)/h_x, 2≤ i ≤ n_x-1, 1/2 (p_3/2,j+p_3/2,j^*)/h_x, i=1, -1/2(p_n_x-1/2,j+p_n_x-1/2,j^*)/h_x, i=n_x, (D_2 ρ)_i,j= (ρ_i,j+1/2-ρ_i,j-1/2)/h_t, 2≤ j ≤ n_t-1, ρ_i,3/2/h_t, j=1, - ρ_i,n_t-1/2/h_t, j=n_t, (D_3 u)_i,j=-1/2∇_L^*(u_i,j-u̅_i,j), 1≤ i ≤ n_x, 1≤ j≤ n_t.The parameter b carries the information of the boundary values ρ^0 and ρ^1. More specifically, b_i,j= ρ^0_i/h_t j=1, -ρ^1_i/h_t j=n_t, 0.§.§ Discretizing the cost functionWe use a combination of a midpoint and a trapezoidal methods to discretize the cost function. On the volume Ω_ij we have ∫_Ω_ij{(pρ^-1p^*)+γ(uρ^-1u^*)} ≈ h_xh_t/4((p_i-1/2,j^*p_i-1/2,j+ p_i+1/2,j^*p_i+1/2,j)(ρ_i,j-1/2^-1+ρ_i,j+1/2^-1)) +γ h_xh_t/2(u_i,j^*u_i,j(ρ_i,j-1/2^-1+ ρ_i,j+1/2^-1)).Let A_1 be the averaging operator over the spatial domain and A_2 be the averaging operator over the time domain (one needs to be careful about the boundaries). Then the cost function(<ref>) may be approximated by ⟨ A_1 (p^*∘ p), A_2 (ρ^-1)+a⟩ h_xh_t+ ⟨ u^*∘ u, A_2 (ρ^-1)+a⟩γ h_xh_t,where a≥ 0 depends only on the boundary values ρ^0 and ρ^1. The inverse operator and the multiplication operator ∘ are applied block-wise. The expressions for A_1, A_2, a are (A_1 (p^*∘ p))_i,j= 1/2(p^*_i-1/2,jp_i-1/2,j +p^*_i+1/2,jp_i+1/2,j), 2≤ i ≤ n_x-1, 1/2p^*_3/2,jp_3/2,j, i=1, 1/2p^*_n_x-1/2,jp_n_x-1/2,j, i=n_x, (A_2 (ρ^-1))_i,j= 1/2(ρ^-1_i,j-1/2 +ρ^-1_i,j+1/2), 2≤ j ≤ n_t-1, 1/2ρ^-1_i,3/2, j=1, 1/2ρ^-1_i,n_t-1/2, j=n_t, a_i,j= 1/2(ρ^0_i)^-1 j=1, 1/2(ρ^1_i)^-1 j=n_t, 0.We remark that it is important to first square then average, and first invert then average, to guarantee stability <cit.>. §.§ Sequential quadratic programming (SQP) Following the above discretization scheme, we obtain the discrete convex optimization problem min f(p, ρ, u) =⟨ A_1 (p^*p), A_2 (ρ^-1)+a⟩ h_xh_t+ ⟨ u^*u, A_2 (ρ^-1)+a⟩γ h_xh_t,s.t. D_1 p+D_2 ρ+D_3 u=b. The Lagrangian of this problem is (p, ρ, u)= f(p, ρ, u)/(h_xh_t)+⟨λ, D_1 p+D_2 ρ+D_3 u-b⟩.The KKT condition <cit.> ∇_p=D_1^* λ+2p∘ A_1^*(A_2(ρ^-1)+a)=0 ∇_ρ =D_2^* λ-ρ^-1∘ A_2^*A_1(p^*p)∘ρ^-1 -γρ^-1∘ A_2^*(u^*u)∘ρ^-1=0 ∇_u=D_3^*λ+2γ u∘ (A_2(ρ^-1)+a)=0 ∇_λ =D_1 p+D_2 ρ+D_3 u-b=0 follow, with ∘ denoting block-wise multiplication.Let w=(p,ρ, u), D=(D_1, D_2, D_3), then at each SQP iteration we solve the system (ÂD^* D0) ([ δ w;δλ ])= -([ ∇_w; ∇_λ ]),and update w,λ using line search. In principle, Problem <ref> can be solved using Newton's method. However, the mixed terms introduce off-diagonal elements in the Hessian, which makes it forbidden for large problems. We adopt an inexact SQP method <cit.>. The matrix  is an approximation of the Hessian of the objective function Â= ( 2Bdiag (A_1^* (A_2(ρ^-1)+a)) 0 0 0 Bdiag (g(p,ρ,u)) 0 0 0 2 γ Bdiag (A_2(ρ^-1)+a) ).Here Bdiag denotes block diagonal operator. More specifically, Bdiag(T_1, T_2,⋯, T_k)= [ T_1 0⋯0 0 T_2⋯0 ⋮ ⋮ ⋱ ⋮ 0 0⋯T_k ]for linear operators T_1, T_2, ⋯, T_k. The operator g(p, ρ, u) is the Hessian of f over ρ with g_i,j+1/2 being the map g_i,j+1/2(X) = ρ^-1_i,j+1/2(A_2^*A_1(p^*p))_i,j+1/2ρ^-1_i,j+1/2Xρ^-1_i,j+1/2+ρ^-1_i,j+1/2X ρ^-1_i,j+1/2(A_2^*A_1(p^*p))_i,j+1/2ρ^-1_i,j+1/2 + γρ^-1_i,j+1/2(A_2^*(u^*u))_i,j+1/2ρ^-1_i,j+1/2Xρ^-1_i,j+1/2+γρ^-1_i,j+1/2X ρ^-1_i,j+1/2(A_2^*(u^*u))_i,j+1/2ρ^-1_i,j+1/2. In each step we solve the linear system (<ref>) in an inexact manner. There are many methods to achieve this. In our approach, we apply the Schur complement and solve the reduced system DÂ^-1 D^* δλ = ∇_λ- DÂ^-1∇_wusing preconditioned conjugated gradients method with incomplete Cholesky factorization <cit.> as a preconditioner. The update for w is then given by δ w = -Â^-1 (D^*δλ+∇_w). In our numerical implementation, we take advantage of the structure of ρ being symmetric, and only save the upper triangular part of it. This is beneficial in terms of both memory andspeed.§.§ 2D and 3D casesIn this section we sketch what happens in higher dimensions, namely 2D and 3D.We begin with the 2D case. Accordingly, we have the discrete convex optimization problem min f(p, ρ, u) =⟨ A_1x (p_x^*p_x)+A_1y(p_y^*p_y), A_2 (ρ^-1)+ a⟩ h_xh_yh_t+ ⟨ u^*u, A_2 (ρ^-1)+a⟩γ h_xh_yh_ts.t. D_1x p_x +D_1y p_y+D_2 ρ+D_3 u=b.The Lagrangian of this problem is (p, ρ, u)= f(p, ρ, u)/(h_xh_yh_t)+⟨λ, D_1x p_x+D_1y p_y+D_2 ρ+D_3 u-b⟩.In the above, a_i,j,k= 1/2(ρ^0_i,j)^-1 k=1, 1/2(ρ^1_i,j)^-1 k=n_t, 0.and b_i,j,k= ρ^0_i,j/h_t k=1, -ρ^1_i,j/h_t k=n_t, 0.It follows that the KKT conditions are ∇_p_x =D_1x^* λ+2p_x∘ A_1x^*(A_2(ρ^-1)+a)=0 ∇_p_y =D_1y^* λ+2p_y∘ A_1y^*(A_2(ρ^-1)+a)=0 ∇_ρ =D_2^* λ-ρ^-1∘ A_2^*(A_1x(p_x^*p_x)+A_1y(p_y^*p_y)) ∘ρ^-1 -γρ^-1∘ A_2^*(u^*u)∘ρ^-1=0 ∇_u=D_3^*λ+2γ u∘ (A_2(ρ^-1)+a)=0 ∇_λ =D_1 p+D_2 ρ+D_3 u-b=0, with ∘ denoting block-wise multiplication as before.Let w=(p_x,p_y,ρ, u). Then at each SQP iteration, we solve the system (ÂD^* D0) ([ δ w;δλ ])= -([ ∇_w; ∇_λ ]),where D=(D_1x,D_1y, D_2, D_3). The matrix  is an approximation of the Hessian of the objective function Â= ( 2Bdiag (A_1x^* (A_2(ρ^-1)+a)) 0 0 0 0 2Bdiag (A_1y^* (A_2(ρ^-1)+a)) 0 0 0 0 Bdiag (g(p,ρ,u)) 0 0 0 0 2 γ Bdiag (A_2(ρ^-1)+a) )The operator g(p, ρ, u) is the Hessian of f over ρ with g_i,j,k+1/2 being the map g_i,j,k+1/2(X) = ρ^-1_i,j,k+1/2(A_2^*(A_1x(p_x^*p_x)+ A_1y(p_y^*p_y)+γ u^*u))_i,j,k+1/2ρ^-1_i,j,k+1/2Xρ^-1_i,j,k+1/2 +ρ^-1_i,j,k+1/2X ρ^-1_i,j,k+1/2(A_2^*(A_1x(p_x^*p_x)+ A_1y(p_y^*p_y)+γ u^*u))_i,j,k+1/2ρ^-1_i,j,k+1/2. The 3D case is quite similar. Now, we have the discrete convex optimization problem min f(p, ρ, u) =⟨ A_1x (p_x^*p_x)+A_1y(p_y^*p_y)+A_1z(p_z^*p_z), A_2 (ρ^-1)+ a⟩ h_xh_yh_zh_t + ⟨ u^*u, A_2 (ρ^-1)+a⟩γ h_xh_yh_zh_ts.t. D_1x p_x +D_1y p_y+D_1zp_z+D_2 ρ+D_3 u=b.The Lagrangian of this problem is (p, ρ, u)= f(p, ρ, u)/(h_xh_yh_zh_t)+⟨λ, D_1x p_x+D_1y p_y+D_1zp_z+D_2 ρ+D_3 u-b⟩.In the above, a_i,j,k,ℓ= 1/2(ρ^0_i,j,k)^-1 ℓ=1, 1/2(ρ^1_i,j,k)^-1 ℓ=n_t, 0.and b_i,j,k,ℓ= ρ^0_i,j,k/h_t ℓ=1, -ρ^1_i,j,k/h_t ℓ=n_t, 0.It follows that the KKT conditions now are ∇_p_x =D_1x^* λ+2p_x∘ A_1x^*(A_2(ρ^-1)+a)=0 ∇_p_y =D_1y^* λ+2p_y∘ A_1y^*(A_2(ρ^-1)+a)=0 ∇_p_z =D_1z^* λ+2p_z∘ A_1z^*(A_2(ρ^-1)+a)=0 ∇_ρ =D_2^* λ-ρ^-1∘ A_2^*(A_1x(p_x^*p_x)+A_1y(p_y^*p_y) +A_1z(p_z^*p_z)) ∘ρ^-1 -γρ^-1∘ A_2^*(u^*u)∘ρ^-1=0 ∇_u=D_3^*λ+2γ u∘ (A_2(ρ^-1)+a)=0 ∇_λ =D_1 p+D_2 ρ+D_3 u-b=0,with ∘ the block-wise multiplication as earlier.Let w=(p_x,p_y,p_z,ρ, u), then at each SQP iteration we solve the system (ÂD^* D0) ([ δ w;δλ ])= -([ ∇_w; ∇_λ ]),where D=(D_1x,D_1y,D_1z, D_2, D_3). The matrix  is an approximation of the Hessian of the objective function ( 2Bdiag (A_1x^* (A_2(ρ^-1)+a)) 0 0 0 0 0 2Bdiag (A_1y^* (A_2(ρ^-1)+a)) 0 0 0 0 0 2Bdiag (A_1z^* (A_2(ρ^-1)+a)) 0 0 0 0 0 Bdiag (g(p,ρ,u)) 0 0 0 0 0 2 γ Bdiag (A_2(ρ^-1)+a) ) The operator g(p, ρ, u) is the Hessian of f over ρ with g_i,j,k,ℓ+1/2 being the map g_i,j,k,ℓ+1/2(X) = ρ^-1_i,j,k,ℓ+1/2(A_2^*(A_1x(p_x^*p_x)+ A_1y(p_y^*p_y)+A_1z(p_z^*p_z)+γ u^*u))_i,j,k,ℓ+1/2ρ^-1_i,j,k,ℓ+1/2Xρ^-1_i,j,k,ℓ+1/2 +ρ^-1_i,j,k,ℓ+1/2X ρ^-1_i,j,k,ℓ+1/2(A_2^*(A_1x(p_x^*p_x)+A_1y(p_y^*p_y)+A_1z(p_z^*p_z) +γ u^*u))_i,j,k,ℓ+1/2ρ^-1_i,j,k,ℓ+1/2 § VECTOR-VALUED OPTIMAL MASS TRANSPORTNext we move to vector-valued optimal transport, which was proposed recently in <cit.>. We briefly review the setup in this section, and refer the reader to <cit.> for details. §.§ Gradients on graphs We consider a connected, positively weighted, undirected graph =(, ,) with n nodes labeled as i, with 1≤ i ≤ n, and N edges. We have that Δ_ =- W^T where Δ_, , W={_1, ⋯, _N} are the graph Laplacian, incidence, and weight matrices, respectively. One can define the Laplacian in terms of a graph gradient and divergence asΔ_ = -∇_^*∇_,where ∇_: ^n →^N, x ↦ W^1/2^T xdenotes the gradient operator and ∇_^*:^N →^n, y ↦ W^1/2 ydenotes its dual.§.§ Vector-valued optimal mass transport We begin by considering a vector-valued density ρ on ^m, i.e., a map from E⊂^m to _+^n such that ∑_i=1^n∫_E ρ_i(x)dx=1.Here the convex compact set E ⊂^m is a domain where the densities are defined, typically the unit n-dimensional cube. To avoid proliferation of symbols, we denote the set of all vector-valued densities and its interior again byand _+, respectively. We refer to the entries of ρ as representing density or mass of individual species/particles that can mutate between one another while maintaining total mass. Mass transfer may only be permissible between specific types of particles. Thus, allowable transfer can be modeled by the existence of a corresponding edge in a graph =(,,) whose vertices incorrespond to those individual species, see <cit.>. The edge weights incan quantify cost, rate, or likelihood of transfer.Following the arguments in <cit.>, this leads to the following (symmetric) Wasssertein 2-metric on vector-valued distributions: Given two given marginals ρ^0, ρ^1 ∈_+ the (square) of the Wasserstein distance is given by: min_ρ, p, u∫_0^1∫_E{p^T(ρ)^-1p+γ u^T[(_2^Tρ)^-1+ (_1^Tρ)^-1]u}dxdt ∂ρ/∂ t+∇_x· p-∇_^*u=0,ρ(0,·)=ρ^0, ρ(1,·)=ρ^1. Here u is the “flux” on graphs, p=[p_1, ⋯, p_n]^T is the “momentum” (mass times velocity vector field), the matrix _1 is the portion of the incidence matrixcontaining 1's (sources), and _2 = _1- (sinks). In what follows, we describe an algorithm for the numerical implementation of this convex optimization problem.§ DISCRETIZATION AND ALGORITHM: VECTOR-VALUED CASE As in the matrix-valued cases, for simplicity of exposition, we consider the discretization in 1D case, and describe the 2D case in Section <ref> below. Thus, we take E=[0,1], and as before our technique extends almost verbatim to the higher dimensional setting; see Section <ref>. We should note that the algorithm presented here in the vector-valued case is very similar to the matrix optimal transport just described in the preceding sections.We discretize the space-time domain [0, 1]×[0, 1] into n_x× n_t rectangular cells. Denote Ω_ij, 1≤ i≤ n_x, 1≤ j≤ n_t as the (i,j) box. We use staggered grid to discretize p and ρ. The variable u is, however, valued at the centers of the cells {Ω_ij}. More specifically, p=(p_i+1/2,j), 0≤ i≤ n_x, 1≤ j≤ n_t ρ=(ρ_i,j+1/2), 1≤ i≤ n_x, 0≤ j≤ n_t u=(u_i,j), 1≤ i ≤ n_x, 1≤ j≤ n_t.Note that the boundary values are p_1/2,j=0, p_n_x+1/2,j=0, 1≤ j≤ n_tand ρ_i,1/2=ρ^0_i, ρ_i,n_t+1/2=ρ^1_i, 1≤ i≤ n_x.We exclude the boundary values from the variables and denote p=(p_i+1/2,j), 1≤ i≤ n_x-1, 1≤ j≤ n_t ρ=(ρ_i,j+1/2), 1≤ i≤ n_x, 1≤ j≤ n_t-1. §.§ Continuity equation We use the preceding discretizing scheme, together with the boundary conditions to rewrite the continuity equation (<ref>) as D_1 p+D_2 ρ+D_3 u=b.Here the linear operators D_1, D_2, D_3 are defined as (D_1 p)_i,j= (p_i+1/2,j -p_i-1/2,j)/h_x, 2≤ i ≤ n_x-1, p_3/2,j/h_x, i=1, -p_n_x-1/2,j/h_x, i=n_x, (D_2 ρ)_i,j= (ρ_i,j+1/2-ρ_i,j-1/2)/h_t, 2≤ j ≤ n_t-1, ρ_i,3/2/h_t, j=1, - ρ_i,n_t-1/2/h_t, j=n_t, (D_3 u)_i,j=-∇_^* u_i,j, 1≤ i ≤ n_x, 1≤ j≤ n_t.The parameter b carries the information of the boundary values ρ^0 and ρ^1. More specifically, b_i,j= ρ^0_i/h_t j=1, -ρ^1_i/h_t j=n_t, 0.§.§ Discretization of the cost function Let A_1 be the averaging operator over the spatial domain and A_2 be the averaging operator over the time domain (as before one needs to be careful about the boundaries). Then the cost function(<ref>) may be approximated by ⟨ A_1 (p^2), A_2 (1/ρ)+a⟩ h_xh_t+ ⟨ u^2, A_2 (1/(_2^Tρ)+1/(_1^Tρ))+c⟩γ h_xh_t,where a≥ 0 depends only on the boundary values ρ^0 and ρ^1. The inverse operator and multiplication operators are applied block-wise. The expressions for A_1, A_2, a are (A_1 (p^2))_i,j= 1/2(p^2_i-1/2,j +p^2_i+1/2,j), 2≤ i ≤ n_x-1, 1/2p^2_3/2,j, i=1, 1/2p^2_n_x-1/2,j, i=n_x, (A_2 (1/ρ))_i,j= 1/2(1/ρ_i,j-1/2 +1/ρ_i,j+1/2), 2≤ j ≤ n_t-1, 1/ρ_i,3/2/2, j=1, 1/ρ_i,n_t-1/2/2, j=n_t, a_i,j= 1/ρ^0_i/2 j=1, 1/ρ^1_i/2 j=n_t, 0, c_i,j= 1/_2^Tρ^0_i/2+1/_1^Tρ^0_i/2 j=1, 1/_2^Tρ^1_i/2+1/_1^Tρ^1_i/2 j=n_t, 0. §.§ Sequential quadratic programming (SQP) From the above discussion, we obtain the discrete convex optimization problem min f(p, ρ, u) =⟨ A_1 (p^2), A_2 (1/ρ)+a⟩ h_xh_t+ ⟨ u^2, A_2 (1/(_2^Tρ)+1/(_1^Tρ))+c⟩γ h_xh_ts.t. D_1 p+D_2 ρ+D_3 u=b. The Lagrangian of this problem is (p, ρ, u)= f(p, ρ, u)/(h_xh_t)+⟨λ, D_1 p+D_2 ρ+D_3 u-b⟩.It follows that the KKT conditions are given by ∇_p=D_1^T λ+2p∘ A_1^T(A_2(1/ρ)+a)=0 ∇_ρ =D_2^T λ-A_2^TA_1(p^2)/ ρ^2 -γ_2(A_2^T(u^2)/(_2^Tρ)^2)-γ_1(A_2^T(u^2)/(_1^Tρ)^2)=0 ∇_u=D_3^Tλ+2γ u∘ (A_2(1/(_2^Tρ)+1/(_1^Tρ))+c)=0 ∇_λ =D_1 p+D_2 ρ+D_3 u-b=0, with ∘ denoting block-wise multiplication.Let w=(p,ρ, u). Then at each SQP iteration, we solve the system (ÂD^T D0) ([ δ w;δλ ])= -([ ∇_w; ∇_λ ]),where D=(D_1, D_2, D_3). Again, the matrix  is an approximation of the Hessian of the objective function Â= ( 2diag (A_1^T (A_2(1/ρ)+a)) 0 0 0 diag (g(p,ρ,u)) 0 0 0 2 γ diag (A_2(1/(_2^Tρ)+1/(_1^Tρ))+c) ).The operator g(p, ρ, u) is the Hessian of f over ρ with g_i,j+1/2 being the map g_i,j+1/2(X) = 2(A_2^TA_1(p^2))_i,j+1/2 /ρ^3_i,j+1/2X+2γ_2[(A_2^T(u^2))_i,j+1/2 /(_2^Tρ)^3_i,j+1/2_2^TX]+2γ_1[(A_2^T(u^2))_i,j+1/2 /(_1^Tρ)^3_i,j+1/2_1^TX].§.§ 2D caseWe concretely work out the 2D case in this section. The higher dimensional cases are very similar, but naturally involve additional indices. We have the discrete convex optimization problem min f(p, ρ, u) =⟨ A_1x (p_x^2)+A_1y(p_y^2), A_2 (1/ρ)+ a⟩ h_xh_yh_t+ ⟨ u^2, A_2 (1/(_2^Tρ)+1/(_1^Tρ))+c⟩γ h_xh_yh_ts.t. D_1x p_x +D_1y p_y+D_2 ρ+D_3 u=b.The Lagrangian of this problem is (p, ρ, u)= f(p, ρ, u)/(h_xh_yh_t)+⟨λ, D_1x p_x+D_1y p_y+D_2 ρ+D_3 u-b⟩.In the above, a_i,j,k= 1/ρ^0_i,j/2 k=1, 1/ρ^1_i,j/2 k=n_t, 0,and b_i,j,k= ρ^0_i,j/h_t k=1, -ρ^1_i,j/h_t k=n_t, 0, c_i,j,k= 1/_2^Tρ^0_i,j/2+1/_1^Tρ^0_i,j/2 k=1, 1/_2^Tρ^1_i,j/2+1/_1^Tρ^1_i,j/2 k=n_t, 0.The KKT conditions now are∇_p_x =D_1x^T λ+2p_x∘ A_1x^T(A_2(1/ρ)+a)=0 ∇_p_y =D_1y^T λ+2p_y∘ A_1y^T(A_2(1/ρ)+a)=0 ∇_ρ =D_2^T λ-A_2^T(A_1x(p_x^2)+A_1y(p_y^2))/ρ^2 -γ_2(A_2^T(u^2)/(_2^Tρ)^2)-γ_1(A_2^T(u^2)/(_1^Tρ)^2)=0 ∇_u=D_3^Tλ+2γ u∘ (A_2(1/(_2^Tρ)+1/(_1^Tρ))+c)=0 ∇_λ =D_1 p+D_2 ρ+D_3 u-b=0,with ∘ denoting block-wise multiplication.Let w=(p_x,p_y,ρ, u), then at each SQP iteration we solve the system (ÂD^* D0) ([ δ w;δλ ])= -([ ∇_w; ∇_λ ]),where D=(D_1x,D_1y, D_2, D_3). The matrix  is an approximation of the Hessian of the objective function( 2diag (A_1x^T (A_2(1/ρ)+a)) 0 0 0 0 2diag (A_1y^T (A_2(1/ρ)+a)) 0 0 0 0 diag (g(p,ρ,u)) 0 0 0 0 2 γ diag (A_2(1/(_2^Tρ)+1/(_1^Tρ))+c) ). The operator g(p, ρ, u) is the Hessian of f over ρ with g_i,j,k+1/2 being the map g_i,j+1/2(X) = 2(A_2^T(A_1x(p_x^2)+A_1y(p_y^2)))_i,j+1/2 /ρ^3_i,j+1/2X+2γ_2[(A_2^T(u^2))_i,j+1/2 /(_2^Tρ)^3_i,j+1/2_2^TX]+2γ_1[(A_2^T(u^2))_i,j+1/2 /(_1^Tρ)^3_i,j+1/2_1^TX].§ NUMERICAL EXPERIMENTSSeveral examples are provided in this section to illustrate the effectiveness of our algorithms. For matrix-valued densities, we present examples in both 2D and 3D settings. In contrast, only 2D examples are studied for vector-valued densities. §.§ Matrix caseOne motivation for matrix-valued optimal mass transport comes from diffusion tensor imaging (DTI). This is a widely used technique in magnetic resonance imaging. In diffusion images, the information at each pixel is captured in a ellipsoid, i.e., a 3 × 3 positive definite matrix, in lieu of a nonnegative number. The ellipsoids describe useful information such as the orientations of the brain fibers.We tested our algorithm on a synthetic data set with n=3. The initial density is a disk positioned at the center of the square domain and all the ellipsoids are isotropic. The terminal density contains four quarter discs located at the corners of the square domain, and the four components have different dominant directions. Both of them are depicted in Figure <ref>. The densities have been smoothed to have low density contrast 10. Here the density contrast is defined to be the maximum of the ratios between the eigenvalues at different locations. In Figure <ref>, we show the optimal density flow with grid size 32× 32 × 10 in space-time and parameter γ=0.01. The masses split into four components and the ellipsoids change gradually from isotropic to anisotropic. To demonstrate the performance of our algorithm, we tested it on the same problem with different mesh grid sizes: 16 × 16 × 10, 32× 32 × 20, 64 × 64 × 40 in space-time. We set the tolerance of the outer SQP iterations to 10^-3, and that of the preconditioning conjugate gradient solver in each iteration to 10^-3. The numbers of SQP iterations for convergence are shown in Table <ref> for different mesh sizes.We then studed the influence of density contrast and the parameter γ on the number of iterations needed to converge. The results for density contrast 50 are shown in Table <ref> with tolerance 10^-2. We can see that the number of iterations increases as we increase the density contrast. Table <ref> showcases the results for different γ values with fixed grid size 32 × 32 × 20. We observe that the number of iterations is positively correlated with the value of γ. Finally, we test our algorithm on a 3D data set. Table <ref> displays the number of iterations for different grid sizes with density contrast 30 and parameter γ=0.1. §.§ Vector caseAn important application of vector-valued optimal mass transport is color image processing. In this cases, the vector-valued densities have three components corresponding to the intensities of the three basic colors red (R), green (G) and blue (B). The masses can transfer from one color channel to another and the cost of transferring is captured using a weighted graph . Here, we treat the three colors equally and take the graph to be a complete graph with unit weights, namely, W=I and = [ 1 1 0 -1 0 1 0 -1 -1 ].The matrices _1,_2 in (<ref>) are then _1 = [ 1 1 0 0 0 1 0 0 0 ],_2 = [ 0 0 0 1 0 0 0 1 1 ]. The two marginal densities are depicted in Figure <ref>. The initial image ρ^0 is a disk located in the center of the square in white color, i.e., all three colors have equal intensity. The terminal distribution ρ^1 is an image of four circle quarters; one at each corner in different colors. Both the images have been smoothed to have density contrast max_k sup_x,yρ_k^i(x)/ρ_k^i(y) ≈ 10. Figure <ref> illustrates the optimal interpolation using vector-valued optimal transport with grid size 128× 128× 10 in space-time and parameter γ=0.01. We observe that the white disk split into four circle quarters and meanwhile the colors change gradually from white to four different colors.We next tested the performance of the algorithm with respect to the grid size. For this, we consider a grid hierarchy from a coarse grid of 32× 32× 10 in space and time through a grid of 64 × 64 × 20 to a grid of 128 × 128 × 40. The parameter γ is set to be 0.01. The tolerance for the outer SQP iteration is set to be 10^-3 and in each iteration the linear equation is solved with a relative residual of 10^-2. The numbers of SQP iterations are recorded in Table <ref>, from which we observe that the number of iterations needed doesn't increase much as we increase the size of the mesh grids. We also applied the same algorithm to images with a higher density contrast 100. The results are shown in Table <ref> for different grid sizes. As can be seen from the table, increasing the density contrast leads to an increasing of the number of SQP iterations. Again, the number of iterations needed to achieve certain precision is affected by the parameter. In Table <ref> we display this change as a function of γ for fixed grid size 64× 64× 20 and density contrast 100.§ CONCLUSIONS AND FUTURE WORKIn this paper, we described a fast algorithm for the numerical implementation of both matrix-valued and vector-valued versions of optimal mass transport. It is straightforward to extend this algorithm to cover matrix-valued transport problems with unequal masses (“unbalanced mass transport”) <cit.>. 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"authors": [
"Yongxin Chen",
"Eldad Haber",
"Kaoru Yamamoto",
"Tryphon T. Georgiou",
"Allen Tannenbaum"
],
"categories": [
"cs.NA",
"cs.DM"
],
"primary_category": "cs.NA",
"published": "20170626170036",
"title": "An Efficient Algorithm for Matrix-Valued and Vector-Valued Optimal Mass Transport"
} |
A.M. Escobar-Ruiz, J.C. López Vieyra and P. Winternitz]Fourth order superintegrable systems separating in Polar Coordinates. I.Exotic [email protected] de recherches mathématiques, and Département de mathématiques et de statistique, Université de Montreal,C.P. 6128, succ. Centre-ville,Montréal (QC) H3C 3J7, Canada [email protected] de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 04510 México, D.F., Mexico [email protected] de recherches mathématiques, and Département de mathématiques et de statistique, Université de Montreal,C.P. 6128, succ. Centre-ville,Montréal (QC) H3C 3J7, Canada We present all real quantum mechanical potentials in a two-dimensional Euclidean space that have the following properties: 1. They allow separation of variables of the Schrödinger equationin polar coordinates, 2. They allow an independent fourth order integral of motion, 3. It turns out that their angular dependent part S(θ) does not satisfy any linear equation. In this case S(θ) satisfies a nonlinear ODE that has the Painlevé property and its solutions can be expressed in terms of the Painlevé transcendent P_6. We also study the corresponding classical analogs of these potentials. The polynomial algebra of the integrals of motion is constructed in the classical case.[ P. Winternitz December 30, 2023 =====================§ INTRODUCTION This article is part of a series devoted to a study of classical and quantum superintegrable systems. Roughly speaking, a Hamiltonian with n degrees of freedom is integrable if it allows n independent well defined integrals of motion in involution. It is minimally superintegrable if it allows n+1 such integrals, maximally superintegrable if it allows 2n-1 integrals (only subsets of n integrals among them can be in involution).The best known superintegrable systems are the harmonic oscillator with its su(n+1) algebra of integrals, and the Kepler-Coulomb system with its o(n+1) algebra (when restricted to fixed bound state energy values).A recent review article gives more precise definitions, a general setting and motivation for studying superintegrable systems <cit.>. It follows from Bertrand's theorem <cit.>that ω r^2 and α/r are the only maximally superintegrable spherically symmetrical potentials in Euclidean real space E_n. Systematic searches for superintegrable classical and quantumsystems in E_2 and E_3 established a connection between second order superintegrability and multiseparability in the Schrödinger or Hamilton-Jacobi equation <cit.>. An extensive literature exists on second order superintegrability in spaces of 2, 3 and n dimensions, Riemannian and pseudo-Riemannian, real or complex, <cit.>.A systematic study of higher order integrability is more recent. Pioneering work is due to Drach <cit.>.For more recent work see <cit.>. The Painlevé transcendents were first introduced in a purely mathematical study by Painlevé<cit.> and Gambier<cit.> and were popularized in books e.g. by Ince<cit.> and Davis<cit.>.They are characterized by the fact that they are solutions of second order nonlinear ODEs that are single valued about any movable singularity of the ODE (movable means that the position of the singularity depends on the initial conditions). We shall call this “the Painlevé property”. Painlevé and Gambier also classified all ODEs with the Painlevé property of the formy” = R(x,y,y’)with R rational in y and y' and analytical in x into 50 equivalence classes under the action of the group preserving the Painlevé property. Of these 50 six give rise to the famous irreducible Painlevé transcendents. The others can either be reduced to one of these six, or integrated in terms of already known functions like elliptic functions, or solutions of linear equations.Linear ODEs have the Painlevé property by default: all the singularities of their solutions are fixed, i.e., they can only occur where the coefficients of the ODEs are themselves singular. In this sense we can say that the nonlinear ODEs with the Painlevé property are the closest ones to linear ODEs. Nonlinear equations with the Painlevé property became important in applications after the discovery of the inverse scattering theory by Kruskal et al.<cit.> and more generally of soliton theory (for reviews see e.g. <cit.> and references therein). A Painlevé test was proposed <cit.>, a simple algorithmic test the passing of which is a necessary condition for an ODE to have the Painlevé property.A Painlevé conjecture was formulated<cit.>, namely that a necessary condition for a PDE to be integrable by inverse scattering techniques is that all of the ODE’s obtained as reductions of the PDE should have the Painlevé property. A systematic search for analytical solutions of many of the PDEsofhydrodynamics, plasma physics, and nonlinear optics lead to various Painlevé transcendents. Painlevé transcendents to our knowledge appeared for the first time in quantum mechanics in articles by Fushchych and Nikitin<cit.> and by Doebner and Zhdanov<cit.>. A systematic search for superintegrable sxystems in E_2 with one integral of motion of order N≥3 and two others of order N ≤2 was started in <cit.> and <cit.> (for N=3). Exotic potentials, by definition not satisfying any linear ODE, were obtained. It turned out that they could always be expressed in terms of the Painlevé transcendents P_1, P_2, P_4 or elliptic functions. The lower order integrals were chosen to be of “Cartesian type” that is they forced the potential to allow separation of variables in Cartesian coordinates. A similar study for N=3 was conducted for second order integrals of polar type <cit.>. Exotic potentials appeared again and this time they were expressed in terms of P_6. For N=4 the situation is similar<cit.>, namely, exotic potentials appear in the Cartesian case, expressed in terms of P_1 ,..., P_5. For N=4, the polar case is the present article, and as we shall see below exotic potentials exist. Unlike the case N=3, they are expressed in terms of the completely general P_6 transcendent.Specific results have also been obtained for N=5 in the Cartesian case<cit.>. New features appear here, namely potentials expressed in terms of solutions of higher order ODEs with the Painlevé property.We conjecture that for all N≥3 exotic potentials will exist and be solutions of ODEs with the Painlevé property.The present article is a contribution to a series <cit.> devoted to superintegrable systems in E_2 with one integral of order n≥ 3 and one of order n≤ 2. In particular, it is a generalization of a paper<cit.> devoted to the case of a third order integral Y. In this article we restrict ourselves to the space E_2.The Hamiltonian has the form H = 1/2(p_x^2+p_y^2) + V(x,y),in classical mechanics p_x and p_y are the momenta conjugate to the Cartesian coordinates x and y. In quantum mechanics they are the corresponding operators p_x = -i ħ∂/∂ x, p_y = -i ħ∂/∂ y. In polar coordinates (x,y)≡ (rcosθ, rsinθ), the classical Hamiltonian reads H = 1/2(p_r^2+ p_θ^2/r^2) +V(r,θ) , V(r,θ) = R(r) + 1/r^2 S(θ) ,here p_r and p_θ are the associated canonical momenta. The corresponding quantum operator takes the formH = -ħ^2/2 (∂^2_r + 1/r∂_r + 1/r^2∂_θ^2)+ V(r,θ). In this article we concentrate on quantum superintegrability and on "exotic" potentials, namely those that do not satisfy any linear differential equations. In all equations we keep the Planck constant ħ explicitly. Classical exotic potentials will be obtained in the limit ħ→ 0. We emphasize that this limit is singular: highest order terms in the equation which defines the potential in (<ref>) vanish, so the classical and quantum cases can differ greatly.In addition to the Hamiltonian H, we have two more conserved quantities which areX= p_θ^2+2 S(θ) , Y= ∑_i+j+k=4A_ijk { L_z^i, p_x^j p_y^k }+{ g_1(x,y), p_x^2} +{ g_2(x,y), p_x p_y} +{ g_3(x,y), p_y^2} + g_4(x,y) , here p_θ=x p_y-y p_x=-i ħ∂/∂θ. The bracket {· , ·} denotes an anticommutator, the set {A_ijk} are real constants and R(r),S(θ),g_1,2,3,4(x,y) are real functions such that [H, Y]=[H, X]=0.The operator Y in (<ref>) is given in Cartesian coordinates for brevity. Puttingp_x=-i ħ (cosθ ∂_r - sinθ/r ∂_θ),p_y=-i ħ (sinθ ∂_r + cosθ/r ∂_θ),we obtain the corresponding expression in polar coordinates. It's leading terms are given explicitly below in (<ref>) and used throughout this article. We have [Y,X]=C ≠ 0 where C is in general a 5th order linear operator. In general, we thus obtain a finitely generated polynomial algebra of integrals of motion <cit.>. We are looking for fourth-order superintegrable systems, so at least one of A_ijk is different from zero. The operator Y is the most general polynomial expression for a fourth-order Hermitian operator of the required form. The commutator [H, Y]contains derivatives of order up to three.Before calculating the commutator [H,Y] we note that three ”trivial” fourth order integrals exist, namely X^2, H^2 and {X, H}. Each of these is a scalar (invariant) under O(2) rotations. By linear combinations of the form Y + u_1X^2 + u_2H^2 + u_3XH, where the u_i are constants, we can eliminate 3 parameters among the {A_ijk} and consequently three terms in Y. Now, we introduce a more convenient set of parameters defined by the relationsA_0 0 4 = 1/2(D_1 - B_1) , A_0 4 0 = 1/2 ( B_1 + D_1) ,A_4 0 0 = 0 ,A_0 2 2 = -3D_1 , A_0 1 3 = B_2 - 2D_2 ,A_0 3 1 = B_2 + 2D_2 ,A_1 0 3 = A_4 - C_1,A_3 0 1 =A_2 ,A_2 2 0 =B_3 ,A_1 2 1 =3 C_1+A_4,A_1 1 2 =A_3 -3 C_2 , A_3 1 0 = A_1 ,A_2 1 1 = 2 B_4 ,A_1 3 0 = C_2+A_3 ,A_2 0 2 = -B_3 .With the above parameters the fourth-order integral (<ref>) takes the following form: Y =A_1 {L_z^3, p_x } +A_2 {L_z^3, p_y } +A_3 {L_z,p_x (p_x^2+p_y^2) } +A_4 {L_z, p_y (p_x^2+p_y^2)} +B_1 (p_x^4-p_y^4)+2 B_2 p_x p_y (p_x^2+p_y^2)+B_3 {L_z^2,p_x^2-p_y^2} +2 B_4 {L_z^2, p_x p_y } +C_1 {L_z, 3 p_x^2 p_y-p_y^3 } + C_2 {L_z, p_x^3 -3 p_y^2 p_x }+D_1 (p_x^4+p_y^4-6 p_x^2 p_y^2)+4 D_2 p_x p_y (p_x^2-p_y^2) + lower order terms .Under rotations around the z-axis, each of the six pairs of parameters(A_1,A_2), (A_3,A_4), (B_1,B_2), (B_3,B_4), (C_1,C_2), (D_1,D_2) ,in (<ref>) forms a doublet (all O(2) singlets have been removed).Under rotations through the angle θ the doublets A_i, B_i, C_i and D_i rotate through θ, 2θ, 3θ and 4θ, respectively. In particular, the doublets (A_1, A_2) and (B_3, B_4) will play a central role in the main equations of the present paper. Explicitly, in polar coordinates, the leading terms of the integral Y are Y =ħ^4 ( ( B_1 cos 2 θ + B_2 sin 2 θ + D_1 cos 4 θ + D_2 sin 4 θ) ∂^4_r +1/ r^4[ D_2 sin 4 θ + D_1 cos 4 θ-2 r (A_1 r^2+A_4) sinθ - (B_2+2 B_4 r^2) sin 2 θ +2 r (A_2 r^2+A_4) cosθ- (B_1+2 B_3 r^2) cos 2 θ-2 r(C_1 cos 3 θ- C_2 sin 3 θ)] ∂^4_θ - 2/r^2[ 3( D_1cos 4θ + D_2sin 4θ ) - r^2 ( B_3cos 2θ + B_4sin 2θ )+ r ( A_3sinθ - A_4 cosθ -3( C_1cos 3θ -C_2 sin 3θ) ) ] ∂^2_r ∂^2_θ- 2/ r[ B_1 sin 2 θ- B_2 cos 2 θ +2 (D_1 sin 4 θ- D_2 cos 4 θ)-r ( C_1 sin 3 θ +C_2 cos 3 θ+ A_3 cosθ + A_4 sinθ) ] ∂^3_r ∂_θ -2/r^3[B_1 sin 2 θ- B_2 cos 2 θ -2(D_1 sin 4 θ + D_2 cos 4 θ) - r (A_3 cosθ+A_4 sinθ -3 (C_1 sin 3 θ + C_2 cos 3 θ ))+ 2 r^2 ( B_3 sin 2 θ - B_4cos 2 θ) -r^3 ( A_1cosθ +A_2 sinθ) ] ∂_r ∂^3_θ ) + … +lower order terms .We introduce the functionsG_1(r, θ)=g_1 cos^2θ + g_2 sin^2θ+g_3 cosθsinθ , G_2(r, θ)= g_1 sin^2θ+ g_2 cos^2θ-g_3 cosθsinθ/r^2 , G_3(r, θ)=- g_1 sin2θ-g_2 sin2θ-g_3 cos2θ/r , G_4(r, θ)=g_4.Then, the quadratic and zero order terms in the integral Y can now be written in polar coordinates as{ g_1(x,y), p_x^2} +{ g_2(x,y), p_x p_y} +{ g_3(x,y), p_y^2} + g_4(x,y) =-ħ^2 ({ G_1(r,θ), ∂_r^2} +{ G_3(r,θ), ∂_r ∂_θ} +{ G_2(r,θ), ∂_θ^2})+ G_4(r,θ) . The structure of this article is as follows. In Section <ref> we derive the determining equations that govern the existence and form of the fourth-order integral Y. In Section <ref> we present a linear compatibility condition that must be satisfied by the potential V(r,θ) in order for a fourth order integral Y to exist. In general this is a fourth order PDE.In Section <ref> we turn to the question of superintegrability. The existence of the second order integral X guarantees that the potentialV(r,θ) has the form given in (<ref>). We rewrite the determining equations and the compatibility condition (<ref>) in polar coordinates. The compatibility condition (<ref>) then reduces to a coupled system of ODEs for R(r) and S(θ). We decouple the equation and solve for R(r). The possible functions R(r) are br^2, a/r and 0 respectively.From Section <ref> on we restrict to exotic potentials which by definition do not satisfy any linear equation. The function R(r) is already determined and is not exotic. The function S(θ) satisfies a linear equation which must be satisfied identically. This requires that all coefficients in Y vanish except A_1,A_2 B_3 and B_4. In Section <ref> we consider the case R(r)=0, i.e. a nonconfining potential (with no bound states).Section <ref>is devoted to confining potentials R(r)=b r^2 and R(r)= a/r.In all cases the function S(θ) is expressed in terms of the Painlevé transcendent P_6 (γ_1, γ_2, γ_3,γ_4;z) whereγ_1, γ_2, γ_3,γ_4 are arbitrary constants. Section <ref> is devoted to classical potentials obtained in the (singular) limit ħ→ 0. The fourth order compatibility condition reduces to a second order non-linear ODE which, interestingly,does not have the Painlevé property.The polynomial algebra generated by the integrals of motion is presented in Section <ref>. The main results are summed up as theorems in the final Section <ref>.§ DETERMINING EQUATIONS FOR A FOURTH ORDER INTEGRAL §.§ Commutator [H, Y] The commutator between the Hamiltonian H (<ref>) and the fourth order integral Y, written in polar coordinates, is a third order differential operator given by [H,Y]=A_rrr ∂^3/∂ r^3+A_rrθ ∂^3/∂ r^2∂θ +A_rθθ ∂^3/∂ r∂θ^2 +A_θθθ ∂^3/∂θ^3 + B_rr ∂^2/∂ r^2 +B_rθ ∂^2/∂ r∂θ +B_θθ ∂^2/∂θ^2+ C_r ∂/∂ r +C_θ ∂/∂θ +C_0= 0, where the coefficients A_rrr,A_rrθ,..., are real functions of r and θ. Terms multiplying derivatives of order five and four vanish identically (they are already accounted for in the form of Y in (<ref>)). In order for Y to be an integral of motion all ten coefficients must vanish simultaneously. The odd order terms in (<ref>) provideus with useful information. The even order terms B_rr,B_θθ,B_r θ and C_0 provide differentialconsequences of the odd order terms and will not be listed below. This difference betweeneven and odd order terms in the commutator [H,Y] is a general feature of the theory<cit.> (for any order N of Y).Vanishing of the coefficients of the third order terms A_rrr, A_θθθ, A_rrθ, A_rθθ in (<ref>) yields, respectively,the following relations: G_1^(1,0) =F_1(θ) V^(1,0)+F_2(r,θ) V^(0,1) ,1/r^2( G_2^(0,1) + 1/r G_3)=F_3(r,θ) V^(1,0) +F_4(r,θ) V^(0,1) , 1/r^2G_1^(0,1) +G_3^(1,0) = 3 F_2(r,θ) V^(1,0) +F_5(r,θ) V^(0,1) ,2/r^3 G_1 + G_2^(1,0) + 1/r^2 G_3^(0,1)= F_5(r,θ) V^(1,0) +3 F_3(r,θ) V^(0,1), Fromthe two equations C_r = 0 and C_θ=0, we obtain 2G_1V^(1,0) + G_3V^(0,1)-1/2 G_4^(1,0)+ ħ^2[G_1^(0,2)/2 r^3 -G_1^(1,0)/r^2 +G_1^(1,2)/2 r^2 +2 G_1^(2,0)/r +G_1^(3,0) -3/2 G_2^(1,0) +1/2 G_2^(1,2) -1/2 r G_2^(2,0) +G_3^(0,3)/4 r^2 -3G_3^(0,1)/4 r^2 +5 G_3^(1,1)/4 r +3/4 G_3^(2,1)]=ħ ^2[F_1 V^(3,0)+F_3 V^(0,3) +F_5 V^(1,2) + F_6 V^(2,1) + F_10 V^(2,0) + 2 F_7 V^(1,1)+F_8 V^(0,2)+ F_11 V^(1,0)+F_12 V^(0,1)],2G_2V^(0,1) + G_3V^(1,0) -G_4^(0,1)/2r^2 +ħ^2[G_1^(1,1)/r^3 +G_1^(2,1)/2 r^2 -G_2^(0,1)/r^2 +G_2^(0,3)/r^2 +1/2 G_2^(2,1) +G_3/4 r^3 +5 G_3^(0,2)/4 r^3 -G_3^(1,0)/4 r^2 +3 G_3^(1,2)/4 r^2 +G_3^(2,0)/2 r +1/4 G_3^(3,0)] =ħ ^2[F_2 V^(3,0) +3 F_3 V^(1,2)+ F_4 V^(0,3)+ F_5 V^(2,1)+ F_7 V^(2,0)+F_9 V^(0,2)+ 2 F_8 V^(1,1) + F_12 V^(1,0)+F_13 V^(0,1)], respectively, where we define V^(i,j)≡∂^i_r ∂^j_θ V(r, θ).The Planck constant ħ is present in the lowest order coefficients C_r and C_θ only (see (<ref>) and (<ref>)).The functions F_1 … F_13 are completely determined by the constants A,B,C,D figuring in the leading part of the integral Y. They are given in Appendix A. §.§ The linear compatibility conditionThe system(<ref>)-(<ref>) viewed as a system of 4 PDE for G_1, G_2 and G_3 is overdetermined and for general potential V(r, θ) has no solutions. The first step towards finding solutions of this system is to establish a necessary linear compatibility condition involving V(r, θ) alone. Such an equation will exist as a consequence of the equality of all mixed derivatives of analytical functions. To obtain the compatibility condition we denote the l.h.s. of the equations (<ref>)-(<ref>) as E_1, … E_4, respectively, and take partial derivatives of these terms (up to third order). The following linear combination of the derivatives vanishes identically 0 = r^6E_2 ^(3,0) +12 r^5E_2^(2,0) +36 r^4E_2^(1,0) - r^4E_4^(2,1) - r^3E_3^(2,0) -6 r^3E_4^(1,1) +24 r^3E_2 -3 r^2E_3^(1,0) -6 r^2E_4^(0,1) + r^2E_3^(1,2) +2rE_3^(0,2) +3r E_1^(1,1) - E_1^(0,1) - E_1^(0,3) hence the same combination of the r.h.s. of (<ref>)-(<ref>) must vanish too and we obtain the compatibility condition: 0= r^6 F_3V^(4,0)+ r^4 (F_4 r^2 - F_5) V^(3,1)-3 r^2 (F_3 r^2 - F_2) V^(2,2)+(F_5 r^2 - F_1) V^(1,3)- F_2 V^(0,4) + r^3 ( 3 r^3 F_3^(1,0)+ 12 r^2 F_3 - r F_5^(0,1) -3 F_2 )V^(3,0)+r ( 3r^5 F_4^(1,0)+12 r^4 F_4-3r^3 F_3^(0,1)-2r^3 F_5^(1,0)-7r^2 F_5+6r F_2^(0,1)+3 F_1)V^(2,1) + ( 3 r^2F_2^(1,0) - 6r^4F_3^(1,0) - 18 r^3F_3 +2 r^2F_5^(0,1) + 9rF_2 - 3 F_1' )V^(1,2) + ( r^2 F_5^(1,0) + 2 r F_5 - 3 F_2^(0,1))V^(0,3) + r (3 r^5 F_3^(2,0) + 24 r^4 F_3^(1,0) + 36 r^3 F_3 - 2r^3 F_5^(1,1) - 6r^2 F_5^(0,1) - 6r^2 F_2^(1,0) + 3r F_2^(0,2) - 9r F_2 + 3F_1' )V^(2,0)+ (3 r^6 F_4^(2,0) + 24 r^5 F_4^(1,0) - 6r^4 F_3^(1,1) + 36 r^4 F_4 -r^4 F_5^(2,0) -18r^3 F_3^(0,1) -8 r^3 F_5^(1,0) +6 r^2 F_2^(1,1) +r^2 F_5^(0,2) - 9r^2 F_5 +15r F_2^(0,1) -F_1 -3 F_1”) V^(1,1) -( + 3 r^4 F_3^(2,0) +18 r^3 F_3^(1,0) +18 r^2 F_3 - 2 r^2 F_5^(1,1) - 3 r F_2^(1,0) -4r F_5^(0,1) + F_2 +3 F_2^(0,2))V^(0,2) + (r^6 F_3^(3,0) + 12 r^5 F_3^(2,0) + 36 r^4 F_3^(1,0) -r^4 F_5^(2,1) - 3r^3 F_2^(2,0) + 24 r^3 F_3 - 6r^3 F_5^(1,1) - 9r^2 F_2^(1,0) + 3r^2 F_2^(1,2) - 6r^2 F_5^(0,1) + 6r F_2^(0,2) -F_1' - F_1^(3)) V^(1,0)+ ( r^6 F_4^(3,0) +12 r^5 F_4^(2,0) -3r^4 F_3^(2,1) +36 r^4 F_4^(1,0) +24 r^3 F_4 -18 r^3 F_3^(1,1) - r^3 F_5^(2,0) -18 r^2 F_3^(0,1) -3r^2 F_5^(1,0) + r^2 F_5^(1,2) +2r F_5^(0,2) +3r F_2^(1,1) -F_2^(0,1) -F_2^(0,3))V^(0,1) . Relation (<ref>) is a fourth order linear PDE for the potential and is a necessary (but not sufficient) condition for the existence of the fourth order integral Y of the form (<ref>). This relation does not contain the Planck constant ħ and is thus the same in classical and in quantum mechanics.§ SUPERINTEGRABILITY: SEPARATION IN POLAR COORDINATES §.§ The determining equations Vanishing of the commutator [H, X]=0 implies that the potential has the separable form of V(r, θ) in (<ref>) and thus allows separation of variables in polar coordinates in the Schrödinger equation (and in the Hamilton-Jacobi equation). In this case the determining equations (<ref>)-(<ref>), coming from the condition [H, Y]=0, take the formG_1^(1,0) = F_1R'-2 F_1/r^3 S +F_2 /r^2S' ,1/r^2( G_2^(0,1)+ 1/r G_3)=F_3R'-2 F_3/r^3 S + F_4 /r^2S',1/r^2G_1^(0,1) +G_3^(1,0) = 3 F_2R'-6 F_2/r^3 S+F_5/r^2S' ,2/r^3 G_1 +G_2^(1,0) + 1/r^2 G_3^(0,1)=F_5R' -2 F_5/r^3 S+3 F_3 /r^2S',G_31/r^2 S' + 2G_1( R'- 2/r^3 S) -1/2 G_4^(1,0) + ħ^2[G_1^(0,2)/2 r^3-3 G_3^(0,1)/4 r^2 +G_3^(0,3)/4 r^2-G_1^(1,0)/r^2-3/2 G_2^(1,0) +5 G_3^(1,1)/4 r +G_1^(1,2)/2 r^2 +1/2 G_2^(1,2) +2 G_1^(2,0)/r -1/2 r G_2^(2,0) +3/4 G_3^(2,1) +G_1^(3,0)]=ħ^2[(6 F_6/r^4-24 F_1/r^5-2 F_11/r^3) S+(6 F_10/r^4-4F_7/r^3+F_12/r^2) S'+(F_8/r^2-2F_5/r^3)S” +F_3/r^2 S”' + F_11R' +F_6R” + F_1R”'] ,G_3( R'- 2/r^3 S) +2G_2 1/r^2 S' -G_4^(0,1)/2 r^2 +ħ^2[G_3/4 r^3 -G_2^(0,1)/r^2 +5 G_3^(0,2)/4 r^3 +G_2^(0,3)/r^2 -G_3^(1,0)/4 r^2 +G_1^(1,1)/r^3 +3 G_3^(1,2)/4 r^2 +G_3^(2,0)/2 r +G_1^(2,1)/2 r^2 +1/2 G_2^(2,1) +1/4 G_3^(3,0)] =ħ^2[ (6F_7/r^4-24F_2/r^5-2F_12/r^3) S +(6F_5/r^4-4F_8/r^3+F_13/r^2)S'+(F_9/r^2-6 F_3/r^3)S” + F_4/r^2S”' + F_12R' +F_7R” +F_2 R”' ], §.§ The linear compatibility conditionSubstituting the separable form (<ref>) of the potential into the compatibility condition (<ref>), and integrating once over r, we obtainΘ(θ) = 288 [ B_1sin 2 θ - B_2cos 2 θ - 8 D_1 sin4 θ + 8 D_2 cos4 θ] S +120 [20 D_1 cos4 θ + 20 D_2 sin4 θ - B_1cos 2 θ - B_2sin 2 θ] S'+60 [ B_1sin2 θ -B_2cos2 θ + 14 D_1 sin 4 θ - 14 D_2 cos 4 θ] S”-30 [ B_1cos2 θ + B_2sin2 θ + 4 D_1 cos4 θ + 4 D_2 sin4 θ] S^(3) -3 [B_1sin2 θ - B_2cos2 θ +2 D_1 sin 4 θ -2 D_2 cos 4 θ]S^(4) - r ( 96 [A_4sinθ +A_3cosθ - 9C_1 sin3 θ - 9C_2 cos3 θ] S-120 [ A_4 cosθ -A_3 sinθ + 9 C_2 sin3 θ - 9 C_1 cos3 θ] S'+480 [C_1sin 3 θ + C_2cos 3 θ]S” +30 [ A_3 sinθ -A_4 cosθ + 3 C_2sin 3 θ - 3 C_1cos 3 θ]S^(3)- 6 [A_4 sinθ + A_3 cosθ + C_1sin3 θ + C_2cos3 θ] S^(4))+576 r^2[D_2cos4 θ -D_1sin4 θ] R+6 r^3[ 6r^3(A_2 sinθ +A_1 cosθ) -4 r^2(B_4cos2 θ - B_3sin2 θ)-r (A_4 sinθ + A_3 cosθ + 9C_1sin3 θ + 9C_2cos3 θ) -3 B_1 sin2 θ +3 B_2 cos2 θ +66 D_1 sin 4 θ -66 D_2 cos 4 θ] R'+6 r^4[6 r^3(A_2 sinθ +A_1 cosθ)- 4 r^2( B_3sin2 θ -B_4cos2 θ)+r (A_4 sinθ+A_3 cosθ + 9 C_1 sin 3 θ + 9 C_2 cos 3 θ) +3 (B_1 sin2 θ -B_2 cos2 θ -6D_1 sin4 θ +6D_2 cos4 θ)] R”+3 r^5[ 2 r^3(A_2 sinθ +A_1 cosθ) + 4 r^2( B_4cos2 θ - B_3sin2 θ) + 2 r ( A_4sinθ + A_3 cosθ - 3C_1sin 3 θ - 3C_2cos 3 θ)-2( B_1 sin2 θ- B_2 cos2 θ+2 D_2 cos4 θ-2 D_1 sin4 θ) ]R^(3) .where Θ(θ) is an arbitrary function of θ. Since S(θ) and R(r) are functions of one variable only, (<ref>) is no longer a PDE. We will obtain several ODEs from it. We differentiate (<ref>) twice with respect to r. This eliminates S(θ) and Θ(θ) from the equation. We then expand in a basis of linearly independent trigonometric functions sinθ, sin2θ, sin3θ, sin4θ, cosθ, cos2θ, cos3θ, cos4θ and obtainthe following set of 8 equations that R(r) must satisfy simultaneously:12 (A_3 - 15r^2A_1) R' - 12 r (A_3 + 27r^2A_1) R” -r^2 (39A_3 + 146r^2A_1) R^(3)-r^3 ( 13A_3 + 22r^2A_1) R^(4) -r^4 (A_3+r^2A_1) R^(5) =0, 2 (9 B_2 - 40r^2B_4) R' - 2 r (9 B_2 - 40r^2B_4) R” - r^2 (B_2 - 128r^2B_4) R^(3) + r^3 (7 B_2 + 32 r^2B_4) R^(4) + r^4 ( B_2 + 2r^2B_4) R^(5) =0, C_2 (36R'-36rR”+3r^2R^(3)+9r^3R^(4)+r^4R^(5))=0,D_2(96R - 6rR'- 42r^2R” + 19r^3R^(3) - r^4R^(4)- r^5R^(5))=0,24(A_4-15r^2A_2) R' -24 r(A_4+27r^2A_2) R” -r^2( 78A_4+ 292r^2A_2) R^(3) -r^3(26A_4 + 44r^2A_2) R^(4) -2 r^4( A_4 +r^2A_2) R^(5) =0, 2 (9 B_1 - 40r^2B_3) R' - 2 r(9 B_1 - 40r^2B_3) R” -r^2( B_1- 128r^2B_3) R^(3) + r^3(7 B_1 + 32 r^2B_3) R^(4) + r^4 (B_1 + 2r^2B_3) R^(5) =0, C_1(36 R'-36r R”+3 r^2 R^(3)+9r^3R^(4)+r^4 R^(5))=0, D_1(96R - 6r R'-42r^2 R”+19r^3 R^(3) - r^4 R^(4)- r^5 R^(5))=0. Taking linear combinations of eqs. (<ref>-<ref>), we get the following Euler-Cauchy type differential equations(A_1 A_4 -A_2 A_3)[ 180R'+324 rR”+146r^2R^(3)+22r^3R^(4)+r^4R^(5)]= 0 , (B_1 B_4 - B_2 B_3)[ 40R'-40rR”-64r^2R^(3)-16r^3R^(4)-r^4R^(5)]= 0 , (C_1^2 + C_2^2)[ 36R'-36rR”+3r^2R^(3)+9r^3R^(4)+r^4R^(5)]= 0 , ( D_1^2+D_2^2 )[ 96R-6rR'-42r^2R”+19r^3R^(3)-r^4R^(4)-r^5R^(5)]= 0 .In particular, the above equations have solutionsR(r) =α _1/r^5+α _2/r^4+α _3/ r^2+α _4/r+α _5,(A_1 A_4 -A_2 A_3) ≠ 0,R(r) =α _1/ r^4+α _2/r^3+α _3/r+r^2 α _4+α _5 ,(B_1 B_4 - B_2 B_3) ≠ 0, R(r) =α _1/r^3+α _2/r^2+r^2 α _3+r^4 α _4+α _5 , C_1^2 + C_2^2 ≠ 0, R(r) =α _1/r^2+α _2/r+r^2 α _3+r^4 α _4+r^6 α _5 ,D_1^2+D_2^2≠ 0, respectively. Otherwise, for(A_1 A_4 -A_2 A_3)= 0 , (B_1 B_4 - B_2 B_3) = 0 , C_1^2 + C_2^2 = 0 , D_1^2+D_2^2 = 0, the equations (<ref>)-(<ref>) are trivially satisfied with arbitrary R(r), but then (<ref>)-(<ref>) are satisfied only when all parameters A_i, B_i, C_i and D_i, vanish (so that no fourth order integral Y exists). The compatibility between common solutions of (<ref>)-(<ref>) and the determining equations (<ref>)-(<ref>), shows that when (<ref>) is satisfied trivially then the most general form of the radial part R(r) of the potential V(r, θ) is* R(r)=b r^2 ;in this case all parameters are zero except B_3, B_4.* R(r)=a/r ; all parameters are zero except A_1, A_2, B_3, B_4 .* R(r)=0 ; all parameters are zero exceptA_1, A_2, B_3, B_4. Exotic potentials V(r, θ) with radial part R(r)=b r^2, a/r or 0 are also the only ones that could allow a third order integral <cit.>.§ NONCONFINING POTENTIALV(R,Θ)=S(Θ)/R^2 This potential corresponds to R(r)=0. In this article we are interested in exotic potentials. Eq. (<ref>) is linear, so it must be satisfied trivially. Hence all parameters in (<ref>) vanish except A_1, A_2, B_3, B_4. It is worth mentioning that the singular potentials of the form V(r,θ)= S(θ)/r^2 require a renormalization scheme in order to obtain a well defined problem with a discrete spectrum <cit.>.The equations (<ref>) - (<ref>) corresponding to the determining equations A_rrr= A_rrθ= A_rθθ= A_θθθ=0, respectively,take the form G_1^(1,0) =0 ,1/r^2( G_2^(0,1)+G_3/r)= - 2 /r^3(A_2 sinθ +A_1 cosθ +2/r( B_4 cos2 θ -B_3 sin 2 θ)) S+ ( 4/r^3( A_2 cosθ- A_1 sinθ) - 4/r^4( B_3 cos2 θ+ B_4 sin 2 θ))S' ,G_3^(1,0) + 1/r^2G_1^(0,1)= 2/r^2( B_3 cos2 θ + B_4 sin2 θ) S' ,1/r^2G_3^(0,1) + G_2^(1,0) +2/r^3G_1 = - 4/r^3(B_3 cos2 θ- B_4 sin 2 θ) S+3 /r^2( A_2 sinθ + A_1 cosθ + 2/r(B_4 cos2 θ - B_3 sin2 θ) ) S' .In particular, the equations (<ref>), (<ref>) and(<ref>)define the r dependence of the functions G_1,2,3. Indeed, from (<ref>) we obtainG_1(r, θ) = β_1(θ) .Substituting (<ref>) into Eq. (<ref>) and integrating we get:G_3(r, θ) = -2/r(B_3 cos 2θ+ B_4 sin 2θ)S' + 1/rβ_1' (θ) + β_3(θ) . Substituting G_1,G_3 into Eq (<ref>), we findG_2(r, θ)= 2/r^2 ( B_3 cos 2θ + B_4 sin 2θ ) S + [ -3/ r(A_1 cos θ +A_2 sin θ ) - 5/ r^2( B_4 cos 2θ - B_3 sin 2θ ) ] S'- 1/r^2 ( B_3 cos 2θ + B_4 sin 2θ )S” +β_2(θ) +1/r β_3' (θ) +1/2 r^2 (2β_1(θ) +β_1” (θ) ) .Let us now determine the functions β_i(θ). Substituting the above functions G_1,G_2,G_3 into (<ref>), i.e. into the determining equation A_θθθ=0, and collecting in powers of r one finds the following three equations which define the functions β_1,2,3: β_2' (θ)=0,β_3”(θ) +β_3 (θ) =-2 (A_1 cosθ +A_2 sinθ) S + 7 (A_2 cosθ-A_1 sinθ)S'+3 (A_1 cosθ +A_2 sinθ)S” ,1/2 β_1”' ( θ) +2 β_1' ( θ)=( B_3 cos2 θ+ B_4 sin2 θ) S”' + 7 (B_4 cos2 θ- B_3 sin2 θ) S” - 14( B_3 cos2 θ+ B_4 sin2 θ)S'- 8 ( B_4 cos2 θ- B_3 sin2 θ)S.Equation (<ref>) implies thatβ_2(θ ) = c_21 ,where c_21 is a constant.Next, replacing S(θ) = T'(θ) , into (<ref>) and solving this equation we find the function β_1(θ): β_1(θ)=2 ( B_4 cos2θ -B_3 sin2 θ) T + 2 ( B_4 sin 2 θ+B_3 cos2 θ) T'+ 1/2 c_11 sin2 θ - 1/2 c_12 cos2 θ + c_13 , where the c's are integration constants.Similarly, the solution to equation (<ref>) provides the function β_3(θ) β_3( θ)= c_31 cosθ +c_32 sinθ +( A_2 cosθ- A_1 sinθ )T + 3 ( A_2 sinθ+A_1 cosθ ) T' .Now let us turn to the equations (<ref>)-(<ref>). From the equation (<ref>), C_r=0, we find the function G_4(r,θ): G_4(r,θ)=[ -1/2 r(2 ( A_1 cosθ + A_2 sinθ) S”' - 6 (A_2 cosθ- A_1 sinθ) S” + 6(A_1 cosθ+A_2 sinθ) S' + β_3”' +β_3' )- 1/r^2(1/2 ( B_4sin2 θ+B_3cos2 θ)S^(4) - 4 (B_3sin2 θ-B_4cos2 θ) S”'- 10 (B_3cos2 θ+B_4sin 2 θ) S” - 8(B_4 cos2 θ- B_3 sin2 θ )S' - 1/4 β_1 ^(4) -β_1”) ] ħ^2 +β_4 -2/rS' β_3+1/r^2(2 (B_4 sin2 θ +B_3 cos2 θ)S'^2-S'β_1'+4 S β_1) . At this point, all eight coefficients A_rrr,A_rrθ,A_rθθ,A_θθθ,B_rr, B_rθ,B_θθ,C_r in (<ref>) vanish. In fact, the main equation to be solved is C_θ=0, presented in (<ref>). Substituting G_1,G_2,G_3,G_4 into the determining equation C_θ=0, (<ref>), and collecting powers of r we get three equations that must be satisfied simultaneously in order for Yin (<ref>) to be an integral of motion:0 =β_4' -4 c_21 T” ,0 =[-4(A_1 cosθ+A_2 sinθ) T'- 8 ( A_1 sinθ - A_2 cosθ)T”+ 6(A_1 cosθ+ A_2 sinθ)T^(3) +5/2(A_1 sinθ - A_2 cosθ)T^(4) -1/2( A_1 cosθ + A_2 sinθ)T^(5)] ħ^2 - 12(A_1 cosθ + A_2 sinθ) (T')^2+ ( 24(A_2 cosθ - A_1 sinθ)T”+6 ( A_1 cosθ+ A_2 sinθ)T^(3)+4(A_1 sinθ- A_2 cosθ) T-4(c_31 cos( θ)+c_32 sin( θ) ) )T'+6 (A_1 cosθ+A_2 sinθ) (T” )^2 + ( 6( A_1 cosθ+A_2 sinθ) T . . -6( c_31 sinθ -c_32 cosθ) )T” + ( 2 ( A_2 cosθ -A_1 sinθ)T . . +2 (c_31 cosθ +c_32 sinθ) )T^(3) ,0 = [ -32 (B_4cos2 θ -B_3sin2 θ) T'-40 (B_4sin2 θ+ B_3cos2 θ)T”+ 20 ( B_4cos2 θ- B_3sin2 θ)T^(3)+ 5 ( B_4sin2 θ+B_3cos2 θ)T^(4) -1/2(B_4cos2 θ- B_3sin2 θ)T^(5)] ħ^2-48 ( B_4cos2 θ - B_3sin 2 θ) ( T')^2+ [-48(B_4sin2 θ+ B_3cos2 θ)T”+ 6(B_4cos2 θ- B_3sin 2 θ)T^(3) + 32 (B_4sin2 θ+ B_3cos2 θ) T-8(c_11 cos2 θ+ c_12 sin2 θ)]T'+ 6(B_4cos2 θ - B_3sin2 θ) (T” )^2+ [24( B_3 sin2 θ-B_4 cos2 θ) T .. -6( c_11 sin2 θ- c_12 cos2 θ)]T”-[4 ( B_4sin2 θ +B_3cos2 θ) T- c_11 cos2 θ- c_12 sin2 θ] T^(3) , At this stage we assume ħ≠ 0. We see that in the classical case (ħ→ 0) equations (<ref>) and (<ref>) simplify greatly. The above non-linear equations (<ref>) and (<ref>) will determine the angular part of the potential. They both pass the Painlevé test.Equation (<ref>) determines the function β_4 β_4(θ) = 4c_21S(θ) + c_41 , together with (<ref>), this defines G_4(r, θ) of (<ref>) completely in terms of S(θ) and some constants. The parameters c_13, c_21 in (<ref>) and (<ref>) can be set equal to zero by linear combinations of H and X. Moreover,c_41 in (<ref>) is simply a constant that commutes with H trivially. Therefore, without loss of generality we choose c_13 =0,c_21 =0,c_41 =0.Equations (<ref>) and (<ref>) depend on mutually exclusive sets of parameters, namely (A_1, A_2, c_31, c_32) and (B_3, B_4, c_11, c_12), respectively. Moreover, for A_1=A_2=0 (<ref>) reduces to a linear equation, as does (<ref>) for B_3=B_4=0. Since we are looking for exotic potentials, all linear equations for T(θ) must be satisfied identically. Hence we have 2 cases to consider * Case (I)A_1^2+A_2^2≠ 0 , B_3=B_4=c_11=c_12=0,By a rotation we can set A_1=0,* Case (II)B_3^2+B_4^2≠ 0 , A_1=A_2=c_31=c_32=0,By a rotation we can set B_4=0.In Case I and II one of the two equations (<ref>)-(<ref>) trivializes, so only one nonlinear equation must be solved. It already passed the Painlevé test. * Case (III) A_1^2+A_2^2≠ 0 ,B_3^2+B_4^2≠ 0In this case the two nonlinear determining equations (<ref>) and (<ref>) remain. Thus they will either be incompatible or T(θ) will be a very special case of the solutions obtained in Case I and Case II. We shall not investigate this case further since it cannot provide any new exotic potentials.We also note that in the quantum case (<ref>) and (<ref>) are fifth order equations. In the classical limit ħ→ 0 they reduce to third order ones, to be considered in Section 6.§.§ Case I, A_2≠0, A_1=B_3=B_4=0 Equation (<ref>) is linear and must be satisfied trivially, so we have c_11=c_12=0. Equation (<ref>) simplifies toA_2 [2sinθT' - 4cosθT” - 3sinθ T”' + 5/4 cosθT^(4) +1/4 sinθT^(5)] ħ^2 + A_2 ( 2cosθ T' + 3 sinθT” -cosθT”') T+ 6 sinθ A_2T'^2 - (12 A_2 cosθT” + 3 sinθA_2 T”' -2 ( c_31 cosθ+c_32 sinθ) ) T' - 3A_2 sinθ( T”)^2 - 3 (c_32 cosθ- c_31 sinθ ) T” -( c_31 cosθ+ c_32 sinθ) T”' =0 .This equation can be integrated once resulting in the 4-th order equationA_2 [2cosθ T' + 2sinθ T” -cosθ T”' -1/4 sinθ T^(4)] ħ^2+A_2 (cosθ T”-2sinθT' ) T + 4 A_2 cosθ ( T') ^2+ (3A_2 sinθ T” +2( c_32 cosθ- c_31 sinθ))T' +( c_31 cosθ+c_32 sinθ)T”+ 1/2K_1 = 0,where K_1 is an arbitrary integration constant. Transforming to the variable z = tan θ ,and dividing by (1/4) (z^2+1)^2, we get: A_2 [ 24 z^2T' + 12 ( 3 z^2+2 ) z T”+4 ( z^2+1 )( 3 z^2+ 1 ) T”' + ( z^2+1 ) ^2z T^(4)] ħ^2 -4 A_2 T”T - A_2 ( 24 z^2+16 )( T' ) ^2 - ( 12 ( z^2+1 )z A_2T” +8 c_32)T'- 4( c_32 z +c_31)T” - 2K_1/( z^2+1 ) ^3/2 = 0. Putting c_31→2 A_2 c_31,c_32→ 2 A_2 c_32 , K_1→2 A_2 K_1, we integrate the above equation, using z as integrating factor to get the following third order non-linear differential equation[2( 1 -3 z^4)T' -2 z ( 3 z^2+1 )( z^2+1 )T”-z^2( z^2+1 )^2T”' ] ħ^2 -2 T ^2+ 4(zT' - 2 c_31) T + 6 z^2( z^2+1 )( T') ^2 + 8 z ( c_32 z + c_31)T' -4 K_1/√(z^2+1)+K_2=0 ,here K_2 is another arbitrary integration constant. The transformation (z,T(z))↦(x,W(x)):z=2 √(x) √(1-x)/1-2 x, T=ħ^2W/√(x)√(1-x) + (3 ħ^2 + 8 c_32) (1-2 x)/ 8 √(x) √(1-x) - 2 c_31 , maps (<ref>)to an equationcontained in a series of papers by C. Cosgrove (see for example [] )on higher order Painlevé equations. Equation (<ref>) is mapped into the third order differential equation Chazy-I.a with parametersq_1=q_4=q_5=q_6 =0, q_2=-q_3=1, q_7= 5 ħ^2 + 16c_32/16 ħ^2, q_8= K_1/ħ^4,q_9=-32 K_1 + 8 K_2 + 3 ħ^4+ 64c_31^2 + 32 ħ^2 c_32 + 64 c_32^2/64 ħ^4 . The equation for W(x) can be integrated, and the resulting non-linear second order differential equation becomes the equation SD-I.a in Cosgrove's paper <cit.> ( W”)^2 =-4/f^2(x)[q_1(x W'-W)^3+q_2 W' (x W'-W)^2 +q_3 ( W')^2 (x W'-W) + q_4 ( W')^3+q_5(x W'-W)^2+q_6 W' (x W'-W)+ q_7 ( W')^2+ q_8(x W'-W)+ q_9 W' + q_10] , The integration constant q_10 is arbitrary and the function f(x) satisfies f(x)=q_1 x^3+q_2 x^2+q_3 x+q_4. Eq. (<ref>) is the first canonical subcase of the more general equation that Cosgrove called the “master Painlevé equation”. Equation SD-I.a is solved by the Backlund correspondenceW(x) =x^2(x-1)^2/4P_6(P_6-1)(P_6-x)[P_6'-P_6(P_6-1)/x(x-1)]^2+1/8(1-√(2γ_1))^2(1-2P_6)-1/4γ_2(1-2x/P_6)-1/4γ_3(1-2(x-1)/P_6-1)+(1/8-γ_4/4)(1-2x(P_6-1)/P_6-x),andW'(x)=-x(x-1)/4P_6(P_6-1)[P_6'-√(2γ_1)P_6(P_6-1)/ x(x-1)]^2-γ_2(P_6-x)/2(x-1)P_6-γ_3(P_6-x)/ 2x(P_6-1) ,where √(2γ_1) can take either sign andγ_1,γ_2,γ_3 and γ_4 are the arbitrary parameters that define the sixth Painlevé transcendent P_6 which satisfies the well known second order differential equation:P_6”=1/2[1/P_6+1/P_6-1+1/P_6-x] (P_6')^2-[1/x+1/x-1+1/P_6-x]P_6'+P_6(P_6-1)(P_6-x)/x^2(x-1)^2[γ_1+γ_2 x/P_6^2 +γ_3 (x-1)/(P_6-1)^2+γ_4 x(x-1)/(P_6-x)^2]. Thus, we have W= W(x ; γ_1, γ_2, γ_3, γ_4) .The parameters γ_1,γ_2,γ_3 and γ_4 are related to the arbitrary constants of integration c_31, c_32, K_1 and K_2 through the relations-4q_7= γ_1-γ_2+γ_3-γ_4-√(2γ_1)+1,-4q_8=(γ_2+γ_3)(γ_1+γ_4-√(2γ_1)), -4q_9=(γ_3-γ_2)(γ_1-γ_4-√(2γ_1)+1)+1/4 (γ_1-γ_2-γ_3+γ_4-√(2γ_1))^2, -4q_10=1/4(γ_3-γ_2)(γ_1+γ_4-√(2γ_1) )^2+1/4(γ_2+γ_3)^2(γ_1-γ_4-√(2γ_1)+1)In particular, (<ref>) together with (<ref>) imply that the constants c_31 and c_32 can be written in terms of the γ's.A superintegrable potential expressed in terms of the Painlevé transcendent P_6 was obtained earlier <cit.>. It allowed a third order integral and required a specific relation between the constants γ_1,...,γ_4. Here we obtain the most general form of P_6.From the inverse transformation x→ z=tanθ in (<ref>) we getx_±=1/2±1/2√(1+z^2)={[ sin^2(θ/2); ; cos^2(θ/2);]we obtain two solutions for S(θ)=T'(θ). For the Case I we obtain two quantum potentials V(r,θ)=∂_θT(x_±)/r^2= ħ^2/r^2( W'(x_±) ∓2 cosθ/sin^2θ W(x_±) + 1/2 sin^2θ Γ),where Γ=(γ_2+γ_4+√(2 γ_1)-γ_1-γ_3-3/8). Both T and W are completely defined through (<ref>)-(<ref>). The integral Y in both cases is Y=ħ^4 {∂_θ^3, sinθ ∂_r }+ħ^4/r{∂_θ^3, cosθ∂_θ}- ħ^2 { G_1(r,θ), ∂_r^2} -ħ^2 { G_3(r,θ), ∂_r ∂_θ} -ħ^2 { G_2(r,θ), ∂_θ^2} + G_4(r,θ), (A_2=1) whereG_1( r,θ) =0 ,G_2( r,θ)=1/r ( 4 cosθ T'+2 c_32 cosθ - ( T + 2 c_31) sinθ) ,G_3( r,θ) = 3 sinθ T' +( T +2 c_31 ) cosθ +2 c_32 sinθ ,G_4( r,θ) = 1/2r(sinθ T^(4) + 4 cosθT^(3) - 3 sinθ T” - 2 cosθ T' ) ħ^2- 2/r( 3 sinθ T' +cosθ T + 2 c_31cosθ + 2 c_32sinθ)T” .here T'=∂_θT(x_±). The integral Y and the corresponding potential V(r,θ) depend on the same constants, namely, the four parameters γ_1,γ_2,γ_3,γ_4 in (<ref>) which define the sixth Painlevé transcendent P_6. §.§ Case II, B_3≠ 0, A_1=A_2=B_4=0 Equation (<ref>) reduces to a linear one that must be satisfied trivially so we have to impose c_31=c_32=0. Equation (<ref>) simplifies toB_3 [ 16sin 2 θ T' -20cos 2 θ T” -10sin2 θT”' +5/2 cos2 θ T^(4) + 1/4 sin2 θ T^(5)] ħ^2 + B_3 (16cos 2 θ T'+ 12 sin2 θ T” -2cos 2 θ T”')T + 24B_3 sin2 θ (T' ) ^2 - (24 B_3cos2 θT” + 3B_3sin2 θT”' + 4 ( c_11 cos2 θ+c_12 sin2 θ) )T'- 3 B_3 sin2 θ ( T”)^2 + 3 (c_12 cos2 θ- c_11 sin 2 θ )T” + 1/2(c_11 cos 2 θ +c_12 sin2 θ)T”'=0,This equationcan be integrated once resulting inB_3 [ 1/4sin2 θ T^(4) + 2 cos2 θ T^(3) - 6 sin2 θ T” -8 cos2 θT' ] ħ^2 + B_3 ( 8sin2 θT' -2cos2 θT”)T -8B_3cos2 θ( T') ^2 +( 2 ( c_12 cos2 θ -c_11 sin2 θ ) - 3B_3 sin2 θ T”)T'+ 1/2( c_11 cos 2 θ + c_12 sin 2 θ ) T”+ 1/2K_1 =0 . Putting z = tan 2θ, (and dividing by the common factor 4 (z^2+1)^3/2) we obtainB_3 ( 4 ( 6 z^2+1 ) T' + ( 36 z^3+ 26 z) T” +4 ( z^2+1 )( 3 z^2+1 ) T^(3)+ ( z^2+1 )^2 zT^(4)) ħ^2 -2 B_3T”T -B_3 ( 12 z^2 + 8 )( T')^2 + (c_12 -6 ( z^2+1 ) z B_3T”)T'+ 1/2( c_12 z+ c_11) T” +1/8 K_1/( z^2 + 1 )^3/2 =0 . We introduce c_11→ 2 B_3 c_11, c_12→ 2 B_3 c_12 and K_1 = 2 B_3 K_1 and again integrate (<ref>) to obtain (2( 3 z^4 +z^2 - 1 )T' +2 z ( 3 z^2+1 )( z^2+1 ) T” +z^2( z^2+1 ) ^2T^(3)) ħ^2 +T^2- (c_11 + 2 zT' ) T-3 z^2( z^2+1 )( T' )^2 + z ( c_12 z + c_11) T'- 1/4 K_1/√(z^2+1)+K_2 =0.The transformation (z,T(z))↦(x,W(x)):z=2√(x)√(1-x)/1-2 x, T=1/4 (1-2x) (ħ ^2-c_12)+2ħ^2W(x)/√(x(1-x))+c_11/2 , maps (<ref>)to an equationcontained in the series of papers by C. Cosgroveon higher order Painlevé equations <cit.>. Equation (<ref>) is mapped into the third order differential equation Chazy-I.awith parametersq_1=q_4=q_5=q_6 =0, q_2=-q_3=1, q_7= 1/16-c_12/8 ħ ^2 , q_8= -K_1/32ħ ^4,q_9= -c_11^2+c_12^2-K_1-4 K_2-ħ ^4/64 ħ ^4 . The solution for the function W(x) is given in (<ref>), however the independent variable is different, namely:x_±=1/2±1/2√(1+z^2)={[ sin^2θ; ; cos^2θ .;]We obtain two solutions for S(θ)=T'(θ). By taking the derivative ∂_θT(x_±) we obtain the quantum potentialsV(r,θ)=∂_θT(x_±)/r^2= ħ^2/r^2( 4 W'(x_±) ∓8 cos 2θ/sin^2 2θ W(x_±) + 1/sin^2 2θ Γ),where Γ=2(γ_2+γ_4+√(2 γ_1)-γ_1-γ_3+3/4), T is now defined through (<ref>)-(<ref>). These potentials correspond to the integral Y=ħ^4 {∂_θ^2, cos 2θ} ∂_r^2-ħ^4/r^2{∂_θ^2, cos 2θ ∂_θ^2} - 2 ħ^4/r {∂_θ^2, sin 2θ ∂_θ} ∂_r+ 2ħ^4/r^2{∂_θ^2,sin 2θ∂_θ} - ħ^4/r{∂_θ^2, cos 2θ} ∂_r -ħ^2 ({ G_1(r,θ), ∂_r^2} +{ G_3(r,θ), ∂_r ∂_θ} +{ G_2(r,θ), ∂_θ^2})+ G_4(r,θ) ,(with B_3=1) where G_1( r,θ) = 2 cos2 θ T' - 2 sin2 θ T +c_11 sin2 θ - c_12 cos2 θ ,G_2( r,θ) = 1/r^2( 2 sin2 θ T - 4 cos2 θ T' - c_11 sin2 θ + c_12 cos2 θ),G_3( r,θ) = 1/r(2 c_11 cos2 θ - 4 cos2 θT - 6 sin2 θT' +2 c_12 sin2 θ),G_4( r,θ) = 1/r^2 ( ( - 1/2 sin2 θT^(4) -4 cos2 θT^(3)+10 sin2 θ T” +8 cos2 θ T') ħ^2+( ( 4 T - 2 c_11) cos2 θ+ ( 6 T' -2 c_12) sin2 θ)T”+( 8 T' - 4 c_12)T' cos2 θ-( 8 T - 4 c_11)T' sin2 θ). § CONFINING POTENTIALS §.§ POTENTIAL V(r,θ)= b r^2 + S(θ)/r^2In this case the compatibility condition (<ref>) is satisfied trivially if all parameters are zero except B_3, B_4. Since we can rotate between these two terms we set B_4=0. The equation for S(θ) corresponds to Case II of section <ref>.The only determining equation to solve is (<ref>) and the solution for the function S(θ) will be the same as for R(r)=0. The only difference with the case R(r)=0 is reflected in the G functions (<ref>) which does not modify the form of the determining equation (<ref>). The corresponding quantum potentials areV(r,θ)=b r^2+∂_θT(x_±)/r^2= b r^2+ħ^2/r^2( 4 W'(x_±) ∓8 cos 2θ/sin^2 2θ W(x_±) + 1/sin^2 2θ Γ),where Γ=2(γ_2+γ_4+√(2 γ_1)-γ_1-γ_3+3/4). The function T is defined through (<ref>)-(<ref>).The integral of motion in this case is Y=ħ^4 {∂_θ^2, cos 2θ} ∂_r^2-ħ^4/r^2{∂_θ^2, cos 2θ ∂_θ^2} - 2 ħ^4/r {∂_θ^2, sin 2θ ∂_θ} ∂_r+ 2 ħ^4/r^2{∂_θ^2,sin 2θ∂_θ} - ħ^4/r{∂_θ^2, cos 2θ} ∂_r-ħ^2 ({ G_1(r,θ), ∂_r^2} +{ G_3(r,θ), ∂_r ∂_θ} +{ G_2(r,θ), ∂_θ^2})+ G_4(r,θ) ,where G_1( r,θ) = 2 cos2 θT' - 2 sin2 θT +c_11 sin2 θ - c_12 cos2 θ ,G_2( r,θ) = 1/r^2( 2 sin2 θ T - 4 cos2 θ T' - c_11 sin2 θ + c_12 cos2 θ)+ 2 b cos2 θ r^2 ,G_3( r,θ) = 1/r(2 c_11 cos2 θ - 4 cos2 θT - 6 sin2 θT' +2c_12 sin2 θ),G_4( r,θ) = 1/r^2 ( ( - 1/2 sin2 θT^(4) -4 cos2 θT^(3)+10 sin2 θ T” +8 cos2 θ T') ħ^2+( ( 4 T - 2 c_11) cos2 θ+ ( 6 T' -2 c_12) sin2 θ)T”+( 8 T' - 4 c_12)T' cos2 θ-( 8 T - 4 c_11)T' sin2 θ)+ b [-8 sin 2 θT +8 cos2 θ T'+4 ( c_11 sin2 θ- c_12 cos2 θ) ] r^2-8 cos2 θb ħ^2 r^2 .§.§ POTENTIAL OF THE FORM V(r,θ)= a/r + S(θ)/r^2In this case the compatibility condition (<ref>) is satisfied trivially if all parameters are zero except A_1, A_2,B_3, B_4 (as in the Case of R(r)=0). From the condition [H,Y]=0 we obtain two 5-th order non-linear equations equations in T(θ) that must be satisfied simultaneously, namely eq. (<ref>) and 0=[-4(A_1 cos θ +A_2 sin θ)T' + 8 ( A_2 cos θ - A_1 sin θ)T” + 6 (A_1 cos θ + A_2 sin θ) T^(3) + 5/2 ( A_1 sin θ -A_2 cos θ)T^(4) - 1/2 (A_1 cos θ + A_2 sin θ)T^(5) + 15 a ( B_3sin 2θ-B_4cos 2θ)]ħ^2 + [4 (A_1 sin θ- A_2 cos θ)T' -6 ( A_1 cos θ + A_2 sin θ)T” + 2 ( A_2 cos θ - A_1 sin θ)T^(3) +24 a ( B_4 sin 2θ+ B_3 cos 2θ)]T -12 (A_1 cos θ +A_2 sin θ) ( T' )^2 + [24 ( A_2 cos θ - A_1 sin θ)T” + 6 (A_2 sin θ+ A_1 cos θ )T^(3) +44 a (B_3 sin 2θ-B_4 cos 2θ)-4 c_31 cos θ -4 c_32 sin θ]T' + 6 (A_2 sin θ +A_1 cos θ) ( T”) ^2 -6 ( 4 a B_4 sin 2θ+4a B_3 cos 2θ -c_32 cos θ + c_31 sin θ)T” -2 ( 2 a B_3 sin 2θ-2 a B_4 cos 2θ- c_31 cos θ - c_32 sin θ) T^(3) -6 a ( c_12 sin 2θ +c_11 cos 2θ).For a=0 (<ref>) coincides with (<ref>).Case I. B_3=B_4=c_11=c_12=0, A_1 and A_2 arbitrary. The non-linear equation (<ref>) is satisfied trivially, while (<ref>) coincides with (<ref>) and thus, in this case, we obtain the quantum potentials V(r,θ)=a/r +∂_θT(x_±)/r^2 =a/r +ħ^2/r^2( W'(x_±) ∓2 cosθ/sin^2θ W(x_±) + 1/2 sin^2θ Γ),where Γ=(γ_2+γ_4+√(2 γ_1)-γ_1-γ_3-3/8), and both T and W are completely defined through (<ref>)-(<ref>). These potentials correspond to the integral (A_2=1) Y =ħ^4 {∂_θ^3, sinθ ∂_r }+ħ^4/r{∂_θ^3, cosθ∂_θ}- ħ^2 { G_1(r,θ), ∂_r^2} -ħ^2 { G_3(r,θ), ∂_r ∂_θ} -ħ^2 { G_2(r,θ), ∂_θ^2} + G_4(r,θ), where G_1( r,θ) =0 ,G_2( r,θ)=1/r ( 4 cosθ T'+2 c_32 cosθ - ( T + 2 c_31) sinθ) + a cosθ ,G_3( r,θ) = 3 sinθ T' +( T +2 c_31) cosθ +2 c_32 sinθ,G_4( r,θ) = 1/2 r(sinθ T^(4) + 4 cosθT^(3) - 3 sinθ T” - 2 cosθ T' ) ħ^2- 2/r( 3 sinθ T' +cosθ T + 2 c_31cosθ + 2 c_32sinθ)T” - 2 a sinθ T+4a cosθ T'-4 a (c_31 sinθ-c_32 cosθ) - a ħ^2 cosθ,here T'=∂_θT(x_±). Case II. For A_1=A_2=c_31=c_32=0, a≠0 and B_3, B_4 arbitrary (<ref>) reduces to a linear equation. For exotic potentials it must be satisfied identically. This implies B_3=B_4=0, so no fourth order integral exists.§ CLASSICAL POTENTIALSThe two determining equations (<ref>)-(<ref>) reduce to third order equations for T(θ) once we impose the condition ħ→ 0. The limit is singular and interestingly, the equations in this case do not pass the Painlevé test. The division into subcases (<ref>) remains. We can always integrate (<ref>)-(<ref>) twice and we obtain a first order nonlinear equation of the form Q_4(z) T'^2 + Q_1(z) T T' + T^2 +Q_2(z) T' + c T + Z(z)=0,where Q_n(z) is a polynomial in z of order n, Z(z) is a rational function and c is a constant. Using the transformationT(z)=m(z) t(z) + n(z),we can factorize (<ref>) as follows(t'-t'_0)(t'+t'_0) = 0,wheret'_0=√((Q_1(mt+n) + Q_2)^2-4Q_4 (Q_0 (mt+n)+(mt+n)^2+Z))/2mQ_4 .and m and n satisfy2Q_4(z)n'(z) + n(z)Q_1(z) + Q_2(z)=0 , 2 Q_4(z) m'(z) + m(z) Q_1(z) = 0, In general, explicit solutions to the equation t'± t'_0=0 are not known. However for special values of the parameters contained in the Q_i and Z, the function t'_0 becomes linear in t and explicit solutions can be constructed.§.§ Case V(r, θ)= S(θ)/r^2 §.§.§ Case IThe classical potential S(θ)=T'(θ) satisfies (<ref>) with ħ↦0. This limit is singular, the order of the equation (<ref>) drops from three to one.The so obtained non-linear first order differential equation reads: T ^2- 2(zT' - 2 c_31) T -3 z^2( z^2+1 ) ( T')^2 - 4 z ( c_32 z + c_31)T' +2 K_1/√(z^2+1) -K_2/2 = 0 , where z=tan θ. Factorization of the l.h.s in (<ref>) in the form of a product of two factors of first order allows us to find particular solutions. These two factors become linear for specific values of the parameters in (<ref>) only.Namely,putting K_1=c_32=0 and K_2=-8 c_31^2 in (<ref>) we obtain the equation 6 z^2(1+z^2) (T'+(2 c_31+T)(1+√(4+3 z^2))/3 z (1+z^2)) (T'+(2 c_31+T)(1-√(4+3 z^2))/3 z (1+z^2)) = 0, from which we derive two particular solutions:T_1 =-2 c_31 + α z^1/3 (3 z^2+2 √(3 z^2+4)+5)^1/6/(√(3 z^2+4)+2)^2/3 , T_2 =-2 c_31 +α (1+z^2)^1/3 (2+√(4+3 z^2))^2/3/z (3 z^2+2 √(3 z^2+4)+5)^1/6 ,where α is an integration constant. By differentiating the preceding results (<ref>) with respect to θ we obtain the classical potentials:V_1(r, θ)=3α^4θ[7+3√(4+3tan^2θ) +cos2θ(1+√(4+3tan^2θ))]/r^2tan^2/3θ√(4+3tan^2θ)(2+√(4+3tan^2θ))^5/3(5+3tan^2θ+2√(4+3tan^2θ))^5/6, andV_2(r, θ)=-α^2/3θ[47+17√( 4+3tan^2θ)+18^2θ(2+√(4+3tan^2θ))+3tan^2θ(5+√( 4+3tan^2θ))]/2r^2√(4+3tan^2θ)(2+√(4+3tan^2θ))^1/3(5+3tan^2θ+2√(4+3tan^2θ))^7/6 .In general, the potentials V(r, θ) associated with (<ref>) possess the integral Y = 2sinθ p_θ^3 p_r +2/r cosθ p_θ^4+ 2G_1(r,θ) p_r^2+ 2G_3(r,θ) p_r p_θ + 2G_2(r,θ) p_θ^2+ G_4(r,θ), (A_2=1) where G_1( r,θ) =0 ,G_2( r,θ)=1/r ( 4 cosθ T'+2 c_32 cosθ - ( T + 2 c_31) sinθ),G_3( r,θ) = 3 sinθ T' +( T +2 c_31) cosθ +2 c_32 sinθ ,G_4( r,θ) =-2/r( 3 sinθ T' +cosθ T + 2 c_31cosθ + 2 c_32sinθ)T” . §.§.§ Case II The classical potential S(θ) satisfies (<ref>) with ħ↦0. This limit is singular, the order of the equation (<ref>) drops from three to one.The so obtained non-linear first order differential equation in T reads: T^2- (c_11 + 2 zT' ) T-3 z^2( z^2+1 )( T' )^2 + z ( c_12 z + c_11) T'- 1/4 K_1/√(z^2+1)+K_2 =0, Factorization of the l.h.s in (<ref>) in the form of a product of two factors of first order allows us to find particular solutions again. The factors are linear for specific values of the parameters in (<ref>) only. These special valuesare K_1=c_12=0 and K_2=1/4c_11^ 2. By substituting these values in (<ref>) we derive two particular solutions T_3 =c_11/2 + α z^1/3 (3 z^2+2 √(3 z^2+4)+5)^1/6/(√(3 z^2+4)+2)^2/3 , T_4 =c_11/2 +α (1+z^2)^1/3 (2+√(4+3 z^2))^2/3/z (3 z^2+2 √(3 z^2+4)+5)^1/6 , where α is an integration constant. By differentiating the preceding results (<ref>) with respect to θ we obtain the classical potentials:V_3(r, θ)=3α^42θ[7+3√(4+3tan^22θ) +cos4θ(1+√(4+3tan^22θ))]/ r^2 tan^2/32θ√(4+3tan^22θ)(2+√(4+3tan^22θ))^5/3(5+3tan^22θ+2√(4+3tan^22θ))^5/6, andV_4(r, θ)=-α^2/32θ[47+17√( 4+3tan^22θ)+18^22θ(2+√(4+3tan^22θ) )+3tan^22θ(5+√(4+3tan^22θ))]/2 r^2 √(4+3tan^22θ)(2+√(4+3tan^22θ))^1/3(5+3tan^22θ+2√(4+3tan^22θ))^7/6 ,where α is a constant.For (<ref>) the potentials V(r, θ) possess the integral Y=2p_r^2p_θ^2 cos 2θ-2/r^2 p_θ^4 cos 2θ - 4/r p_r p_θ^3 sin 2θ +2G_1(r,θ) p_r^2+ 2G_3(r,θ) p_r p_θ + 2G_2(r,θ) p_θ^2+ G_4(r,θ), (with B_3=1) where G_1( r,θ) = 2 cos2 θT' - 2 sin2 θ T +c_11 sin2 θ - c_12 cos2 θ ,G_2( r,θ) = 1/r^2( 2 sin2 θ T - 4 cos2 θ T' - c_11 sin2 θ + c_12 cos2 θ) ,G_3( r,θ) = 1/r(2 c_11 cos2 θ - 4 cos2 θT - 6 sin2 θT' +2 c_12 sin2 θ),G_4( r,θ) = 1/r^2 (( ( 4 T - 2 c_11) cos2 θ+ ( 6 T' -2 c_12) sin2 θ)T”+( 8 T' - 4 c_12) T' cos2 θ-( 8 T - 4 c_11) T' sin2 θ) . §.§ Potential V(r, θ) = b r^2 + S(θ)/r^2 The classical potentials are given by V(r, θ) =b r^2 + T'(θ)/r^2 , T from (<ref>), and they correspond to the integral Y= 2p_r^2p_θ^2 cos 2θ-2/r^2 p_θ^4 cos 2θ - 4/r p_r p_θ^3 sin 2θ +2 G_1(r,θ) p_r^2 + 2G_3(r,θ) p_r p_θ + 2G_2(r,θ) p_θ^2+ G_4(r,θ) . (with B_3=1) where G_1( r,θ) = 2 cos2 θ T' - 2 sin2 θ T +c_11 sin2 θ - c_12 cos2 θ ,G_2( r,θ) = 1/r^2( 2 sin2 θ T - 4 cos2 θ T' - c_11 sin2 θ + c_12 cos2 θ) + 2 b r^2 cos 2 θ ,G_3( r,θ) = 1/r(2 c_11 cos2 θ - 4 cos2 θT - 6 sin2 θT' +2 c_12 sin2 θ),G_4( r,θ) = 1/r^2 (( ( 4 T - 2 c_11) cos2 θ+ ( 6 T' -2 c_12) sin2 θ)T”+( 8T' - 4 c_12)T' cos2 θ-( 8 T - 4 c_11)T' sin2 θ) + 4 b r^2 (2 cos2 θ T'-2 sin2 θ T+c_11 sin 2 θ-c_12 cos2 θ). §.§ Potential V(r, θ) = a/r + S(θ)/r^2 Similarly, the classical potentials are given by V(r, θ) = a/r + T'(θ)/r^2 , T from (<ref>), and they corresponds to the integralY=2sinθ p_θ^3 p_r +2/r cosθ p_θ^4+ 2G_1(r,θ) p_r^2+ 2G_3(r,θ) p_r p_θ++2G_2(r,θ) p_θ^2+ G_4(r,θ),(A_2=1) where G_1( r,θ) =0 ,G_2( r,θ)=1/r ( 4 cosθT'+2 cosθ c_32 - ( T + 2 c_31) sinθ) +a cosθ ,G_3( r,θ) = 3 sinθ T' +( T +2 c_31) cosθ +2 sinθ c_32 ,G_4( r,θ) =-2/r( 3 sinθ T' +cosθ T + 2 c_31cosθ + 2 c_32sinθ)T”+ 2 a (2 cosθ T'-sinθ T)-4 a (c_31 sinθ-c_32 cosθ) . § POLYNOMIAL ALGEBRAIn this section we discuss the algebra of the integrals of motion in the classical case <cit.>.Take the second order integral X and the fourth order ones Y, (<ref>) and (<ref>) respectively. Let us define, via their Poisson bracket {}__PB, the fifth order polynomial in momenta C≡{ Y, X }__PB , which by construction is also an integral of motion. Now we study the algebra generated by the four quantities H, X, Y and C. The relevant (non vanishing) Poisson brackets are { X, C }__PB and { Y, C }__PB only.First we consider the case of the extended harmonic oscillator potential V(r ,θ)=b r^2 + T'(θ)/r^2 . For the particular solutions (<ref>), T_3 and T_4, the algebra generated by the integrals is given by { X, C }__PB= 16 X Y,{ Y, C }__PB= 8 [ 48 H^2X^2-128 X^3 b -Y^2 + b σ_3,4 ],where σ_3=512/9 α^3, σ_4=-512/3 α^3 and α a non zero constant, respectively. At b=0 this algebra reduces to that of the Case II, R(r)=0. For an arbitrary solution of (<ref>), in order to the algebra to be closed the function T must satisfy a sixth order polynomial equation presented in the Appendix B. Then the algebra takes the form { X, C }__PB= 16 X Y-32 K_1 H,{ Y, C }__PB= 8 [ 48 H^2X^2 -Y^2 -128 b X^3 -64 c_12 H^2 X +16(c_11^2+c_12^2-4 K_2)H^2 +192 b c_12 X^2 ]- 512 b (c_11^2+c_12^2-4 K_2) X - b λ , where λ is an arbitrary constant. It is a quartic polynomial algebra.For the extended Coulomb potentialV(r ,θ)= a/r + T'(θ)/r^2 ,with the particular solutions T_1 and T_2 we have that { X, C }__PB= 4 X Y,{ Y, C }__PB= 2 [ 32 H X^3 + 12 a^2 X^2 - Y^2 + σ_1,2 H],where σ_1=-16/9 α^3 and σ_2=16/3 α^3, respectively. At a=0 this algebra corresponds to the Case I, R(r)=0. Similarly, for a general solution of (<ref>) the function T must also satisfy a sixth order polynomial equation and the corresponding algebra reads { X, C }__PB= 4 X Y+ 8 a K_1,{ Y, C }__PB= 2 [ 32 HX^3 -Y^2 + 12 a^2 X^2 + 96c_32 H X^2 + 64(c_31^2+c_32^2+K_2/8)H X + 32 a^2 c_32 X ]- λ H + 32 a^2 (c_31^2+c_32^2+K_2/8),In the classical case the algebra of H, X, Y and C is useful to obtain and classify the trajectories. In full generality, namely for general solutions of (<ref>) and (<ref>), an algebraic equation for the non-trivial part T(θ) of the potential can be derived by requiring the algebra to be closed.In the quantum case, once the functions G_1,...,G_4 (<ref>) figuring in the integral Y (<ref>) are calculated, it is possible to express the two commutators [X, C] and [Y, C] as polynomials in X,Y and H. As a matter of fact, the condition that the algebra of the integrals of motion should close leads directly to the fifth order equations (<ref>) and (<ref>) for T. Moreover, this closure also provides the integrals of these equations such as e.g. eq. (<ref>). § CONCLUSIONSWe studied superintegrability in a two-dimensional Euclidean space. Classical and quantum fourth-order superintegrable potentials separating in polar coordinates were derived. We can summarize the main results via the following TheoremsTheorem 1. In quantum mechanics, the confining superintegrable systems correspond to V(r, θ)=a/r + ħ^2/r^2( W'(x_±) ∓2 cosθ/sin^2θ W(x_±) + 8(γ_2+γ_4+√(2 γ_1)-γ_1-γ_3)-3/16 sin^2θ),here x_±=sin^2(θ/2),cos^2(θ/2) andV(r, θ)=b r^2 +ħ^2/r^2( 4 W'(x_±) ∓8 cos 2θ/sin^2 2θ W(x_±) + 4(γ_2+γ_4+√(2 γ_1)-γ_1+γ_3)-3/2 sin^2 2θ),x_±=sin^2θ,cos^2θ where W(x) is given by (<ref>) in both cases. The leading term of the integral Y in (<ref>) is {L_z^3, p_y} and {L_z^2, p^2_x-p_y^2}, respectively.The non-confining potentials are given by (<ref>) with integral (<ref>), and (<ref>) with integral (<ref>).The functionW =W(x, P_6(x); γ_1, γ_2, γ_3, γ_4)is expressed in terms of the sixth Painlevé transcendent P_6 (<ref>) in full generality. In the case of a third order superintegrable system, not all four (γ_1, γ_2, γ_3, γ_4) but three constants in (<ref>) are arbitrary only. Moreover, the third order system does not allow any confining potentials. Theorem 2. In classical mechanics, the superintegrable confining systems correspond to V(r, θ)=a/r + T'(θ)/r^2 ,where T' satisfies (<ref>) and a is an arbitrary constant. The leading term of the integral Y in (<ref>) is {L_z^3, p_y}, andV(r, θ)=b r^2 + T'(θ)/r^2 ,here T' satisfies (<ref>), b is constant, and the leading term of Y is given by{L_z^3, p^2_x-p_y^2}.Particular solutions of (<ref>) and (<ref>) were presented in (<ref>) and (<ref>), respectively .The non-confining superintegrable systems are given by (<ref>) and (<ref>) with integral (<ref>), and (<ref>), (<ref>) with integral (<ref>), respectively.Work is currently in progress on a continuation of this article. We will add a general investigation of the polynomial algebra generated by the integrals of motion in the classical and quantum cases. We also plan to present figures of the classical trajectories and to use the algebra of integrals to calculate the energy spectrum and the wave functions in the quantum case. Another part of the project is to determine allcorresponding non-exotic potentials.§ ACKNOWLEDGMENTS The research of P. W. was partiallysupported by a research grant from NSERC of Canada. J.C.L.V. thanks PASPA grant (UNAM, Mexico) and the Centre de Recherches Mathématiques, Université de Montréal for the kind hospitality while on sabbatical leave during which this work was done.The research of A.M.E. was partially supported by a fellowship awarded by the Laboratory of Mathematical Physics of the CRM and by CONACyT grant 250881 (Mexico) for postdoctoral research. § FUNCTIONS F_I Explicitly, the functions F_1, … ,F_13 in (<ref>)-(<ref>) are given by F_1 = 2 ( B_1 cos 2 θ +B_2 sin2 θ +D_1 cos4 θ +D_2 sin4 θ) , F_2 = 1/r(B_2 cos 2 θ -B_1 sin 2 θ - 2 D_1 sin 4 θ + 2 D_2 cos 4 θ) + A_3 cos θ +A_4 sin θ + C_1 sin 3 θ +C_2 cos 3 θ , F_3=1/r^3(B_2 cos2θ -B_1 sin2θ + 2D_1 sin4θ - 2D_2 cos4θ)+ 1/r^2( A_4 sinθ + A_3 cosθ - 3 C_2cos3 θ - 3 C_1sin3 θ)- 2/r( B_3 sin 2 θ -B_4 cos 2 θ) + A_2 sinθ + A_1 cosθ , F_4= 2/r^4(D_1 cos 4 θ +D_2 sin 4 θ -B_1 cos 2 θ -B_2 sin 2 θ)+ 4/r^3( A_4 cosθ - A_3 sinθ - C_1 cos 3 θ + C_2 sin 3 θ)- 4/r^2( B_3 cos2 θ + B_4 sin2 θ) + 4/r( A_2 cosθ - A_1 sinθ) , F_5= - 6/r^2(D_1 cos4 θ +D_2 sin4 θ) + 2/r( A_4 cosθ - A_3 sinθ + 3( C_1cos3 θ - C_2sin3 θ) ) + 2 (B_3 cos2 θ + B_4 sin2 θ), F_6 = 3/r( B_2cos 2θ -B_1sin 2θ +2 D_2cos 4θ -2D_1sin 4θ)+3 (A_4sinθ + A_3cosθ + C_1sin 3θ + C_2cos 3θ) , F_7 = 12/r^2(D_1sin 4θ -D_2cos 4θ)+1/2r( A_4sinθ +A_3cosθ.. -30C_1sin 3θ -30C_2cos 3θ) -2 (B_3sin 2θ -B_4cos 2θ) ,F_8 = -3/r^3( B_1cos 2θ + B_2sin 2θ -5D_1cos 4θ -5D_2sin 4θ) + 3/2 r^2(A_4cosθ -A_3sinθ - 9 C_1cos 3θ + 9 C_2sin 3θ)-5/r( B_3cos 2θ + B_4sin 2θ) +3/2(A_2cosθ -A_1sinθ) , F_9 =9/r^4(B_1sin 2θ -B_2cos 2θ -2D_1sin 4θ +2D_2cos 4θ) +15/2 r^3( 3 C_1sin 3θ +3 C_2cos 3θ - A_3cosθ - A_4sinθ)+12/r^2(B_3sin 2θ -B_4cos 2θ)-9/2r(A_1cosθ+ A_2sinθ),F_10= -9/r (D_1 cos 4θ +D_2 sin 4θ) +3/2 (A_4 cosθ -A_3 sinθ + 3C_1 cos 3θ - 3C_2 sin 3θ), F_11=-3/r^2( B_1 cos 2θ + B_2sin 2θ - 5D_1cos 4θ - 5D_2sin 4θ) - 4 ( B_3cos 2θ + B_4sin 2θ) + 1/r(A_4cosθ- A_3sinθ - 9 C_1cos 3θ + 9 C_2sin 3θ) ,F_12= 2/r^3( B_1sin 2θ - B_2cos 2θ - 14D_1sin 4θ + 14 D_2cos 4θ)+3/2r^2(11 C_1 sin 3θ + 11 C_2 cos 3θ -A_3cosθ - A_4sinθ)+6/r( B_3sin 2θ - B_4cos 2θ) - 3/2( A_1cosθ + A_2sinθ) ,F_13=4/r^4(2B_1 cos 2θ +2B_2sin 2θ -11D_1cos 4θ -11D_2sin 4θ)-2/r^3( A_4 cosθ - A_3 sinθ - 17 C_1cos 3θ + 17 C_2sin 3θ)+12/r^2(B_3 cos 2θ +B_4 sin 2θ) - 3/r(A_2cosθ -A_1sinθ), For a non-confining potential R(r)=0, the functions F_i (<ref>)-(<ref>) reduce to F_1 =0 ,F_2 =0 ,F_3 =A_1cosθ + A_2sinθ +2/r( B_4cos 2θ - B_3sin 2θ), F_4 = 4/r(A_2cosθ - A_1sinθ) -4/r^2(B_3cos2θ+ B_4sin2θ) ,F_5 = 2( B_3cos2θ+ B_4sin2θ), F_6 = 0, F_7 = 2 (B_4cos 2θ- B_3sin 2θ) , F_8 = 3/2(A_2 cosθ- A_1sinθ) -5/ r(B_3 cos 2θ + B_4sin 2θ) , F_9 = 12/r^2( B_3sin 2θ- B_4cos 2θ)-9/2r(A_1cosθ + A_2sinθ), F_10= 0, F_11=-4 (B_3 cos 2θ + B_4sin 2θ) , F_12= 6/r(B_3sin 2θ - B_4cos 2θ) - 3/2( A_1cosθ + A_2 sinθ), F_13= 12/r^2(B_3 cos 2θ + B_4sin 2θ) -3/ r(A_2 cosθ -A_1sinθ),§ ALGEBRA OF INTEGRALS OF MOTION IN CLASSICAL LIMIT An algebraic equation for the non-trivial part T(θ) of the potential was derived by requiring the algebra generated by the integrals of motion H, X, Y and C to be closed. For the extended harmonic oscillator potential V(r ,θ)=b r^2 + T'(θ)/r^2 , the corresponding algebraic equation is a sixth order polynomial equation in T given byτ_0(z) + τ_1(z) T(z) + τ_2(z) T^2(z)+ τ_3(z) T^3(z)+ τ_4(z) T^4(z)+ τ_5(z) T^5(z)+ τ_6(z) T^6(z)=0 , where z=tan 2 θ and τ_0=786432(c_11^4 (f K_1-4 K_2)+2 c_11^2 (K_1-4 f K_2)^2+f K_1^3+48 f K_1 K_2^2-64 K_2^3-12 K_1^2 K_2) z^2+ 3072 c_11(-c_11^2 (256 c_12(16 K_2-3 f K_1)+λ)-(K_1-4 f K_2) (256 c_12(16 f K_2+5 K_1)+9 f λ)) z^3+ 3(512 [-128 c_12^2 (-128 f^2 K_2^2-80 f K_2 K_1+K_1^2)-6 c_11^2 (-256 c_12^2 (f K_1-8 K_2)+c_12λ -768 K_1 (K_1-4 f K_2)) +9 c_12 f λ(8 f K_2+K_1)-2048 K_2 (-12 f K_2 K_1+3 K_1^2+32 K_2^2)]+27 λ ^2) z^4-1536 c_11(-512 c_12^3 (f K_1-16 K_2)+2304 c_12 K_1 (8 f K_2+K_1)+6 c_12^2 λ +27 f λK_1) z^5 3(27 (65536 c_11^2 K_1^2+λ ^2)-1024 (c_12^3 λ +1024 c_12^4 K_2+16384 K_2^3)) z^6 , τ_1= 3145728 c_11(2 c_11^2 (f K_1-4 K_2)+c_11^4-8 f K_1 K_2+K_1^2+16 K_2^2) z^2 -6144 [c_11^2 (256 c_12(11 f K_1+16 K_2)+15 λ) +f K_1 (1024 c_12 K_2-9 λ) +4 K_2 (4096 c_12 K_2+9 λ)-1280 c_12 K_1^2-2048 c_12 c_11^4] z^3- 24576 c_11 [-256 c_12^2 (3 c_11^2-4 f K_1+4 K_2) -128 (c_11^2 (9 f K_1-16 K_2)+12 f K_1 K_2-6 K_1^2+64 K_2^2)+3 c_12λ ] z^4 + 3072 [4 c_11^2 (256 c_12(4 c_12^2+9 f K_1-32 K_2)-9 λ)-512 c_12^3 (f K_1-16 K_2) +256 c_12(72 f K_2 K_1+9 K_1^2-128 K_2^2)+6 c_12^2 λ +9 λ(3 f K_1-8 K_2)] z^5 -12288 c_11(9 c_12λ +4096 c_12^2 K_2-256 c_12^4+192 (9 K_1^2-64 K_2^2)) z^6 , τ_2=-3145728(2 c_11^2 (f K_1-4 K_2)+c_11^4-8 f K_1 K_2+K_1^2+16 K_2^2) z^2+ 294912c_11(128 c_12(-c_11^2+f K_1+4 K_2)+λ) z^3+ 24576 [3 c_12λ -128 (c_11^2 (22 c_12^2+33 f K_1+16 K_2)+8 c_12^2 (K_2-f K_1)-8 c_11^4+12 f K_1 K_2-6 K_1^2+64 K_2^2)] z^4 +12288 c_11 [256 c_12(16 c_11^2-12 c_12^2-27 f K_1+96 K_2) +27 λ ] z^5 +12288 [9 c_12λ +2048 c_12^2 (c_11^2+2 K_2)-192 (64 K_2 (c_11^2+K_2), -9 K_1^2)-256 c_12^4] z^6τ_3=-196608(128 c_12(-c_11^2+f K_1+4 K_2)+λ) z^3 + 50331648c_11(-c_11^2+2 c_12^2+3 f K_1+4 K_2) z^4-24576 (256 c_12(32 c_11^2-4 c_12^2-9 f K_1+32 K_2)+9 λ) z^5 + 50331648c_11(c_11^2-c_12^2+6 K_2) z^6τ_4=-25165824 z^4 (c_11^2 (6 z^2-1)-c_12^2 (z^2-2)-10 c_12 c_11 z+3 f K_1+2 K_2 (3 z^2+2)) ,τ_5= 50331648z^5 (3 c_11 z-2 c_12),τ_6= -50331648 z^6.(f=√(1+z^2)). For the special values K_1=c_12=0 and K_2=1/4c_11^ 2, the algebraic equation (<ref>) becomes 262144z^3 (c_11-2 T)^6-1024λ(9 z^2+8) (c_11-2 T)^3-27λ ^2 z (z^2+1)=0. solutions of which coincide with T_3,4 (<ref>), as it should be. For the extended Coulomb potential V(r ,θ)=a/r + T'(θ)/r^2 , the corresponding algebraic equation is also a sixth order polynomial equation in T given byυ_0(z) + υ_1(z) T(z) + υ_2(z) T^2(z)+ υ_3(z) T^3(z)+ υ_4(z) T^4(z)+ υ_5(z) T^5(z)+ υ_6(z) T^6(z)=0 , where z=tanθ and υ_0= 128 (64 c_31^4 (K_2-4 f K_1)+16 c_31^2 (f K_2-4 K_1)^2-64 f K_1^3-12 f K_1 K_2^2+K_2^3+48 K_1^2 K_2) z^2-64 c_31(8 c_31^2 (64 c_32(3 f K_1-K_2)+λ)+(4 K_1-f K_2) (64 c_32(f K_2+5 K_1)-9 f λ)) z^3+ (64 [32 c_32^2 (f^2 K_2^2+10 f K_2 K_1-2 K_1^2)-24 c_31^2 (32 c_32^2 (2 f K_1-K_2)+c_32λ +24 K_1 (f K_2-4 K_1)) -9 c_32 f λ(f K_2+2 K_1)+4 K_2 (-6 f K_2 K_1+24 K_1^2+K_2^2)]+27 λ ^2)z^4 -128 c_31(256 c_32^3 (f K_1-K_2)+288 c_32 K_1 (f K_2+2 K_1)+12 c_32^2 λ -27 f λK_1) z^5 + (27 (4096 c_31^2 K_1^2+λ ^2)+128 (-4 c_32^3 λ +64 c_32^4 K_2+K_2^3)) z^6, υ_1=-1024 fc_31(16 c_31^2 f (K_2-4 f K_1)+64 c_31^4 f+16 f K_1^2+f K_2^2-8 K_1 K_2) z^2 -32f[f^3 K_2 (9 λ -64 c_32 K_2)-4 K_1 (16 c_32 K_2+9 f^2 λ)+8 c_31^2 (64 c_32(f K_2+11 K_1)-15 f λ)+ 1280c_32fK_1^2 +8192c_32 c_31^4f] z^3+ 1024f c_31 [3 c_32 f λ +16 c_32^2 f (-24 c_31^2-16 f K_1+K_2) +96 f K_1 (3 c_31^2 f+K_1)-4 K_2 (8 c_31^2 f+K_1)-4 f K_2^2]z^4 +64 f[-12 c_32^2 f λ +8 c_31^2(9 f λ-32 c_32 (4 f (4 c_32^2+K_2) -7 K_1))+ 256 c_32^3 f (K_2-f K_1)-64 c_32 K_1 (9 f K_1+4 K_2)+27 f^2 λK_1 ]z^5+ 512f c_31 (f (9 c_32λ -128 c_32^4+216 K_1^2)-8 K_2 (8 c_32^2 f+3 K_1)-6 f K_2^2) z^6 18432 fc_32 K_1 (16 c_31^2-K_2) z^7, υ_2=256 f (16 c_31^2 f (4 f K_1-K_2)-64 c_31^4 f-16 f K_1^2-f K_2^2+8 K_1 K_2) z^2+ 3072 fc_31 (f λ -8 c_32(8 c_31^2 f+f K_2+4 K_1)) z^3 + 256 f[3 c_32 f λ +32 c_31^2 f (-44 c_32^2+33 f K_1+K_2)+16 c_32^2 f (K_2-16 f K_1)+512 c_31^4 f+96 f K_1^2 -4 f K_2^2-4 K_1 K_2] z^4 + 128 fc_31(64 c_32(32 c_31^2 f-24 c_32^2 f-6 f K_2+15 K_1)+27 f λ) z^5+ 128 f (9 c_32 f λ +64 c_32^2 f (16 c_31^2-K_2)-24 K_2 (K_1-8 c_31^2 f)-128 c_32^4 f+216 f K_1^2-6 f K_2^2) z^6221184 fc_31c_32K_1 z^7,υ_3=512 f (f λ -8 c_32(8 c_31^2 f+f K_2+4 K_1)) z^3 + 8192 c_31 f^2 (8 c_31^2-16 c_32^2+12 f K_1+K_2) z^4+64 f (64 c_32(-2 f (4 c_32^2+K_2)+64 c_31^2 f+5 K_1)+9 f λ) z^5-4096 c_31 f^2 (16 c_31^2-16 c_32^2-3 K_2)z^6+ 36864 fc_32K_1 z^7 ,υ_4= -512f^2z^4 (16 c_31^2 (6 z^2-1)-16 c_32^2 (z^2-2)-160 c_32 c_31 z-24 f K_1-K_2 (3 z^2+2)) ,υ_5= -4096z^5 (3 c_31 z-2 c_32) ,υ_6= -1024z^6 . For the special values K_1=c_32=0 and K_2=-8 c_31^ 2, the algebraic equation (<ref>) becomes 256 z^3 (-36 z^3 (2 c_31+T)^6-4 α ^3 (9 z^2+8) (2 c_31+T)^3+3 α ^6 (z^3+z))=0, in agreement with the particular solutions T_1,2 (<ref>).aipauth4-1 | http://arxiv.org/abs/1706.08655v3 | {
"authors": [
"Adrian M. Escobar-Ruiz",
"J. C. López Vieyra",
"P. Winternitz"
],
"categories": [
"math-ph",
"math.MP"
],
"primary_category": "math-ph",
"published": "20170627030437",
"title": "Fourth order superintegrable systems separating in Polar Coordinates. I. Exotic Potentials"
} |
Detecting Approximate Reflection Symmetry in a Point Set using Optimization on Manifold Rajendra Nagar1 and Shanmuganathan Raman2 Electrical Engineering, Indian Institute of Technology Gandhinagar, India, 382355 Email: [email protected], [email protected] December 30, 2023 ==================================================================================================================================================================================================== We propose an algorithm to detect approximate reflection symmetry present in a set of volumetrically distributed points belonging to ℝ^d containing a distorted reflection symmetry pattern. We pose the problem of detecting approximate reflection symmetry as the problem of establishing correspondences between the points which are reflections of each other and we determine the reflection symmetry transformation. We formulate an optimization framework in which the problem of establishing the correspondences amounts to solving a linear assignment problem and the problem of determining the reflection symmetry transformation amounts to solving an optimization problem on a smooth Riemannian product manifold. The proposed approach estimates the symmetry from the geometry of the points and is descriptor independent. We evaluate the performance of the proposed approach on the standard benchmark dataset and achieve the state-of-the-art performance. We further show the robustness of our approach by varying the amount of distortion in a perfect reflection symmetry pattern where we perturb each point by a different amount of perturbation. We demonstrate the effectiveness of the method by applying it to the problem of 2-D and 3-D reflection symmetry detection along with comparisons.Symmetry, Optimization, Manifolds. § INTRODUCTIONSymmetry present in natural and man-made objects enriches the objects to be physically balanced, beautiful, easy to recognize, and easy to understand. Characterizing and finding the symmetry has been an active topic of research in computer vision and computer graphics as physical objects form the basis for these research areas. The digitized objects are mainly represented in the form of meshes, volumes, sets of points, and images. The primary objective has been to detect symmetry in objects depicted through these different representations. We particularly aim to detect reflection symmetry present in objects represented by a set of finite number of points belonging to ℝ^d. In Fig. <ref>, we present an example result of the proposed approach for illustration. The motivation behind detecting symmetry in higher dimensional spaces (d>3) is inspired by the fact that many physical data points reside in the space of dimensions greater than three. For example, an RGB-D image captured using a Kinect sensor, which has become a major tool for interaction of human with machine, has four dimensions at each pixel location. Another example is the embedding of feature points or shapes into a higher-dimensional space. In the scale invariant feature transform (SIFT) algorithm, each keypoint is represented in a 128-dimensional space <cit.>. We not only target data residing in 2-D (image) and 3-D (point cloud), but also develop a genericframework to detect symmetry in higher dimensional data. The problem of establishing correspondences between reflection symmetry points and determining the hyperplane of reflection symmetry has been extensively studied due to its astounding applications such as compression of objects, symmetrization, shape matching, and symmetry aware segmentation of shapes <cit.>. Most of the existing algorithms attempt this problem by using surface signatures such as Gaussian curvature, eigenbases of the Laplace-Beltrami operator, and heat kernels for the points sampled on a given surface (<cit.>). The challenge we face is that, we only have a set of discrete points in ℝ^d. We can not take benefits from local surface signatures by fitting a surface over these points. For the case d=2, an explanation could be that the prominent surface signatures, such as Gaussian curvatures, are meaningful only if the surface is non-linear. For the case d≥ 3, an explanation could be that if the point set represents a volumetric shape, fitting a surface could be hard and eigenbases of Laplace-Beltrami operator are not defined for a set of finite points since it is not a compact manifold without the boundary <cit.>. Prominent methods such as <cit.> and <cit.> are independent of surface features and employ randomized algorithms to establish correspondences between the reflection symmetry points. However, they require fine tuning of a hyper-parameter to handle the reflection symmetry patterns perturbed by an unknown source of noise and an improper choice of this parameter could lead to higher time complexity. Both these categories of algorithms are sequential in the sense that they first establish the correspondences between the reflection symmetry points and then determine the reflection symmetry hyperplane. Therefore, many outlier correspondences could be detected along with the correct correspondences. In summary, detecting symmetry in a set containing a finite number of points is a non-trivial problem.In this work, we propose an optimization framework where we jointly establish correspondences between reflection symmetry points and determine the reflection symmetry hyperplane in a set of points containing a distorted reflection symmetry pattern. In order to design the cost function, we introduce an affine transformation to obtain the reflection point of a point in ℝ^d. The main intuition behind forming this cost function is that the reflection point of a point obtained through the optimal reflection hyperplane should be present closest to its ground-truth reflection point.The primary contributions of this work are listed below. * We propose an optimization based algorithm to establish correspondences between the reflection symmetry points and determine the reflection symmetry transformation in a set of discrete points residing in ℝ^d containing a distorted reflection symmetry pattern. * We show that the proposed optimization framework is convex in translation and correspondences matrix, and locally convex in each of the rotation matrices. * The proposed approach is shown to not use any shape descriptors and can be applied to point sets obtained by sampling any shape residing inℝ^d. * We demonstrate the effectiveness of the proposed approach by detecting symmetry in 2-D images and 3-D point clouds.We organize the remainder of the paper as follows. In <ref>, we present the related works to our approach. In <ref>, we formulate the energy minimization problem. In <ref>, we find the optimal rotations and translation. In <ref>, we find the optimal mirror symmetric correspondences. In <ref>, we prove the convergence properties. In <ref>, we report the computational complexity of our algorithm. In <ref>, we report the results and the evaluation of the proposed approach. In <ref>, we conclude the work with future directions. § RELATED WORKS The problem of characterizing and detecting the reflection symmetry in digitized objects has been extensively studied. The works <cit.> and <cit.> provide a survey of the symmetry detection algorithms. The symmetry detection algorithms can be categorized based on either the form of the input data or whether the algorithm is dependent or independent of the surface features. General forms of the input data are: set of points, mesh, volume, and image. Most of the methods for symmetry detection in meshes first extract salient keypoints on the surface and then describe each point using local surface features. The prominent surface features are: Gaussian curvatures, slippage features, moments, geodesic distances, and extended Gaussian images(<cit.>).Symmetry detection in a set of points without features. These algorithms detect reflection symmetry in a set of points without using surface features. Our work also falls in this category. In the work by Zabrodsky et al., the authors find the closest shape to a given shape represented by a set of points in ℝ^2and it requires point correspondences <cit.>. However, our goal is different in the sense that we find reflection correspondences within the given set of points in ℝ^d. In the work by Lipman et al., the authors propose the concept of symmetry factored embedding where they represent pairs of points which are in the same orbit in a new space and propose the concept of symmetry factored distance to find the distance between such pairs <cit.>. In the work by Xu et al., the authors detect multi-scale symmetry <cit.>. The authors use a randomized algorithm to detect the correspondences efficiently. However, performance degrades as the perfect pattern gets perturbed due to noisy measurements. We compare the correspondences established by our method to that of this method and show that our method performs better than this method when the patterns are perturbed. It is fair to compare with this method on the perturbed patterns because most of the real world patterns are not perfectly symmetric, e.g., human face and butterfly wings. In the works by Combès et al. <cit.>, Speciale et al. <cit.>, Ecins et al. <cit.>,Cicconet et al. <cit.>, Li et al. <cit.>, and Sipiran et al. <cit.>, the authors automatically detect the symmetry plane in a point cloud. But, the methods in <cit.>, <cit.>, <cit.>, <cit.>, and <cit.> do not establish correspondences. However, correspondences are an important aspect as shown in (<cit.>). Ecins et al. proposed an ICP based approach <cit.> where they used the normals at each point to determine the symmetry. However, this method is applicable only to non-volumetric point clouds, i.e., points sampled from a surface.Symmetry detection in meshes using surface features.These algorithms either directly use surface patches described using local features or first detect the salient keypoints on the surface and describe them using the local surface features. Here, we review only the salient works to give an idea of these algorithms. Mitra et al. detect partial and approximate symmetries in 3D models <cit.>. They start with sampling salient keypoints on the surface and pair them up using their local principal curvatures. Then using the Hough transformation, they find the pairs of reflection symmetry points. Then in the Hough transformation space, they perform the clustering of the pairs to determine all the partial symmetries.Martinet et al. detect symmetries by generalized moment functions where the shape symmetry gets inherited as symmetry in these functions <cit.>. Raviv et al. detect symmetry in non-rigid shapes by observing that the intrinsic geometry of a shape is invariant under non-rigid shape transformations <cit.>. Berner et al. start with constructing a graph based on the similarity of slippage features detected on the surface <cit.>. Then, they detect the structural regularities by matching the sub-graphs. Cohen et al. detect symmetry usinggeometric and image cues <cit.>. They use it to reconstruct accurate 3D models. We refer the reader to some of the pioneering works for more details on this category (<cit.>). There exist algorithms which find symmetry in meshes and volume without sampling keypoints. Theworks described in(<cit.>) belong to this category.Symmetry detection algorithm for real images. These algorithms primarily rely on the local image features such as edge orientations, curvatures, and gradients. The recent works such as (<cit.>) present excellent algorithms for reflection symmetry detection in images. Given the accurate detection of keypoints, the algorithm developed in this work can be used to detect reflection symmetry in images without using local features.Our algorithm is similar toIterative Closest Point (ICP) algorithm (<cit.>) only in the sense that we also follow the alternation between the optimization of reflection transformation (rotation and translation in ICP) and the correspondences between the mirror symmetric points (correspondences between the points of two different shapes in ICP). Our algorithm differs from the ICP algorithmsince ICP has a different error function in the transformation parameters than the error function of our problem. Furthermore, our matching is bijective since we impose the bijectivity constraints in our optimization framework. These constraints ensure that each point has exactly one mirror image point. § PROPOSED APPROACH Consider a set 𝒮={𝐱_i}_i=1^nof points, where 𝐱_i∈ℝ^d, containing a distorted reflection symmetry pattern. Our goal is to determine the reflection symmetry transformation and establish the correspondences between the reflection symmetry points. In Fig. <ref>, we show the graphical representation of our problem. We formulate an optimization framework in which both the correspondences between reflection symmetry points and the reflection symmetry transformation are variables as described below. We use the notation [k] for the set {1,2,…,k}, where k is a natural number.§.§ Problem Formulation We introduce reflection transformation in ℝ^d in order to obtain the reflection of a point through a hyperplane π, not necessarily passing through the origin. The intuition is based on the fact that any hyperplane in ℝ^d is a d-1 dimensional subspace. Therefore, it can be made parallel to the subspace spanned by any d-1 coordinate axes by translating the origin of the coordinate system on the hyperplane π and then rotating these d-1 axes sequentially (by the angle between the hyperplane π and the axis). In this new coordinate system, the reflection of a point through the hyperplane π can be obtained by multiplying the coordinate corresponding to the remaining axis of the point by -1. Then the reflection in the original coordinate system is obtained by applying the inverse procedure.The reflection point 𝐱_i^'∈ℝ^d of a point 𝐱_i∈ℝ^d through the reflection symmetry hyperplane π is determined by an affine transformation as shown in Equation <ref>.𝐱_i^' = (∏_u=1^d-1𝐑_u)𝐄(∏_u=1^d-1𝐑_u)^⊤(𝐱_i-𝐭)+𝐭. Here, i,i^'∈[n], 𝐭∈ℝ^d is the translation vector which translates the origin of the coordinate system on the hyperplane π, 𝐑_u is a rotation matrix of size d× d that rotates the u^th axis about the origin such that it becomes perpendicular to the normal of the hyperplane π, and the matrix 𝐄 is defined as 𝐄=[ 𝐈_d-1 0_d-1; 0_d-1^⊤-1 ] and satisfies 𝐄^⊤=𝐄, 𝐄^⊤𝐄=𝐈_d. The matrix 𝐑_u is an orthogonal matrix (𝐑_u^⊤𝐑_u=𝐑_u𝐑_u^⊤=𝐈_d) with determinant equal to +1, ∀ u∈{1,2,…,d-1}.Here, 0_d-1 is a vector of size (d-1)× 1 with all the coordinates equal to zero and 𝐈_d-1 is an identity matrix of size (d-1)×(d-1).Now, we introduce the essential properties of this transformation in order to formulate the problem. We show that the rotation matrices (𝐑_1,…,𝐑_d-1) and the translation vector 𝐭 uniquely determine the reflection hyper-plane π. We let 𝐓=∏_u=1^d-1𝐑_u throughout this paper and note that it is again an orthogonal matrix with determinant equal to +1.Theorem 1: The point 𝐱_i^' is the reflection of the point 𝐱_i through the hyperplane π if and only if the point 𝐱_i is the reflection of the point 𝐱_i^' through the hyperplane π.Proof: We prove the forward direction of the Theorem 1, since the backward direction can be proved in a similar way. Let us assume that the point 𝐱_i^' is the reflection of the point 𝐱_i. Therefore, Equation (<ref>) holds true. Now, we multiply Equation (<ref>) by 𝐓𝐄𝐓^⊤from left and use the identities 𝐄^⊤=𝐄, 𝐄𝐄=𝐈_d,𝐓^⊤𝐓=𝐓𝐓^⊤=𝐈_d to achieve,𝐓𝐄𝐓^⊤𝐱_i^'=𝐱_i-𝐭+𝐓𝐄𝐓^⊤𝐭 ⇒𝐱_i=𝐓𝐄𝐓^⊤(𝐱_i^'-𝐭)+𝐭. Theorem 2: The normal vector of the reflection hyper-plane π lies in the null space of the matrix 𝐈_d+𝐓𝐄𝐓^⊤, the hyper-plane π passes through 𝐭, and the null space of the matrix 𝐈_d+𝐓𝐄𝐓^⊤ is an one-dimensional subspace of ℝ^d.Proof: We subtract Equation (<ref>) from Equation (<ref>) to achieve𝐱_i-𝐱_i^'=𝐓𝐄𝐓^⊤(𝐱_i^'-𝐱_i)⇒(𝐈_d+𝐓𝐄𝐓^⊤)(𝐱_i-𝐱_i^')=0.Therefore, the normal to the reflection hyperplane π, which is in the direction of the vector (𝐱_i-𝐱_i^'), lies in the null space of the matrix (𝐈_d+𝐓𝐄𝐓^⊤). It is easy to show that the reflection hyperplane π passes through the translation 𝐭 by noting that the reflection point of the point 𝐭 is 𝐭. This is possible only if the point 𝐭 lies on the reflection hyperplane π. We prove that the null space of the matrix 𝐈_d+𝐓𝐄𝐓^⊤ is an one-dimensional subspace of ℝ^d in order to show that there exists an unique hyperplane π. The nullspace of a matrix is the space spanned by the eigenvectors corresponding to the zero eigenvalue. Let 𝐩=[ p_1 p_2 … p_d ]^⊤∈ℝ^d be any vector. If 𝐩 is an eigenvector corresponding to the zero eigenvalue of the matrix 𝐈_d+𝐓𝐄𝐓^⊤, then we must have𝐩^⊤(𝐈_d+𝐓𝐄𝐓^⊤)𝐩=0 ⇒𝐩^⊤𝐈_d𝐩+(𝐓^⊤𝐩)^⊤𝐄(𝐓^⊤𝐩)=0 ⇒𝐩^⊤𝐈_d𝐩+𝐛^⊤𝐄𝐛=0 ⇒∑_u=1^dp_u^2+∑_u=1^d-1b_u^2-b_d^2=0 ⇒∑_u=1^dp_u^2+∑_u=1^db_u^2-2b_d^2=0. Here, 𝐛=𝐓^⊤𝐩. We note that 𝐛_2^2=(𝐓^⊤𝐩)^⊤(𝐓^⊤𝐩)=𝐩^⊤𝐩=𝐩_2^2.Therefore, from Equation (<ref>) we have∑_u=1^dp_u^2+∑_u=1^dp_u^2-2b^2_d=0⇒∑_u=1^dp_u^2=b^2_d⇒∑_u=1^d-1b_u^2=0.Therefore, b_1=b_2=…=b_d-1=0 and b_d∈ℝ. Hence, the vector 𝐛 lies in the one dimensional space {𝐪:q_1=q_2=…=q_d-1=0, q_d∈ℝ}. Since 𝐛=𝐓^⊤𝐩⇒𝐩=𝐓𝐛. Since the rotation does not change the dimension of a linear space, the vector 𝐩 also lies in one dimensional space. Given the set 𝒮, our goal is to find all the correct reflection correspondences (i,i^')∈[n]×[n] and the matrices (𝐑_1,𝐑_2,…,𝐑_d-1,𝐭) which define thereflection symmetry hyperplane π. We represent all the correspondences by a permutation matrix 𝐏∈{0,1}^n× n, such that 𝐏_ii^'=1 if the point 𝐱_i^' is the reflection point of the point 𝐱_i and 𝐏_ii^'=0, otherwise. Here, we note from Theorem 1 that 𝐏_ii^'=1⇔𝐏_i^' i=1. Now, we let 𝐑=(𝐑_1,𝐑_2,…,𝐑_d-1)∈𝕍. Here, 𝕍=ℝ^d× d×ℝ^d× d×…×ℝ^d× d. Let 𝐗=[ 𝐱_1 𝐱_2 … 𝐱_n ]∈ℝ^d× n be the matrix containing all the points of the set 𝒮 as its columns. Since the i^th column of the matrix 𝐗𝐏 is the reflection point of the point 𝐱_i, the reflection transformation (𝐑,𝐭) maps the matrix 𝐗 to the reflected points matrix 𝐗𝐏. Using Equation <ref>, we write the reflected points in the form of the matrix 𝐓𝐄𝐓^⊤(𝐗-𝐭𝐞^⊤)+𝐭𝐞^⊤, where 𝐞=[ 1 1 … 1 ]^⊤ is a vector of size n×1. Therefore, Equation (<ref>) holds truewhen the input set contains a perfect reflection symmetry pattern.𝐓𝐄𝐓^⊤(𝐗-𝐭𝐞^⊤)+𝐭𝐞^⊤=𝐗𝐏.In practice, a reflection symmetry pattern might have been distorted. Therefore, we would be able to find only the approximate reflection symmetry. We find the reflection transformation (𝐑,𝐭) and the correspondences matrix 𝐏 in such a way that the symmetry error, which we define as 𝐓𝐄𝐓^⊤(𝐗-𝐭𝐞^⊤)+𝐭𝐞^⊤-𝐗𝐏_F^2, is minimized. Here ._F is the Frobenius norm operator. We frame this problem in an optimization framework as shown in Equation (<ref>). 𝐑,𝐭,𝐏min (∏_u=1^d-1𝐑_u)𝐄(∏_u=1^d-1𝐑_u)^⊤(𝐗-𝐭𝐞^⊤)+𝐭𝐞^⊤-𝐗𝐏_F^2 s.t. 𝐏𝐞=𝐞,𝐏^⊤𝐞=𝐞,𝐏∈{0,1}^n× n,𝐑_u^⊤𝐑_u=𝐈_d=𝐑_u𝐑_u^⊤,(𝐑_u)=1,𝐑_u∈ℝ^d× d,∀ u∈[d-1],𝐭∈ℝ^d.By imposing the constraints 𝐏𝐞=𝐞 and 𝐏^⊤𝐞=𝐞, we ensure that each point has only one reflection point. We adopt an alternating optimization approach to solve the problem defined in Equation (<ref>). We start with initializing the reflection transformation (𝐑,𝐭) and solve for the optimal correspondences 𝐏 and then for this optimal 𝐏, we solve for optimal the (𝐑,𝐭). We continue to alternate till convergence.Once 𝐏 is fixed, if we minimize the cost over the set 𝕍×ℝ^d, then we have to make sure that the orthogonality and the unit determinant constraints hold true for the matrices 𝐑_u, ∀ u∈[d-1]. One approach could be the Lagrange augmentation which requires us to handle 3d-3 additional Lagrange multipliers. However, we observe that the set ℳ={(𝐑_1,…,𝐑_d-1,𝐭):𝐑_u^⊤𝐑_u=𝐑_u𝐑_u^⊤=𝐈_d,(𝐑_u)=1,𝐑_u∈ℝ^d× d, ∀ u∈[d-1],𝐭∈ℝ^d} of constraints is a smooth Riemannian product manifold over which the optimization algorithms are well studied <cit.>. We solve the sub-optimization problem for optimal (𝐑,𝐭) on a manifold which we discuss in Section <ref>. We observe that the optimization of Equation (<ref>) for 𝐏 is a standard linear assignment problem for which we formulate an integer linear program which we discuss in Section <ref>. §.§ Optimizing reflection transformation (𝐑,𝐭) In this step, we fix the correspondences matrix 𝐏 and find the optimal reflection transformation (𝐑,𝐭) by taking advantages from the differential structure of the set ℳ. We shall now briefly introduce the differential geometry of the setℳ. Differential geometry of the set ℳ of constraints. In order to introduce the essential differential geometry of the set ℳ, we follow <cit.>. The elements of the set ℳ are of the form (𝐑,𝐭)≃(𝐑_1,…,𝐑_𝐝-1,𝐭). All the orthogonal matrices (each for rotation along a single axis) with determinant +1 form a Lie group, also known as special orthogonal group, which is a smooth Riemannian manifold. The Euclidean space ℝ^d is also a smooth Riemannian manifold. Therefore the set ℳ is a product manifold, 𝒮𝒪(2,d)×…×𝒮𝒪(2,d)×ℝ^d, the product of d-1 special orthogonal groups 𝒮𝒪(2,d) and an Euclidean space ℝ^d. Each rotation matrixperforms rotation about a single axis. Therefore, all the possible rotation matrices about a particular axis form a 𝒮𝒪(2) embedded in the Euclidean space ℝ^d× d. We denote this group as 𝒮𝒪(2,d).The tangent space 𝒯_(𝐑,𝐭)ℳ at the point (𝐑,𝐭)∈ℳ is {(𝐑Ω,𝐭):Ω_u^⊤=-Ω_u,Ω_u∈ℝ^d× d,∀ u∈[d-1],𝐭∈ℝ^d}.Here, 𝐑Ω=(𝐑_1Ω_1,…,𝐑_d-1Ω_d-1). The Riemannian metric ⟨ ., .⟩_(𝐑,𝐭) on the product manifold ℳ, which gives the intrinsic distance between two elements (𝐑Ω,η_𝐭) and (𝐑Ω^',η^'_𝐭) of the tangent space at the point (𝐑,𝐭) of the manifold ℳ, is defined in Equation (<ref>). ⟨(𝐑Ω,η_𝐭),(𝐑Ω^',η^'_𝐭)⟩_(𝐑,𝐭)=η^⊤_𝐭η^'_𝐭+∑_u=1^d-1trace(Ω_u^⊤Ω_u^').Let f̅:𝕍×ℝ^d→ℝ be a scalar function. Let the function f=f̅|_ℳ be the restriction of the function f̅ on the product manifold ℳ. Since the product manifold ℳ is a submanifold of the Riemannian manifold 𝕍×ℝ^d,the Riemannian gradient of the function f at the point (𝐑,𝐭) is obtained by projecting the Riemannian gradient of the function f̅ at the point (𝐑,𝐭)∈𝕍×ℝ^d on the tangent space at the point (𝐑,𝐭)∈ℳ. Therefore, the Riemannian gradient of the function fat the point (𝐑,𝐭) is defined in Equation (<ref>).grad f(𝐑,𝐭)=(ℙ_𝐑(∇_𝐑f̅),ℙ_𝐭(∇_𝐭f̅))∈𝒯_(𝐑,𝐭)ℳ. Since the tangent space at a point in an Euclidean space is again an Euclidean space, the second component is given by ℙ_𝐭(∇_𝐭f̅)=∇_𝐭f̅. The first component is defined asℙ_𝐑(∇_𝐑f̅)=(ℙ_𝐑_1(∇_𝐑_1f̅),…,ℙ_𝐑_d-1(∇_𝐑_d-1f̅)).Here,ℙ_𝐑_j(∇_𝐑_jf̅)=𝐑_jskew(𝐑_j^⊤∇_𝐑_jf̅),where skew(𝐀)=0.5(𝐀-𝐀^⊤). We define ξ_𝐑_j(𝐑_j)=ℙ_𝐑_j(∇_𝐑_jf̅). The Riemannian Hessian of the function f at a point (𝐑,𝐭) is a linear map, Hess f:𝒯_(𝐑,𝐭)ℳ→𝒯_(𝐑,𝐭)ℳ and is defined as shown in Equation (<ref>).Hess f(𝐑,𝐭)[η_𝐑,η_𝐭]=(ℙ_𝐑(Dξ_𝐑(𝐑)[η_𝐑]),ℙ_𝐭(Dξ_𝐭(𝐭)[η_𝐭])).Here, the first component ℙ_𝐑(Dξ_𝐑(𝐑)[η_𝐑]) is equal to(ℙ_𝐑_1(Dξ_𝐑_1(𝐑_1)[η_𝐑_1]),…, ℙ_𝐑_d-1(Dξ_𝐑_d-1(𝐑_d-1)[η_𝐑_d-1])),where η_𝐑_j=𝐑_jΩ_j. The termDξ_𝐱(𝐱)[η_𝐱]=lim_t→ 0ξ(𝐱+tη_𝐱)-ξ(𝐱)/tis the classical derivative of the vector field ξ(𝐱) in the direction η_𝐱.The Riemannian trust region method. Our goal is to minimize the function f(𝐑,𝐭) over the product manifold ℳ. There exists a generalization of the popular optimization methods on the Riemannian manifolds. Since our problem is locally convex in each variable 𝐑_j, which we prove in Theorem 6, we employ the Riemannian trust region approach <cit.>. It requires the Riemannian gradient and the Riemannian Hessian operator for the function f, which we find as follows. Let f̅ be a function from the set 𝕍×ℝ^d to ℝ and defined as f̅(𝐑,𝐭)=𝐓𝐄𝐓^⊤(𝐗-𝐭𝐞^⊤)+𝐭𝐞^⊤-𝐗𝐏_F^2. Its classical gradients with respect to both the variables are given in the Equations (<ref>) and (<ref>). The detailed derivation is given in the Appendices A1 and A2.∇_𝐭f̅=2(𝐈_d-𝐓𝐄𝐓^⊤)(2𝐞^⊤𝐞𝐭-𝐗𝐞-𝐗𝐏𝐞). ∇_𝐑_jf̅=-2(∏_u=1^j-1𝐑_u)^⊤𝐀(∏_u=1^d-1𝐑_u)𝐄(∏_u=j+1^d-1𝐑_u)^⊤.Here,𝐀=(𝐗𝐏-𝐭𝐞^⊤)(𝐗-𝐭𝐞^⊤)^⊤+(𝐗-𝐭𝐞^⊤)(𝐗𝐏-𝐭𝐞^⊤)^⊤which satisfies 𝐀^⊤=𝐀. Now let the function f=f̅|_ℳ be the restriction of the function f̅ on the set ℳ.We obtain the Riemannian gradient of the function f at a point (𝐑,𝐭) by projecting the Riemannian gradient of the function f̅ over the tangent space 𝒯_(𝐑,𝐭) at the point (𝐑,𝐭). Since the manifold 𝕍×ℝ^d is an Euclidean space, the Riemannian gradient of the function f̅ is equal to its classical gradient. Therefore, we apply the definition given in Equation (<ref>) in order to find the Riemannian gradient grad f(𝐑,𝐭) of the function f which we denote as (ξ_𝐑_1(𝐑_1),…,ξ_𝐑_d-1(𝐑_d-1),ξ_𝐭) and define in Equations (<ref>) and (<ref>). The detailed derivation is given in the Appendices A3 and A4.ξ_𝐭(𝐭)=2(𝐈_d-𝐓𝐄𝐓^⊤)(2𝐞^⊤𝐞𝐭-𝐗𝐞-𝐗𝐏𝐞), ξ_𝐑_j(𝐑_j)=-𝐑_j(∏_u=1^j𝐑_u)^⊤𝐀(∏_u=1^d-1𝐑_u)𝐄(∏_u=j+1^d-1𝐑_u)^⊤ +𝐑_j(∏_u=j+1^d-1𝐑_u)𝐄(∏_u=1^d-1𝐑_u)^⊤𝐀^⊤(∏_u=1^j𝐑_u). We determine the Riemannian Hessian of the function f using the definition given in Equation (<ref>). In order to determine the j^th component Hess_𝐑_j(f(𝐑,𝐭))[𝐑_jΩ_j] of the Riemannian Hessian, which is equal to ℙ_𝐑_j(Dξ_𝐑_j(𝐑_j)[𝐑_jΩ_j]), we first find the classical derivative Dξ_𝐑_j(𝐑_j)[𝐑_jΩ_j] of the Riemannian gradient ξ_𝐑_j(𝐑_j) in the direction 𝐑_jΩ_j and then apply the projection operator ℙ_𝐑_j. Therefore, the j^th component Hess_𝐑_j(f(𝐑,𝐭))[𝐑_jΩ_j] of the Riemannian Hessian is equal to1/2𝐑_j([𝐁_1,[𝐑_j^⊤𝐁_2𝐑_j, Ω_j]]+[[Ω_j,𝐁_1],𝐑_j^⊤𝐁_2𝐑_j]). The detailed derivation is given in the Appendix A5. Here [.,.] is the Lie bracket and defined as [𝐔,𝐕]=𝐔𝐕-𝐕𝐔 for any two matrices 𝐔 and 𝐕,𝐁_1=(∏_u=j+1^d-1𝐑_u)𝐄(∏_u=j+1^d-1𝐑_u)^⊤,and 𝐁_2=(∏_u=1^j-1𝐑_u)^⊤𝐀(∏_u=1^j-1𝐑_u).In a similar way, we determine the component, ℙ_𝐭(Dξ_𝐭(𝐭)[η_𝐭]), of the Riemannian Hessian which is shown in Equation (<ref>).Hess_𝐭(f(𝐑,𝐭))[η_𝐭]=4n(𝐈_d-𝐓𝐄𝐓^⊤)η_𝐭. The detailed derivation is given in the Appendix A6. Now, we apply the Riemannian-trust-region method using the Riemannian gradient and Hessian defined in Equations (<ref>), (<ref>), (<ref>), and (<ref>) in order to obtain the optimal solution.We use the manopt toolbox in order to implement the optimization problem given in Equation (<ref>) for a fixed 𝐏 <cit.>.Determining the reflection symmetry hyperplane π. In order to determine the reflection hyperplane π, we use Theorem 2 which states that the normal vector of π lies in the null space of the matrix 𝐈_d+(∏_u=1^d-1𝐑_u)𝐄(∏_u=1^d-1𝐑_u)^⊤ and the optimal translation 𝐭 lies on the hyperplane. §.§ Optimizing Correspondences 𝐏 After obtaining the current estimate of the reflection transformation (𝐑,𝐭), we improve the correspondences matrix 𝐏 by solving the problem given in Equation (<ref>) while fixing (𝐑,𝐭). We show that this sub-problem is equivalent to a linear assignment problem, where an assignment is a pair (i,i^') of reflection symmetry points.Claim 1: The optimization problem given in Equation (<ref>) is a linear assignment problem in 𝐏, for a fixed(𝐑,𝐭).Proof: Let us consider the cost function in Equation (<ref>) and let 𝐗_m=𝐓𝐄𝐓^⊤(𝐗-𝐭𝐞^⊤)+𝐭𝐞^⊤. We have 𝐗_m-𝐗𝐏_F^2=trace((𝐗_m-𝐗𝐏)^⊤(𝐗_m-𝐗𝐏))=trace(𝐗_m^⊤𝐗_m-2𝐗_m^⊤𝐗𝐏+𝐗^⊤𝐗𝐏𝐏^⊤).Since, the first and the third terms (using the fact that the permutation matrices are orthogonal) are not the functions of 𝐏,the problem of finding the point of minimum of the function 𝐗_m-𝐗𝐏_F^2 is identical to the problem of finding the point of maximum of the function trace(𝐗_m^⊤𝐗𝐏). Using the identity trace(𝐀^⊤𝐁)=𝚟𝚎𝚌(𝐀)^⊤𝚟𝚎𝚌(𝐁),we have that trace(𝐗_m^⊤𝐗𝐏)=𝚟𝚎𝚌(𝐗^⊤𝐗_m)^⊤𝚟𝚎𝚌(𝐏), where the operator 𝚟𝚎𝚌 vectorizes a matrix by stacking all the columns successively in a column vector.Therefore, for a fixed reflection transformation, the problem defined in Equation (<ref>) is equivalent to the problem defined in Equation (<ref>).𝐏∈{0,1}^n× nmax trace(𝐗_m^⊤𝐗𝐏)=𝚟𝚎𝚌(𝐗_m^⊤𝐗)^⊤𝚟𝚎𝚌(𝐏) subject to 𝐏𝐞≤𝐞,𝐏^⊤𝐞≤𝐞,which is a standard linear assignment problem.□Claim 2: The problem defined in Equation (<ref>) is an integer linear program.Proof: Let 𝐯_1 be a vector of size n^2× 1 with the first n coordinates equal to one and the last n(n-1) coordinates equal to zero. Let 𝐞_1be a vector of size n× 1 with all the coordinates equal to zero except the first coordinate which is equal to one. Let 𝐯_2=[ 𝐞_1^⊤ 𝐞_1^⊤ … 𝐞_1^⊤ ]^⊤ be a vector of size n^2× 1. Now let us construct the matrices 𝐀_1 and 𝐀_2, each of size n× n^2, such that the i^th row of the matrix 𝐀_1 is equal to the row vector 𝚌𝚜(𝐯_1^⊤,n(i-1)) and the i^th row of the matrix 𝐀_2 is equal to the row vector 𝚌𝚜(𝐯_2^⊤,i-1). Here 𝚌𝚜(𝐯^⊤,i) is a row vector obtained by circularly shifting any row vector 𝐯^⊤ right by i coordinates.Now, it is trivial to verify that the constraint 𝐏^⊤𝐞≤𝐞 is equivalent to 𝐀_1𝚟𝚎𝚌(𝐏)≤𝐞 and the constraint 𝐏𝐞≤𝐞 is equivalent to 𝐀_2𝚟𝚎𝚌(𝐏)≤𝐞. Therefore, the problem defined in Equation (<ref>) is equivalent to the problem defined in Equation (<ref>).𝐚∈{0,1}^n^2× 1max 𝚟𝚎𝚌(𝐗_m^⊤𝐗)^⊤𝐚 subject to [ 𝐀_1^⊤ 𝐀_2^⊤ ]^⊤𝐚≤[ 𝐞^⊤ 𝐞^⊤ ]^⊤which is an integer linear program with 𝐚=𝚟𝚎𝚌(𝐏).□Solving the ILP. SinceILP is an NP-complete problem, there may not exist a polynomial time algorithm to find the optimal solution. We relax this ILP to a linear program by converting the constraint 𝐚∈{0,1}^n^2× 1 into 𝐚∈[0,1]^n^2× 1. Now, the above ILP becomes a linear program. We first solve this LP using the Karmarkar's algorithm in <cit.> which takes O(n^3.5) time. The solution 𝐚^⋆=[ a^⋆_1 a^⋆_2 … a^⋆_n^2 ]^⊤ of this LP belongs to [0,1]^n^2× 1 which is a continuous solution. However, our final solution 𝐚^f=[ a^f_1 a^f_2 … a^f_n^2 ]^⊤ of the proposed ILP should be a discrete solution. We follow the rounding approach, as explained in(<cit.>, ch. 11). The i-th element a^f_i of the final solution is equal to 1, if a^⋆_i≥0.5 and equal to 0, if a^⋆_i<0.5. This solution 𝐚^f may not be the optimal solution because according to <cit.>, 𝚟𝚎𝚌(𝐗_m^⊤𝐗)^⊤𝐚^f≥1/2×𝚟𝚎𝚌(𝐗_m^⊤𝐗)^⊤𝐚^OPT. Here, 𝐚^OPT is the optimal solution of the above ILP.§.§ Convergence Analysis We derive the essential results in order to prove that the alternating optimization framework converges.Theorem 3:The cost function f(𝐑,𝐭,𝐏) is convex in the variable 𝐭. Proof: In order to prove this, we prove that the Riemannian Hessian of the function f with respect to the variable 𝐭 is a positive semi-definite (PSD) matrix. Let η_𝐭=[ η_1 η_2 … η_d ]^⊤∈ℝ^d. Then using the definition of Riemannian metric, we have⟨η_𝐭,Hess_𝐭(f)[η_𝐭]⟩_𝐭=η_𝐭^⊤Hess_𝐭(f)[η_𝐭].Now, using the Riemannian Hessian Hess_𝐭(f)[η_𝐭] defined in Equation (<ref>), we have thatη_𝐭^⊤Hess_𝐭(f)[η_𝐭]=η_𝐭^⊤η_𝐭-(𝐓^⊤η_𝐭)^⊤𝐄(𝐓^⊤η_𝐭)Now let𝐪=𝐓^⊤η_𝐭. Then, we obtainη_𝐭^⊤Hess_𝐭(f)[η_𝐭]=η_𝐭^⊤η_𝐭-𝐪^⊤𝐄𝐪 =η_𝐭_2^2-∑_u=1^d-1q_u+q^2_d=η_𝐭_2^2-𝐪_2^2+2q_d^2. Now, we know that 𝐓𝐓^⊤=𝐈. Hence, we have𝐪_2^2=𝐪^⊤𝐪=η_𝐭^⊤𝐓𝐓^⊤η_𝐭=η_𝐭^⊤η_𝐭=η_𝐭_2^2.Therefore,η_𝐭_2^2-𝐪_2^2=0 ⇒η_𝐭^⊤Hess_𝐭(f)[η_𝐭]=2q_d^2≥ 0.□Theorem 4:At the critical point, the matrix 𝐓^⋆=∏_u=1^d𝐑_u^⋆ contains the eigenvectors of the matrix 𝐀 as columns.Proof: At the critical point, the Riemannian gradient given in Equation (<ref>) vanishes. Therefore, ξ_𝐑_j(𝐑_j)=0_d× d. Now pre-multiplying it with (∏_u=1^j𝐑_u)𝐑_j^⊤ and then post-multiplying with (∏_u=j+1^d-1𝐑_u), we achieve𝐀𝐓^⋆𝐄=𝐓^⋆𝐄(𝐓^⋆)^⊤𝐀𝐓^⋆⇒(𝐓^⋆)^⊤𝐀𝐓^⋆𝐄=𝐄(𝐓^⋆)^⊤𝐀𝐓^⋆.Now, let 𝐐=[ 𝐐_1 𝐪_2; 𝐪^⊤_3 q_4 ]=(𝐓^⋆)^⊤𝐀𝐓^⋆ be a matrix. Then, we have 𝐐𝐄=𝐄𝐐. Therefore,[ 𝐐_1 𝐪_2; 𝐪^⊤_3 q_4 ][ 𝐈_d-1 0_d-1; 0^⊤_d-1-1 ]=[ 𝐈_d-1 0_d-1; 0^⊤_d-1-1 ][ 𝐐_1 𝐪_2; 𝐪^⊤_3 q_4 ] ⇒𝐪_2=0_d-1,𝐪_3=0_d-1,𝐐_1𝐈_d-1=𝐈_d-1𝐐_1.Since, 𝐈_d-1 is a diagonal matrix and the equality 𝐐_1𝐈_d-1=𝐈_d-1𝐐_1 holds true, it is easy to prove that 𝐐_1 is a diagonal matrix.Therefore, the matrix 𝐐 is also diagonal. The spectral theorem states that every real symmetric matrix has eigenvalue decomposition with real eigenvalues and orthogonal eigenvectors. Here, we have observed that the matrix 𝐀 is a real symmetric matrix and satisfies 𝐐=(𝐓^⋆)^⊤𝐀𝐓^⋆, where the matrix 𝐐 is a diagonal matrix and the matrix 𝐓^⋆ is an orthogonal matrix. Therefore, the matrix 𝐓^⋆ is the matrix containing the eigenvectors of the matrix 𝐀. In Theorem 5, we prove that the order of stacking eigenvectors of 𝐀 as columns of 𝐓^⋆ affects the convexity of the problem. □ Theorem 5: The cost function f(𝐑,𝐭,𝐏) is locally convex in each rotation matrix 𝐑_j. Proof: In order to show the local convexity in𝐑_j, we have to show that the value ⟨𝐑_jΩ_j, 𝐇[𝐑_jΩ_j]⟩_𝐑_j≥0 in the neighborhood of the optimal angle θ_j^⋆. Here, 𝐇[𝐑_jΩ_j]=Hess_𝐑_j(f(𝐑,𝐭))[𝐑_jΩ_j]. By using the Riemannian metric defined in Equation (<ref>), we have ⟨𝐑_jΩ_j, 𝐇[𝐑_jΩ_j]⟩_𝐑_j=trace(Ω_j^⊤𝐑_j^⊤𝐇[𝐑_jΩ_j]). By using Equation (<ref>), the matrix𝐑_j^⊤𝐇[𝐑_jΩ_j] is equal to0.5[𝐁_1,[𝐑_j^⊤𝐁_2𝐑_j, Ω_j]]+0.5[[Ω_j,𝐁_1],𝐑_j^⊤𝐁_2𝐑_j].In the Appendix A7, we show that the trace(Ω_j^⊤𝐑_j^⊤𝐇[𝐑_jΩ_j]) is equal to4×trace(𝐑_j^⊤𝐁_2𝐑_j(Ω_j𝐁_1Ω_j-Ω_jΩ_j𝐁_1)).We visualize this term for d=2. For d=2, the matrix Ω=[0 -θ;θ0 ],𝐄=[10;0 -1 ],and let 𝐀=[ a_1 a_2; a_2 a_3 ] and 𝐑=[cosθ -sinθ;sinθcosθ ]. We have that⟨𝐑Ω, 𝐇[𝐑Ω]⟩_𝐑=8a_2θ^2sin(2θ)+4θ^2cos(2θ)(a_1-a_3).In Fig. <ref>, we plot the value ⟨𝐑Ω, 𝐇[𝐑Ω]⟩_𝐑/θ^2 against the initialization angle θ for six reflection symmetry patterns having different orientations for symmetry axis. We observe that the PSD values are positive in the proximity of the optimal angles. Therefore, it is locally convex. We further observe that this quantity is maximum at the optimal angle. We also observe that, if θ is the symmetry axis orientation, then the PSD value becomes positive in the proximity of θ andθ+180^∘. The reason for the second range is that, if θ is the slope of a line, then θ+180^∘ is also the slope of the same line.In Theorem 4, we claimed that the order in which the eigenvectors are stackedas columns of the matrix 𝐑 affects the local convexity. We prove it as follows. At the critical point, we have that 𝐑^⊤𝐀𝐑=diag(d_1,d_2). We note that Ω_j𝐁_1Ω_j-Ω_jΩ_j𝐁_1=𝐄 for d=2. Now from Equation (<ref>), we achieve⟨𝐑Ω, 𝐇[𝐑Ω]⟩_𝐑=d_1-d_2⇒ d_1≥ d_2.Therefore, the first column of the matrix 𝐑^⋆ should be the eigenvector corresponding to the maximum eigenvalue and the second column of the matrix 𝐑^⋆ should be the eigenvector corresponding to the minimum eigenvalue of the matrix 𝐀.□ Theorem 6:The proposed alternating framework converges to the global minimum if the initialization of the rotation matrices 𝐑_1, …, 𝐑_d-1 are within the proximity of the optimal rotation matrices and initialization of the translation 𝐭 is any random vector.Proof: We observe that the proposed alternation framework is basically the block coordinate descent (BCD) method, where (𝐑_1,…,𝐑_d-1,𝐭) and 𝐏 are two blocks of coordinates. According to <cit.>, the BCD method converges if the cost function is convex in each block of coordinates. Here, we have seen that the cost function is convex in the coordinates 𝐭 (Theorem 3), convex in the coordinates 𝐏 on the relaxed domain [0,1]^n× n, and locally convex in the coordinates (𝐑_1,…,𝐑_d-1) (Theorem5). This implies that if the initialization of (𝐑_1,…,𝐑_d-1) is within the proximity of the optimal solution, then the alternating framework converges to the global minimum. We experimentally show this theorem for the case d=2. We use the dataset for d=2 with σ=0 as mentioned in <ref>. In Fig. <ref>, we plot the error (averaged over all optimal angles) at the convergence point against the initialization angles for the case d=2 (we shift the error vectors for different optimal angles so that the optimal angle is always 90^∘). We observe that the variance becomes zero for initialization angle θ_0∈(90^∘-12^∘,90^∘+9^∘) and θ_0∈(270^∘-12^∘,270^∘+9^∘). The reason for the second range is that, if θ is the slope of a line, then θ+180^∘ is also the slope of the same line.□ In summary, in order to obtain the optimal (𝐑^⋆,𝐭^⋆,𝐏^⋆), we follow Algorithm <ref>. Initialization Strategy: In the Theorem 5, we have shown that f(𝐑,𝐭,𝐏) is locally convex in rotation matrix 𝐑. Therefore, Algorithm 1 converges to the global minimum if we initialize the rotation matrix in the proximity of the global solution. Hence, we approximate the initial 𝐑 by finding a small set of candidate pairs of mirror symmetric points. We discuss the proposed approach for finding a small set of candidate pairs of mirror symmetric points as follows.Let us consider the input set 𝒮={𝐱_i}_i=1^n. We propose a randomized approach to find a small set of candidate pairs of mirror symmetric points. We select two points, 𝐱_p and 𝐱_q, uniformly at random from the set 𝒮. Let 𝐱_p^' and 𝐱_q^' be their actual mirror images, respectively. We then construct two sets, 𝒫={(𝐱_p,𝐱_i)}_i=1,i≠ p,q^n and 𝒬={(𝐱_q,𝐱_i)}_i=1,i≠ q,p^n of pairs of points. Given the sets 𝒫 and 𝒬, our goal is to find the pairs (𝐱_p,𝐱_p^') and (𝐱_q,𝐱_q^'). It is trivial to observe that (𝐱_p,𝐱_p^')∈𝒫 and (𝐱_q,𝐱_q^')∈𝒬. We note that each pair of points define its own symmetry plane, the one which is perpendicular to the line segment joining the two points and passing through the mid-point of this line segment. Now, if the pairs (𝐱_p,𝐱_p^') and (𝐱_q,𝐱_q^') are true pairs then both the reflection planes, defined by these two pairs, should be the same. For each pair (𝐱_p,𝐱_i)∈𝒫, we keep sampling a pair (𝐱_q,𝐱_j)∈𝒬 uniformly at random without replacement until the reflection planes defined by these two pairs are the same. We determine whether the two reflection planes, defined by these two pairs, π_pi:η_pi^⊤𝐱-c_pi=0 and π_qj:η_qj^⊤𝐱-c_qj=0 are the same if the conditions, cos^-1(η_pi^⊤η_qj)≤ϵ_θ and min{d_q,d_j}/max{d_q,d_j}≥1-ϵ_d are true.Here, η_pi=𝐱_p-𝐱_i/𝐱_p-𝐱_i_2 is the normal vector to the plane π_pi, c_pi=η_pi^⊤(𝐱_p+𝐱_i/2) is thedistance of the origin from the plane π_pi, η_qj=𝐱_q-𝐱_j/𝐱_q-𝐱_j_2 is the normal vector to the plane π_qj, c_qj=η_qj^⊤(𝐱_q+𝐱_j/2) is the distance of the origin from the plane π_qj, d_q=|η_pi^⊤𝐱_q-c_pi|, and d_j=|η_pi^⊤𝐱_j-c_pi|.We repeat the above procedure ten times. With this, we get a set of 20 (2 for each run) candidate pairs of mirror symmetric points. Since we consider the case where only a single symmetric object is present in the input set, we consider the median plane of the 20 planes defined by the above computed 20 candidate pairs. Now, we use the normal η to this median plane for initialization.We also initialize the initial translation vector 𝐭 as the median of the mid-points of the line segment joining the points of the candidate pairs of the mirror symmetric points.First, we subtract each data point of the point cloud from the estimated 𝐭 of the point cloud. This ensures that the reflection symmetry plane passes through the origin. Now, we know the unit normal to the reflection symmetry plane. Therefore, we use the Householder transform to reflect each point which is 𝐱_i^'=(𝐈-2ηη^⊤)𝐱_i. Therefore, we have the matrix 𝐗 containing the original point cloud and the matrix 𝐗_m containing the reflected point cloud about the estimated reflection symmetry plane. Now, using 𝐗 and 𝐗_m, we solve the linear assignment problem, defined in Equation (<ref>) to find the matrix 𝐏. Now, we use these approximate correspondences to estimate the reflection symmetry plane as step 4 of Algorithm 1.§ TIME COMPLEXITY There are two main steps involved in our algorithm. The first one is to solve for reflection symmetry transformation matrices 𝐑_1,𝐑_2,…,𝐑_d-1,𝐭using the Riemannian trust region <cit.>. The second step is to find the pairs of reflective symmetric points using an integer linear program. The time complexity of Riemannian trust region method is O(nd^2). Since solving integer linear program is an NP-complete problem, we first relax it to a linear program (as discussed at the end of <ref>).The time complexity of solving a linear program is polynomial in the number of points in the point cloud. We use the Karmarkar's algorithm in <cit.> which has the time complexity of O(n^3.5). Therefore, the overall complexity of our approach is polynomial in the number of points in the point cloud which is equal to O(nd^2)+O(n^3.5)≈ O(n^3.5), since d<<n. It takes around 38.4 seconds (d=3) to find the symmetry plane and all the pairs of mirror symmetric points in a point cloud with 500 points using MATLAB on a Linux machine with i7-7500U CPU @ 2.70GHz, and 16GB RAM. § EVALUATION AND RESULTS§.§ Evaluation of Reflection Symmetry PlaneIn order to evaluate the performance of reflection symmetry plane detection, we compare the performance of our approach with the performance of the methods in <cit.>, <cit.>, and <cit.>. We compare the detected plane of reflection symmetry to that of these methods on the dataset in <cit.> with F-score as the evaluation metric proposed in <cit.>.The dataset given in <cit.> contains models of 1354 3D real world objects in which the ground-truth plane of reflection symmetry is provided for all the objects.Speciale et al. proposed a Hough transform voting based approach <cit.>. Ecins et al. proposed an ICP based approach <cit.>. First, they initialize the reflection symmetry plane and then iteratively update the reflection symmetry plane using the Levenberg-Marquardt solver till convergence. Theyhave further used the normals at each point to reject outliers points. Therefore, they need oriented point clouds, i.e., normal at each point be given.Cicconet et al. first reflected the original point cloud about an arbitrary reflection plane and then used the ICP algorithm to align the original point cloud and the reflected point cloud <cit.>. Then, they determine the reflection symmetry plane. In order to evaluate the accuracy of detecting reflection symmetry plane for each method, we find the precision and recall ratesand the F-score. According to <cit.>, the precision and the recall rates are defined as P=TP/TP+FP, R=TP/TP+FN, respectively. The F-Score is defined as F=2RP/R+P. According to <cit.>, TP is equal to the number of correctlyestimated reflection symmetry planes, FP isequal to the number of incorrectly estimated reflection symmetry planes, and FN is equal to the number ofground-truth reflection symmetry planes which are not detected. According to <cit.>, a detected plane of reflective symmetry is declared to be correct or incorrect as follows. Let 𝐱_1^e, 𝐱_2^e, and 𝐱_3^e be three points on the detected plane of reflection symmetry. Let𝐱_1^g, 𝐱_2^g, and 𝐱_3^gbe three points onthe ground truth plane of reflection symmetry of the underlying symmetric object. These three points on the plane of reflection symmetry planes are any three points from the four points of intersection of the plane of reflection symmetry with the bounding box of the underlying reflective symmetric object. Now, according to <cit.>, the detected plane of reflection symmetry is declared correct if the angle between the normal of the detected plane of reflection symmetry, which is defined as η_e=(𝐱_1^e-𝐱_2^e)×(𝐱_1^e-𝐱_3^e), and the normal of the ground truth plane of reflection symmetry, which is defined as η_g=(𝐱_1^g-𝐱_2^g)×(𝐱_1^g-𝐱_3^g), is less than a predefined threshold, i.e., cos^-1(|η_e^⊤η_g|/η_e_2η_g_2)<t_θ. Furthermore, according to <cit.>, the distance between the center of the detected plane of reflection symmetry, which defined as𝐜_e=𝐱_1^e+𝐱_2^e/2,from the ground truth plane of reflection symmetry is less than a predefined threshold, i.e., |𝐜_e^⊤η_g-η_g^⊤𝐱_1^g|/η_g_2<t_d. In order to find the precision vs. recall curve,we change thethreshold for angle as t_θ∈[0,45^∘] and the threshold for distance as t_d∈[0,2s]. Here, s=min{𝐱_1^e-𝐱_2^e_2,𝐱_1^e-𝐱_3^e_2,𝐱_1^g-𝐱_2^g_2,𝐱_1^g-𝐱_3^g_2}.In Fig. <ref>, we plot the recall vs. precision curves for the methods in <cit.>, <cit.>, <cit.>, and the proposed approach on the dataset given in <cit.>. We show the maximum F-score for each method in the legends. The maximum F-score for <cit.> is equal to 0.83, for<cit.> is equal to 0.67, for<cit.> is equal to 0.73, and for the proposed approach is equal to 0.86. §.§ Robustness to PerturbationsIn order to measure the qualitative performance of the proposed approach, we investigate the following two errors which are functions of the perturbation radius σ^2: e_d=1/n∑_i=1^n|⟨𝐳̂_i,𝐯̂⟩| ande_m=1/n∑_i=1^n|𝐯̂^⊤𝐱_i^m+w_0|. The error e_d represents how well the vectors, along the line segments joining the estimated reflection symmetry points, align with the normal to the hyperplane π at convergence. The error e_m represents how well the mid-points of line segments joining reflection symmetry points lie on the estimated hyperplane π. Here, 𝐳̂_i=𝐱_i-𝐱_i^'/𝐱_i-𝐱_i^'_2, 𝐯̂ is the unit normal to the hyperplane π,𝐱_i^m=𝐱_i+𝐱_i^'/2, and w_0 is the distance of the hyperplane π from the origin. In Fig. <ref>, we show the errors e_d and e_m against the perturbation radius σ^2.We observe that the values e_m and e_d for the proposed approach are close to that of the ground-truth reflection symmetry even as the value of σ^2 increases.We construct the following dataset to perform the above experiment. Let {𝐱_1,𝐱_2,…,𝐱_n/2}be the randomly chosenn/2 points. Given the reflection transformations{𝐑_1,…,𝐑_d-1,𝐭}, we reflect these points using the Definition 1 in order to get the final symmetric set𝒮={𝐱_1,𝐱_2,…,𝐱_n/2,𝐱_1^',𝐱_2^',…,𝐱_n/2^'}. Then, we perturb each point with random noiseas 𝐱𝐱+𝒩(0_d,diag(σ^2,σ^2,…,σ^2)),∀𝐱∈𝒮. Here, σ^2 is the perturbation radius and the perturbation is different for each point. For the case d=2, we create sets containing reflection symmetry patterns with n∈{50, 100, 150, 200, 250, 300} with 0≤ x,y≤ 1. For each n, we take 19 different symmetry axis orientations in the range from -90^∘ to 90^∘ with step size of 10^∘. We choose σ^2∈{0,0.01,0.02,…,0.1} to get 11 different perturbations. In total, we have 1254 sets for the evaluation. In Fig. <ref>, we show an example point set from this dataset.For the case d=3, we create reflective symmetric setswith n∈{50, 100, 150, 200, 250, 300} with 0≤ x,y≤ 1. For each n, we take 16 different symmetry plane orientations by considering θ_1∈{-30^∘, 0^∘, 35^∘, 80^∘} andθ_2∈{-30^∘, 0^∘, 35^∘, 80^∘}. We choose σ^2∈{0,0.01,…,0.1}. In total, we obtain 1056 point sets. §.§ Evaluation in Higher Dimensional DataDatasets. Since datasets for higher dimensions (d>3) are not available with ground-truth reflection symmetry, we synthetically createdatasets as follows. For the case d=6 and d=8, we create mirror symmetric point clouds using Definition 1, with n∈{50, 100, 150, 200, 250, 300} and 0≤ x,y≤ 1. For each n, we take 20 random symmetry plane normals. We choose σ^2∈{0,0.02,0.04,…,0.1} to get 6 different perturbations. In total, we have 720 sets for evaluation. For all these point clouds, we have the ground-truth correspondences between the symmetric points and the normals to the ground-truthsymmetry planes.Evaluation of correspondences. In order to evaluate the performance, we measure the correspondence rate which is the number of correct correspondences out of the estimated correspondences.Let (i,i_e^') be the estimated correspondence and let (i,i_g^') be the ground-truth correspondence. Then, we decide if the estimated correspondence (i,i_e^') is correct based on a distance threshold τ_d. If thedistance 𝐱_i_e^'-𝐱_i_g^'_2 between the points 𝐱_i_e^' and 𝐱_i_g^' is less than the distance threshold τ_d, then the correspondence (i,i_e^') is correct and otherwise, incorrect. For a given threshold τ_d, we count the correspondences (i,i_e^') for which the condition 𝐱_i_e^'-𝐱_i_g^'_2<τ_d holds true. In Fig. <ref>, we show the correspondence rate vs the distance threshold curves for the different perturbation radius σ^2∈{0,0.02,0.04,…,0.1} and for d=6 and d=8. We vary the distance threshold as τ_d∈{0,0.01,0.02,…,0.34}.We observe that the correspondence rate increases as the distance threshold increases and the correspondence rate decreases as the perturbation radius increases for both d=6 and d=8.Evaluation of symmetry plane. To evaluate the performance of the reflection plane detection in higher dimensional point clouds(d>3), instead of finding d-1 points on the estimated hyperplane (since finding d-1 points could be difficult), we measure the distance between their normals. Without loss of generality, we prepare the datasetsuch that the reflection symmetry plane passes through the origin. Now, let η^g and η^e be the unit normals to the ground-truth and the estimated reflection symmetry planes, respectively. Then, we declare the estimated reflection symmetry plane to be correct, if cos^-1(|(η^g)^⊤η^e|)<τ_θ. We vary the angle threshold τ_θ in the range [0^∘,5^∘] with a step size of 0.01^∘. In Fig. <ref>, we show the precision rate vs the angle threshold τ_θ curves fordifferent perturbation radius σ^2∈{0,0.02,0.04,…,0.1} and for d=6 and d=8. We observe that the precision rate increases as the angle threshold increases anddecreases as the perturbation radius increases for both d=6 and d=8. §.§ Results In Fig. <ref>,we show the detected reflection symmetry for two real 3D scans of buildings from the dataset <cit.>. In Fig. <ref>, we present the results for the case d=3.The point cloud in Fig. <ref>(a) contains 912045 points and the point cloud in Fig. <ref>(b) contains 767474 points. Since the computational complexity is O(n^3.5)+O(nd^2), the computation time and space requirement (storing the matrices 𝐀_1 and 𝐀_2) are very high. Therefore, in order to compute the reflection symmetry in these scans, we randomly sample around 600 points. In both cases, we show the reflection symmetry plane by the blue color and estimated pairs of reflective symmetric points by the red colored line segment joining them.In order to make our algorithm robust to the part removal, we simply put the extra constraint 𝐞^⊤𝐏𝐞≤ 2k in ILP defined in Equation (<ref>) which limits the number of pairs to at most k.For d=2, we detect reflection symmetry in the set of corner points in a real image. In order to determine the symmetry axis, we use Theorem 2. For d=3, we use existing standard 3D models dataset <cit.>. In order to calculate the symmetry axis in an image using the proposed approach, we first find the set of corner points <cit.>. This set may contain the corners not lying on the symmetric object. Therefore, we apply the proposed approach with RANSAC <cit.>. We compare the proposed results withthe results of two descriptor based methods <cit.> and <cit.>. We evaluate on real and synthetic images containing single symmetric object from the dataset <cit.>. In TABLE <ref>, we present the precision and the recall rates. We observe that for synthetic images, the precision rate is very high for the proposed approach because most of the corner points lie on the symmetric object. Whereas, in real images, the set of corner points contains many outlier corners which leads to the degraded performance. Precision rates for the proposed approach are higher than that for the methods <cit.> and <cit.>. The recall rates are better than that of the method <cit.> and comparable to that of the method <cit.>. This leads to the conclusion that symmetry detection can be performed even when the feature descriptors are not available. In Fig. <ref>, we show the results on the datasets <cit.>, and <cit.>. The last two images show the failure cases from the datasets <cit.>. The reason could be that the pixels which are responsible for symmetry detection such as pixels on eyes and ear tips in the second image are not detected in the corner point detection step. Influence of Different Initializations.We first create the following dataset of 3D point clouds. We create 5000 point clouds {S_i}_i=1^5000 with known ground-truth symmetries as discussed in 5.2. We keep 500 points in each point cloud. Without loss of generality, we choose the reflection symmetry plane such that it makes 90^∘ angle with x-axis and y-axis, i.e., the x-y plane. For each point cloud, we initialize the variable 𝐭_i^0=mean(S_i) and (θ^0_x,θ^0_y) on every point of the grid domain {-90^∘,-80^∘,…,+80^∘,+90^∘}×{-90^∘,-80^∘,…,+80^∘,+90^∘}. We then run our approach and measure the error at the convergence e_i(θ_x^0,θ_y^0)=𝐑^⋆_x𝐑_y^⋆𝐄(𝐑^⋆_x𝐑_y^⋆)^⊤(𝐗_i-𝐭^⋆𝐞^⊤)+𝐭^⋆𝐞^⊤-𝐗_i𝐏^⋆_F^2 for each initialization (θ_x^0,θ_y^0). Then, we find the average error e(θ_x^0,θ_y^0)=1/5000∑_i=1^5000e_i(θ_x^0,θ_y^0). Here, 𝐑_x and 𝐑_y are defined as follows.𝐑_x=[ 1 0 0; 0cosθ^0_x -sinθ^0_x; 0sinθ^0_xcosθ^0_x ],𝐑_y=[cosθ^0_y 0 -sinθ^0_y; 0 1 0;sinθ^0_y 0cosθ^0_y ].In Fig. <ref>, we show the average error e(θ_x^0,θ_y^0). We observe that if the initialization (θ_x^0,θ_y^0) is far away from the global optimum (0^∘,0^∘), then the erroris very high. As the distance between the initialization angles (θ_x^0,θ_y^0) and the global optimum angles (0^∘,0^∘) decrease, the error e(θ_x^0,θ_y^0) remains approximately constant and suddenly drops to near zero after a particular distance. This indicates that, if the initialization angles are within a particular distance from the global optimum, then our approach always find the global optimum solution. This empirical result concurs with the result we already proved in Theorems 5 and 6.§ CONCLUSIONIn this work, we have developed a theory for establishing the correspondences between the mirror symmetric points in ℝ^d. We, further, determine the reflection symmetry transformation in a volumetric set of points in ℝ^d containing a perturbed reflection symmetry pattern using optimization on Riemannian manifold. We have shown that our method is robust to a significant amount of perturbation and achieves 100% accuracy for no perturbation. We have further shown the significance of this work by detecting reflection symmetry in real images and comparing with state-of-the-art methods. The proposed approach is particularly suitable for detecting reflection symmetry of objects in applications where obtaining a robust local descriptor is highly challenging. The linear assignment problem is a time consuming step which restricts us to apply it on the large point sets. However, a proper sampling method can be employed to reduce the size of the point set without losing the symmetry present in the point set. We believe that the fundamental theory and algorithm developed in this work will pave the way for researchers to exploit them for scenarios where estimating feature descriptors is a challenging task. Our approach detects single reflection symmetry plane of an object. Consider the third row of Fig. 11 in which there are multiple reflection symmetry planes present. In such cases, the detected reflection symmetry plane will be the one to which the initialized plane is the closest. For example, in the third row of Fig. 11, we have shown both the reflection symmetry planes detected depending on different initializations. This may not be a proper way of detecting multiple symmetries, though this is an interesting direction. We would like to extend our approach for the detection of multiple reflection symmetry planes of a symmetric object exhibiting multiple symmetries or a point cloud containing more than one symmetric objects. § APPENDIX §.§ A1. Euclidean gradient of the function f̅ with respect to the variable𝐭 (Equation (10)) We write the cost function as follows.f̅(𝐑,𝐭,𝐏) = 𝐓𝐄𝐓^⊤(𝐗-𝐭𝐞^⊤)-(𝐗𝐏-𝐭𝐞^⊤)_F^2= (𝐓𝐄𝐓^⊤𝐗-𝐗𝐏)+(𝐈_d-𝐓𝐄𝐓^⊤)𝐭𝐞^⊤_F^2. We notethat(𝐈_d-𝐓𝐄𝐓^⊤)^⊤(𝐈_d-𝐓𝐄𝐓^⊤) = 2(𝐈_d-𝐓𝐄𝐓^⊤).Therefore, we have (the terms which are not functions of 𝐭 are not shown)f̅(𝐑,𝐭,𝐏)=trace(2𝐞𝐭^⊤(𝐈_d-𝐓𝐄𝐓^⊤)𝐭𝐞^⊤ +2(𝐗^⊤𝐓𝐄𝐓^⊤-𝐏^⊤𝐗^⊤)(𝐈_d-𝐓𝐄𝐓^⊤)𝐭𝐞^⊤). Now taking the derivative with respect to 𝐭 we have,∇_𝐭f̅=2(𝐈_d-𝐓𝐄𝐓^⊤)𝐭𝐞^⊤𝐞+2(𝐞^⊤𝐞𝐭^⊤(𝐈_d-𝐓𝐄𝐓^⊤))^⊤ +2(𝐞^⊤(𝐗^⊤𝐓𝐄𝐓^⊤-𝐏^⊤𝐗^⊤)(𝐈_d-𝐓𝐄𝐓^⊤))^⊤ =4(𝐈_d-𝐓𝐄𝐓^⊤)𝐭𝐞^⊤𝐞+2(𝐈_d-𝐓𝐄𝐓^⊤)(𝐓𝐄𝐓^⊤𝐗-𝐗𝐏)𝐞 Here we have that (𝐈_d-𝐓𝐄𝐓^⊤)𝐓𝐄𝐓^⊤=-(𝐈_d-𝐓𝐄𝐓^⊤). Therefore,∇_𝐭f̅ = 4(𝐈_d-𝐓𝐄𝐓^⊤)𝐭𝐞^⊤𝐞-2(𝐈_d-𝐓𝐄𝐓^⊤)(𝐗+𝐗𝐏)𝐞= 2(𝐈_d-𝐓𝐄𝐓^⊤)(2𝐭𝐞^⊤𝐞-𝐗𝐞-𝐗𝐏𝐞). §.§ A2. Euclidean gradient of the function f̅ with respect to the variable 𝐑_j (Equation (11)) Let us consider the cost function as defined in equation (5) (in main manuscript):f̅(𝐑,𝐭,𝐏)=(∏_u=1^d-1𝐑_u)𝐄(∏_u=1^d-1𝐑_u)^⊤(𝐗-𝐭𝐞^⊤)+𝐭𝐞^⊤-𝐗𝐏_F^2.Now, let us define𝐓=∏_u=1^d-1𝐑_u, 𝐔=𝐗-𝐭𝐞^⊤ and 𝐕=𝐗𝐏-𝐭𝐞^⊤.Then the cost function becomes.f̅(𝐑,𝐭,𝐏) = 𝐓𝐄𝐓^⊤𝐔-𝐕_F^2= trace((𝐓𝐄𝐓^⊤𝐔-𝐕)^⊤(𝐓𝐄𝐓^⊤𝐔-𝐕))= trace((𝐔^⊤𝐓𝐄𝐓^⊤-𝐕^⊤)(𝐓𝐄𝐓^⊤𝐔-𝐕))= trace(𝐔^⊤𝐓𝐄𝐓^⊤𝐓𝐄𝐓^⊤𝐔-2𝐔^⊤𝐓𝐄𝐓^⊤𝐕 +𝐕^⊤𝐕) Here we note that 𝐓𝐄𝐓^⊤𝐓𝐄𝐓^⊤=𝐈_d, thereforef̅(𝐑,𝐭,𝐏) = trace(𝐔^⊤𝐔-2𝐔^⊤𝐓𝐄𝐓^⊤𝐕+𝐕^⊤𝐕).Now taking the classical gradient of f̅ with respect to 𝐑_j we have. (We follow <cit.> for the necessary properties.)∂f̅/∂𝐑_j=-2∂/∂𝐑_jtrace(𝐔^⊤(∏_u=1^d-1𝐑_u)𝐄(∏_u=1^d-1𝐑_u)^⊤𝐕) =-2∂/∂𝐑_jtrace(𝐔^⊤(∏_u=1^j-1𝐑_u)𝐑_j(∏_u=j+1^d-1𝐑_u)𝐄 (∏_j+1^d-1𝐑_u)^⊤𝐑_j^⊤(∏_u=1^j-1𝐑_u)^⊤𝐕) =-2((∏_j+1^d-1𝐑_u)𝐄(∏_u=1^d-1𝐑_u)^⊤𝐕𝐔^⊤(∏_u=1^j-1𝐑_u))^⊤ -2(∏_u=1^j-1𝐑_u)^⊤𝐕𝐔^⊤(∏_u=1^d-1𝐑_u)𝐄(∏_u=j+1^d-1𝐑_u)^⊤ =-2(∏_u=1^j-1𝐑_u)^⊤𝐔𝐕^⊤(∏_u=1^d-1𝐑_u)𝐄(∏_j+1^d-1𝐑_u)^⊤ -2(∏_u=1^j-1𝐑_u)^⊤𝐕𝐔^⊤(∏_u=1^d-1𝐑_u)𝐄(∏_u=j+1^d-1𝐑_u)^⊤ =-2(∏_u=1^j-1𝐑_u)^⊤(𝐔𝐕^⊤+𝐕𝐔^⊤)(∏_u=1^d-1𝐑_u)𝐄(∏_u=j+1^d-1𝐑_u)^⊤ =-2(∏_u=1^j-1𝐑_u)^⊤𝐀(∏_u=1^d-1𝐑_u)𝐄(∏_u=j+1^d-1𝐑_u)^⊤Where 𝐀 = (𝐕𝐔^⊤+𝐔𝐕^⊤)= (𝐗𝐏-𝐭𝐞^⊤)(𝐗-𝐭𝐞^⊤)^⊤+(𝐗-𝐭𝐞^⊤)(𝐗𝐏-𝐭𝐞^⊤)^⊤. §.§ A3. The Riemannian gradient of the function f with respect to the variable𝐭 (Equation (12))Using the definition, as defined in main paper, of Riemannian gradient ξ_𝐭(𝐭) of the function f with respect to the variable 𝐭 we haveξ_𝐭(𝐭)=ℙ_𝐭(∇_𝐭f̅)=∇_𝐭f̅ =2(𝐈_d-(∏_u=1^d-1𝐑_u)𝐄(∏_u=1^d-1𝐑_u)^⊤)(2n𝐭-𝐗𝐞-𝐗𝐏𝐞). §.§ A4. The Riemannian gradient of the function f with respect to the variable 𝐑_j (Equation (13)) Using the definition, as defined in main paper, of Riemannian gradient ξ_𝐑_j(𝐑_j) of the function f with respect to the variable 𝐑_j we haveξ_𝐑_j(𝐑_j)=ℙ_𝐑_j(∇_𝐑_jf̅)=𝐑_jskew(𝐑_j^⊤∇_𝐑_jf̅). ξ_𝐑_j(𝐑_j)=𝐑_jskew(𝐑_j^⊤∇_𝐑_jf̅). 𝐑_j^⊤∇_𝐑_jf̅=-2(∏_u=1^j𝐑_u)^⊤𝐀(∏_u=1^d-1𝐑_u)𝐄(∏_u=j+1^d-1𝐑_u)^⊤ ∇_𝐑_jf̅^⊤𝐑_j=-2(∏_u=j+1^d-1𝐑_u)𝐄(∏_u=1^d-1𝐑_u)^⊤𝐀^⊤(∏_u=1^j𝐑_u).Therefore, ξ_𝐑_j(𝐑_j)=𝐑_j𝐑_j^⊤∇_𝐑_jf̅-∇_𝐑_jf̅^⊤𝐑_j/2 =-𝐑_j(∏_u=1^j𝐑_u)^⊤𝐀(∏_u=1^d-1𝐑_u)𝐄(∏_u=j+1^d-1𝐑_u)^⊤ +𝐑_j(∏_u=j+1^d-1𝐑_u)𝐄(∏_u=1^d-1𝐑_u)^⊤𝐀^⊤(∏_u=1^j𝐑_u). §.§ A5. The Riemannian Hessian of the function f with respect to 𝐑_j (Equation (14))Next, we determine the Riemannian Hessian of the function f. In order to determine the j^th component Hess_𝐑_j(f(𝐑,𝐭))[𝐑_jΩ_j]=ℙ_𝐑_j(Dξ_𝐑_j(𝐑_j)[𝐑_jΩ_j]), of the Riemannian Hessian, we first find the classical derivative Dξ_𝐑_j(𝐑_j)[𝐑_jΩ_j] of the Riemannian gradient ξ_𝐑_j(𝐑_j) in the direction 𝐑_jΩ_j and then we apply the projection operator ℙ_𝐑_j. Now using Equation <ref> we haveξ_𝐑_j(𝐑_j)=-𝐑_j(∏_u=1^j𝐑_u)^⊤𝐀(∏_u=1^d-1𝐑_u)𝐄(∏_u=j+1^d-1𝐑_u)^⊤ +𝐑_j(∏_u=j+1^d-1𝐑_u)𝐄(∏_u=1^d-1𝐑_u)^⊤𝐀^⊤(∏_u=1^j𝐑_u) =-𝐑_j𝐑_j^⊤(∏_u=1^j-1𝐑_u)^⊤𝐀(∏_u=1^j-1𝐑_u)𝐑_j(∏_j+1^d-1𝐑_u)𝐄(∏_j+1^d-1𝐑_u)^⊤ +𝐑_j(∏_u=j+1^d-1𝐑_u)𝐄(∏_u=j+1^d-1𝐑_u)^⊤𝐑_j^⊤(∏_u=1^j-1𝐑_u)^⊤𝐀^⊤(∏_u=1^j-1𝐑_u)𝐑_j =-𝐑_j𝐑_j^⊤𝐁_2𝐑_j𝐁_1+𝐑_j𝐁_1𝐑_j^⊤𝐁_2𝐑_j.Here,𝐁_1=(∏_u=j+1^d-1𝐑_u)𝐄(∏_u=j+1^d-1𝐑_u)^⊤, 𝐁_2=(∏_u=1^j-1𝐑_u)^⊤𝐀^⊤(∏_u=1^j-1𝐑_u). Now Dξ_𝐑_j(𝐑_j)[𝐑_jΩ_j]=d/dtξ_𝐑_j(𝐑_j+t𝐑_jΩ_j)|_t=0 =d/dt(-(𝐑_j+t𝐑_jΩ_j)(𝐑_j+t𝐑_jΩ_j)^⊤𝐁_2(𝐑_j+t𝐑_jΩ_j)𝐁_1)|_t=0 +d/dt((𝐑_j+t𝐑_jΩ_j)𝐁_1(𝐑_j+t𝐑_jΩ_j)^⊤𝐁_2(𝐑_j+t𝐑_jΩ_j))|_t=0.The first term is equal to -𝐑_j𝐑_j^⊤𝐁_2𝐑_jΩ_j𝐁_1.The second term is equal to 𝐑_j𝐁_1𝐑_j^⊤𝐁_2𝐑_jΩ_j-𝐑_j𝐁_1Ω_j𝐑_j^⊤𝐁_2𝐑_j+𝐑_jΩ_j𝐁_1𝐑_j^⊤𝐁_2𝐑_j.Therefore,Dξ_𝐑_j(𝐑_j)[𝐑_jΩ_j]=-𝐑_j𝐑_j^⊤𝐁_2𝐑_jΩ_j𝐁_1+ (𝐑_j𝐁_1𝐑_j^⊤𝐁_2𝐑_jΩ_j -𝐑_j𝐁_1Ω_j𝐑_j^⊤𝐁_2𝐑_j+𝐑_jΩ_j𝐁_1𝐑_j^⊤𝐁_2𝐑_j) 𝐑_j^⊤Dξ_𝐑_j(𝐑_j)[𝐑_jΩ_j] =-𝐑_j^⊤𝐑_j𝐑_j^⊤𝐁_2𝐑_jΩ_j𝐁_1+𝐑_j^⊤(𝐑_j𝐁_1𝐑_j^⊤𝐁_2𝐑_jΩ_j -𝐑_j𝐁_1Ω_j𝐑_j^⊤𝐁_2𝐑_j+𝐑_jΩ_j𝐁_1𝐑_j^⊤𝐁_2𝐑_j) =-𝐑_j^⊤𝐁_2𝐑_jΩ_j𝐁_1+ (𝐁_1𝐑_j^⊤𝐁_2𝐑_jΩ_j -𝐁_1Ω_j𝐑_j^⊤𝐁_2𝐑_j+Ω_j𝐁_1𝐑_j^⊤𝐁_2𝐑_j) (Dξ_𝐑_j(𝐑_j)[𝐑_jΩ_j])^⊤𝐑_j =𝐁_1^⊤Ω_j𝐑_j^⊤𝐁_2^⊤𝐑_j𝐑_j^⊤𝐑_j+ (-Ω_j𝐑_j^⊤𝐁_2^⊤𝐑_j𝐁_1^⊤𝐑_j^⊤ +𝐑_j^⊤𝐁_2^⊤𝐑_jΩ_j𝐁_1^⊤𝐑_j^⊤-𝐑_j^⊤𝐁_2^⊤𝐑_j𝐁_1^⊤Ω_j𝐑_j^⊤)𝐑_j =𝐁_1^⊤Ω_j𝐑_j^⊤𝐁_2^⊤𝐑_j+ (-Ω_j𝐑_j^⊤𝐁_2^⊤𝐑_j𝐁_1^⊤ +𝐑_j^⊤𝐁_2^⊤𝐑_jΩ_j𝐁_1^⊤-𝐑_j^⊤𝐁_2^⊤𝐑_j𝐁_1^⊤Ω_j)𝐑_j^⊤Dξ_𝐑_j(𝐑_j)[𝐑_jΩ_j]-(Dξ_𝐑_j(𝐑_j)[𝐑_jΩ_j])^⊤𝐑_j =-𝐑_j^⊤𝐁_2𝐑_jΩ_j𝐁_1+𝐁_1𝐑_j^⊤𝐁_2𝐑_jΩ_j-𝐁_1Ω_j𝐑_j^⊤𝐁_2𝐑_j +Ω_j𝐁_1𝐑_j^⊤𝐁_2𝐑_j-𝐁_1^⊤Ω_j𝐑_j^⊤𝐁_2^⊤𝐑_j+ Ω_j𝐑_j^⊤𝐁_2^⊤𝐑_j𝐁_1^⊤ -𝐑_j^⊤𝐁_2^⊤𝐑_jΩ_j𝐁_1^⊤+𝐑_j^⊤𝐁_2^⊤𝐑_j𝐁_1^⊤Ω_j =𝐁_1[𝐑_j^⊤𝐁_2^⊤𝐑_j, Ω_j]-[𝐑_j^⊤𝐁_2^⊤𝐑_j, Ω_j]𝐁_1+ [Ω_j,𝐁_1]𝐑_j^⊤𝐁_2𝐑_j-𝐑_j^⊤𝐁_2^⊤𝐑_j[Ω_j,𝐁_1] =[𝐁_1,[𝐑_j^⊤𝐁_2𝐑_j, Ω_j]]+[[Ω_j,𝐁_1],𝐑_j^⊤𝐁_2𝐑_j].Here [.,.] is the Lie bracket and defined as [𝐔,𝐕]=𝐔𝐕-𝐕𝐔 for any two matrices 𝐔 and 𝐕. Hess_𝐑_j(f(𝐑,𝐭))[𝐑_jΩ_j]=ℙ_𝐑_j(Dξ_𝐑_j(𝐑_j)[𝐑_jΩ_j]) =𝐑_jskew(𝐑_j^⊤Dξ_𝐑_j(𝐑_j)[𝐑_jΩ_j]) =1/2𝐑_j(𝐑_j^⊤Dξ_𝐑_j(𝐑_j)[𝐑_jΩ_j]-(Dξ_𝐑_j(𝐑_j)[𝐑_jΩ_j])^⊤𝐑_j) =1/2𝐑_j( [𝐁_1,[𝐑_j^⊤𝐁_2𝐑_j, Ω_j]]+[[Ω_j,𝐁_1],𝐑_j^⊤𝐁_2𝐑_j]).§.§ A6. The Riemannian Hessian of the function f with respect to 𝐭 (Equation (15))In a similar way, we determine the second component, ℙ_𝐭(Dξ_𝐭(𝐭)[η_𝐭]), of the Riemannian Hessian. Since ℝ^d is a vector space we have ℙ_𝐭(Dξ_𝐭(𝐭)[η_𝐭])=Dξ_𝐭(𝐭)[η_𝐭] Dξ_𝐭(𝐭)[η_𝐭]=d/dqξ_𝐭(𝐭+qη_𝐭)|_q=0=4n(𝐈_d-(∏_u=1^d-1𝐑_u)𝐄(∏_u=1^d-1𝐑_u)^⊤)η_𝐭.ThereforeHess_𝐭(f(𝐑,𝐭))[η_𝐭]=4n(𝐈_d-(∏_u=1^d-1𝐑_u)𝐄(∏_u=1^d-1𝐑_u)^⊤)η_𝐭. §.§ A7. Steps of Theorem 5Showing thefacttrace(Ω_j^⊤𝐑_j^⊤𝐇[𝐑_jΩ_j])=4trace(𝐑_j^⊤𝐁_2𝐑_jΩ_j(𝐁_1Ω_j-Ω_j𝐁_1)).Nowtrace(Ω_j^⊤𝐑_j^⊤𝐇[𝐑_jΩ_j])=trace(-Ω_j^⊤𝐑_j^⊤𝐁_2𝐑_jΩ_j𝐁_1 +Ω_j^⊤𝐁_1𝐑_j^⊤𝐁_2𝐑_jΩ_j-Ω_j^⊤𝐁_1Ω_j𝐑_j^⊤𝐁_2𝐑_j +Ω_j^⊤Ω_j𝐁_1𝐑_j^⊤𝐁_2𝐑_j -Ω_j^⊤𝐁_1^⊤Ω_j𝐑_j^⊤𝐁_2^⊤𝐑_j+Ω_j^⊤Ω_j𝐑_j^⊤𝐁_2^⊤𝐑_j𝐁_1^⊤ -Ω_j^⊤𝐑_j^⊤𝐁_2^⊤𝐑_jΩ_j𝐁_1^⊤+Ω_j^⊤𝐑_j^⊤𝐁_2^⊤𝐑_j𝐁_1^⊤Ω_j) =trace(Ω_j𝐑_j^⊤𝐁_2𝐑_jΩ_j𝐁_1-Ω_j𝐁_1𝐑_j^⊤𝐁_2𝐑_jΩ_j +Ω_j𝐁_1Ω_j𝐑_j^⊤𝐁_2𝐑_j-Ω_jΩ_j𝐁_1𝐑_j^⊤𝐁_2𝐑_j +Ω_j𝐁_1Ω_j𝐑_j^⊤𝐁_2𝐑_j- Ω_jΩ_j𝐑_j^⊤𝐁_2𝐑_j𝐁_1 +Ω_j𝐑_j^⊤𝐁_2𝐑_jΩ_j𝐁_1-Ω_j𝐑_j^⊤𝐁_2𝐑_j𝐁_1Ω_j) =trace(4𝐑_j^⊤𝐁_2𝐑_jΩ_j𝐁_1Ω_j-2𝐑_j^⊤𝐁_2𝐑_jΩ_jΩ_j𝐁_1 -2𝐑_j^⊤𝐁_2𝐑_j𝐁_1Ω_jΩ_j)Since,trace(𝐑_j^⊤𝐁_2𝐑_j𝐁_1Ω_jΩ_j)=trace((𝐁_1Ω_jΩ_j)^⊤𝐑_j^⊤𝐁_2𝐑_j) =trace(Ω_jΩ_j𝐁_1𝐑_j^⊤𝐁_2𝐑_j) =trace(𝐑_j^⊤𝐁_2𝐑_jΩ_jΩ_j𝐁_1), we havetrace(Ω_j^⊤𝐑_j^⊤𝐇[𝐑_jΩ_j])=trace(4𝐑_j^⊤𝐁_2𝐑_jΩ_j𝐁_1Ω_j -4𝐑_j^⊤𝐁_2𝐑_jΩ_jΩ_j𝐁_1) =4 trace(𝐑_j^⊤𝐁_2𝐑_j(Ω_j𝐁_1Ω_j-Ω_jΩ_j𝐁_1)). 10 url@samestyle lowe2004distinctive D. G. 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Bolles, “Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography,” Communications of the ACM, vol. 24, no. 6, pp. 381–395, 1981. rauschert2011symmetry1 I. Rauschert, J. Liu, and Y. Liu, “Symmetry detection competition: A summary of how the competition is carried out,” in IEEE CVPR Workshop Symmetry Detection Real World Images, 2011, pp. 1–66. cicconet2016convolutional M. Cicconet, V. Birodkar, M. Lund, M. Werman, and D. Geiger, “A convolutional approach to reflection symmetry,” arXiv preprint arXiv:1609.05257, 2016. liu2013symmetry J. Liu, G. Slota, G. Zheng, Z. Wu, M. Park, S. Lee, I. Rauschert, and Y. Liu, “Symmetry detection from realworld images competition 2013: Summary and results,” in IEEE CVPR Workshops, 2013, pp. 200–205. petersen2008matrix K. B. Petersen, M. S. Pedersen et al., “The matrix cookbook,” Technical University of Denmark, vol. 7, p. 15, 2008. | http://arxiv.org/abs/1706.08801v6 | {
"authors": [
"Rajendra Nagar",
"Shanmuganathan Raman"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170627120339",
"title": "Detecting Approximate Reflection Symmetry in a Point Set using Optimization on Manifold"
} |
=1shapes.geometric, arrowsPf⟨⟩Łℒ_ωℒ_ω^∨ℒ_ω̅∇_ω∇_-ω𝕀 ∧ 𝖢𝖢regℳ𝖯𝖳showonlyrefstheoremTheorem[section] claimClaim[section] lemmaLemma[section] *remarkRemarkdefinitionDefinition[section] exampleExample[section] *acknowledgementsAcknowledgements*outlineOutlinefigures/Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, CanadaDepartment of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, [email protected] revisit the relations between open and closed string scattering amplitudes discovered by Kawai, Lewellen, and Tye (KLT). We show that they emerge from the underlying algebro-topological identities known as the twisted period relations. In order to do so, we formulate tree-level string theory amplitudes in the language of twisted de Rham theory. There, open string amplitudes are understood as pairings between twisted cycles and cocycles. Similarly, closed string amplitudes are given as a pairing between two twisted cocycles. Finally, objects relating the two types of string amplitudes are the α'-corrected bi-adjoint scalar amplitudes recently defined by the author <cit.>. We show that they naturally arise as intersection numbers of twisted cycles. In this work we focus on the combinatorial and topological description of twisted cycles relevant for string theory amplitudes. In this setting, each twisted cycle is a polytope, known in combinatorics as the associahedron, together with an additional structure encoding monodromy properties of string integrals. In fact, this additional structure is given by higher-dimensional generalizations of the Pochhammer contour. An open string amplitude is then computed as an integral of a logarithmic form over an associahedron. We show that the inverse of the KLT kernel can be calculated from the knowledge of how pairs of associahedra intersect one another in the moduli space. In the field theory limit, contributions from these intersections localize to vertices of the associahedra, giving rise to the bi-adjoint scalar partial amplitudes.Combinatoricsand Topologyof Kawai–Lewellen–Tye Relations Sebastian Mizera December 30, 2023 ============================================================ We thank Freddy Cachazo for insightful comments on this work. We are grateful to Oliver Schlotterer for bringing the paper <cit.> to our attention, which has inspired this line of research. We also thank Paolo Benincasa, Lauren Williams, and Karen Yeats for useful discussions. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. § INTRODUCTION Recent years have seen a vast improvement in our understanding of quantum field theories through the study of scattering amplitudes <cit.>. Such advancements were often made possible by considering a generalization of ordinary field theories into string theories. The main advantage of this approach is that strings—as extended objects—provide a way of smoothing out interactions between the scattering states. More precisely, the moduli space of a string worldsheet continuously connects its different factorization channels. As a result, a sum over discrete objects—such as Feynman <cit.> or on-shell <cit.> diagrams—in field theory is replaced by an integral over a continuous worldsheet in string theory. In the infinite tension limit, where strings become point-like, this integral localizes to disconnected corners of the moduli space, which give rise to the field theory amplitudes. In this way, thinking of field theory amplitudes as a limit of the string theory ones provides a way of unifying all factorization channels under a single object.The prime example of usefulness of string theory in the study of field theory amplitudes are the Kawai–Lewellen–Tye (KLT) relations discovered in 1985 <cit.>. They give a way of writing the amplitudes for scattering of closed strings entirely in terms of a quadratic combination of open string amplitudes. In the field theory limit, where closed strings reduce to gravitons—particle excitations of General Relativity—and open strings reduce to gluons—excitations of the Yang–Mills theory—KLT relations give a connection between graviton and gluon scattering amplitudes. Such a relationship not only hints at a fundamental interplay between the two types of theories, but also provides enormous simplifications for practical calculations, both in string and field theory.KLT relations have been most thoroughly studied in the field theory limit. In its modern form found by Cachazo, He, and Yuan (CHY) they read <cit.>: 𝒜^GR = ∑_β, γ 𝒜^YM(β)m^-1(β | γ) 𝒜^YM(γ).Here, 𝒜^GR is an n-point graviton amplitude, while 𝒜^YM(β) is an n-point gluon partial amplitude with ordering β. The sum proceeds over two sets of (n-3)! permutations β and γ forming a basis for the Yang–Mills amplitudes. The object m(β | γ) is a double-partial amplitude of a bi-adjoint scalar theory <cit.>. It is convenient to think of the relation (<ref>) as a matrix product of a transposed vector, inverse of a matrix, and another vector, where rows and columns are labelled by permutations.It was not always clear that coefficients of the KLT expansion can be written in the form (<ref>) as the inverse of a matrix. In their original work, Kawai, Lewellen, and Tye used contour deformation arguments to arrive at these coefficients as coming from monodromy factors around vertex operators on the boundary of a worldsheet <cit.>. They evaluated explicit form of the quadratic relations for low-point examples. A closed-form expression for the KLT relations to arbitrary number of particles in field theory was later given in Appendix A of <cit.> by Bern, Dixon, Perelstein, and Rozowsky. Properties of this expansion were systematically studied and proven in a series of papers <cit.> by Bjerrum-Bohr, Damgaard, Feng, Søndergaard, and Vanhove, who also generalized the allowed bases of permutations to a larger set. They introduced the matrix S[β|γ] called a KLT kernel, which allows to write the KLT relations as a matrix product. Finally, Cachazo, He, and Yuan recognized <cit.> that the KLT kernel can be understood as the inverse matrix of bi-adjoint scalar amplitudes, i.e., S[β | γ] = m^-1(β | γ), ultimately leading to the form given in (<ref>). This also allowed to construct the kernel from the most general sets of permutations labelling the columns and rows of m(β | γ), so that coefficients of the KLT expansion are not necessarily polynomials in the kinematic invariants.At this point one could ask: Where do KLT relations come from? It turns out that a fruitful path to consider is to go back to the string theory case, where these relations were first conceived. It was proposed by the author <cit.> that the full string theory KLT relations can be rewritten in a form analogous to (<ref>) as follows: 𝒜^closed = ∑_β, γ 𝒜^open(β)m_α'^-1(β | γ) 𝒜^open(γ).Here, 𝒜^closed and 𝒜^open(β) are the n-point closed and open string amplitudes respectively. The role of the string theory KLT kernel is played by the inverse of a matrix m_α'(β | γ), which is constructed out of the bi-adjoint scalar amplitudes with α' corrections. Recall that α' is a parameter inversely proportional to the string tension, such that α' → 0 corresponds to the field theory limit. In this way, (<ref>) is a direct analogue of (<ref>), where every piece of the puzzle receives string corrections. By evaluating explicit examples of m_α'(β | γ), which from now on we will refer to as the inverse KLT kernel, we found that they have a surprisingly simple structure, giving rise to compact expressions in terms of trigonometric functions. Moreover, they can be calculated using Feynman-like diagrammatic rules <cit.>, hinting at an underlying combinatorial underpinnings. In this work we show that string theory KLT relations in the form (<ref>) are in fact a result of a deep connection between string theory amplitudes, algebraic topology, and combinatorics.Practically at the same time as the initial work on the KLT relations, on the other side of the globe, mathematicians Aomoto, Cho, Kita, Matsumoto, Mimachi, Yoshida, and collaborators were developing a seemingly unrelated theory of hypergeometric functions <cit.>. It eventually led to the formulation of twisted de Rham theory, which is a generalization of the conventional de Rham theory to integrals of multi-valued functions <cit.>. Let us first intuitively explain its key ingredients, leaving precise definitions for later sections. A twisted homology group H_m(X,ℒ_ω) on some manifold X is a space of twisted cycles, which are regions of X together with an additional information about branches of a multi-valued function. Similarly, a twisted cohomology group H^m(X,∇_ω) is a space of twisted cocycles, which are differential forms on X satisfying certain conditions. A pairing between a twisted cycle and a cocycle is then simply an integral of a differential form over a given region of X which is sensitive to the branch structure of the integrand. Twist measures multi-valuedness of the integrand.One can also define a natural set of a dual twisted homology H_m(X,ℒ^∨_ω) and a dual twisted cohomology H^m(X,∇_ω^∨). For the purpose of this work, the duality is roughly speaking given by complex conjugation. One can define a pairing between these two dual spaces too, giving rise to another integral of a multi-valued function. Having defined two different pairs of twisted homologies and cohomologies, we would like to calculate invariants between them as well. As it turns out, it is possible to pair two twisted cycles belong to a twisted homology and its dual. The resulting object is called an intersection number of twisted cycles<cit.>. It is computed from the information of how these cycles intersect one another in X, as well as their associated branch structure. Similarly, one can also define an intersection number of twisted cocycles<cit.>. What is more, in 1994 Cho and Matsumoto found identities—known as the twisted period relations—between pairings computed from different twisted homologies and cohomologies described above <cit.>.In this work we show that Kawai–Lewellen–Tye relations are a consequence of twisted period relations. In order to do so, we first formulate string theory tree-level amplitudes in the language of twisted de Rham theory. Open string partial amplitudes 𝒜^open(β) are given as pairings between twisted cycles and twisted cocycles, while closed string amplitudes 𝒜^closed come from intersection numbers of twisted cocycles. Finally, inverse of the KLT kernel m_α'(β | γ) is calculated from intersection numbers of twisted cycles. We can schematically summarize these pairings in the following diagram: [row sep = 6em, column sep = 10em] H^m(X, ∇_ω) [leftrightarrow]r𝒜^closed[leftrightarrow]d[swap]𝒜^open(β)H^m(X,∇^∨_ω) [leftrightarrow]d 𝒜^open(γ) H_m(X, L_ω) [leftrightarrow]r[swap] m_α'(β | γ)H_m(X, L^∨_ω)Twisted period relations for the above pairings become KLT relations in exactly the same form as (<ref>). We give a proof of this statement in Section <ref>, where we also define bases of twisted cycles and cocycles relevant for string amplitudes.These twisted cycles and cocycles turn out to have interesting combinatorial properties. It is known that an n-point tree-level open string partial amplitude is given by an integral of a differential form over a simplex Δ_n-3, belonging to the moduli space of genus-zero Riemann surfaces with n punctures <cit.>. However, in order to resolve degenerate points close to the vertices of the simplex, one considers a blowup of the moduli space, π^-1(ℳ_0,n) = ℳ_0,n<cit.>. On this space, the simplex becomes a different polytope known as the associahedron, K_n-1<cit.>. An example of this procedure is given below: < g r a p h i c s >Twisted cycles are then given by (n-3)-dimensional associahedra with an additional structure keeping track of the branches of the integrand. This structure is most conveniently summarized by introducing an additional regularization of twisted cycles based on the Pochhammer contour <cit.> and its higher-dimensional generalizations. We give details of this construction in Section <ref>. In Section <ref> we also find a basis of twisted cocycles for string amplitudes and show they are given by logarithmic (n-3)-forms. With these constructions, an open string partial amplitude becomes an integration of a logarithmic form over an associahedron. It is interesting to see how physical properties arise in this formulation. Unitarity is made manifest from the fact that facets of the associahedra are given by products of two lower-dimensional associahedra. Locality is manifest from the fact that a higher-dimensional Pochhammer contour yields only simple poles in all factorization channels. Similarly, each lower-dimensional face of the associahedron has an associated factorization diagram. For instance, contact terms come from the bulk of the polytope, while trivalent diagrams come from its vertices. Since each propagator comes with a power of α', it means that in the field theory limit only the regions of the moduli space around the vertices of the associahedra contribute. We show how to construct this limit explicitly in Appendix <ref>.The most novel concept studied in this work, however, is the evaluation of the intersection numbers of twisted cycles. We show how to calculate them on explicit examples and in general in Section <ref>. There, we also prove that combinatorial rules for finding intersection numbers are equivalent to the diagrammatic expansion found empirically in <cit.>, establishing that the α'-corrected bi-adjoint scalar amplitudes m_α'(β | γ) are given by intersection numbers of twisted cycles. Geometric and topological meaning of these objects can be easily pictured. The real section of the moduli space ℳ_0,n is tiled by (n-1)!/2 associahedra K_n-1(β)<cit.>, each labelled with some permutation β. The problem of calculating m_α'(β | γ) reduces to finding the intersection of two associahedra K_n-1(β) and K_n-1(γ) in the moduli space:< g r a p h i c s >The intersection number then receives contributions from all the (0,1,2,…)-dimensional faces belonging to the intersection K_n-1(β) ∩ K_n-1(γ). In the above example, these are five vertices, five edges, and one polygon. Once again, in the field theory limit these contributions localize to vertices only, and hence can be written as a sum over trivalent diagrams. Since the intersection region belongs to both associahedra K_n-1(β) and K_n-1(γ) at the same time, the trivalent diagrams have to be compatible with both planar orderings β and γ. This is indeed the standard definition of the field theory bi-adjoint scalar double-partial amplitude m(β | γ). It is quite surprising that a scattering amplitude in a quantum field theory can be understood as arising from such an abstract mathematical object as an intersection number of twisted cycles.This paper is structured as follows. In Section <ref> we give an introduction to the topics of twisted de Rham theory, as well as string theory amplitudes. In Section <ref> we define the twisted cycles and cocycles that are relevant for string theory amplitude computations. There, we also establish the equivalence between Kawai–Lewellen–Tye relations and twisted period relations. In Section <ref> we discuss the interpretation of the inverse KLT kernel as intersection numbers of twisted cycles. We give a combinatorial description of the blowup procedure leading to the associahedron, and present the regularization of twisted cycles using a generalized Pochhammer contour. After giving explicit examples of the evaluation of intersection numbers for lower-point cases, we prove that they are equivalent to the diagrammatic rules for the computation of m_α'(β | γ) in general. We conclude with the summary of the results and a discussion of open questions in Section <ref>. In Appendix <ref> we discuss how to obtain the field theory limit of open string amplitudes from contributions localized around the vertices of the associahedra. § MATHEMATICAL & PHYSICAL PRELIMINARIES This section is meant to give an informal introduction to both mathematics of twisted de Rham theory and physics of string theory amplitudes for the readers not familiar with these topics. §.§ Twisted de Rham Theory In the study of hypergeometric functions one encounters integrals of multi-valued functions. In order to analyze properties of such objects, it is useful to reformulate the problem in the language of algebraic topology, where integrals are understood as pairings between integration cycles and corresponding cocycles as the integrands. In the case when the integrand is a single-valued object, the problem is governed by de Rham theory and its homology and cohomology groups <cit.>. In the case of multi-valued integrands, one needs to keep track of additional information about the branch structure along the integration region. Study of such objects leads to a generalization of de Rham theory into its twisted version.Twisted de Rham theory dates back to the work of Aomoto <cit.>, Deligne <cit.>, Kita <cit.>, and Gelfand <cit.> who laid out foundations for this theory, which later grew into a field of research developed by various authors, see, e.g., <cit.>. Overview of these results is presented in textbooks by Aomoto and Kita <cit.>, as well as Yoshida <cit.>.[See also textbooks by Haraoka <cit.> and Kimura <cit.> in Japanese, as well as one by Orlik and Terao <cit.>, who discuss hypergeometric functions from the viewpoint of arrangements of hyperplanes.] In this section we outline the basics of twisted de Rham theory that should serve as intuition for the remainder of the paper. We follow the discussion in <cit.>.Despite initial motivation coming from hypergeometric functions, twisted de Rham theory extends to more general objects. We will consider integrals of the form: ∫_γ u(z) φ(z),where u(z) and φ(z) are a multi-valued function and a single-valued differential form respectively. Let us define the function u(z) asu(z) := ∏_i=1^k f_i(z)^α_iwithα_i ∈ℂ∖ℤ,where f_i(z) = f_i(z_1,z_2,…,z_m) are linear polynomials defined on an m-dimensional complex space minus the singular locus of u(z), called a divisor, D:X := ℂ^m ∖ D with D := ⋃_i=1^k { f_i(z) = 0 }.The function u(z) and the m-form φ(z), together with the m-dimensional region γ are defined on the same manifold X. We demand that γ has endpoints only on the divisor D, which implies that it does not have any boundaries on X. Hence, γ can be called a topological cycle.In order to give a more precise definition of (<ref>) let us introduce a smooth triangulation of X that will serve as an intuitive example. We take the cycle γ to be an m-simplex Δ. Since u(z) is multi-valued, we need to specify its branch on Δ. We use the notation Δ⊗ u_Δ(z) to signify the choice of a branch u_Δ(z) of u(z) on Δ. With this definition (<ref>) becomes: ∫_Δ⊗ u_Δφ(z):= ∫_Δ{ u(z) on the branchu_Δ(z) } φ(z).We say that Δ is loaded with u_Δ(z). Since on a small neighbourhood around Δ the form u_Δ(z) φ(z) is single-valued, we can apply the ordinary Stokes theorem to find: ∫_∂Δ u_Δ(z) φ(z) = ∫_Δ d (u_Δ(z) φ(z) ) = ∫_Δ u_Δ(z) ( d + ω) φ(z),where ω := d log u(z) = ∑_i=1^kα_i d f_i(z)/f_i (z) is a single-valued 1-form on Δ. The combination in the brackets defines a differential operator := d + ω, called a connection. It is straightforward to check that · = 0, which makesan integrable connection <cit.>. With these definitions, (<ref>) becomes: ∫_Δ⊗ u_Δφ(z)= ∫_∂_ω( Δ⊗ u_Δ)φ(z),where the remaining part is to specify how the boundary operator ∂_ω acts on Δ⊗ u_Δ (z). Let us illustrate it with a couple of examples. We use the standard notation <cit.> for an m-simplex, Δ =01 ⋯ m. In the one-dimensional case Δ =01 we have: ∂ 01=1-0 and similarly∂_ω(01 ⊗ u_ 01 (z) ) =1 ⊗ u_ 1 (z) -0 ⊗ u_ 0 (z).Here the branch u_ 1 (z) is induced from u_ 01 (z) at the boundary of 1 of 01, and similarly for u_ 0 (z). Therefore, the twisted Stokes theorem (<ref>) in this case becomes: ∫_ 01 ⊗ u_ 01 φ(z)= ∫_ 1 ⊗ u_ 1 φ(z)- ∫_ 0 ⊗ u_ 0 φ(z),where each contribution gives u(z) φ(z) evaluated at an appropriate branch at points z=0 and 1. Similarly, in the two-dimensional case, where Δ =012 we have: ∂_ω(012 ⊗ u_ 012 (z) ) =12 ⊗ u_ 12 (z) +20 ⊗ u_ 20 (z) +01 ⊗ u_ 01 (z).Here, the twisted cycles associated to the boundaries 12, 20, and 01 are determined by u_ 012(z), which naturally translates to the twisted Stokes theorem: ∫_ 012 ⊗ u_ 012 φ(z)= ∫_ 12 ⊗ u_ 12 φ(z)+ ∫_ 20 ⊗ u_ 20 φ(z)+ ∫_ 01 ⊗ u_ 01 φ(z).A generalization to higher-dimensional simplices is now clear. The twisted boundary operator acts on an m-simplex as: ∂_ω(01⋯ m ⊗ u_ 01⋯ m (z) ) = ∑_i=0^m (-1)^i 01⋯î⋯ m ⊗ u_ 01⋯î⋯ m(z),where the hat denotes a removed label. For every triangulable manifold this definition can be used to compute the action of the boundary operator by gluing simplices together.Let us interpret the above analysis in the language of algebraic topology. In order to track the information about branches we define homology with coefficients in a local system ℒ_ω^∨ defined by the differential equation ∇_ωξ = dξ + ωξ = 0.It admits a formal solution for ξ of the form ξ(z) = c / u(z), where c ∈ℂ is a constant. The space generated by local solutions of (<ref>) is therefore one-dimensional. Let us cover the manifold X with a locally finite open cover, such that X=⋃_i U_i, and fix a solution ξ_i on each of the open sets U_i. On the intersection of two of them, U_i and U_j, we have: ξ_i(z) = ζ_ij ξ_j(z) for z∈ U_i ∩ U_j,where ζ_ij is a constant on U_i ∩ U_j. Given that a solution ξ(z) on U_i ∩ U_j can be expressed as ξ(z) = c̃_i ξ_i(z) = c̃_j ξ_j(z) for constants c̃_i, c̃_j ∈ℂ, we have c̃_i = ζ_ij^-1c̃_j. Therefore, the set of local solutions of (<ref>) defines a flat line bundle, denoted by ℒ_ω^∨, obtained by gluing the fibers {c̃_i} by transition functions {ζ_ij^-1}. Similarly, we can define a dual line bundle ℒ_ω, which corresponds to the transition functions {ζ_ij}. It is generated by local solutions of the differential equation ∇_-ωξ = dξ - ωξ = 0.Since the boundary operator (<ref>) coincides with the above system generated by ℒ_ω, we can define a twisted chain group C_m(X,ℒ_ω) with the basis of Δ⊗ u_Δ(z). The boundary operator is given by a map:C_m(X,ℒ_ω)C_m-1(X,ℒ_ω),for which one can show ∂_ω∘∂_ω = 0. The definition of the m-th twisted de Rham homology group is given by a natural generalization the usual homology group:[Twisted homology groups H_k(X,ℒ_ω) with k<m generically vanish <cit.>. For the purpose of this work, we will be only interested in top homologies and cohomologies.] H_m(X,ℒ_ω) := ∂_ω im ∂_ω.In other words, twisted homology is a space of boundary-less topological cycles with a loading, γ⊗ u_γ(z), which are not boundaries themselves. We call these elements twisted (or loaded) cycles.Let us turn to the associated twisted cohomology, which now has a straightforward definition. Since the function u(z) vanishes at the boundaries of the cycles, the right-hand side of (<ref>) is equal to zero. This implies that adding a combination ∇_ωξ(z) to φ(z) does not affect the result of the integration. In other words, φ(z) and φ(z) + ∇_ωξ(z) are in the same cohomology class for any smooth (m-1)-form ξ(z). This leads to the definition of the m-th twisted cohomology:H^m(X,∇_ω) := ∇_ω im ∇_ω,which means it is a space of cocycles which are closed but not exact with respect to ∇_ω. We call these elements twisted cocycles. In similarity to the twisted homology case, one can also define a dual twisted cohomology with the connection ∇_ω^∨ = ∇_-ω. We will make use of this fact in the remainder of the paper. We can now use the twisted homology (<ref>) and twisted cohomology (<ref>) to define a non-degenerate pairing:H_m(X,ℒ_ω) × H^m(X,∇_ω)⟶ ℂ,given by γ⊗ u_γ, φ(z):= ∫_γ⊗ u_γφ(z).This is a way of formulating the initial integral (<ref>) in the language of twisted de Rham theory.Manifolds considered in this work will generically be non-compact. In this case, one ought to consider the locally finite twisted homology group H^lf_m(X,ℒ_ω) defined using a locally finite cover of X. Pairings between different twisted cycles and cocycles require at least one of them to be compact or with compact support <cit.>. We will explicitly construct a map from H_m^lf(X,ℒ_ω) to the space of compact twisted cycles, H_m(X,ℒ_ω), in Section <ref>. We will discuss the use of an inclusion map from H^m(X,∇_ω) to the compactly supported twisted cohomology H_c^m(X,∇_ω) in Section <ref>. For mathematically rigorous definitions of these statements see, e.g., <cit.>.[For a treatment of non-compact topological spaces in general, see the textbooks on algebraic topology <cit.>.] §.§ String Theory Scattering Amplitudes Much of the structure of quantum field theories and their generalizations are encapsulated in scattering amplitudes. Physically, they calculate the probabilities of given scattering states—such as particles or strings—to interact with each other. Despite the fact that efficient calculation of scattering amplitudes is indispensable in experimentally testing predictions of current models of physics at particle colliders <cit.>, we will be mainly interested in their mathematical structure. Let us focus the discussion on string theory amplitudes.The very first examples of string amplitudes appeared in the pioneering papers of Veneziano <cit.>, Virasoro <cit.>, Shapiro <cit.>, as well as Koba and Nielsen <cit.>, long before the formulation of string theory. Since then, calculation of string theory scattering amplitudes developed into a rich field of research of its own, see, e.g., <cit.>. For historical account of the developments of string theory see <cit.>. Here, we will give a brief review of the topics relevant for this paper. Great introduction to the subject is given in the classic textbooks by Green, Schwarz, and Witten <cit.>, as well as Polchinski <cit.>.Strings come in two types: open and closed. Evolution of strings in spacetime creates a two-dimensional surface called the worldsheet. Using the underlying conformal symmetry, we can map the worldsheet into a Riemann surface, which takes the scattering states into vertex operators. Scattering amplitude is then given as an integration of vertex operator correlation function over all their inequivalent positions. The n-point open string amplitude takes the form: 𝒜^open_full = Tr(T^a_1 T^a_2⋯ T^a_n) ∫_𝔇(12⋯ n)d^n z/vol SL(2,ℝ)∏_i<j(z_j - z_i)^α' s_ijF(z)+ ….Let us dissect this formula one-by-one. Each string has an associated spacetime momentum k_i^μ for μ = 0,1,…,d, where d+1 is the spacetime dimension. We take all momenta to be incoming and impose momentum conservation ∑_i k_i^μ = 0. Each string also has a coloura_i associated to a generator T^a_i of the unitary group U(N). Other possible quantum numbers, such as polarization vectors, are all enclosed in the rational function F(z). In fact, F(z) is a part of the correlation function of vertex operators which depends on the type of string theory used. The multi-valued function ∏_i<j(z_j - z_i)^α' s_ij, called the Koba–Nielsen factor <cit.>, is common to all types of strings. In the exponent we used the parameter α', which is proportional to the inverse of the string tension and serves as a coupling constant of the string amplitude (<ref>). Here s_ij = k_i · k_j is an inner product of the momenta known as the Mandelstam invariant<cit.>. The integration variables { z_1, z_2, …, z_n } are the positions of the vertex operators, which we associate to marked points—or punctures—on the boundary of a genus-zero Riemann surface. Due to the inherit SL(2,ℝ) redundancy of the correlator, one needs to quotient out the action of this group, which is denoted by division by vol SL(2,ℝ). In practice, it boils down to fixing positions of three punctures, which by convention is taken to be (z_1, z_n-1, z_n) = (0,1,∞). In doing so, one picks up a constant factor (z_1-z_n-1)(z_n-1-z_n)(z_n - z_1) due to the Faddeev–Popov Jacobian <cit.>. The disk ordering𝔇(12⋯ n) denotes a region of integration given by {z_1 < z_2 < … < z_n} after gauge-fixing. It comes with the associated trace Tr(T^a_1 T^a_2⋯ T^a_n) of Chan–Paton factors <cit.> due to the colour structure of the strings. The ellipsis in (<ref>) denote a sum over all (n-1)! cyclically-inequivalent permutations of the vertex operators, each decorated with a trace factor.It is important to mention that in (<ref>) we have only displayed contributions from the genus-zero Riemann surface. In order to obtain the full string theory amplitude, one sums over all possible genera of Riemann surfaces. Genus-zero terms correspond to the tree-level—or classical—scattering amplitudes, while the genus-one and higher terms give quantum corrections. For the purpose of this work we will restrict ourselves to tree-level amplitudes only.The amplitude (<ref>) admits a natural splitting into partial (or colour-ordered) amplitudes 𝒜^open(β) defined as coefficients of a Chan–Paton trace with the permutation β. These will be the objects of our interest. We have: 𝒜^open(β) := ∫_𝔇(β)d^n z/vol SL(2,ℝ)∏_i<j (z_β(j) - z_β(i))^α' s_β(i), β(j)F(z),where the only information about the permuation β comes from the disk ordering 𝔇(β) and the choice of the branch for the Koba–Nielsen factor. Scattering amplitudes of closed strings are defined similarly. For their n-point scattering we have: 𝒜^closed := ∫d^2n z/vol SL(2,ℂ)∏_i<j|z_i - z_j|^2α' s_ijF(z)F(z̅).Here, the integration proceeds over the full moduli space of a genus-zero Riemann surface with n punctures, ℳ_0,n. The SL(2,ℂ) redundancy is fixed by choosing positions of three punctures. The integrand of (<ref>) factors into two functions, a holomorphic and an anti-holomorphic one, which once again depend on the type of string theory under consideration. The common piece is given by the Koba–Nielsen factor. Note that in both (<ref>) and (<ref>) we have omitted coupling constants that give rise to an overall normalization factor, see, e.g., <cit.>.The precise form of the integrands of (<ref>) and (<ref>) will not be important for our purposes and can be found, for instance, in <cit.>. It was shown by Mafra, Schlotterer, and Stieberger <cit.> that open string partial amplitudes can be expanded in a basis of the so-called Z-theory amplitudes <cit.> as follows: 𝒜^open(β) = ∑_γ∈𝒞 n(γ)Z_β(γ),whereZ_β(γ) := ∫_𝔇(β)d^n z/vol SL(2,ℝ) ∏_i<j(z_β(j) - z_β(i))^α' s_β(i), β(j)/(z_γ(1)-z_γ(2))(z_γ(2)-z_γ(3))⋯(z_γ(n)-z_γ(1)).The coefficients of the expansion, n(γ), are only a function of kinematic invariants, polarization vectors, and possibly Grassmann variables in the supersymmetric case. The entire dependence on the string parameter α' and the colour ordering β is encapsulated in the Z-theory amplitude (<ref>). The sum is over a set 𝒞 of (n-3)! permutations.[In fact, (<ref>) is another instance of a field theory KLT relation <cit.>. This fact, however, will not play any role in this work.] Such a set is called a Bern–Carrasco–Johansson (BCJ) basis <cit.> originally found for Yang–Mills amplitudes and later generalized to the open string ones by Stieberger <cit.>. The Z-integral (<ref>) depends on two permutations, β serving as a disk ordering, and γ which determines the form of the integrand function. Because all the string theoretic properties of open string amplitudes are determined by the Z-theory amplitudes, it will be sufficient to study the integrals (<ref>) as the primary ingredients in our work. A similar decomposition can be performed in the closed string case (<ref>). It reads: 𝒜^closed = ∑_β∈ℬ, γ∈𝒞 n(β)n(γ)J(β | γ), whereJ(β | γ) := ∫d^2n z/vol SL(2,ℂ)∏_i<j|z_i - z_j|^2α' s_ij/(z_β(1)-z_β(2))⋯(z_β(n)-z_β(1))(z_γ(1) - z_γ(2)) ⋯(z_γ(n) - z_γ(1)).The sum in (<ref>) proceeds over two sets of permutations ℬ and 𝒞, each of length (n-3)!. The coefficients n(γ) are the same as in the open string case (<ref>). The object (<ref>) is an integral over the moduli space ℳ_0,n<cit.>, with the integrand composed of two pieces called the Parke–Taylor factors[The name comes due to the resemblence to the scattering amplitude of gluons with MHV helicity configuration found by Parke and Taylor <cit.>.] which also appear in (<ref>). In contrast with (<ref>), however, (<ref>) is symmetric under the exchange of the two permutations β and γ.Calculation of the above string integrals is a difficult problem that has been approached in many different ways, see, e.g. <cit.>. The general approach is to perform an expansion around α' = 0. In particular, it is known that in the α' → 0 limit, the integrals Z_β(γ), J(β | γ), as well as entries of the inverse of the string theory KLT kernel m_α'(β | γ) all approach the same answer, up to global scaling: lim_α' → 0 Z_β(γ) = lim_α' → 0 J(β|γ) = lim_α' → 0 m_α'(β|γ) = α'^3-nm(β | γ).Here m(β | γ) are double-partial amplitudes of the so-called bi-adjoint scalar<cit.>. This amplitude is given by a sum over all trivalent Feynman diagrams 𝒯 which are planar with respect to both β and γ:m(β | γ) := (-1)^w(β | γ)+1∑_𝒯∈𝒢_β∩𝒢_γ1/∏_e ∈𝒯 s_e,where 𝒢_β denotes the space of all trivalent Feynman diagrams planar with respect to the ordering β, and e ∈𝒯 means the set of internal edges of a given diagram 𝒯. The Mandelstam invariant s_e equals p_e^2/2, where p_e is the momentum flowing through the edge e. We have included a sign factor <cit.> featuring the relative winding number between the two permutations, w(β|γ). The amplitudes m(β|γ) are the entries of the inverse of the field theory KLT kernel matrix. § KAWAI–LEWELLEN–TYE RELATIONS AS TWISTED PERIOD RELATIONS Twisted de Rham theory developed primarily in Japan towards the end of twentieth century has been motivated by trying to understand properties of hypergeometric functions. In particular, an interest lies in finding algebraic relations between different hypergeometric functions. The simplest instance of such an identity is a quadratic relation between Euler beta functions, B(a,b):B(a,b) B(-a,-b) = -π(1/a + 1/b) ( 1/tanπ a + 1/tanπ b), where B(a,b) := ∫_0^1 z^a-1 (1-z)^b-1 dz.In pursuit of generalizing this relation to other integrals of multi-valued functions, Cho and Matsumoto discovered identities called the twisted period relations<cit.>. In this section we discuss how to apply these relations to the case of string theory scattering amplitudes and show their equivalence with the Kawai–Lewellen–Tye relations <cit.>.We first review the statement of twisted period relations. Let us consider a twisted homology H_m(X,ℒ_ω) and the associated twisted cohomology H^m(X,∇_ω) on an m-dimensional manifold X and choose a basis of twisted cycles γ_i ⊗ u_γ_i(z) and twisted cocycles φ_j(z) with i,j = 1,2,…, d. Recall that due to a twisted version of de Rham theorem, dimensions of both spaces are equal <cit.>, i.e., d :=H_m(X,ℒ_ω) =H^m(X,∇_ω). We can organize the bilinears between the bases of twisted cycles and cocycles into a d × d matrix with elements:𝐏_ij := γ_i ⊗ u_γ_i, φ_j(z)= ∫_γ_i ⊗ u_γ_iφ_j(z).This defines a twisted period matrix𝐏.[Recall that a period is an integral of an algebraic function over a domain specified by polynomial inequalities <cit.>. Twisted period is a natural extension of this definition <cit.>.] Similarly, we can choose the dual twisted homology H_m(X,ℒ_ω^∨) together with its associated twisted cohomology H^m(X,∇_ω^∨) on the same manifold. Recall that that dual here means that the homology is defined with a multi-valued function u^-1(z) insted of u(z). Once again, we choose bases of twisted cycles γ_i^∨⊗ u_γ_i^∨^-1(z), as well as twisted cocycles φ_j^∨(z) with i,j = 1,2,…, d of the same dimension d as above. This leads to the definition of a dual twisted period matrix 𝐏^∨ with elements: 𝐏_ij^∨ := γ_i^∨⊗ u_γ_i^∨^-1, φ_j^∨(z)= ∫_γ_i^∨⊗ u_γ_i^∨^-1φ^∨_j(z).Relating these two matrices requires a definition of additional pairings between twisted homology and cohomology groups. It turns out one can define a non-degenerate pairingH_m(X,ℒ_ω) × H_m(X,ℒ_ω^∨)⟶ ℂ,called the intersection number of twisted cycles<cit.>. In (<ref>) at least one of the twisted cycles ought to be compact. It owes its name to the fact that evaluation of this pairing requires the knowledge of how twisted cycles intersect one another topologically, aided with an information of the branch structure of both twisted cycles. Intersection theory of twisted cycles was originally developed by Kita and Yoshida <cit.>. We will give precise definition of (<ref>) in Section <ref>, together with the discussion of how to construct a regularization map from H^lf_m(X,ℒ_ω) to H_m(X,ℒ_ω). For the time being, let us define a d × d matrix 𝐇 built out of the pairings (<ref>): 𝐇_ij = γ_i ⊗ u_γ_i, γ_j^∨⊗ u_γ_j^∨^-1.Similarly, there exists a pairing between the two twisted cohomologies,H^m(X,∇_ω) × H^m(X,∇_ω^∨)⟶ ℂ,known as the intersection number of twisted cocycles<cit.>. In (<ref>) at least one of the twisted cocycles needs to be with compact support. Different ways of evaluating this pairing were given by Deligne and Mostow <cit.>, Cho and Matsumoto <cit.>, as well as Ohara <cit.>. We can now define another d × d matrix 𝐂 with elements: 𝐂_ij = φ_i(z), φ_j^∨(z) .Cho and Matsumoto showed <cit.> that the matrices defined above can be related by: 𝐂 = 𝐏^⊺ (𝐇^-1)^⊺𝐏^∨or equivalently𝐇 = 𝐏 (𝐂^-1)^⊺ (𝐏^∨)^⊺ These are the twisted Riemann period relations.[The name comes due to the resemblance of (<ref>) to the standard period relations on Riemann surfaces, see, e.g., <cit.>.] As long as the matrices 𝐏,𝐏^∨,𝐇,𝐂 are defined by bases of their respective homologies and cohomologies, they are invertible. By 𝐏^⊺ we denote a transpose of the matrix 𝐏. The relations (<ref>) hold under the condition that the cocycles in the bases φ_i(z) and φ_j^∨(z) are logarithmic <cit.>.Note that the dual twisted homology and cohomology are defined with a multi-valued function u^-1(z). In order to apply the above relations to string theory amplitudes, we need to consider a different set of spaces defined with a complex conjugate function u(z) instead. Such a setting was first considered by Hanamura and Yoshida <cit.>, and later studied in the context of Selberg-type integrals by Mimachi and Yoshida <cit.>, see also <cit.>. Indeed, a canonical isomorphism ℒ_-ω≅ℒ_ω can be defined when the exponents α_i in u(z) are real and sufficiently generic. From now on we will implicitly use such an isomorphism and work with the dual twisted homology defined by the system ℒ_ω^∨ = ℒ_ω and a dual twisted cohomology defined with the connection ∇_ω^∨ = ∇_ω. See <cit.> for details of this construction. The pairing (<ref>) then takes the form:[The name intersection number of twisted cocycles is justified only in the case of the dual cohomology defined with ∇_ω^∨ = ∇_-ω, where the pairing receives contributions only from certain regions of the moduli space. We discuss it in Section <ref>. In the case ∇_ω^∨ = ∇_ω there is nothing to intersect. To author's best knowledge, the intersection form of cohomology groups (<ref>) is poorly understood beyond the one-dimensional case.]φ_i(z), φ_j^∨(z):= ∫_X |u(z)|^2 φ_i(z) φ_j^∨(z),such that the integral converges. Study of the Hodge structure of such integrals was initiated in <cit.>. Let us now turn to the problem of formulating tree-level string theory amplitudes in the language of twisted de Rham theory. §.§ Twisted Cycles for String Amplitudes Open string scattering amplitudes are defined on the moduli space of genus-zero Riemann surfaces with n punctures, X=ℳ_0,n. After gauge fixing the positions of three of them to (z_1, z_n-1, z_n) = (0,1,∞), the regions of integration of the open string amplitudes are given by a disk ordering 𝔇(β) with a permutation β, which is an (n-3)-simplex labelled by β: Δ_n-3(β) := { 0 < z_β(2) < z_β(3) < ⋯ < z_β(n-2) < 1 }.It is embedded in the real section of the moduli space, ℳ_0,n(ℝ). Here the overbar denotes a closure of the space. Twisted cycles are then defined as follows. Twisted cycle on X = _0,n labelled by a permutation β is given by (β) := Δ^o_n-3(β) ⊗ 𝖲𝖫_β[u(z)], whose topological part is the interior of the simplex Δ_n-3(β). The branch of u(z) for a given twisted cycle is chosen according to the so-called standard loading, denoted by 𝖲𝖫. We define it as 𝖲𝖫_β[u(z)] = ∏_i<j( z_β(j) - z_β(i))^α' s_β(i),β(j). The set {(β)| β∈ (1,𝔖_n-3(2,3,…,n-2),n-1,n)} of cardinality (n-3)! forms a basis of twisted cycles. Here, 𝔖_n-3 denotes permutations of a set of n-3 labels.Twisted cycles are elements of H_n-3^lf(X,ℒ_ω). The size of the basis is known to be (n-3)! from the study of Selberg integrals by Aomoto, see, e.g., <cit.>, as well as the BCJ basis for open string amplitudes <cit.>, or equivalently size of the KLT matrix <cit.>. Of course, one can also choose different bases of twisted cycles labelled by different sets of (n-3)! orderings, not necessarily being related by a permutation operator. The multi-valued function u(z) is given by the Koba–Nielsen factor:u(z) := ∏_i<j(z_i - z_j)^α' s_ij = ∏_i=2^n-2(0 - z_i)^α' s_1i∏_i=2^n-2(z_i - 1)^α' s_i,n-1∏_2 ≤ i < j ≤ n-2(z_i - z_j)^α' s_ij.The twist 1-form ω then becomes: ω = d log∏_i<j (z_i - z_j)^α' s_ij = α' ∑_i<j s_ijd log (z_i - z_j) = α' ∑_i=2^n-2( ∑_j ≠ is_ij/z_i - z_j) dz_i = α' ∑_i=2^n-2 E_idz_i,where E_i := ∑_j ≠ i s_ij/(z_i - z_j) are the so-called scattering equations<cit.>. The divisor D is defined by the singular locus of u(z), i.e.,D := ⋃_i=2^n-2 { z_i = 0 } ⋃_i=2^n-2 { z_i - 1 = 0 }⋃_2 ≤ i < j ≤ n-2{ z_i - z_j = 0 }.Since D does not belong to the manifold X, the objects (<ref>) have no boundaries in X, that is ∂ (β) = ∅ for any β. This justifies the use of the name twisted cycle.The above definition is not fully satisfactory, as it contains singular points when more than two punctures coalesce at once. In order to resolve this issue, we consider a Deligne–Mumford–Knudsen compactification <cit.> of ℳ_0,n, given by the so-called minimal blowup, π^-1(ℳ_0,n) = ℳ_0,n<cit.>. A blowup of a simplex (<ref>) is a polytope called the associahedron<cit.>. We will study this object more closely in Section <ref>. An additional regularization from locally finite twisted homology H_n-3^lf(X,ℒ_ω) into H_n-3(X,ℒ_ω), where the twisted cycles are compact can be constructed by considering Pochhammer contour and its higher-dimensional generalizations, see, e.g., <cit.>. We will show how to obtain it in Section <ref>, and how to use it the study of the field theory limit of open string amplitudes in Appendix <ref>. §.§ Twisted Cocycles for String Amplitudes The dimension of the twisted cohomology group H^n-3(X,∇_ω) is also (n-3)!. A convenient basis for this space studied in the string amplitudes literature <cit.> is given by the so-called Parke–Taylor factors:[Note that in order to be consistent with the literature, we have not permuted the differential form in the numerator. As a consequence, Parke–Taylor factors for different permutations are related by relabelling and an additional change of sign.](β) = dz_1dz_2 dz_3⋯ dz_n-2 dz_n-1 dz_n/(z_β(1) - z_β(2))(z_β(2) - z_β(3)) ⋯ (z_β(n-1) - z_β(n))(z_β(n) - z_β(1))/ vol SL(2,ℝ).Here one needs to fix the SL(2,ℝ) redundancy in the same way as for twisted cycles by taking (z_1, z_n-1, z_n) = (0,1,∞) and compensating with a constant Faddeev–Popov factor: vol SL(2,ℝ) = dz_1dz_n-1 dz_n/(z_1 - z_n-1)(z_n-1 - z_n)(z_n - z_1),This leads to the following definition for twisted cocycles. Twisted cocycle on X = ℳ_0,n labelled by a permutation β is given by (β) := dz_2 dz_3⋯ dz_n-2/(0 - z_β(2))(z_β(2) - z_β(3)) ⋯ (z_β(n-2) - 1). The set {(β)| β∈ (1,𝔖_n-3(2,3,…,n-2),n-1,n)} of cardinality (n-3)! forms a basis of twisted cocycles.Twisted cocycles are elements of H^n-3(X,∇_ω). Once again, it is often necessary to consider a blowup of (<ref>) defined on ℳ_0,n. We illustrate how to perform it in practice in the Appendix <ref>, see also <cit.>. In order to satisfy the assumptions of the twisted period relations (<ref>), it is required that the twisted cycles (<ref>) are logarithmic. In the following we prove by construction that (<ref>) is a logarithmic differential form. The Parke–Taylor factor (<ref>) can be represented as a logarithmic (n-3)-form: (β) = (-1)^n(β)dlog(0 - z_β(2)/z_β(2),β(3))∧dlog(z_β(2),β(3)/z_β(3),β(4))∧ ⋯ ∧dlog(z_β(n-3),β(n-2)/z_β(n-2) - 1), where z_ab := z_a - z_b. Note that the prefactor is a constant. We will prove the claim inductively in n. For clarity of notation let us specialize to the canonical permutation _n = (12 ⋯ n) without loss of generality. The cases n=3,4 can be checked explicitly: (_3) = -1, (_4) = d log0 - z_2/z_2 - 1 = ( 1/z_2 - 1/z_2 - 1) dz_2 = dz_2/(0-z_2)(z_2-1). Assuming that the statement is true for n-2 and n-1, for n ≥ 5 we find: (_n)= (-1)^ndlog0 - z_2/z_23⋯ dlogz_n-4,n-3/z_n-3,n-2 dlogz_n-3,n-2/z_n-2 - 1 = (-1)^n( dlog0 - z_2/z_23⋯ dlogz_n-4,n-3/z_n-3,n-2) ( dz_n-3/z_n-3,n-2 - (z_n-3-1)dz_n-2/z_n-3,n-2(z_n-2-1)). The term in the first pair of brackets is almost proportional to (_n-1), however it includes an additional variable z_n-2 which is a constant in the definition of (_n-1). We need to take it into account by adding an extra differential with respect to z_n-2. Then the term in the first brackets becomes: (_n-1)/(-1)^n-1 + ( dlog0-z_2/z_23⋯ dlogz_n-5,n-4/z_n-4,n-3) dz_n-2/z_n-3,n-2. Once again, the term in the brackets is proportional to the lower-point case (_n-2) plus terms including the 1-form dz_n-3. However, the additional terms give rise to the 2-form dz_n-3 dz_n-2 in the expression (<ref>), and hence vanish in the full expression for (_n), since they are wedged with the second bracket in (<ref>). Up to these terms (<ref>) equals (_n-1)/(-1)^n-1 + (_n-2)/(-1)^n-2dz_n-2/z_n-3,n-2. Note that in both (_n-1) and (_n-2) we have set the punctures fixed at 1 to have an arbitrary position, denoted by z_n-3 and z_n-4 respectively. Plugging the above expression into (<ref>) we find: (_n)= (-1)^n( (_n-1)/(-1)^n-1 + (_n-2)/(-1)^n-2dz_n-2/z_n-3,n-2) ( dz_n-3/z_n-3,n-2 - (z_n-3-1)dz_n-2/z_n-3,n-2(z_n-2-1)) = 1/z_n-3,n-2( z_n-3-1/z_n-2-1 (_n-1)dz_n-2 - 1/z_n-3,n-2 (_n-2)dz_n-3 dz_n-2). We now use the inductive assumption to obtain: (_n)= 1/z_n-3 - z_n-2( z_n-3-1/z_n-2-1 1/(0-z_2) (z_2 - z_3)⋯ (z_n-4-z_n-3) (z_n-3 - z_n-2) - 1/z_n-3 - z_n-2 1/(0-z_2) (z_2 - z_3)⋯ (z_n-4 - z_n-3))dz_2 ⋯ dz_n-2 = dz_2dz_3 ⋯ dz_n-2/(0-z_2)(z_2 - z_3)⋯ (z_n-3 - z_n-2)(z_n-2 - 1)( z_n-3 - 1/z_n-3 - z_n-2 - z_n-2 - 1/z_n-3 - z_n-2) = dz_2dz_3 ⋯ dz_n-2/(0-z_2)(z_2 - z_3)⋯ (z_n-3 - z_n-2)(z_n-2 - 1), which completes the proof.Note how due to its recursive nature, (_n) in its logarithmic form contains Fibonacci number of terms, F_n-2<cit.>, that all collapse to a single one (<ref>) once summed over. One may wonder if generalizations of the Parke–Taylor factor used to describe multi-trace amplitudes <cit.> or general scalar theories <cit.> are logarithmic. In the following we show that they are not, and therefore cannot enter the bases of the twisted homologies used in twisted period relations (<ref>). The multi-trace Parke–Taylor factors of the form (β | γ | ⋯) = dz_1dz_2 ⋯ dz_n/(z_β(1) - z_β(2)) ⋯ (z_β(|β|) -z_β(1)) (z_γ(1) - z_γ(2)) ⋯ (z_γ(|γ|) - z_γ(1)) ⋯ / vol SL(2,ℝ), where the permutations β, γ, … are a partition of (12⋯ n), are not logarithmic on _0,n. Recall that a differential form is logarithmic if it has no higher-order poles along the divisor of X given by the singular locus of u(z). We will show that for the multi-trace Parke–Taylor factor, there always exists a higher-order pole, and therefore it cannot be logarithmic. Let us focus on the subpermutation β, which without loss of generality we can choose to be β = (12⋯ m) for some 2 ≤ m ≤ n-2. Since |β| < n-1, we can fix two of the punctures in the remaining permutations to be 1 and ∞. Let us also take z_1 = 0. We then perform a blowup along the face {z_1 = z_2 = ⋯ = z_m} by taking z_i = τ y_i for i=1,2,…,m, as well as set y_1 = 0, y_m = 1. Changing the variables of integration from {z_2, z_3, …, z_m} to {τ,y_2,y_3,…,y_m-1}, the differentials in the numerator scale as dz_2dz_3 … dz_m∼ τ^m-2 dτ, while the denominator scales as (0 - z_2)(z_2 - z_3) ⋯ (z_m - 0)∼ τ^m. Hence the contour given by {|τ| = ε} receives the contribution proportional to dτ / τ^2, which is a double pole. We conclude that multi-trace Parke–Taylor factors are not logarithmic.§.§ KLT Relations Revisited With the definitions of twisted cycles (<ref>) and twisted cocycles (<ref>) we can study their pairings. We also have analogous definitions for the dual spaces H_n-3(X,ℒ_ω^∨) and H^n-3(X,∇_ω^∨), whose bases we label with (β)^∨ and (β)^∨ respectively. Elements of the period matrices (<ref>) and (<ref>) then become: (β), (γ)= Z_β(γ) and(β)^∨, (γ)^∨ = Z_β(γ),Both of these bilinears give the Z-integrals as defined in (<ref>). Similarly, a pairing between two twisted cocycles is given by the J-integral (<ref>): (β), (γ)^∨ = J(β | γ).In Section <ref> we will prove that: (β), (γ)^∨ = (i/2)^n-3 m_α'(β | γ),for an appropriately defined pairing between the two twisted cycles computed by their intersection number. Here, m_α'(β | γ) denotes the α'-corrected bi-adjoint scalar amplitudes introduced in <cit.>. We can build matrices out of the above pairings and apply the twisted period relation (<ref>) in order to obtain the relation:[By m_α'^-1(β | γ) we denote the inverse of a matrix m_α'(γ | β) with rows labelled by γ∈𝒞 and columns labelled by β∈ℬ.] J(δ | ϵ) = ∑_β∈ℬ, γ∈𝒞 Z_β(δ)m_α'^-1(β | γ)Z_γ(ϵ).Note that here we have absorbed the constant factor (i/2)^n-3 from (<ref>) into the definition a coupling constant of J(δ | ϵ). Using the fact that open string amplitudes can be expanded in the basis of Z-integrals, and closed string amplitudes can be expanded in the basis of J-integrals as: 𝒜^open(β) = ∑_δ∈𝒟 n(δ)Z_β(δ) and𝒜^closed = ∑_δ∈𝒟, ϵ∈ℰ n(δ)n(ϵ)J(δ | ϵ), we find 𝒜^closed = ∑_β∈ℬ, γ∈𝒞𝒜^open(β)m_α'^-1(β | γ) 𝒜^open(γ).These are the Kawai–Lewellen–Tye relations <cit.>. We conclude that twisted period relations for string theory amplitudes are equivalent to KLT relations. In fact, similar identities can be written for any other string-like models having BCJ representations of the form (<ref>). Let us illustrate (<ref>) with an example for n=4. The sizes of the bases are 1, and we can choose them to be ℬ=𝒞={(1234)}. The KLT relations (<ref>) then read: 𝒜^closed_4 = 𝒜^open(1234) (1/tanπα' s + 1/tanπα' t)^-1𝒜^open(1234), where we used the notation s = s_12 and t = s_23. We will give a method of calculating the above coefficient of KLT expansion in Section <ref>. The four-point open string amplitude is given by the Veneziano amplitude <cit.> proportional to the beta function, B(α' s, α' t). Plugging it into (<ref>) and using trigonometric identities, we find: 𝒜^closed_4 = sinπα' s sinπα' t/sinπα'(s+t) B(α' s, α' t)^2 = -πα'^2 u^2 Γ(α' s) Γ(α' t) Γ(α' u)/Γ(1-α' s) Γ(1-α' t) Γ(1-α' u), where u = s_13 = -s - t by momentum conservation. This expression is indeed proportional to the Virasoro–Shapiro amplitude for four-point closed string scattering <cit.>. For n=5 the size of the basis is 2. Let us take ℬ={(12345),(12435)} and 𝒞={(13254),(14253)}. The KLT relations (<ref>) become: 𝒜^closed_5= [ [ 𝒜^open(12345); 𝒜^open(12435) ]]^⊺[ [ 1sinπα' s_23 sinπα' s_450;0 1sinπα' s_24 sinπα' s_35 ]]^-1[ [ 𝒜^open(13254); 𝒜^open(14253) ]] = sinπα' s_23 sinπα' s_45 𝒜^open(12345) 𝒜^open(13254)+(3 ↔ 4). In Section <ref> we will discuss how to calculate entries of the above inverse of the KLT kernel. For more examples of KLT relations we refer the reader to <cit.>.§.§ Basis Expansion and the Circuit Matrix A related question in twisted de Rham theory is how monodromy group acts on the period matrix (<ref>), see, e.g. <cit.>. More precisely, the problem translates to finding a matrix 𝐌 which relates two period matrices 𝐏' and 𝐏 with different choices of bases for twisted cycles, say 𝒟 and 𝒟': 𝐏' = 𝐌𝐏.The representative of the monodromy group, 𝐌, is called a circuit matrix<cit.>. Following the derivation given in <cit.>, we can show how to construct the entries of the matrix 𝐌 from intersection numbers of twisted cycles in the following way. Let 𝐇 be a d × d matrix of intersection numbers of twisted cycles defined with bases 𝒞 and 𝒟 for the rows and columns respectively, and 𝐇_k' be an (n-3)! vector of intersection numbers of a given twisted cycle labelled by k and the basis 𝒞. Similarly, let 𝐏_l be a vector of pairings between the basis 𝒟 and a given twisted cocycle labelled by l, and 𝐏_kl be a pairing between the twisted cycle k and twisted cocycle l. We can organize these objects into a (d+1) × (d+1) matrix, whose determinant equals <cit.>: [ [ 𝐇𝐇_k'; 𝐏_l^⊺ 𝐏_kl'; ]] = ( 𝐏_kl' - (𝐇_k')^⊺ (𝐇^-1)^⊺𝐏_l ) 𝐇 = 0.This expression vanishes because the final column of the matrix is linearly dependent of the remaining d columns, which form a basis. Since 𝐇 is non-vanishing, the term in the brackets ought to be equal to zero. Repeating the same procedure d × d times, we can build a new period matrix 𝐏', whose basis of twisted cycles is 𝒟', while 𝐏 has a basis 𝒟. Note that both matrices have the same bases of twisted cocycles. Rows of 𝐇' are labelled by the basis 𝒞, while its columns are in 𝒟'. The final expression reads: 𝐏' = (𝐇')^⊺ (𝐇^-1)^⊺𝐏implying𝐌 = (𝐇')^⊺ (𝐇^-1)^⊺,which gives an explicit realization of the circuit matrix 𝐌. Note that this expression is independent of the choice of the basis 𝒞, whose labels are contracted in the expression for 𝐌. When 𝒟=𝒟' we have 𝐌=, as expected. It would be interesting to understand how (<ref>) arises directly from twisted de Rham theory.In the case of string amplitudes, this expression translates to: 𝒜^open(β) = ∑_γ∈𝒞, δ∈𝒟 m_α'(β | γ)m_α'^-1(γ | δ) 𝒜^open(δ),which gives a way of expressing a given open string partial amplitude in a BCJ basis <cit.> given by a set 𝒟 of (n-3)! partial amplitudes 𝒜^open(δ) for δ∈𝒟. For examples of how to evaluate (<ref>) see <cit.>.§ INVERSE KLT KERNEL AS INTERSECTION NUMBERS OF TWISTED CYCLES Intersection theory for twisted cycles was introduced by Kita and Yoshida in 1992 <cit.>, who later developed it further in a series of papers <cit.>. Since then, intersection numbers have been evaluated for a large family of different types of hypergeometric functions <cit.>, including Selberg-type integrals <cit.>. For our purposes, intersection numbers of twisted cycles play a central role in the KLT relations by computing entries of the inverse of the KLT kernel. It is therefore important to understand how to evaluate them in the setting of string intergrals. In this section we discuss a combinatorial way for computing intersection numbers of twisted cycles and prove its equivalence to the diagrammatic rules for calculating m_α'(β | γ) given in <cit.>.Let us first review the key aspects of the intersection numbers of twisted cycles. Let H^lf_m(X,ℒ_ω) be the m-th locally finite twisted homology group on a non-compact m-dimensional manifold X = ℂ^m ∖ D, where the divisor D is the singular locus of a multi-valued function u(z) = ∏_i=1^k f_i(z)^α_i. The twist 1-form ω = d log u(z) defines an integrable connection ∇_ω = d + ω. The twisted homology has coefficients in ℒ_ω, the local system of solutions to the differential equation dξ = ωξ. Twisted cycles are then elements of H^lf_m(X,ℒ_ω). Working under the assumption that the exponents α_i ∈ℝ∖ℤ of u(z) are sufficiently generic, one can define an isomorphismH_m^lf(X,ℒ_ω)H_m(X,ℒ_ω),which is the inverse of the natural map from H_m(X,ℒ_ω) to H_m^lf(X,ℒ_ω). We refer to the map (<ref>) as regularization<cit.>. We will give plenty of explicit examples of regularized twisted cycles in the following sections.Similarly, we have a dual m-th locally finite twisted homology group H_m^lf(X,ℒ_ω^∨) with the coefficients in the local system ℒ_ω^∨ defined with d ξ = - ωξ. Kita and Yoshida showed <cit.> that there exists a non-degenerate pairing,H_m(X,ℒ_ω) × H_m^lf(X,ℒ_ω^∨) ℂ,known as the intersection form. Together with the regularization map (<ref>), it defines the intersection number of two twisted cycles, 𝖢 = γ⊗ u_γ(z) ∈ H_m^lf(X,Ł) and𝖢^∨ = γ^∨⊗ u^-1_γ^∨(z) ∈ H_m^lf(X,ℒ_-ω)as reg 𝖢∙𝖢^∨ = ∑_z ∈γ ∩ γ^∨𝖨𝗇𝗍_z(γ, γ^∨)u_γ(z)u_γ^∨^-1(z).Here, 𝖨𝗇𝗍_z(γ, γ^∨) is the topological intersection number of two topological cycles γ and γ^∨ at point z. The sum proceeds over all intersections between the two cycles. When they intersect non-tangentially—which will be the case throughout this work—the topological intersection number 𝖨𝗇𝗍 is equal to +1 or -1 depending on their relative orientation, as follows: -.35 < g r a p h i c s >=+1 or-.35 < g r a p h i c s >=-1. Despite the fact that most of the literature on intersection numbers of twisted cycles has been focused on studying the pairing with the dual homology defined with ℒ^∨ = ℒ_-ω, one can also apply these ideas to the case complex conjugate case ℒ^∨ = ℒ_ω which is more relevant to physics. Hanamura and Yoshida <cit.> considered an isomorphism ℒ_-ω≅ℒ_ω which can be canonically defined if all α_i are real and sufficiently generic. Then, for two twisted cycles given by 𝖢 = γ⊗ u_γ(z) ∈ H_m^lf(X,Ł) and𝖢^∨ = γ^∨⊗u_γ^∨(z)∈ H_m^lf(X,ℒ_ω)the intersection number is defined as:𝖢∙𝖢^∨ = ∑_z ∈γ ∩ γ^∨𝖨𝗇𝗍_z(γ, γ^∨)u_γ(z) u_γ^∨(z) /|u(z)|^2,which is analogous to (<ref>). Indeed, when the exponents α_i are real, both definitions agree with each other. For this reason, for considerations of intersection numbers of twisted cycles it will not be important to make distinction between the two cases ℒ_-ω and ℒ_ω, and hence we will denote twisted cycles belonging to both twisted homologies with same symbols. We will also not distinguish betweenand ^∨, as they are given by the same definition (<ref>).Let us focus on the twisted cycles relevant to open string scattering amplitudes. Recall that the multi-valued function defining the local system Ł is given by the Koba–Nielsen factor:u(z) = ∏_i<j (z_i - z_j)^α' s_ij.Here, the Mandelstam invariants s_ij = k_i · k_j in the exponents are chosen in such a way that none of the invariants s_ij… = (k_i + k_j + …)^2/2 is an integer. In the following we will set α' = 1 for clarity of notation. The manifold X is the moduli space of genus-zero Riemann surfaces with n punctures, ℳ_0,n. Twisted cycles (β) on this space were defined in (<ref>) with the standard loading operator 𝖲𝖫, which chooses the branch of the Koba–Nielsen factor for a given permutation β in a canonical way. Using this definition, in the case of n=4 we have: (1234) = { 0 < z_2 < 1 }⊗ z_2^s_12(1-z_2)^s_23 = (0,1)⊗ z^s (1-z)^t,where we denote the only manifold coordinate as z = z_2 and the exponents with the usual notation s = s_12 and t = s_23. In the case of n=5 the basis has two elements: (12345) = { 0 < z_2 < z_3 < 1 }⊗ z_2^s_12 (1-z_2)^s_24 (z_3 - z_2)^s_23 z_3^ s_34 (1-z_3)^ s_34, (13245) = { 0 < z_3 < z_2 < 1 }⊗ z_2^s_12 (1-z_2)^ s_24 (z_2 - z_3)^ s_23 z_3^ s_34 (1-z_3)^s_34.One can also define other bases of twisted cycles (β). They have a straightforward definition analogous to (<ref>). For instance, in the next section we will make us of the four-point twisted cycles: (2134) = (-∞,0)⊗ (-z)^s (1-z)^t and(1324) = (1,∞)⊗ z^s (z-1)^t.Before evaluating intersection numbers let us give an explicit construction of the regularization map (<ref>) for twisted cycles (β), as well as discuss how they are affected by the blowup procedure <cit.>. §.§ Regularization of Twisted Cycles The cycles relevant for string amplitudes (<ref>) are non-compact. Since the definition of the intersection number requires at least one of the twisted cycles to be compact, we need to employ a regularization. In this section we discuss an explicit construction of such a map, based on the Pochhammer contour and its higher-dimensional generalizations, see, e.g., <cit.>.Let us review how the standard Pochhammer contour is constructed. We start by considering the integral:I := ∫_0^1 z^s (1-z)^t φ(z),where s,t ∉ℤ and φ(z) is any single-valued 1-form. As defined, the integral converges only for sufficiently positive values of s and t. In order to make the it convergent for all values of these parameters, one can employ an alternative contour of integration γ, known as the Pochhammer contour: γ:=< g r a p h i c s >This contour winds around the two branch points z=0,1 once in both directions. We picture the branch cuts as extending from z=0,1 downwards to -i∞. Let us track how this contour is related to the one used in (<ref>). Starting from the point P and moving right, we first obtain the contribution equal to I. After winding around z=1 in a positive direction along C_1, one picks up a phase factor e^2π i t, so that the next stretch towards z=0 equals to -e^2π i t I, where the minus comes from a different orientation that (<ref>). Next, winding around z=0 gives an additional factor of e^2π i s from C_0, so that the following contribution becomes e^2π i (s+t) I. Winding around z=1, this time in a negative direction C_1^', takes the phase factor back to e^2π i t, so that the final contribution is e^2π i t I. After performing another turn around z=0 in a negative direction given by C_0^', we land at the point P on the original branch. Summing up the contributions, we have: ∮_γ z^s (1-z)^tφ(z) = ( 1 - e^2π i t + e^2π i (s+t) - e^2π i s) ∫_0^1 z^s (1-z)^tφ(z),or equivalently ∫_0^1 z^s (1-z)^t φ(z) = ∮_γ' z^s (1-z)^tφ(z) withγ' := γ/( e^2π i s - 1) ( e^2π i t - 1).Let us split the contour γ' into three parts: regions near the two branch points z=0,1, and the interval (ε, 1-ε). In order to be precise, we will use a small parameter ε as the radius of the circular contours. The contributions near the branch point at z=0 give: C_0 + C_0^'/(e^2π i s-1)(e^2π i t-1) = ( e^2π i t - 1 ) S(ε,0)/(e^2π i s-1)(e^2π i t-1) = S(ε,0)/e^2π i s-1,where by S(a,z) we denote a positively oriented circular contour with centre at z and starting at a point a. Similarly, around z=1 we find the contribution C_1 + C_1^'/(e^2π i s-1)(e^2π i t-1) = (1 - e^2π i s) S(1-ε, 1)/(e^2π i s-1)(e^2π i t-1) = - S(1-ε,1)/e^2π i t-1.Finally, the contours along the real axis simply give (ε, 1-ε). Putting everything together, we find the regularization of the original cycle (0,1) to be:(0,1):= γ^'= S(ε,0)/e^2π i s-1 + (ε, 1-ε) - S(1-ε,1)/e^2π i t-1=< g r a p h i c s >.Here, we have introduced a graphical notation to denote the regularized cycle. It is understood that the circular parts of the contour come multiplied with the additional factors 1/(e^2π i s-1) and -1/(e^2π i t-1) that are not represented explicitly. We will make a repeated use of this regularization in the following sections. Note that we have been implicitly working with a twisted cycle (0,1)⊗ z^s (1-z)^t relevant for string amplitude calculations.Generalizations to higher-dimensional cycles can be made by performing a similar regularization <cit.>. Since locally we can describe a manifold X as a direct product of lower-dimensional spaces, we can employ the regularization (<ref>) near the singularities on these product spaces. In the case of X = ℳ_0,n with n ≥ 5, however, there is an additional difficulty coming from the fact that the singular locus of u(z) is not normally crossing. For example, in the case of n=5 the function u(z) defining the local system Ł is singular at { z_2 = 0}∪{ z_2 - 1 = 0 }∪{ z_2 - z_3 = 0 }∪{ z_3 = 0 }∪{ z_3 - 1 = 0},which has degenerate points at (z_2, z_3) = (0,0), (1,1), and also (∞,∞). The way forward is to consider a blowup of this space <cit.>, denoted by ℳ_0,5 = π^-1(ℳ_0,5), where all triple singular points get resolved.In Figure <ref> we have illustrated the real section of _0,5, denoted by _0,5(ℝ), where the twisted cycles live before the blowup, as well as its image, _0,5(ℝ). Note that in this representation we brought the point at infinity to a finite position for convenience. The resulting space is divided into twelve chambers separated by the singular lines. Each of the lines has an associated label corresponding to the exponent of the given zero in u(z), or equivalently a phase factor that one picks up upon crossing the branch line. For example, the line defined by {z_2 - 1 = 0} is labelled with (24), since it corresponds to the factor (z_2 - 1)^s_24 in u(z).Blowup has been performed in the neighbourhood of the points (0,0), (1,1), and (∞,∞), resulting in three new locally-defined curves labelled by (123), (234), and (235). For example, near (z_2,z_3) = (0,0) the blowup introduced a line (123) corresponding to the factor z_2^s_12(z_2 - z_3)^s_23z_3^s_13, whose exponents sum up to s_12 + s_23 + s_13 = s_123. Points labelled with the same symbol on these new curves are identified, and so are the segments between them. Each of the vertices can be uniquely specified as intersection of two lines, for instance the point (z_2,z_3)=(0,1) is written as (12) ∩ (34).Each of the twelve chambers after the blowup forms a polygon known as the associahedron, K_4.[Historically, skeleton of the associahedron first appeared the doctoral thesis of Tamari in 1951 <cit.>. In 1963, Stasheff gave a realization of the associahedron as a cell complex in his work on associativity of H-spaces <cit.>. For this reason, associahedron is often referred to as the Stasheff polytope. Since then, many realizations of the polytope have been constructed, see, e.g., <cit.>. For a historical account see, e.g., the introduction of <cit.>. Connection to the moduli space _0,n was first found by Kapranov in <cit.>, and from a combinatorial point of view later by Devadoss <cit.>. It was also independently rediscovered by Yoshida in the context of hypergeometric functions <cit.>.] Since twisted cycles (β) are defined as associahedra with a uniquely specified standard loading given by (<ref>), we will sometimes not distinguish between the two. Notice that the canonical twisted cycle (12345) is neighbouring all the other cycles, except for (14253), by either an edge or a vertex. Adjacency relations between different associahedra will be important in the evaluation of intersection numbers of twisted cycles. Regularized twisted cycles have a natural definition analogous to (<ref>). For example, (12345) in a small neighbourhood of the edge (23) can be represented as:[Orientations of the cycles are naturally induced from the right-handed manifold X. More precisely, a multi-dimensional residue { |g_i(z)| = ε} is oriented by ⋀_i dg_i > 0<cit.>. For the purpose of this section, however, this fact will not be important as we will consider pairings between twisted cycles, for which possible signs due to orientations always cancel out.] (12345) |_(23) = { < g r a p h i c s > }{ < g r a p h i c s > },where in both cases the horizontal direction is embedded in the real part of _0,5 illustrated in Figure <ref>. Similarly, near the point (12) ∩ (123) we have:(12345) |_(12) ∩ (123) = { < g r a p h i c s > }{ < g r a p h i c s > }. In general, one considers the Deligne–Mumford–Knudsen compactification <cit.> of ℳ_0,n, denoted by _0,n, in which the singular locus of u(z)=0 is normally crossing. It is given by the procedure called the minimal blowup<cit.>. It is known that real part of each chamber of _0,n is isomorphic to an associahedron K_n-1, see, e.g., <cit.>. We will give properties of associahedra for general n in Section <ref>, after studying examples of intersection numbers for n=4,5, which will illustrate how they are connected to adjacency relations between different associahedra. Generalized Pochhammer contour for K_n-1 is defined analogously to (<ref>) and (<ref>). We can now give a precise definition of the pairing between twisted cycles, which gives rise to the entries of 𝐇. Non-degenerate pairing between two twisted cycles is given by ⟨𝖢(β), 𝖢(γ) ⟩ :=(β) ∙(γ),where (β) and (γ) are two, not necessarily distinct, twisted cycles defined as a blowup of (<ref>). For simplicity we will use the same notation for n=4, even though in this case there is no need for a blowup.§.§ Four-point Examples We start evaluation of intersection numbers with the simplest example of n=4, which will illustrate most of the core ideas at play. We first consider the case of the self-intersection number of the twisted cycle (1234). In order to avoid degeneracy on the interval (ε, 1-ε), let us make a small deformation of one of the cycles into a sine-like curve, on top of the regularization (<ref>) for the other cycle:⟨(1234), (1234) ⟩ = ((0,1)⊗ z^s (1-z)^t ) ∙( (0,1)_sin⊗ z^s (1-z)^t )=< g r a p h i c s >= -1/e^2π i s-1 - 1 - 1/e^2π i t-1.There are three intersection points: near z=0, at z=1/2, and near z=1. The first contribution gives 1/(e^2π i s-1) from the definition (<ref>) times -1 arising from the topological intersection number (<ref>) for this relative orientation of the cycles. Similarly, the second factor is simply -1 due to the relative orientation at the intersection point at z=1/2. The final factor is -1/(e^2π i t-1) times +1 due to the orientation.Intersection numbers are independent of the deformation of the second twisted cycle <cit.>. For instance, we can calculate it with one of the cycles deformed into a small upside-down sine curve to obtain:⟨(1234), (1234) ⟩ = ((0,1)⊗ z^s (1-z)^t ) ∙( (0,1)_-sin⊗ z^s (1-z)^t )=< g r a p h i c s >= -e^2π i s/e^2π i s-1 + 1 - e^2π i t/e^2π i t-1.This time, the two end-point intersection numbers have picked up monodromy factors. Near z=0 we have 1/(e^2π i s-1) from the definition of (<ref>) times the phase factor e^2π i s times -1 due to the orientation. Similar reasoning gives the contribution from the neighbourhood of z=1. The mid-point intersection has changed orientation and hence give the contribution +1. Another choice is a deformation into an arc curve:⟨(1234), (1234) ⟩ = ((0,1)⊗ z^s (1-z)^t ) ∙( (0,1)_arc⊗ z^s (1-z)^t )=< g r a p h i c s >= -1/e^2π i s-1 - e^2π i t/e^2π i t-1,which receives contributions from only two intersection points, which we have analyzed before separately. Finally, we have the deformation:⟨(1234), (1234) ⟩ = ((0,1)⊗ z^s (1-z)^t ) ∙( (0,1)_-arc⊗ z^s (1-z)^t )=< g r a p h i c s >= -e^2π i s/e^2π i s-1 - 1/e^2π i t-1.It is straightforward to show that all the above calculations (<ref>), (<ref>), (<ref>), and (<ref>) give the same answer: ⟨(1234), (1234) ⟩ = i/2( 1/tanπ s + 1/tanπ t). Let us now turn to studying intersection numbers of two distinct twisted cycles. Intersecting (1234) with (2134) one obtains:⟨(1234), (2134) ⟩ = ((0,1)⊗ z^s (1-z)^t ) ∙( (-∞,0)⊗ (-z)^s (1-z)^t )=< g r a p h i c s >= e^π i s/e^2π i s - 1 = i/2( -1/sinπ s).This time, there is only one intersection point near z=0 giving 1/(e^2π i s - 1) times the monodromy factor e^π i s. The topological intersection number gives +1. In the remaining case of intersecting twisted cycles (1234) and (1324) we have:⟨(1234), (1324) ⟩ = ((0,1)⊗ z^s (1-z)^t ) ∙( (1,∞)⊗ (z)^s (z-1)^t )=< g r a p h i c s >= e^π i t/e^2π i t - 1 = i/2( -1/sinπ t),which comes from the contribution near z=1. Note that in both cases (<ref>) and (<ref>), the minus sign in the final answer can be tracked down to the fact that the two cycles involved induce opposite orientation on the boundaries at z=0 and z=1 respectively. For example, in the case (<ref>) the boundary of the first cycle (1234) is ∂(0,1) = {1} - {0}, and for the second cycle (1324) is ∂(1,∞) = {∞} - {1}, so the boundaries at z=1 contributing to the intersection number have opposite orientations.All of the remaining combinations of four-point twisted cycles can be obtained by relabelling the cases considered above. As we will see, the four-point cases (<ref>) and (<ref>) from this section will also serve as building blocks for intersection numbers for higher multiplicities. §.§ Five-point Examples Before moving on to the most general case, let us study several five-point examples to gain some intuition about higher-dimensional twisted cycles. Without loss of generality we can fix the first twisted cycle to be (12345) and consider its intersections with other cycles. A representation of the real section of the moduli space _0,5 was given in Figure <ref>, from which one can read off the adjacency of different twisted cycles.We first consider the self-intersection number ⟨(12345), (12345) ⟩. Kita and Yoshida showed <cit.> that one can define a deformation of the second cycle similar to the sinusoid we used in the n=4 case (<ref>). The deformation is made in such a way that the self-intersection number receives contributions from neighbourhoods of the barycenter of the associahedron K_4 itself, barycenters of all its facets, and its vertices. Due to the regularization employed, locally near a vertex given by H_1 ∩ H_2, where H_1 and H_2 are two facets, we receive a contribution 1/(e^2π i s_H_1-1)(e^2π i s_H_2-1), near the barycenter of each facet H_1 we get 1/(e^2π i s_H_1-1), and near the barycenter of the whole associahedron we get 1. Explicitly, we have:⟨(12345), (12345) ⟩ = 1 + 1/e^2π i s_12-1 + 1/e^2π i s_23-1 + 1/e^2π i s_34-1 + 1/e^2π i s_45-1 + 1/e^2π i s_51-1+ 1/(e^2π i s_12-1)(e^2π i s_34-1) + 1/(e^2π i s_23-1)(e^2π i s_45-1)+ 1/(e^2π i s_34-1)(e^2π i s_51-1) + 1/(e^2π i s_45-1)(e^2π i s_12-1)+ 1/(e^2π i s_51-1)(e^2π i s_23-1),which is a sum over contributions from five vertices, five facets, and one polygon. See <cit.> for details of the derivation. As a matter of fact, (<ref>) admits an alternative, more concise form:⟨(12345), (12345) ⟩ = ( i/2)^2 ( 1 + 1/tanπ s_12 tanπ s_34 + 1/tanπ s_23 tanπ s_45 + 1/tanπ s_34 tanπ s_51 + 1/tanπ s_45 tanπ s_12 + 1/tanπ s_51 tanπ s_23). Other cases can be obtained by reducing to previously calculated results. For example, two twisted cycles (12345) and (13245) share the facet (23). Working locally in its neighbourhood, we can write the intersection number as a product of the one in the real direction orthogonal to (23) times the boundary (23) itself:⟨(12345), (13245) ⟩ =< g r a p h i c s >= e^π i s_23/e^2π i s_23 - 1((12345) |_(23)∙ (13245) |_(23)) = -i/21/sinπ s_23(< g r a p h i c s > ) = - (i/2)^2 1/sinπ s_23( 1/tanπ s_45 + 1/tanπ s_51).Other cases follow the same algorithm. The remaining topology to consider is that of the intersection of (12345) with (12453). This time, these two cycles intersect at a vertex point (12) ∩ (45). We first consider the real direction orthogonal to (12) and then the intersection within (12). Being careful about the orientations of the cycles we find:⟨(12345), (12453) ⟩ =< g r a p h i c s >= e^π i s_12/e^2π i s_12 - 1((12345) |_(12)∙ (12453) |_(12)) = -i/21/sinπ s_12(< g r a p h i c s > ) = -(i/2)^2 1/sinπ s_12 1/sinπ s_45.Here, in the second step, both cycles induce the same orientation on the facet (123), giving an overall plus sign contribution. We will come back to the point of orientations induced on boundaries in the next section.There is only one chamber in _0,5(ℝ) which is not adjacent to (12345), pictured near the top of Figure <ref>. It corresponds to the twisted cycle (13524). Because the two cycles do not intersect, we have ⟨(12345), (13524) ⟩ = 0.In general, if two chambers are not adjacent, the corresponding intersection number vanishes.Having studied several examples, the general strategy for evaluating intersection numbers is now clear: after identifying the intersection face F of the two cycles, we project onto facets containing F one by one until the problem reduces to calculating self-intersection numbers for smaller twisted cycles. We will now prove that for general n this procedure reproduces the results of <cit.> and can be streamlined using simple diagrammatic rules. §.§ Proof of the General Case Let us review the structure of _0,n(ℝ) in the general case. It is known that this space is divided into (n-1)!/2 chambers, each isomorphic to an (n-3)-dimensional associahedron K_n-1, see, e.g., <cit.>. The space is divided by 2^n-1 - n - 1<cit.> hyperplanes bounding the associahedra, including the ones at infinity. For concreteness, let us specialize to the associahedron defined with the identity permutation, _n := (12⋯ n), which we denote byK_n-1(_n) := π^-1{ 0 < z_2 < z_3 < ⋯ < z_n-2 < 1 },where the overbar means closure of this space, so that bounding facets are also included. Associahedra for other permutations are defined analogously. Twisted cycles on the blowup space _0,n are then given as interior of the associahedron loaded with the function u(z) with an appropriate phase given by the standard loading (<ref>): (β) = K^o_n-1(β) ⊗𝖲𝖫_β [u(z)].This is a blowup of the definition (<ref>). Note that in this way we have identified only half of the (n-1)! cyclically-inequivalent permutations. This is because each associahedron comes with an orientation induced from the right-handed space _0,n(ℝ). The remaining half of the twisted cycles with reversed permutation β^⊺, for example _n^⊺ = (n ⋯ 21), can be related to the (n-1)!/2 set by (β^⊺) = (-1)^n (β),which means that when n is even, twisted cycles corresponding to associahedra with both orientations are identified.[An alternative is to consider an orientable double cover of _0,n(ℝ), whose combinatorics has been studied in <cit.>. Such space also has a known decomposition into (n-2)!permutohedra<cit.>, which in the language of amplitudes corresponds to the choice of a Del Duca–Dixon–Maltoni basis <cit.>.] In the odd case, the minus sign arises because of the change of integration region and gauge fixing condition for {z_1, z_n-1, z_n}. Note that a given permutation corresponds to a right-handed associahedron if the labels {z_1, z_n-1, z_n} come in the canonical ordering, and to a left-handed one otherwise.Following <cit.>, we will label the n(n-3)/2 facets bounding the chamber K_n-1(_n) with:(12 ⋯i), (23⋯i+1),…, (n-i,n-i+1, ⋯,n-1) for i=2,3,…,n-2. Each of these facets is a direct product of two other associahedra <cit.>. In our notation, for a face labelled by (ω) we have:(ω)≅K_|ω|(ω,-I) × K_n-|ω|(_n ∖ω,I),where by _n ∖ω we mean the complement of ω in _n. We have introduced a new label I, which can be thought of as corresponding to a puncture at infinity in both disk orderings. It inherits the phases from the Koba–Nielsen factor u(z) corresponding particles in the set ω, and hence the puncture with the label I can be represented as having associated momentumk_I^μ := ∑_i ∈ω k_i^μ = - ∑_i ∈_n ∖ω k_i^μ.Similarly, in the second associahedron, the label -I has an associated momentum -k_I^μ. Note that when |ω|=2, K_2 is a point and hence the corresponding facets can be thought of as being isomorphic to a single associahedron.Every codimension-k face F of K_n-1(_n), for k=1,2,…,n-3, can be uniquely specified as an intersection of k facets H_1,H_2,…,H_k from the above set (<ref>), i.e., F=H_1 ∩ H_2 ∩⋯∩ H_k. For each face F, the condition disjoint/contained is satisfied for all pairs of facets H_i = (ab⋯) and H_j=(cd⋯). It says that their labels are either disjoint, ab⋯∩ cd⋯ = ∅, or one contains the other, i.e., ab⋯⊂ cd⋯ or cd⋯⊂ ab⋯. For example, for the associahedron K_4(_5) we have five facets:(12),(23),(34),(123),(234),and its five codimension-2 faces, or vertices, are given by:(12) ∩ (34),(12) ∩ (123),(23) ∩ (123),(23) ∩ (234),(34) ∩ (234),which can be read off from the Figure <ref>.It is known that two associahedra sharing a facet H from the family (<ref>) have permutations that differ by a transposition of the labels of H<cit.>. For example, K_4(12345) and K_4(14325) share the facet (234). Whenever two associahedra are adjacent through a codimension-k face, they can be reached by a series of k such transposition moves, up to a change of orientation. Conversely, if such a series does not exist, then two associahedra are not adjacent. For instance, K_4(12345) and K_4(12453) and adjacent through the vertex (123) ∩ (23), which means that one can pass between them by crossing through (12) and (123) in either order. At the same time, K_4(12345) is not adjacent to K_4(13524), since they do not share any facets, see Figure <ref>. As a generalization of (<ref>), a codimension-k face is isomorphic to a product of k+1 associahedra <cit.>.Finally, let us remark on orientations that associahedra induce on their faces. For each facet (ω) from the set (<ref>), K_n-1(_n) and its neighbour induce the same orientation on (ω) if |ω| is odd, and an opposite one if |ω| is even <cit.>. For example, K_4(12345) and K_4(14325) induce the same orientation on (234), while K_4(12345) and K_4(13245) induce on opposite orientation on (23), as can be seen from Figure <ref>. Orientations of the lower-dimensional faces can be deduced from applying the same rules recursively. For a combinatorial description of the boundary operator acting on associahedra see <cit.>.In combinatorics, associahedron K_n-1 is a convex polytope whose vertices correspond to all legal ways of inserting bracketings in an word of length n-1 in the following way. A pair of brackets is assigned to each of the n(n-3)/2 facets (<ref>). Then, two facets meet if and only if their bracketings are compatible, i.e., satisfy the disjoint/contained condition. Repeating this procedure, every codimension-k face F corresponds to a correct insertion of k pairs of brackets. The number of such faces is given by T(n-2,k+1)<cit.>. Another interpretation, originally due to Loday <cit.>, is that of rooted trees with n-1 leaves, where a face F is a tree with k+1 nodes. We illustrate this in Figure <ref>.[For more visualizations of associahedra and tiling of moduli spaces see the work of Devadoss, e.g., <cit.>.]. We will think of the rooted trees as Feynman diagrams <cit.>. Let us now prove the statement that intersection numbers of twisted cycles give rise to the rules for evaluating the inverse KLT kernel m_α'(β|γ) given in <cit.>. We split the arguments into two parts. Firstly, we show that self-intersection numbers are proportional to diagonal amplitudes m_α'(_n|_n) in Lemma <ref>. Secondly, we show that the rules for evaluating arbitrary intersection numbers reduce to the self-intersections as building blocks according to the graphical rules of <cit.> in Theorem <ref>. The self-intersection number of the twisted cycle with identity permutation, (_n), is equal to the diagonal α'-corrected bi-adjoint scalar amplitude m_α'(_n | _n) given in <cit.> up to a global factor, (_n), (_n)= (i/2)^n-3 m_α'(_n | _n).Kita and Yoshida showed <cit.> that for general n the self-intersection number is given as a sum over contributions from barycenters of all the codimension-(0,1,…,n-3) faces of the associahedron. The contribution coming from a codimension-k face F = H_1 ∩ H_2 ∩⋯ H_k is a product of k terms 1/( e^2π i s_H_i - 1 ) for every facet H_i intersecting at F. More explicitly, we have: (_n), (_n)= (-1)^n-1∑_k=0^n-3 ∑_F = H_1 ∩⋯∩ H_k1/(e^2π i s_H_1-1) (e^2π i s_H_2-1) ⋯(e^2π i s_H_k-1),where we have also included the global sign <cit.>. The term in the sum corresponding to k=0, i.e., the one coming from the barycenter of the whole associahedron is regarded as 1. Examples of the evaluation of (<ref>) were given in (<ref>) for n=4, and in (<ref>) for n=5.Another way of thinking about the self-intersection number (<ref>) is using the interpretation of the associahedron described by Feynman diagrams, as illustrated in Figure <ref>. In this way, the sum (<ref>) proceeds over all possible Feynman diagrams in an auxiliary theory described by the following rules. Every internal edge with momentum p^μ gives rise to a propagator -1/(e^iπ p^2 - 1). The theory also has an infinite number of Feynman vertices with valency p=3,4,5,…, each coming with a factor of (-1)^p-1. It is straightforward to check that the signs give rise to the correct prefactor in (<ref>). It is useful to construct equations of motion for such a theory. Let us use a normalization such that factors of i/2 from (<ref>) are absorbed into propagators and vertices:-i/2( e^i π - 1 ) ϕ = ∑_p=3^∞(-2/i)^p-3ϕ^p-1 = ϕ^2/1 - 2 i ϕ.Here, ϕ is a real scalar matrix-valued field, and we have denoted := ∂_μ∂^μ. The left-hand side gives a normalized propagator -2/(i(e^i π p^2 - 1)), while the terms on right-hand side give rise to p-valent vertices with factors (-2/i)^p-3. The additional minus signs are responsible for the factor of (-1)^n-3 in (<ref>). One can verify that such a change of normalization yields a global factor for an amplitude that counterweights the prefactor of (<ref>).It was also conjectured in <cit.> that the α'-corrected bi-adjoint scalar amplitudes m_α'(_n | _n) can be expanded using another auxiliary theory with propagators given by 1/tanπ/2. The equation of motion then reads: (tanπ/2) φ = V'[φ], where the scattering field is denoted by φ and the functional V'[φ] describes Feynman rules for the vertices. The goal is to prove this theory yields the same amplitudes as the one described by (<ref>), with the function V'[φ] generating Catalan numbers as proposed in <cit.>.Scattering amplitudes can be obtained from both (<ref>) and (<ref>) using the following standard procedure, see, e.g., <cit.>. One introduces a coupling constant g, such that when g=0 the theory becomes free, i.e., the scattering field has no self-interactions. It is then possible to Taylor expand the field around g=0, so that equations of motion can be solved iteratively. On the support of this solution, the left-hand sides of (<ref>) and (<ref>) become generating functions of integrated scattering amplitudes. It is important that the left-hand sides contain the inverse propagator, which is responsible for striping away the only remaining external propagator. The bottom line is that in order for both equations of motion to produce the same amplitudes, the right-hand sides of both (<ref>) and (<ref>) have to be equal on the support of both equations of motion.Given this knowledge, let us find V'[φ] that gives the same amplitudes in both theories. Equating right-hand sides of (<ref>) and (<ref>) gives: (tanπ/2) φ = -i/2( e^i π - 1 ) ϕand V'[φ] = ϕ^2/1 - 2 i ϕ.Let us expand the tangent in the first equation to get: ( e^i π - 1 ) ( 1/i( e^i π + 1 )^-1φ + i/2ϕ) = 0.Since the term in the second brackets is not in the kernel of ( e^i π - 1 ) for generic momenta, it has to vanish. Formally multiplying the term in the second brackets by the operator i ( e^i π + 1 ) from the left, we obtain:0 = φ - 1/2( e^i π + 1 ) ϕ = φ - 1/2( e^i π - 1 ) ϕ - ϕ.We recognize the second term as being proportional to the left-hand side of the equation of motion (<ref>). On the support of (<ref>) we have: φ - i V'[φ] - ϕ = 0.Finally, using the second equation in (<ref>), we can eliminate ϕ in order to get a constraint on the functional V'[φ]:V'[φ]^2 - V'[φ] + φ^2 = 0. This equation has two solutions for V'[φ], however one of them has a constant independent of φ, which would not have an interpretation as a Feynman vertex in (<ref>). The other solution isV'[φ] = 1/2( 1 - √(1 - 4φ^2)) = ∑_p=3 odd^∞ C_(p-3)/2 φ^p-1 which is a generating function for the Catalan numbers C_(p-3)/2<cit.>. There is an infinite number of vertices of odd valency p=3,5,7,9,…, each contributing to a factor C_(p-3)/2 equal to 1,1,2,5,… respectively. This verifies the conjecture posed in <cit.> and concludes the proof of (<ref>).Total number of terms in the 1/(e^i π p^2 - 1) representation is given by the Schröder–Hipparchus number <cit.>. Total number of terms in the 1/tanπ/2 representation is given by the series <cit.>. For n ≥ 4 the latter is always smaller.The intersection number of two twisted cycles (β) and (γ) is equal to the α'-corrected bi-adjoint scalar amplitude m_α'(β | γ) given in <cit.> up to a global factor, (β), (γ)= (i/2)^n-3 m_α'(β | γ).We will show that evaluation of intersection numbers gives rise to a recursion relation which is the same as the graphical rules found in <cit.>.Let (β) and (γ) be two twisted cycles on _0,n. A codimension-h intersection face F of the corresponding associahedra K_n-1(β) and K_n-1(γ) can be written asF := K_n-1(β) ∩ K_n-1(γ) = H_1 ∩ H_2 ∩…∩ H_h.If F = ∅ then (β), (γ)= 0, since the cycles are not adjacent. If F = K_n-1(β) = K_n-1(γ), then necessarily (β) = (γ) up to an orientation, which gives the self-intersection number ±(β), (β) reducing to the case proven in Lemma <ref>.Otherwise, let us first fix orientations of both twisted cycles to be the same. If a change of orientation was needed and n is odd then the intersection number picks up a minus sign. Since F is also a codimension-h face belonging to both K_n-1(β) and K_n-1(γ), the labels of the facets in the set { H_1, H_2, …, H_h } are necessarily pairwise disjoint/contained. Let us pick one such facet H, such that all other H_i either contain H or are disjoint with H. Permutation β then splits naturally into two parts, β = (H,β∖ H), where β∖ H denotes the complement of H in β. Similarly, for γ we have γ = (H, γ∖ H).Let us consider intersection of these two twisted cycles locally as a product of the one in the real orthogonal direction H^⊥ times the one within H. Since the intersection in H^⊥ reduces to the previously studied case (<ref>), we have:(β), (γ)= i/2(-1)^|H|-1/sinπ s_H (β) |_H, (γ)|_H= i/2(-1)^|H|-1/sinπ s_H < (H,-I), (H,-I) > ×< (β∖ H,I), (γ∖ H,I) >,where we have used that the facet H is a product of two smaller twisted cycles according to (<ref>). The new twisted cycles have loading naturally induced from the one of (β) and (γ). A potential minus sign arises since β and γ induce different orientations on H when |H| is even. The new label I corresponds to momentum k_I^μ = ∑_i ∈ H k_i^μ. Since |H| ≥ 2, the right-hand side of (<ref>) is a product of a self-intersection number and an intersection number for a cycle with smaller n. Thus, it provides a recursion relation that can be used to evaluate an arbitrary intersection number. Simple arithmetic reveals that the overall prefactor becomes (i/2)^n-3 after performing all the recursive steps.It follows that the intersection number (<ref>) receives contributions from the intersection face F = H_1 ∩ H_2 ∩⋯∩ H_h constructed out of the factors 1/sinπ s_H_i and 1/tanπ s_H' for H' ⊂ H_i.Let us demonstrate how to conveniently calculate (<ref>) using graphical rules of <cit.>.[Similar combinatorial rules for studying adjacency of associahedra in _0,n(ℝ) were described in <cit.> using an interpretation of the associahedron K_n-1 as a configuration space of triangulations of n-sided polygons.] We will do so in two steps, first giving the rule for the absolute value of (<ref>) and then its sign. One can arrive at the permutation γ by a series of h label flips H_1, H_2, …, H_h, which are the same as in (<ref>), up to a final orientation change. Let us illustrate both permutations as circles connecting the labels (1,2,…,n) in the corresponding orders. We start with two orderings β, and perform a series of flips H_1, H_2, …, H_h on the second permutation to arrive at the ordering γ, possibly up to a global orientation change, as follows:< g r a p h i c s >H< g r a p h i c s > {H_1,H_2,…,H_h}∖H < g r a p h i c s >Here we have arranged the flips so that H used in (<ref>) is the first one performed. The remaining flips {H_1, H_2, …, H_h}∖{H} do not change the fact that there exists a crossing point I when the labels in H are brought arbitrarily close together. The rule is then to associate a self-intersection number < (H,-I) , (H,-I) > to the polygon created by labels (H,-I), a factor 1/sinπ s_H = 1/sinπ s_I to the intermediate edge, and an intersection number < (β∖ H,I) , (γ∖ H,I) > to the remainder of the diagram. Repeating this procedure recursively, one obtains the full intersection number (<ref>).Finally, we prove that the sign of (<ref>) is given by (-1)^w(β|γ)+1, where w(β | γ) is the relative winding number of the two permutations, following the prescription of <cit.>. Keeping track of signs in the above algorithm, we start with two identical permutations β, for which w(β|β) = 1 gives a plus sign, as expected. Then applying a single flip H gives a sign (-1)^|H|-1. Similarly, the winding number changes by |H|-1, giving the same contribution. In the final step, a potential orientation flip would contribute (-1)^n, while the winding number changes by n-2, thus giving an identical sign contribution. This shows that (-1)^w(β|γ)+1 calculates the correct sign of (<ref>) and hence concludes the proof.Let us illustrate this procedure in practice by calculating (12345), (13245), or equivalently m_α'(12345|13245). After drawing a circle diagram with both permutations (12345) and (13245), we find the dual of the polygon created by the red loop. This results in a diagram connecting two subamplitudes, m_α'(23,-I|23,-I) and m_α'(451I | 451I) with k_I^μ = k_2^μ + k_3^μ, with the propagator given by 1/sinπ s_23. The first subamplitude is equal to 1, while the second one is a sum of two propagators given by tangents in the factorization channels s_34 and s_51, according to (<ref>). To summarize, we have:< g r a p h i c s >=< g r a p h i c s >=-(i/2)^21/sinπ s_23( 1/tanπ s_45 + 1/tanπ s_51).The minus sign arises because of the winding number w(12345|13245) = 2. The above result encodes the fact that the two associahedra K_4(12345) and K_4(13245) share the face (23), as well as two vertices (23)∩ (123) and (23) ∩ (234), see Figure <ref>.Let us consider another example by evaluating (12345), (12453), or equivalently m_α'(12345|12453). This time, the dual graph is composed of three subamplitudes, each of which is trivalent and hence contributes the contact term 1. The two propagators in the s_12 and s_45 channels are given by sine factors, as follows:< g r a p h i c s >=< g r a p h i c s >=-(i/2)^21/sinπ s_12 1/sinπ s_45.The winding number is w(12345|12453) = 2, giving an overall minus sign. Since there is only one trivalent graph contributing, it means that the associahedra K_4(12345) and K_4(12453) share only a single vertex (12) ∩ (123), see Figure <ref>.Next, let us consider a six-point example of (123456), (124365), or equivalently m_α'(123456|124365). The dual diagram reduces to three subamplitudes, two of which give 1, while the third contributes a four-point self-intersection number (<ref>). Recall that self-intersection numbers themselves are given by diagrams with vertices of odd valency and propagators built out of tangent functions <cit.>. In our case, we have:< g r a p h i c s >=< g r a p h i c s >= (i/2)^31/sinπ s_34 1/sinπ s_56( 1/tanπ s_12 + 1/tanπ s_234).The plus sign arises because w(123456|124365)=3. The above answer also encodes the fact that the two associahedra K_5(123456) and K_5(124365) share the vertices (34) ∩ (1234) ∩ (12) and (34) ∩ (1234) ∩ (234), as well as the codimension-2 edge (34) ∩ (1234) between them.Finally, let us consider a 12-point example of the intersection number of twisted cycles (123456789,10,11,12) and (127856,11,12,9,10,34). For these two permutations the dual diagram is trivalent and hence given as a product of propagators of the sine type as follows:< g r a p h i c s >=< g r a p h i c s >= -(i/2)^91/sinπ s_12 1/sinπ s_34 1/sinπ s_56 1/sinπ s_78 1/sinπ s_9,10[-3em]×1/sinπ s_11,12 1/sinπ s_1234 1/sinπ s_5678 1/sinπ s_9,10,11,12.The winding number equals to 4, which gives an overall minus sign. The corresponding nine-dimensional associahedra intersect only at a single vertex (12) ∩ (34) ∩ (56) ∩ (78) ∩ (9,10) ∩ (123456789,10) ∩ (1234) ∩ (5678) ∩ (12345678) in the moduli space. For more examples of how to evaluate m_α'(β | γ) we refer the reader to <cit.>. In the field theory limit, α' → 0, only the faces of maximal codimension contribute. In other words, a given intersection number reduces to a sum over trivalent diagrams. Since the intersection number has only terms coming from the intersection face, F = K_n-1(β) ∩ K_n-1(γ), these diagrams are necessarily planar with respect to both orderings, β and γ. This gives rise to the usual definition of the bi-adjoint scalar double-partial amplitudes m(β | γ)<cit.>.In general, since each facet of a given associahedron can be written as a product of lower-dimensional associahedra according to (<ref>), the corresponding scattering amplitude factors into two lower-point amplitudes connected by a label I. Physically, this is the statement of unitarity. Since the regularized twisted cycles are given by the generalized Pochhammer contour, near the facet it receives the contribution of 1/(e^2π i s_I - 1), which contains an infinite number of simple poles at s_I = 0, ± 1, ± 2, …, allowing propagation of massless, massive, and tachyonic modes. Physically, this corresponds to the statement of locality. Both of these properties are associated to twisted cycles. In the case of pairings between twisted cycles and twisted cocycles, such as the ones giving rise to open string amplitudes (<ref>), it is the role of the cocycle to select if a given factorization channel is utilized or not.§ CONCLUSION In this work we have shed new light on the Kawai–Lewellen–Tye relations <cit.>. By applying the tools of twisted de Rham theory, we have shown that they follow from the underlying algebro-topological identities known as the twisted period relations. On the way, we have formulated tree-level string theory amplitudes in a way that makes connections to combinatorics and topology. In particular, we have explored the relation to the polytope called the associahedron. We have shown that the inverse of the KLT kernel can be computed from the knowledge of how associahedra intersect one another in the moduli space. From this perspective, the inverse of the KLT kernel appears to be a more fundamental object than the kernel itself, in both string and field theory. Introduction of twisted de Rham theory in the study of string integrals opens new directions not only for the KLT relations, but also scattering amplitudes in a more general setting.Since the formalism of twisted de Rham theory applies to a broad spectrum of topological spaces and multi-valued functions, one may wonder about generalizations of the calculations presented in this work to other cases. Indeed, the most natural extension is to consider higher-genus amplitudes in string theory. This direction looks particularly promising in the light of the recent analysis of monodromy properties of higher-genus string integrals by Tourkine and Vanhove <cit.>, as well as Hohenegger and Stieberger <cit.>. A related construction of KLT relations for one-loop field theory integrands has been recently given by He, Schlotterer, and Zhang <cit.>. We leave the study of these intriguing questions for future research. Aside of string theory, intersection numbers of twisted cycles have been previously calculated in the context of conformal field theories by Mimachi and Yoshida <cit.> in order to explain identities between correlation functions found by Dotsenko and Fateev <cit.>. It would be interesting to investigate whether these identities can be further generalized, perhaps even to the case of the conjectured conformal field theory on the null boundary of asymptotically-flat spacetimes <cit.>.The Deligne–Mumford–Knudsen compactification of the moduli space ℳ_0,n can be constructed as a Chow quotient of the Grassmannian Gr(2,n)<cit.>. It is also known that planar amplitudes in the 𝒩=4 super Yang–Mills (SYM) theory—which are the field theory limit of superstring amplitudes—can be defined on the positive Grassmannian <cit.>. Additionally, they also have a description in terms of a geometric object found by Arkani-Hamed and Trnka <cit.>. It is natural to expect it to be related to the associahedron described in this work. The problem is somewhat akin to the so-called Grassmanian dualities <cit.> with an additional complication of the α' corrections. The main obstacle comes from the fact that our results are largely independent of the spacetime dimension and precise theory under consideration, as it only suffices that its amplitudes have a BCJ representation in terms of Z-integrals. Associahedra also appear in a slightly different context of cluster algebras used to study 𝒩=4 SYM amplitudes <cit.>.[Some steps in the direction of applying twisted de Rham theory to 𝒩=4 SYM amplitudes have been taken in <cit.>.]One may then wonder about seeing the field theory limit from a different point of view. Indeed, one such possibility coming naturally from twisted de Rham theory is to consider the dual twisted homology and cohomology defined with the multi-valued function u^-1(z) instead of u(z). This corresponds to reversing the string parameter α' in one of the amplitudes. An equivalent procedure has been recently studied in the context of string theory by Siegel and collaborators <cit.>. In particular, Huang, Siegel, and Yuan showed that considering a chiral version of KLT relations, one obtains the field theory amplitude as follows <cit.>: 𝒜^GR = ∑_β∈ℬ, γ∈𝒞𝒜^open(β)m_α'^-1(β | γ) 𝒜^open_chiral(γ).where we have used the same notation as in (<ref>), except 𝒜^open_chiral(γ) denotes a string amplitude with a flipped spacetime signature η_μν→ -η_μν, or equivalently a replacement α' → -α'. The equation (<ref>) is in fact equivalent to the original twisted Riemann period relation found in <cit.>. In this interpretation, the field theory amplitude 𝒜^GR is an intersection number of twisted cocycles, given by the pairing:H^n-3_c(X,ℒ_ω) × H^n-3(X,ℒ_ω^∨)⟶ ℂ,where the dual system is defined through ℒ_ω^∨ = ℒ_-ω and one of the cohomologies is with compact support <cit.>. Methods for evaluation of this pairing have been given in <cit.>. In particular, Matsumoto gave a simple proof <cit.> of the fact that these intersection numbers localize on the intersections of the singular locus of u(z). Translating to the string theory case, with the bases of twisted cocycles given by (<ref>), the intersection numbers are given by a sum of multi-dimensional residues around the vertices of all the associahedra in the moduli space ℳ_0,n. A given vertex contributes if only if the differential form (β) (γ) has a double pole at the place corresponding to this vertex. Since both differential forms are logarithmic, double poles arise only if the two Parke–Taylor factors share a factorization channel. In this way, the sum over all residues receives contributions only from the vertices laying on the intersection K_n-1(β) ∩ K_n-1(γ) in the moduli space. This gives rise to the bi-adjoint scalar amplitude m(β | γ) as a sum over all Feynamn diagrams compatible with both permutations β and γ. It is also possible to consecutively apply the global residue theorem (GRT) <cit.>—in a way analogous to the one considered by Dolan and Goddard <cit.>—in order to obtain a dual description that localizes on the residues around the scattering equations⋀_i=2^n-2{|E_i| = ε}. In pictures, the duality translates between different types of residues as follows:< g r a p h i c s >It is known that when all the exponents of the Koba–Nielsen factor (<ref>) are positive, there are (n-3)! solutions of the constraint ∏_i=2^n-2δ(E_i) laying in the hypercube (0,1)^n-3⊂ℳ_0,n(ℝ) with one solution per associahedron <cit.>. Of course, the advantage of this approach is the fact that the residues are computed far away from the faces, so no blowup is necessary for explicit computations. The resulting formula for the intersection number of twisted cocycles reads, up to normalization factors: ∮_⋀_i=2^n-2{|E_i| = ε}(β) (γ)/∏_i=2^n-2 E_i This is the so-called Cachazo–He–Yuan formula <cit.> for the bi-adjoint scalar amplitude m(β | γ). Other amplitudes, such as 𝒜^GR, can be expanded in the basis of m(β | γ), as in (<ref>). Notice that the field theory limit is obtained in the limit of vanishing twist ω = α' ∑_i=2^n-2 E_idz_i. It is tempting to suggest that there could exist an alternative derivation of the CHY formulae by imposing a constraint on ω, perhaps in relation to Morse theory discussed in <cit.>. It would also be interesting to find out how it relates to other approaches of connecting CHY formalism and string theory, see, e.g., <cit.>, in particular in the context of ambitwistor strings <cit.>. We leave the study of these connections for future investigations.Also in the field theory limit, there exists another instance of relations between gravity and Yang–Mills amplitudes known as the BCJ double-copy<cit.>. In this context, Carrasco studies the space of trivalent graphs and its relation to associahedra and permutohedra <cit.>. It would be interesting to see how this story fits with ours. Here, twisted cycles play the role of colour factors, while twisted cocycles play the role of kinematics factors. Moreover, blowup of the moduli space ℳ_0,n—or its double cover <cit.>—provides a natural way of understanding the configuration space of trivalent diagrams as its limit. We hope this language could contribute to deeper understanding of colour-kinematics duality and its connection to KLT relations, particularly at higher loops.Last but not least, it is important to understand questions arising from this work on the level of rigour of mathematics. Such issues involve, for example, study of the Hodge structure of the intersection form for twisted cohomology groups (<ref>), finding an algebro-topological derivation of the form of the circuit matrix given in (<ref>), or study of the formula (<ref>) in the context of Morse theory. From the point of view of combinatorics, further study of the moduli spaces of marked bordered Riemann surfaces of higher genus and their tilings, along the lines of <cit.>, is important in understanding higher-loop generalizations of KLT relations. It is also known that string amplitudes have a rich motivic structure, see, e.g., <cit.>. In particular, J-integrals (<ref>) and Z-integrals (<ref>) can be related by <cit.>:J(β | γ) = [ Z_β(γ) ],whereis the single-valued projection introduced by Brown <cit.>. This relation bears resemblance to the twisted period relations for the basis of twisted cycles and cocycles (<ref>). It would interesting to study the connection between motivic structure and twisted de Rham theory in this context.name=§[display] Appendix 0pt [ .6pt]§ FIELD THEORY LIMIT FROM THE GENERALIZED POCHHAMMER CONTOUR As was shown in Section <ref>, tree-level open string partial amplitudes can be understood as pairings between twisted cycles and twisted cocycles. In this appendix, we show how to obtain its field theory limit, α' → 0, by utilizing the generalized Pochhammer contour and blowup of the moduli space described in Section <ref>.Recall that open string amplitudes can be expanded in the basis of Z-theory amplitudes (<ref>). They in turn are given by the pairing:H_n-3(X,ℒ_ω) × H^n-3(X,∇_ω)⟶ ℂ,denoted by Z_β(γ) = (β), (γ), using the basis of twisted cycles (<ref>) and cocycles (<ref>) for string amplitudes. We also employ the regularization (β) in order to make twisted cycles compact, and work on the blowup of the moduli space, _0,n. Notice that the information about the factors of α' of Z_β(γ) is entirely contained in the regularized twisted cycle (β). In order to take the field theory limit, let us count the powers of α' contributing to different pieces of the generalized Pochhammer contour based on the associahedron K_n-1(β).Each face F of codimension k can be written as F = H_1 ∩ H_2 ∩⋯∩ H_k. Near each facet H_i, (β) picks up a factor 1/(e^2π i α' s_H_i - 1), which in the α' → 0 limit scales as 1/α'. We conclude that the string integral in the α' → 0 limit receives leading contributions from the faces F of maximal codimension n-3, or in other words, vertices of the associahedron K_n-1(β). Since all the singularities of the string amplitude are encapsulated in the choice of the generalized Pochhammer contour, and the integrals to be performed are finite when α' → 0, we can takeu(z) → 1 in the same limit. To summarize, we have: lim_α' → 0(β), (γ)= 1/(2π i α')^n-3∑_v = H_1 ∩⋯∩ H_n-31/∏_i=1^n-3 (± s_H_i)∮_|H_i|=εi=1,…,n-3(γ).where the sum proceeds over all the Catalan number C_n-2<cit.> of vertices v of K_n-1(β). The integrals are performed along an appropriately oriented tubular neighbourhood of each vertex v.[For a reference on the computation of multi-dimensional contour integrals see, e.g., Chapter 5 of <cit.>.]Let us work out explicit examples of the evaluation of (<ref>). One needs to be extra careful about sign factors coming from orientation induced by the associahedron on the vertices. Let us illustrate this fact for n=4. In the α' → 0 limit, the regularized twisted cycle defined in (<ref>) becomes lim_α' → 0(0,1) = 1/α'( S(ε,0)/2π i s - S(1-ε,1)/2π i t).Recall that we use S(a,z) to denote a positively-oriented circular contour starting at a and with a centre at z. The contours around the two vertices of K_3(_4) come with different signs due to different orientations induced from (0,1). Let us now evaluate (<ref>) for a four-point example (1234), (1234). From the pole around z=0 we obtain: lim_α' → 0(1234), (1234) |_z=0 = 1/2π i α' s∮_|z| = εdz/(0-z)(z-1) = 1/α' s,and from around z=1 we find the contribution: lim_α' → 0(1234), (1234) |_z=1 = -1/2π i α' t∮_|z-1| = εdz/(0-z)(z-1) = 1/α' t.Hence we find the answer which is a sum over two Feynman diagrams in the s and t channels. The two contributions worked out to give the same sign. In general, all the vertices contributing to (<ref>) will give the same sign. For another choice of the twisted cocycle, (2134), we have: lim_α' → 0(1234), (2134)= 1/2π i α' s∮_|z| = εdz/(z-0)(0-1) = -1/α' s.Notice that contribution from the vertex z=1 vanishes, since (2134) does not have a pole at z=1. Similarly, for (1324) we obtain: lim_α' → 0(1234), (1324)= -1/2π i α' t∮_|z-1| = εdz/(0-1)(1-z) = -1/α' t,since there are no poles at z=0.For higher-point cases one needs to consider a blowup of the integrals (<ref>). Let us illustrate the procedure with an n=5 example for (12345), (12345). The corresponding associahedron K_4(_5) has five vertices. The contribution from (z_2, z_3) = (0,1) can be calculated straightforwardly:lim_α' → 0(12345), (12345) |_(12) ∩ (34) = -1/(2π i α')^21/s_12s_34∮_|z_2| = ε|z_3 - 1| = εdz_2dz_3/(0-z_2)(z_2 - z_3)(z_3 - 1) = -1/α'^2s_12s_34.Here we have used the tubular contour given by { |z_2| = ε}{ |z_3 - 1| = ε}. Next, near (z_2, z_3) = (0,0) we perform a blowup using the change of variables from {z_2, z_3} into {y_2, τ} given by z_2 = τ y_2 and z_3 = τ. Since dz_2dz_3 = τ dy_2dτ, we have the contribution:lim_α' → 0(12345), (12345) |_(123) = 1/(2π i α')^2∑_H ∈{(12), (23)}1/s_123 (± s_H)∮_|H| = ε|τ| = ετ dy_2dτ/(0-τ y_2)(τ y_2 - τ)(τ - 1) = -1/2π i α'^2∑_H ∈{(12), (23)}1/s_123 (± s_H)∮_|H| = εdy_2/(0-y_2)(y_2 - 1) = - 1/α'^2 s_123( 1/s_12 + 1/s_23).In the first line, the powers of τ add up to create a simple pole dτ / τ, over which we have integrated. In the second line we have used the result of the four-point computations (<ref>) and (<ref>) with the appropriate signs for s_H, H ∈{(12),(23)}. For the remaining two vertices near (z_2,z_3) = (1,1) we use the variables τ, y_3 defined through z_2 = 1 - τ and z_3 = 1 - τ y_3, so that dz_2dz_3 = τ dτ dy_3. A similar calculation reveals:lim_α' → 0(12345), (12345) |_(234) = 1/(2π i α')^2∑_H ∈{(23), (34)}1/s_234 (± s_H)∮_|τ| = ε|H| = ετ dτ dy_3/(-1+τ)(-τ + τ y_3)(-τ y_3) = -1/2π i α'^2∑_H ∈{(23), (34)}1/s_234 (± s_H)∮_|H| = εdy_3/(0-y_3)(y_3 - 1) = - 1/α'^2 s_234( 1/s_23 + 1/s_34),where once again we have used a residue theorem to integrate over the simple pole dτ / τ. Summing up all the contributions and using momentum conservation, we have lim_α' → 0(12345), (12345)= -1/α'^2( 1/s_12s_34 + 1/s_23s_45 + 1/s_34s_51 + 1/s_45s_12 + 1/s_51s_23).Using the same procedure with different cocycles, it is straightforward to verify other examples, for instance: lim_α' → 0(12345), (13245)= 1/α'^2 s_23( 1/s_45 + 1/s_51), lim_α' → 0(12345), (12453)= 1/α'^2 s_12 s_45. In general, in the α' → 0 limit one finds that Z-integrals (<ref>) collapse to the bi-adjoint scalar partial-amplitudes <cit.>: lim_α' → 0(β), (γ)= - (-α')^3-nm(β | γ),where we have included the normalization factor. The method of computing this result using the generalized Pochhammer contour presented above, despite having a simple geometrical interpretation in terms of the associahedron, is not particularly efficient. In this light, it would be interesting to study systematic ways of evaluating (<ref>) and its higher-order terms, which could provide a new way of performing the α' expansion.JHEP | http://arxiv.org/abs/1706.08527v1 | {
"authors": [
"Sebastian Mizera"
],
"categories": [
"hep-th",
"math-ph",
"math.AG",
"math.CO",
"math.MP"
],
"primary_category": "hep-th",
"published": "20170626180002",
"title": "Combinatorics and Topology of Kawai-Lewellen-Tye Relations"
} |
x y zlemLemma proProposition defiDefinition teoTheorem corCorollary exaExample remRemark [#1]⌊#1⌋♯Computing denumerants innumerical 3–semigroupsThe second author is supported by the projects MTM2014-55367-P, FQM-343 and FEDER funds. The first author is supported by the project MTM2014-60127-P. F. Aguiló–Gost D. Llena January 16, 2017 ====================================================================================================================================================================================================== As far as we know, usual computer algebra packages can not compute denumerants for almost medium (about a hundred digits) or almost medium–large (about a thousand digits) input data in a reasonably time cost on an ordinary computer. Implemented algorithms can manage numerical n–semigroups for small input data.Here we are interested in denumerants of numerical 3–semigroups which have almost medium input data. A new algorithm for computing denumerants is given for this task. It can manage almost medium input data in the worst case and medium–large or even large input data in some cases. Keywords: Denumerant, numerical semigroup, L–shape.§ INTRODUCTION Letbe the set of non negative integers. We denote the equivalence class of k modulo m as [k]_m. Given n_1,…,s_k∈, 1<n_1<⋯<n_k and (n_1,…,n_k)=1, the numerical k–semigroup T generated by G={n_1,…,n_k} is defined byT=n_1,…,n_k={x_1n_1+⋯+x_kn_k: x_1,…,x_k∈}.The generating set G has not necessarily be minimal. The cardinality of a minimal generating set is the embedding dimension, e(T), of the semigroup. Given an element m∈ T∖{0}, the Apéry set of T with respect to m is the set (m,T)={s∈ T: s-m∉ T}. It is well known the equivalence s∈(m,T)⇔ s=min([s]_m∩ T) and so, (m,T)={s_0,…,s_m-1} with s_i≡ im.Given s∈ T, a vector (x_1,…,x_n)∈^k such that x_1n_1+…+x_kn_k=s is called a factorization of s in T. Let us denote the set of factorizations of s in T by(s,T)={(x_1,…,x_k)∈^k: x_1s_1+⋯+x_ks_k=s}.The denumerant of s in T is defined as the cardinality of the set (s,T), denoted by (s,T)=|(s,T)|. The Frobenius number of T is defined by (T)=max(∖ T). Detailed results on numerical semigroups can be found in the book of J. C. Rosales and P. A. García–Sánchez <cit.>. It is also interesting the book of Ramírez Alfonsín <cit.> where it can be found a complet source of results related to Frobenius number.Sylvester <cit.> in 1882 gave the generating function ϕ(z) of (m,n_1,…,n_k)ϕ(z)=1/(1-z^n_1)(1-z^n_2)⋯(1-z^n_k).Schur <cit.> in 1926 studied the asymptotic behaviour of the denumerant,lim sup_m→∞(m,n_1,…,n_k)/m^k-1/n_1⋯ n_k(k-1)!=1.Sylvester <cit.> in 1857 and Cayley <cit.> in 1860 gave the expression (m,n_1,…,n_k)=P_k(m)+Q_k(m) where P_k(m) is a polynomial of degree k-1 and Q_k(m) is a periodic function in the variable m. Beck, Gessel and Komatsu <cit.> in 2001 found an expression for P_k(m) that depends upon Bernoulli numbers.Popoviciu <cit.> in 1953 found an efficient semi–closed expression[This expression only requires O(logmax{p,q}) arithmetic operations to be applied.] for n=2 of (m,p,q)(m,p,q)=m+pf(m)+qg(m)/(p,q)-1,where f(m)≡-mp^-1q with 1≤ f(m)≤ q and g(m)≡-mq^-1p with 1≤ g(m)≤ p.Ehrhart <cit.> in 1967 and Sertöz and Özlük in 1991 gave recursive denumerant formulae for 2≤ k≤4. You can find an exhaustive set of results on denumerants in the book of J. Ramírez Alfonsín <cit.>.No similar efficient semi–closed expressions are known for k≥3, however there are some known numerical algorithms to find the set of factorizations (m,T) in the general case. Unfortunately, as far as we know, usual computer algebra systems have implemented no command for denumerant. Thus, the calculation of denumerant turns to be a time consuming task. Taking for instance, n_1=7^k, n_2=11^k, n_2=(7^k,11^k), P_k=n_1n_2n_3, S_k=n_1+n_2+n_3 and m_k=P_k-S_k-k, we obtain the figures of Table <ref> for (m_k,T_k) and T_k=n_1,n_2,n_3. The reason why we choose m_k is clear by Theorem <ref>.Table <ref> shows how popular CAS programs[The commands for computing the denumerant (m,a,b,c) arefor Mathematica 8,for Sage 7.3 andfor GAP 1.5.1] can not manage almost medium (about half a hundred digits). Clearly the Gap package takes advantage for these input instances. From now on, we focus our attention to denumerants of numerical 3–semigroups and the notation n_1=a, n_2=b, n_3=c and T=a,b,c will be used here.Popoviciu <cit.> gave an O(clog c) algorithm, in the worst case, for computing (m,T) when {a,b,c} are pairwise coprime numbers (pcn). Lisoněk <cit.> in 1995 gave an O(ablog b) algorithm, in the worst case for pcn (this time costcan be reduced to O(ab) provided that a number of max{O(a^2b^2), O(abc)} precomputed values, related to T, can be stored in the computer memory for later usage). Brown, Chou and Shiue in 2003 <cit.> gave an O(ablog c) algorithm, in the worst case. This last work also contains interesting results on denumerants that can be taken into account for numerical calculations. We refer to these algorithms as , and , respectively. Notice that the speed of Algorithm versus Algorithm depends on the ratio clog c/ablog b.Algorithms ,andcalculate the denumerants of Table <ref> significantly faster. A non-compiled Sage 7.3 implementations of them give the figures in Table <ref> (using the same processor of Table <ref>).Nonetheless, these algorithms do not reach the necessary efficiency for managing almost medium input. The goal of this work is to provide a reasonably efficient new algorithm which allows such kind of inputs when working on ordinary computers.Our algorithm has a theoretical time cost of O(b+log c), in the worst case. However, numerical evidences suggest that, in some cases, it can have a smaller cost[As an instance, the same data of Table <ref> for k=10^3 is calculated in 0.009836 seconds and for k=10^5 in 2.110464 seconds.]. This algorithm is based on a semi-closed denumerant expression given in <cit.> which is included here in Theorem <ref>.The summary of the paper is the following: Section 2 contains the basic known tools, mainly Theorem <ref> and expression (<ref>). Section 3 developes expression (<ref>) to be used for numerical purposes. In this developing it is apparent that the main computation depends on the so called S^± discrete sums. Some tools to calculate S^± sums are developed in Section 4, mainly the so called hS-type sets. Section 5 contains the main algorithm and Section 6 analyzes the time cost, in the worst case. Finally, in Section 7, several instances of time tests are given.§ SOME DEFINITIONS AND KNOWN RESULTS In this section we give the main known results that allow us to reach our goal. The usual notation for semigroups will be T=a,b,c with 1≤ a<b<c and (a,b,c)=1. Also the product P=abc and sum S=a+b+c of the generators are used.Although algorithms P and L act over pairwise coprime generators, this condition can be removed by the following result due to Brown, Shou and Shiue <cit.>. Here the integer u'_v(t) is defined to be the unique integer value 1≤ u'_v(t)≤ v such that uu'_v(t)≡-tv with u,v≥1 and (u,v)=1. Consider the semigroup T=a,b,c with (a,b,c)=1. Set g_a=(b,c), g_b=(a,c) and g_c=(a,b). For any integer n>0, the integer value n'=n-(g_a-a'_g_a(n))a-(g_b-b'_g_b(n))b-(g_c-c'_g_c(n))c is multiple of g_ag_bg_c and the denumerant's identity (n,T)=(n'/g_ag_bg_c,T') holds with T'=a/g_bg_c,b/g_ag_c,c/g_ag_b. Here it is understood that (0,T')=1 and (n'/g_ag_bg_c,T')=0 whenever n'<0.By the following theorem, due to Ehrhart in 1967, we only need to compute denumerants in the range of values m∈{0,1,…,P-1}.[Ehrhart 1967 <cit.>] Consider T=a,b,c witha, b and c pcn. Set P=abc, S=a+b+c and m=qP+r with 0≤ r<P. Then, (m,T)=(r,T)+q(m+r+S)/2.In particular,(P,T)=P+S/2+1.The range {0,…,P-1} can be reduced to {0,…,P-S} by the following theorem due to Sertöz and Özluk in 1991.[Sertöz and Özlük 1991 <cit.>] Consider T=a,b,c with a, b and c pcn. Set P=abc and S=a+b+c. Then, for 1≤ x≤ S-1 we have(P-x,T)=P+S/2-x.In particular,(P-S+1)=P-S/2+1.The time cost, in the worst case, of the algorithms P, L and BCS for computing the denumerant (m,a,b,c) have been given for the largest value of m (by theorems <ref> and <ref>), that is m≈ P=abc.We use the concept of L–shape as a main tool for the new algorithm. Thus, we include here some known results for this geometrical discrete structure. Denote the interval [s,t)={x∈: s≤ x<t}, the unitary square m,n=[m,m+1)×[n,n+1)∈^2 and the discrete backwards cone Δ(u,v)={m,n: (m,n)∈^2,0≤ m≤ u, 0≤ n≤ v} for each u,v∈. We also denote the equivalence class of u modulo v by [u]_v.Consider each unitary square m,n, for (m,n)∈^2, labelled by the equivalence class [ma+nb]_c. Define the minimum valuesM_n=min{sa+tb: (s,t)∈^2, [sa+tb]=[n]_c}. [Minimum distance diagram] Consider a numerical 3–semigroup T=⟨ a,[0] b,[0] c⟩. A minimum distance diagram (MDD), , related to T is a set of c unitary squares that fulfils the following properties (a) for each n∈{0,…,c-1}, there is some unitary square s,t∈ such that [sa+tb]_c=[n]_c,(b) Δ(s,t)⊆ for each s,t∈,(c) if s,t∈, then sa+tb=M_n with [sa+tb]_c=[n]_c and M_n defined by (<ref>).Minimum distance diagrams related to numerical 3–semigroups are known to be L–shapes or rectangles (that will be considered as degenerated L-shapes). For this reason we also refer to MDD as L-shapes and they are denoted by the lengths of their sides (l,h,w,y), see Figure <ref>, with 0≤ w<l, 0≤ y<h and lh-wy=c. An L–shape tessellates the plane by translation through the vectors =(l,-y) and =(-w,h). The following result characterizes the L-shapes related to T=a,b,c. From now on we assume 0<a<b<c and (a,b,c)=1.[A. and Marijuán 2014 <cit.>] Consider the numerical 3–semigroup T=a,b,c. An L-shape =(l,h,w,y) is related to T if and only if (a) lh-wy=c and (l,h,w,y)=1,(b) la-yb≡0c and hb-wa≡0c,(c) la-yb≥0, hb-wa≥0 and both expressions can't vanish at the same time.Each numerical 3–semigroup has two related L-shapes at most (either one if (la-yb)(hb-wa)>0 or two whenever (la-yb)(hb-wa)=0, see <cit.>). L-shapes contain main information of the related semigroup. For instance, if a semigroup T=a,b,c has related the L-shape , we have (c,T)={ia+jb: i,j∈}.A classification of 3–semigroups was given in terms of its related L–shapes in <cit.>. The tessellation of the plane associated with each L–shape was used to derive the semi-closed expression (<ref>) for the denumerant in <cit.>. Given T=a,b,c and a related L–shape =(l,h,w,y), let us denote δ=(la-yb)/c and θ=(hb-wa)/c. From the definition ofand Theorem <ref>, it follows that * a=hδ+yθ and b=wδ+lθ,* δ,θ∈ and δ+θ>0,* δ=0⇒ y>0 and θ=b/l=a/y>0,* θ=0⇒ w>0 and δ=a/h=b/w>0,* w=0⇒θ=b/l>0,* y=0⇒δ=a/h>0>0.All these properties will be used along this work.Given m∈ T, it is called the basic factorization of m with respect to , (x_0,y_0,z_0)∈(m,T), the unique factorization such that x_0,y_0∈. This factorization can be computed in time cost O(log c) <cit.>.[A. and P.A. García Sánchez 2010 <cit.>] Given T=a,b,c and a related L–shape =(l,h,w,y), assume m∈ T. Define A_m=⌊z_0/δ+θ⌋, where (x_0,y_0,z_0) is the basic factorization of m wrt . For each 0≤ k≤ A_m, setS_k={[ ⌊y_0+k(h-y)/y⌋if δ=0,; ⌊z_0-k(δ+θ)/δ⌋if y=0,; min{⌊y_0+k(h-y)/y⌋,⌊z_0-k(δ+θ)/δ⌋}if δ y≠0, ].andT_k={[ ⌊x_0+k(l-w)/w⌋if θ=0,; ⌊z_0-k(δ+θ)/θ⌋if w=0,; min{⌊x_0+k(l-w)/w⌋,⌊z_0-k(δ+θ)/θ⌋}if θ w≠0. ].Then, the denumerant of m in T is(m,T)=1+A_m+∑_k=0^A_m(S_k+T_k). The sum appearing in this theorem is known as the basic sum of the denumerant with respect to the L–shape . The direct computation of this sum does not give an efficient algorithm for calculating the denumerant. However, as it will be seen later, a detailed analysis of this expression does it.Take T=5,7,11 and m=87. A related L–shape is =(5,3,2,2), with δ=θ=1. The basic factorization of 87 in T is (x_0,y_0,z_0)=(2,0,7). Thus, we have A_m=3. Then, it follows that S_0+T_0=0+1=1, S_1+T_1=0+2=2, S_2+T_2=1+3=5, S_3+T_3=1+1=2 and so(87,T)=1+3+(1+2+5+2)=13.A geometric representation of the plane projection of the set (87,T), π((87,T)), is depicted in Figure <ref>. It has a tree-like structure, given by the vectors ,and + (and so, it follows the tessellation of the plane by ). Each unitary square s,t is labelled with the value 5s+7t (notice that values corresponding to unitary squares in the gray L–shape form the Apéry set (87,T)={0,12,24,14,15,5,17,7,19,20,10}). The unitary squares corresponding to the first two coordinates of each factorization are circled. From the coordinates of a circled unitary square s,t follows the related factorization (s,t,87-3s-7t/11). The set of factorizations is(87,T)= {(2,0,7),(0,3,6),(5,1,5),(3,4,4),(1,7,3),(8,2,3),(13,0,2),(6,5,2),(4,8,1),(2,11,0),(11,3,1),(16,1,0),(9,6,0)}.§ DEVELOPING THE BASIC SUMAs it has been commented before, the basic sum (<ref>) does not provide a direct efficient algorithm for calculating denumerants. Thus, a detailed analysis is needed. We consider three main cases: case (i) δ=0, case (ii) θ=0 and case (iii) δθ>0.The analysis of these cases reveals that the basic sum depends on several sums of the same kind. These sums will be referred to as S^± sums and will be studied in the next section. These sums have the form S^±(s,t,q,N)=∑_k=0^N⌊s± kt/q⌋ with 0≤ s,t<q.In this section we assume that =(l,h,w,y) is an L–shape related to the numerical 3–semigroup T=a,b,c. We also assume that m∈ T and (x_0,y_0,z_0) is the basic factorization of m with respect to T. §.§ Case (i) δ=0 This case leads to the following expressions of the denumerant. Let us assume δ=0. Set A_m=⌊z_0/θ⌋. Then, (i.1) if w=0, then(m,T)=(1+A_m)(1+A_m+y_0)+(h-2)A_m(1+A_m)/2+∑_k=0^A_m⌊y_0+kh/y⌋,where y_0=y_0y+y_0 with 0≤y_0<y and h=hy+h with 0≤h<y.(i.2) if w>0, set k_0=⌈z_0w-x_0θ/b⌉. Then, (i.2.1) if k_0=0, then (m,T) has the same expression as in (<ref>).(i.2.2) if 1≤ k_0≤ A_m,(m,T) =(1+A_m)(1+A_m+y_0)+k_0(x_0-A_m)+(h-2)A_m(1+A_m)/2+l(k_0-1)k_0/2+∑_k=0^A_m⌊y_0+kh/y⌋+∑_k=0^k_0-1⌊x_0+kl/w⌋,where y_0,y_0,h,h are defined as in the previous case, x_0=x_0w+x_0 with 0≤x_0<w and l=lw+l with 0≤l<w.(i.2.3) if k_0>A_m,(m,T) =(1+A_m)(1+x_0+y_0)+(l+h-2)A_m(1+A_m)/2+∑_k=0^A_m⌊x_0+kl/w⌋+∑_k=0^A_m⌊y_0+kh/y⌋,where y_0,y_0,h,h,x_0,x_0,l and l are defined as in the previous case.Notice that k_0≥0. Indeed, let us see -1<z_0w-x_0θ/b (recall that we have w>0). From b=lθ (recall that δ=0), we have (recalling b=lθ)-1<z_0w-x_0θ/b⇔ 0<z_0w+(l-x_0)θ.Now, as x_0,y_0∈, it follows that 0≤ x_0<l and so the inequality 0<z_0w+(l-x_0)θ holds. Proof of Theorem <ref>: If δ=0, then we have y≠0, θ>0. From Theorem <ref>, we have A_m=⌊z_0/θ⌋ and S_k=⌊y_0+k(h-y)/y⌋=⌊y_0+kh/y⌋-k for all 0≤ k≤ A_m. Now two subcases appear, (i.1) w=0 and (i.2) w>0. (i.1) Assume w=0. From (<ref>), T_k=⌊z_0-kθ/θ⌋=A_m-k for all 0≤ k≤ A_m. From (<ref>),(m,T)=1+A_m+∑_k=0^A_m(⌊y_0+kh/y⌋+A_m-2k).Setting y_0=y_0y+y_0 with 0≤y_0<y and h=hy+h with 0≤h<y, the above expression of (m,T) turns to be(m,T) =1+A_m+∑_k=0^A_m(⌊y_0+kh/y⌋+y_0+k(h-2)+A_m)=(1+A_m)(1+A_m+y_0)+(h-2)A_m(1+A_m)/2+∑_k=0^A_m⌊y_0+kh/y⌋. (i.2) Assume now w>0. Then,T_k=min{⌊x_0+k(l-w)/w⌋,⌊z_0-kθ/θ⌋}=min{⌊x_0+kl/w⌋,⌊z_0/θ⌋}-k.The inequality ⌊x_0+kl/w⌋≥⌊z_0/θ⌋ holds when either x_0+kl/w≥z_0/θ or n≤x_0+kl/w<z_0/θ<n+1 for some n∈. The former holds when k≥ k_0=⌈z_0w-x_0θ/b⌉, the latter holds whenever 0<z_0/θ-x_0+kl/w<1⇔z_0w-x_0θ/b-w/l<k<z_0w-x_0θ/b (and, in this case, ⌊x_0+kl/w⌋=⌊z_0/θ⌋ holds). Notice that, from 0<w/l<1, if there exists some k_1∈ such that z_0w-x_0θ/b-w/l<k_1<z_0w-x_0θ/b, this value k_1 must be unique and equality ⌊x_0+k_1l/w⌋=⌊z_0/θ⌋ holds. Thus, it follows thatT_k=⌊x_0+kl/w⌋-k if k<k_0,A_m-k if k≥ k_0.By Remark <ref> we have k_0≥0 and we consider three possible options. (i.2.1) Assume k_0=0. Then, k≥ k_0 for all k and T_k=A_m-k for all k. Therefore, the expression of T_k is the same as in the previous case for all k. Thus the denumerant has the same expression as the previous case. (i.2.2) Assume 1≤ k_0≤ A_m. Now, the expression of T_k changes upon the value of k<k_0 and k≥ k_0 according to (<ref>). Thus,(m,T) =1+A_m+∑_k=0^A_m(⌊y_0+kh/y⌋-k)+∑_k=0^k_0-1(⌊x_0+kl/w⌋-k)+∑_k=k_0^A_m(A_m-k)=1+A_m+∑_k=0^A_m⌊y_0+kh/y⌋+∑_k=0^A_m(y_0+k(h-2))+∑_k=0^k_0-1⌊x_0+kl/w⌋+∑_k=0^k_0-1(x_0+kl)+∑_k=k_0^A_mA_m=(1+A_m)(1+A_m+y_0)+k_0(x_0-A_m)+(h-2)A_m(1+A_m)/2+l(k_0-1)k_0/2+∑_k=0^A_m⌊y_0+kh/y⌋+∑_k=0^k_0-1⌊x_0+kl/w⌋,where y_0,y_0,h,h,x_0,x_0,l and l are those parameters defined in the statement (i.2.2) of the theorem. (i.2.3) Assume k_0>A_m. Now, following (<ref>), we have T_k=⌊x_0+kl/w⌋-k for all k. Then,(m,T) =1+A_m+∑_k=0^A_m(⌊y_0+kh/y⌋-k+⌊x_0+kl/w⌋-k)=1+A_m+∑_k=0^A_m(y_0+kh+x_0+kl-2k)+∑_k=0^A_m⌊y_0+kh/y⌋+∑_k=0^A_m⌊x_0+kl/w⌋=(1+A_m)(1+x_0+y_0)+(l+h-2)A_m(1+A_m)/2+∑_k=0^A_m⌊x_0+kl/w⌋+∑_k=0^A_m⌊y_0+kh/y⌋,where y_0,y_0,h,h,x_0,x_0,l and l are the same as those defined in (i.2.2).□Notice how all the expressions of denumerant given by Theorem <ref> contain sums of type S^±. §.§ Case (ii) θ=0 This case is similar to the case (i). Now we have w≠0, δ≠0, A_m=⌊z_0/δ⌋ and T_k=⌊x_0+kl/w⌋-k for all k. As in the previous case, we consider the following parametersx_0 =x_0w+x_0,0≤x_0<w,l =lw+l,0≤l<w,y_0 =y_0y+y_0,0≤y_0<y,h =hy+h,0≤h<y.Defining k_1=⌈z_0y-y_0δ/a⌉ (recall that a=hδ) and using similar arguments of (i.2) in the proof of Theorem <ref>, we have S_k in (<ref>) turns to beS_k=⌊y_0+kh/y⌋-k if k<k_1,A_m-k if k≥ k_1. The following result can be obtained using similar arguments like in the proof of Theorem <ref>. Let us assume θ=0. Set A_m=⌊z_0/δ⌋. Then, (ii.1) if y=0, then(m,T)=(1+A_m)(1+A_m+x_0)+(l-2)A_m(1+A_m)/2+∑_k=0^A_m⌊x_0+kl/w⌋,(ii.2) if y>0, set k_1=⌈z_0y-y_0δ/a⌉. Then, (ii.2.1) if k_1=0, then (m,T) has the same expression as in (<ref>).(ii.2.2) if 1≤ k_1≤ A_m,(m,T) =(1+A_m)(1+A_m+x_0)+k_1(y_0-A_m)+(l-2)A_m(1+A_m)/2+h(k_1-1)k_1/2+∑_k=0^A_m⌊x_0+kl/w⌋+∑_k=0^k_1-1⌊y_0+kh/y⌋,(ii.2.3) if k_1>A_m, then the denumerant has the same expression as (<ref>). Although some expressions of Theorem <ref> seem to be the same as some expressions of Theorem <ref>, the value A_m is not the same. In the former case we have A_m=⌊z_0/θ⌋ and A_m=⌊z_0/δ⌋ in the latter.In Theorem <ref> we have k_1≥0. Indeed, k_1≥0⇔-1<z_0y-y_0δ/hδ (recall that hδ=a) and -1<z_0y-y_0δ/hδ⇔0<z_0y+δ(h-y_0). The factorization (x_0,y_0,z_0) is the basic one of m with respect to the L–shape . Thus, x_0,y_0∈⇒ h>y_0.§.§ Case (iii) δθ>0Now we have A_m=⌊z_0/δ+θ⌋, a=hδ+yθ and b=wδ+lθ. There are four different options that give different expressions of A=∑_k=0^A_mS_k and B=∑_k=0^A_mT_k in (<ref>), (iii.1) w=y=0,(iii.2) w≠0 and y=0,(iii.3) w=0 and y≠0,(iii.4) wy≠0.Here we use the same notation as in (<ref>) plus the following onez_0 =z_0,1δ+z_0,1, 0≤z_0,1<δ,z_0 =z_0,2θ+z_0,2, 0≤z_0,2<θ, δ =δθ+δ, 0≤δ<θ, θ =θδ+θ, 0≤θ<δ,Let us assume the numerical 3–semigroup T=a,b,c has related the L–shape =(l,h,w,y). Consider m=x_0a+y_0b+z_0c, where (x_0,y_0,z_0) is the basic factorization of m with respect toin T. Then,(iii.1) if w=y=0,(m,T) =(1+A_m)(1+z_0,1+z_0,2)-(δ+θ+2)A_m(1+A_m)/2+∑_k=0^A_m⌊z_0,1-kθ/δ⌋+∑_k=0^A_m⌊z_0,2-kδ/θ⌋, (iii.2) if w≠0 and y=0, set k_0=⌈z_0w-x_0θ/b⌉; then,(iii.2.1) if k_0=0, the denumerant has the same expression as in (<ref>). Otherwise, when k_0>0, we have(m,T)=(1+A_m)(1+z_0,1)-(1+θ)A_m(1+A_m)/2+∑_k=0^A_m⌊z_0,1-kθ/δ⌋+Bwhere B is defined through the rules of (iii.2.2) or (iii.2.3). (iii.2.2) if 1≤ k_0≤ A_m, we haveB =(1+A_m)z_0,2+k_0(x_0-z_0,2)+(l+δ)(k_0-1)k_0/2-(δ+1)A_m(1+A_m)/2+∑_k=0^k_0-1⌊x_0+kl/w⌋+∑_k=k_0^A_m⌊z_0,2-kδ/θ⌋(iii.2.3) if k_0>A_m, thenB=(1+A_m)x_0+(l-1)A_m(1+A_m)/2+∑_k=0^A_m⌊x_0+kl/w⌋,(iii.3) if w=0 and y≠0, define k_1=⌈z_0y-y_0δ/a⌉; then,(iii.3.1) if k_1=0, the denumerant has the same expression as in (<ref>).Otherwise, when k_1>0, we have(m,T)=(1+A_m)(1+z_0,2)-(δ+1)A_m(1+A_m)/2+∑_k=0^A_m⌊z_0,2-kδ/θ⌋+Awhere A is defined by (iii.3.2) or (iii.3.3). (iii.3.2) if 1≤ k_1≤ A_m, thenA =(1+A_m)z_0,1+k_1(y_0-z_0,1)+(θ+h)(k_1-1)k_1/2-(θ+1)A_m(1+A_m)/2+∑_k=0^k_1-1⌊y_0+kh/y⌋+∑_k=k_1^A_m⌊z_0,1-kθ/δ⌋ (iii.3.3) if k_1>A_m, thenA=(1+A_m)y_0+(h-1)A_m(1+A_m)/2+∑_k=0^A_m⌊y_0+kh/y⌋,(iii.4) if wy≠0, define k_0 and k_1 as in (iii.2) and (iii.3), respectively; then(m,T)=1+A_m+A+B,where A and B are ruled by the following expressions, depending on k_0 and k_1. - If k_1=0, thenA=(1+A_m)z_0,1-(θ+1)A_m(1+A_m)/2+∑_k=0^A_m⌊z_0,1-kθ/δ⌋. - If 1≤ k_1≤ A_m, thenA =(1+A_m)z_0,1-(θ+1)A_m(1+A_m)/2+k_1(y_0-z_0,1)+(h+θ)(k_1-1)k_1/2+∑_k=0^k_1-1⌊y_0+kh/y⌋+∑_k=k_1^A_m⌊z_0,1-kθ/δ⌋. - If k_1>A_m, then A has the same expression as in (<ref>).- If k_0=0, thenB=(1+A_m)z_0,2-(δ+1)A_m(1+A_m)/2+∑_k=0^A_m⌊z_0,2-kδ/θ⌋. - If 1≤ k_0≤ A_m, then B has the same expression as that in (<ref>).- If k_0>A_m, then B has the same expression as in (<ref>).Proof: For the stated values of k_0 and k_1 (recall that now we have a=hδ+yθ and b=wδ+lθ), from (<ref>) and (<ref>), we still haveS_k=⌊y_0+kh/y⌋-k if k<k_1,A_m-k if k≥ k_1,andT_k=⌊x_0+kl/w⌋-k if k<k_0,A_m-k if k≥ k_0.Then, all the expressions of the statement are obtained using the same arguments of the proof of Theorem <ref>.□Similar arguments of remarks <ref> and <ref> leave to k_0≥0 and k_1≥0.Although in the statement of Theorem <ref> appear sums like s=∑_k=k_0^A_m⌊z_0,2-kδ/θ⌋ that is not of type S^± (the sum do not begin at k=0), we can reduce it to one sum of type S^±. Indeed, taking a generic sum ∑_k=n_1^n_2⌊s± kt/q⌋ with 0≤ s,t<q, and changing the summation index, u=k-n_1, we obtain an S^± sum∑_u=0^n_2-n_1⌊s± n_1t± ut/q⌋=α(1+n_2-n_1)+∑_u=0^n_2-n_1⌊α± ut/q⌋with α=s± n_1t, α=αq+α and 0≤α<q. § DISCRETE SUMS S^± Let us denote the discrete sum S^± byS^±(s,t,q,N)=∑_k=0^N⌊s± kt/q⌋,0≤ s,t<q.These type of sums appear to be a main tool for computing denumerants, as it has been seen in the previous section. In this section we study some properties of S^± in order to obtain an efficient numerical calculation of it. This calculation will be done in a discrete Lebesgue–like sense. §.§ S^+ sumsConsider the function f(x)=⌊s+xt/q⌋ that defines the general term of an S^+(s,t,q,N) sum. Let us define the k–interval I_k⊂[0,N] by I_k={x∈[0,N]: f(x)=k}. A k–interval I_k is called an hS-type interval if |I_k∩|=⌈q/t⌉.Given a k–interval I_k, we have (i) I_k=[x_k,x_k+1) with x_k=kq-s/t.(ii) ⌊q/t⌋≤|I_k∩|≤⌈q/t⌉ holds except, eventually, the first and/or last intervals. Proof: Item (i) comes directly from the expression of f. A real interval, I=[α,β) of length ℓ=β-α, contains at least ⌊ℓ⌋ integers and it contains at most ⌈ℓ⌉ integers. Item (ii) comes from the length of |I_k|=x_k+1-x_k=q/t. □We discuss the value of S^+(s,t,q,N) depending on the following three subcases (a) t| q,(b) t∤ q and (t,q)=1,(c) t∤ q and (t,q)=g>1.§.§.§ Assume t| q The maximum value attained by f in [0,N] isM=⌊s+Nt/q⌋at x_M=Mq-s/t. Figure <ref> shows the case s=1, t=3, q=12 and N=30. All k-intervals are hS-type ones. The distribution of integrals values is the same in each interval.Let assume t| q. Then,S^+(s,t,q,N)=q/t M(M-1)/2+M(N-⌈ x_M⌉+1). Proof: Let us denote q=tq. By Lemma <ref>, each I_k interval is a hS–type interval, that is |I_k|=q (except, perhaps, the first I_0 and the last one I_M). We divide the interval I=[0,N] in three regions I=I_0∪ J∪ I_M, where M is the maximum value attained by the function f in I.Let us denote n=x_M-x_1/q. Then, there are n k-intervals different from I_0 and I_M in I, i.e. I_1=[x_1,x_2),…,I_M-1=[x_M-1,x_M) (here n=M-1 holds). The last interval I_M can be eventually one point (that is I_M={N}). The sum isS^+(s,t,q,N)=∑_k=0^N f(k)=0+∑_k=x_1^x_M-1f(k)+∑_k=x_M^N f(k).Now we add these values like a discrete Lebesgue–like sum∑_k=x_1^x_M-1f(k)=∑_j=1^M-1|I_j∩|j=∑_j=1^M-1qj=q (M-1)M/2.Finally, we have to add ∑_k=x_M^N f(k)=|I_M∩|M. The number of integral points in I_M is |I_M∩|=N-⌈ x_M⌉+1. Thus, the value of S^+(s,t,q,N) is the stated one. □§.§.§ Assume t∤ q and (t,q)=1 Assume t∤ q. We use the notationq =qt+q̂, 1≤q̂<t,s =st+ŝ, 0≤ŝ<t,S(s,t,q,N) =q(M-1)M/2+M(N-⌈ x_M⌉+1).Given a set A⊂, a subset J⊂ A of hS indices is a set of ordered indices in A of hS–type intervals. We define S_J=∑_k∈ Jk.Assume t∤ q. Then, I_k⊂[0,N] is an hS–type interval if and only if(ŝ-kq̂)t<q̂. Proof: The modulo in the statement is taken from the set of residues {0,1,…,t-1}. Notice that |I_k|=x_k+1-x_k=q+q̂/t. The interval I_k is hS–type if and only if ⌈ x_k⌉-x_k<q̂/t (so, the maximum number of integral values are located in I_k). This condition can be restated in a more numerically stable relation. Fromx_k=kq-s/t=kq-s-ŝ-kq̂/t,putting ŝ-kq̂=α t+β with 0≤β<t, we have x_k=n-β/t. Thus, inequality ⌈ x_k⌉-x_k<q̂/t holds if and only if β<q̂. Equivalently (ŝ-kq̂)t<q̂. □Let us consider s=0, t=3, q=11 and N=30. Figure <ref> shows all k-intervals in this case. Notice the hS-type intervals for k∈{0,1,3,4,6,7} (when 0≤ k≤ M=8 is a solution of -2k3<2). Thus, only the intervals I_2 and I_5 are not of type hS in Figure <ref>. The distribution pattern of integral values inside the k-intervals is also ruled modulo t. This fact is detailed in the following result. Let I_k and I_k+T, T>0, be two intervals with the same distribution of integral values. Then, (i) T≡0t.(ii) The minimum value of T is t. Proof: In particular, ⌈ x_k+T⌉-x_k+T=⌈ x_k⌉-x_k holds. Then, x_k+T-x_k∈, that is (recalling that (q,t)=1)x_k+T-x_k=T q/t∈⇔ T≡0t.The minimum T>0 for the value Tq/t to be an integer is T=t. □ The distribution of integral values in the k-intervals has period t. Although Lemma <ref> and Lemma <ref> give a characterization of hS–intervals, we need a more accurate description of these intervals. This description will be used to efficiently obtain a subset of hS–indices J of Definition <ref>. Indeed, from Lemma <ref>, the set J can be parameterized byJ={q^-1(s-i)t| 0≤ i≤q-1}.Notice that J≠∅ because q≥1 (t∤ q and (t,q)=1) and q^-1s∈ J always. This parameterization is useless, from the point of view of numerical efficiency, whenever we need the elements of J to be sorted. Noting that elements of J are sorted by a rule defined by two moduli, q and t, we can obtain a sorted parameterization of J. For instance (notice that (t,q)=1)J={[q^-1(s-(s+iu)q)]t| 0≤ i≤q-1}, u≡ tqis an example of such parameterization. Assume t∤ q and (t,q)=1. Consider the set of hS-type indices J={j_0,…,j_q-1}⊂{0,…,t-1} and S(s,t,q,N) given by expression (<ref>). Then, (a) If j_0≥ M, then S^+(s,t,q,N)=S(s,t,q,N) holds.(b) If j_0<M, there are three different cases. (b.1) If j_q-1≥ M, consider the subset of hS indices K⊂{0,…,M-1}. Then,S^+(s,t,q,N)=S(s,t,q,N)+S_K holds.(b.2) If j_q-1<M and j_0+t≥ M, then S^+(s,t,q,N)=S(s,t,q,N)+S_J holds.(b.3) If j_q-1<M and j_0+t<M, set u=⌊M-1/t⌋ and consider the set of hS-type indices K⊂{j_0+ut,…,M-1}. Then,S^+(s,t,q,N)=S(s,t,q,N)+uS_J+qt(u-1)u/2+S_K. Proof: S^+(s,t,q,N) can be calculated from S plus all additional summands corresponding to hS-type intervals. That is, each hS-type interval I_j has an additional value j which must be added to S.(a) When j_0≥ M, there is no hS-type interval in [0,M). Thus, all k-intervals in this region has q integral values. Then, S^+ has the same expression as in Theorem <ref> replacing q/t by q, that is S^+(s,t,q,N)=S(s,t,q,N).(b) When j_0<M there are hS-type intervals in [0,M). So, we also have to add all hS-type indices contained in [0,M) for obtaining S^+. (b.1) Assume j_q-1≥ M. Consider the set of hS-type indices K⊂{0,…,M-1}. There are no more hS-type indices to consider and S^+(s,t,q,N)=S(s,t,q,N)+S_K. (b.2) Assume j_q-1<M and j_0+t≥ M. Then, by Lemma <ref>, all hS-type indices are J. Thus, S^+(s,t,q,N)=S(s,t,q,N)+S_J holds. (b.3) Assume j_q-1<M and j_0+t<M. By Lemma <ref>, the behaviour of the k-intervals is t-periodic. The maximum number of periods included in the set of indices A={0,…,M-1} is u=⌊M-1/t⌋. That is, all the elements in {j_0,…,j_q-1,j_0+t,…,j_q-1+t,…,j_0+(u-1)t,…,j_q-1+(u-1)t} are hS-type indices. The remaining hS-type indices are located in the set of hS indices K⊂{ut,…,M-1}. Therefore,S^+(s,t,q,N) =S+∑_l=0^u-1(∑_j∈ J(lt+j))+S_K=S+∑_l=0^u-1(lt|J|+S_J)+S_K=S+|J|t∑_l=0^u-1l+uS_J+S_K=S+|J|t(u-1)u/2+uS_J+S_K.The statement follows from the identity |J|=q. □ The sets of hS-type indices J and K of Theorem <ref> are obtained at time cost O(q), in the worst case. The first and last elements of J, j_0 and j_q-1, can be obtained at constant time cost from the (sorted) parameterization (<ref>) of J. §.§.§ Assume t∤ q and (t,q)=g>1 When t∤ q and (t,q)=g>1, we have x_h+1-x_k=q/t=q̃/t̃, where t̃=t/g and q̃=q/g. Assume t∤ q and (t,q)=g>1. Let's assume I_k and I_k+T, T>0, are two intervals with the same distribution of integral values. Then, (i) T≡0t̃.(ii) The minimum value of T is t̃. Proof: This lemma follows from the proof of Lemma <ref> with the additional identity q/t=q̃/t̃, (t̃,q̃)=1. □ In particular, Lemma <ref> ensures that the distribution of integrals values of k-intervals in [0,N] has period t̃. Now, by Lemma <ref>, detecting hS-type intervals is done as follows. Sets =sg+s_g, 0≤ s_g<g, s =st̃+ŝ̂, 0≤ŝ̂<t̃, q̃ =qt̃+q̂̂̂, 1≤q̂̂̂<t̃.Then, I_k is an hS interval if and only if(ŝ̂-kq̂̂̂)t̃<q̂̂̂,that is similar to the characterization given in Lemma <ref>.Figure <ref> shows k-intervals in the case s=3, t=4, q=14 and N=30. Here the distribution of integrals values in k-intervals has period t̃=2, that is hS-type intervals follow the rule (<ref>), i.e. the set hS-type indices in [0,8) is {1,3,5,7}. Sorted and non sorted parameterization of J can also be obtained as in (<ref>) and (<ref>). The same expressions holdreplacing t by t̃, s by ŝ̂ and q by q̂̂̂. Now, we denoteS(s,t,q,N)=⌊q̃/t̃⌋ M(M-1)/2+M(N-⌈ x_M⌉+1)and similar results are obtained from the t̃–periodicity of the hS-type intervals ruled by (<ref>). Assume t∤ q and (t,q)=g>1. Set t̃=t/g and q̃=q/g. Then, interchanging t by t̃ and q by q̂̂̂, statements of Theorem <ref> hold. Notice that M and x_M are also calculated like in (<ref>), i.e. using t and q (not t̃ and q̃).Now, the sets of hS indices J and K are computed using (<ref>) at time cost O(t̃). §.§ S^- sums The minus sumsS^-(s,t,q,N)=∑_k=0^N⌊s-kt/q⌋share some behaviour with plus sums S^+(s,t,q,N). We can define by analogy k-intervals I_k=(x_k,x_k+1]⊂ (those intervals such that g(x)=⌊s-xt/q⌋=k for x∈ I_k) with x_k=s-(k+1)q/t and x_k+1=s-kq/t. hS-type intervals are also defined to be those I_k with |I_k∩|=⌈q/t⌉. We denote nowM=-⌊s-Nt/q⌋and x_M=s+(M-1)q/t,that are the analog to (<ref>) for S^+. Also three cases are taken into account now, i.e. t| q, t∤ q with (t,q)=1 and t∤ q with (t,q)=g>1. We give here, without proof, the main results for computing S^- sums. When t| q, all intervals are hS-type ones and have the same distribution of integral values. The following result can be proved using similar arguments as in Theorem <ref>. Assume t| q. Then,S^-(s,t,q,N)=-q/t (M-1)M/2-M(N-⌊ x_M⌋). When t∤ q and (t,q)=1, we also use the notation q and s defined in (<ref>) and (<ref>). Assume t∤ q. Then, I_k⊂[0,N] is an hS–type interval if and only if(ŝ+kq̂)t<q̂.Lemma <ref> allows a non sorted parameterization of the set of hS-type indices J⊂{0,…,t-1}, that isJ={q^-1(i-s)t| 0≤ i≤q-1},which is an analogous expressions to (<ref>) for plus sums. A sorted parameterization of J is given by (now u≡-tq)J={q^-1[(s+iu)q-s]t| 0≤ i≤q-1} if s<qandJ={q^-1[(s+(i+1)u)q-s]t| 0≤ i≤q-1} if s≥q.In any case, as it has been done before, the sorted elements of J will be denoted by J={j_0,…,j_q-1}.The distribution of integral values in k-intervals also has period t on the indices k like in the plus sums. Let us denote the sumS(s,t,q,N)=-q (M-1)M/2-M(N-⌊ x_M⌋)which corresponds to S^- when there is no hS-type k–interval, similar to (<ref>) for S^+. The following result is the analog of Theorem <ref> for S^+. Assume t∤ q and (t,q)=1. Consider the set of hS-type indices J={j_0,…,j_q-1}⊂{0,…,t-1} and S(s,t,q,N) given by expression (<ref>). Then, (a) If j_0≥ M, then S^+(s,t,q,N)=S(s,t,q,N) holds.(b) If j_0<M, there are three different cases: (b.1) If j_q-1≥ M, consider the subset of hS indices K⊂{0,…,M-1}. Then,S^+(s,t,q,N)=S(s,t,q,N)-S_K holds.(b.2) If j_q-1<M and j_0+t≥ M, then S^+(s,t,q,N)=S(s,t,q,N)-S_J holds.(b.3) If j_q-1<M and j_0+t<M, set u=⌊M-1/t⌋ and consider the set of hS-type indices K⊂{j_0+ut,…,M-1}. Then,S^+(s,t,q,N)=S(s,t,q,N)-uS_J-qt(u-1)u/2-S_K.When t∤ q and (t,q)=g>1, we denote t̃=t/g and q̃=q/g. The value q in (<ref>) is the same. i.e. q=⌊q̃/t̃⌋=⌊ q/t⌋. Using the same notation as in (<ref>), the analog to Lemma <ref> is(ŝ̂+kq̂̂̂)t̃<q̂̂̂and non sorted and sorted characterizations of J, (<ref>),(<ref>) and (<ref>), have the same expressions by replacing u by u≡-t̃q̂̂̂, t by t̃,s by ŝ̂ and q by q̂̂̂. Assume t∤ q and (t,q)=g>1. Set t̃=t/g and q̃=q/g. Then, interchanging t by t̃ and q by q̂̂̂, statements of Theorem <ref> hold.Remarks <ref> and <ref> have their analogs here.§ ALGORITHM Let us consider any numerical 3–semigroup N=n_1,n_2,n_3 and n∈ N. By Lemma <ref>, there is another semigroup T=a,b,c, with 1≤ a<b<c and (a,b)=(a,c)=(b,c)=1, and m∈ T such that (n,N)=(m,T). Lemma <ref> only requires a time cost of O(log n_3), in the worst case. Moreover, by Theorem <ref> and Theorem <ref>, it can be assumed that m∈{0,…,P-S} with P=abc and S=a+b+c.There are three possible cases for the semigroup T, (1) a>1 and c∉a,b,(2) a>1 and c∈a,b,(3) a=1.Now we analyze each case for finding the related L-shapes. Then, the time cost of the related S^± sums will be studied. §.§ Case 1: a>1 and c∉a,bIn this case we have e(T)=3. From c∉a,b, we also have c≤(a,b)<(a-1)(b-1)<ab. Let a,b,c be a numerical 3–semigroup with 1<a<b<c. Assume that =(l,h,w,y) is related to T with wy≠0. Then,M_c=(l-w)a+(h-y)b=(δ+θ)c=min{kc: k≥1, kc∈a,b}. Proof. Assume k_0c<M_c for some k_0∈ with k_0≥1. Then, k_0c=α a+β b with α,β∈ and α+β≥2 (the identity α+β=1 leads to k_0c=a or k_0c=b, a contradiction to a<b<c).Assume α≥1. Then, the squares α-1,β and l-w-1,h-y represent the same equivalence class [0]_c. From l-w-1,h-y∈ (because of y>0) and α-1,β∉ (only one square infor each equivalence class), we have (l-w-1)a+(h-y)b≤(α-1)a+β b. Thus, M_c≤ k_0c holds and makes a contradiction.The case β≥1 also makes a contradiction by similar arguments. □ Let T=a,b,c be a numerical 3-semigroup with 1<a<b<c, (a,b)=(a,c)=(b,c)=1 and c∉a,b. Then, only one L-shape (l,h,w,y) is related to T and wy≠0.Proof. Assume w=0. Then hb=0+θ c holds with θ≥1. Thus, c| h ((b,c)=1) and b|θ. Now, from c=lh, it follows that h=c and l=1. So, a=yb+δ c holds and makes a contradiction. Indeed, either δ=0 we have y≠0 and a| b or δ>0 and a≥ c, a contradiction. The case y=0 also leads to contradiction by similar arguments.According to <cit.>, T has only one related L-shape =(l,h,w,y) iff (la-yb)(hb-wa)>0. Assume la=yb holds. Then, a| y and b| l ((a,b)=1) and hb=wa+θ c with θ≥1 (recall that δ+θ≥1). From c=lh-wy=yb/ah-wy=y/a(hb-wa), we have ac=yθ c. So, y| a holds and so y=a. Therefore, θ=1 and l=b hold.Let us consider now M_c defined in Lemma <ref>. Then,M_c=(l-w)a+(h-y)b=(b-w)a+(h-a)b=hb-wa=θ c=c.So, c∈a,b holds and makes a contradiction. The assumption hb=wa also leads to contradiction by similar arguments. □This lemma ensures that yb<la and wa<hb hold for (l,h,w,y) related to T. A direct consequence of Lemma <ref> is the non-symmetry of T. Let T=a,b,c be a numerical 3-semigroup with 1<a<b<c and (b,c)=(a,c)=1. Assume =(l,h,w,y) is an L-shape related to T. Then,la= min{ka: k≥1, ka∈b,c},hb= min{kb: k≥1, kb∈a,c}. Proof. Here we prove the first equality. The second one is proved by similar arguments.Asis an L-shape related to T, we have(c,T)={ia+jb: i,j∈}. In particular, it follows that la=min{ka: ka∉(c,T)}. Using the same notation of <cit.>, we have c_1a=r_12b+r_13c with r_12,r_13>0, where c_1a=min{ka: k≥1,ka∈b,c}.As la∉(c,T), we have la-c∈ T and so la-c=x_1a+x_2b+x_3c. Assuming x_1≠0, (l-x_1)a-c=x_2b+x_3c∈ T holds and then (l-x_1)a∉(c,T). This is a contradiction to the minimality of la. Therefore, x_1=0 holds. Thus, la-c∈b,c and l≥ c_1 from the minimality of c_1a.Now, from c_1a=r_12b+r_13c with r_13>0, it follows that c_1a-c∈ T. Then, c_1a∉(c,T) and l≤ c_1 from the minimality of la. □ Let T=a,b,c be a numerical 3-semigroup with 1<a<b<c, (a,b)=(a,c)=(b,c)=1 and c∉a,b. Assume T has only one related L-shape (l,h,w,y). Then, h<a and l<b.Proof. By Lemma <ref>, it follows that la≤ ab and l≤ b holds. Similarly, h≤ a also holds.Assume l=b. So, ab=la=yb+δ c with δ≥1 (by Lemma <ref> we have la>yb). Then, b(a-y)=δ c holds and thus c|(a-y) ((b,c)=1). That is, a=y+α c with α≥1 which contradicts inequality a<c. Similar arguments lead to contradiction assuming h=a. □In this case the sides of the L-shape are bounded by w<l<b and y<h<a. §.§ Case 2: a>1 and c∈a,b Identities (a,b)=(b,c)=(a,c)=1 also hold. Assume c=λ a+μ b, λ,μ∈, 0<μ<a, (a,μ)=(b,λ)=1. Then, λ≠ b and there are two L-shapes related to T=a,b,c, _1=(λ+b,a,b,a-μ) with (δ,θ)=(1,0) and _2 with (δ,θ)=(0,1) given by_2=(b,a+μ,b-λ,a) if λ<b, (b,(1+⌊λ/b⌋)a+μ,b-s,a) if λ>b where λ=⌊λ/b⌋ b+s, 0≤ s<b. §.§ Case 3: a=1 Consider a semigroup T=1,b,c with 1<b<c and (b,c)=1. Consider the numerical semigroup T=1,b,c with (b,c)=1. Then, there are two related L-shapes _1=(c,1,b,0) with parameters (δ,θ)=(1,0) and _2 with parameters (δ,θ)=(0,1) given by_2=(b,2,2b-c,1) if c<2b, (b,1+⌊ c/b⌋,b-r,1) if c>2b where c=⌊ c/b⌋ b+r, 0≤ r<b. Proof. _1 is related to T by Theorem <ref>. As θ=0, using the transformation of L-shapes defined in <cit.>, we obtain _2 from _1. □ § TIME COST Let us analyze now the time cost, in the worst case. This analysis will be done under the assumption of m≈ P=abc. This is the same assumption as the one made in the analysis of algorithms ,and .Applying Theorem <ref>, Theorem <ref> or Theorem <ref> requires the calculation of the L-shape =(l,h,w,y), the related basic factorization (x_0,y_0,z_0) and all the related S^± sums. The first two calculations have a time cost of O(log c) <cit.>. Then, all S^± have to be calculated.Consider a generic sum S^±(s,t,q,N)=∑_k=0^N⌊s± kt/q⌋, with 0≤ s,t<q. Using the same notation of Section <ref>, we have * If t| q, Theorem <ref> for S^+ and Theorem <ref> for S^- ensure a constant time cost.* If t∤ q and (t,q)=1, Theorem <ref> for S^+ or Theorem <ref> for S^- has to be applied. All computations are focused on finding the subsets of hS indices K_1⊂{0,…,M-1}, K_2⊂{j0+ut,…, M-1} or J⊂{0,…,t-1}. As |K_1|,|K_2|≤|J|, the cost has order O(q)≤ O(t), in the worst case.* If t∤ q and (t,q)=g>1, consider t̃=t/g. Then, Theorem <ref> or Theorem <ref> and similar arguments as in the previous case ensure a time cost upper bounded by O(t̃).Previous comments point to the fact that the higher cost of computation is reached when t∤ q and (t,q)=1. In this case, the time cost is upperbounded by O(t). Given a semigroup S=n_1,n_2,n_3 and n∈ S, apply Lemma <ref> at constant time cost for obtaining T=a,b,c with 1≤ a<b<c and (a,b)=(b,c)=(a,c)=1 and m∈ T such that (n,S) can be calculated from (m,T). Then, we have to analyze the time cost of each case given in the previous section.The worst case for the calculation of S^±(s,t,q,N), as it is highlighted in Remark <ref>, appears when t∤ q and (t,q)=1. This case will be assumed in all cases in the following analysis. Thus, the resulting worst case order will be a pessimistic estimation.* Case 1 (a>1, c∉a,b). By Lemma <ref>, the L-shapebelongs to the case (iii) δθ>0, subcase (iii.4) wy≠0. The following sums have to be evaluated * If k_1=0, there is one sum S^-_1=∑_k=0^A_m⌊z_0,1-kθ/δ⌋. As 0≤θ<δ=la-yb/c<la/c<ab/c<a, the cost of calculating S_1^- is upperbounded by O(a).* If 1≤ k_1≤ A_m, there are two sums S^+_1=∑_k=0^k_1-1⌊y_0+kh/y⌋ and S^-_1=∑_k=0^A_m⌊z_0,1-kθ/δ⌋. The calculation of S^-_1 is O(a). As h<y<a, the order for calculating S^+_1 is also upperbounded by O(a). Thus, the worst case order of this case is O(a).* If k_1>A_m, there is one sum S_2^+=∑_k=0^A_m⌊y_0+kh/y⌋. This sum has the same order as S_1^+, that is O(a).* If k_0=0, there is one sum S_2^-=∑_k=0^A_m⌊z_0,2-kδ/θ⌋. From δ<θ=hb-wa/c<hb/c<ab/c<a, the order is upperbounded by O(a).* If 1≤ k_0≤ A_m, there are two sums S_3^+=∑_k=0^k_0-1⌊x_0+kl/w⌋ and S^-_2. From l<w<b, the cost of both calculations is O(a)+O(b)=O(b).* If k_0>A_m, we have S_4^+=∑_k=0^A_m⌊x_0+kl/w⌋ with the same cost of S_3^+, that is O(b).Therefore, the overall cost of the Case 1 is O(b). * Case 2 (a>1, c∈a,b). Let us consider c=λ a+μ b with 1≤μ<a. By Lemma <ref> there are three possible cases to be examined.Consider the L-shape _1=(λ+b,a,b,a-μ) with δ=1 and θ=0. Then, A_m=z_0 and k_1=⌈z_0a-y_0/a⌉≤ z_0=A_m. Thus, the case k_1>A_m never appears. So, * If k_1=0, there is one sum S_1^+=∑_k=0^A_m⌊x_0+kl/w⌋ with w=b. Then, the order is upperbounded by O(b).* If 1≤ k_1≤ A_m, there are two sums S_1^+ and S_2^+=∑_k=0^k_1-1⌊y_0+kh/y⌋, From y=a-μ<a, the order is upperbounded by O(a)+O(b)=O(b).Let us examine the other related L-shape _2 which have an expression depending on λ. Assume λ<b. Then, we have _2=(b,a+μ,b-λ,a) with δ=0 and θ=1. As δ=0, this is the Case-(i) with w=b-λ>0. So, A_m=z_0 holds and the case k_0>A_m never appears. Then, * If k_0=0, there is one sum S_1^+=∑_k=0^A_m⌊y_0+kh/y⌋. From y=a, we have order upperbounded by O(a).* If 1≤ k_0≤ A_m, there are two sums S_1^+ and S_2^+=∑_k=0^k_0-1⌊x_0+kl/w⌋. From w=b-λ<b, the order is upperbounded by O(a)+O(b)=O(b).Assume now λ>b. Then, the related L-shape is _2=(b,(1+⌊λ/b⌋)a+μ,b-s,a) with 0≤ s<b, δ=0 and θ=1. We also are in the Case (i) with w=b-s>0. So, A_m=z_0 and k_0≤ A_m always holds. Then, * If k_0=0, there is one sum S_1^+=∑_k=0^A_m⌊y_0+kh/y⌋. From y=a, the order is upperbounded by O(a).* If 1≤ k_0≤ A_m, there are two sums S_1^+ and S_2^+=∑_k=0^k_0-1⌊x_0+kl/w⌋. The order is upperbounded by O(a)+O(b)=O(b).In any case, using either _1 or _2, the overall order is upperbounded by O(b). * Case 3 (a=1). We have (b,c)=1. By Lemma <ref>, there are three possibilities to analyze.Consider _1=(c,1,b,0), with δ=1 and θ=0. This L-shape can be used in the two cases c<2b and c>2b (note that c≠2b). Look at the Case (ii) in Section <ref> and Theorem <ref>. As y=0, from (<ref>) we have to calculate one sum S_1^+=∑_k=0^A_m⌊x_0+kl/w⌋. From l<w=b, the order is upperbounded by O(b).Let us consider now the case c<2b. Again by Lemma <ref>, we can use the L-shape _2=(b,2,2b-c,1) with δ=0 and θ=1. Note that A_m=z_0. We have to look at Theorem <ref>-(i.2) (y=1>0). We have k_0≤ A_m, then the case (i.2.3) never appears. Then, * If k_0=0, there is only one sum given by (<ref>) S_1^+=∑_k=0^A_m⌊y_0+kh/y⌋. From y=1, the cost is constant.* If 1≤ k_0≤ A_m, there are two sums S_1^+ and S_2^+=∑_k=0^k_0-1⌊x_0+kl/w⌋. From l<w=2b-c<b, the cost is upperbounded by O(b). Then, the total cost is upperbounded by O(b)+O(1)=O(b).When c>2b, we can use the L-shape _2=(b,1+⌊ c/b⌋,b-r,1) with c=⌊ c/b⌋ b+r and 0≤ r<b. The related parameters are δ=0 and θ=1. Using the same arguments of the previous case, it follows that the overall order is upperbounded by O(b).Therefore, using any admissible L-shape, the total cost of this case is also O(b). So, we need a cost of O(log c) to calculate the related L-shape and the basic coordinates plus O(b) to calculate the involved S^± sums. Hence, our algorithm has a time cost of O(b+log c). Common instances of semigroups are such that O(log c)≪ O(b). Then, we have the following result. The time cost, in the worst case, for computing the denumerant (m,T) is upperbounded by O(b+log c). When many instances of m∈ T are given and the semigrup T is fixed, the related L-shape is computed only once. Thus, the first calculated denumerant has a time cost of O(b+log c). The subsequent instances only need a time cost of O(b). § SOME TIME TESTS All the computations of this section have been made using SageMath 7.3 <cit.> and non compiled code on aprocessor. Here we test our algorithm, denoted by , versus the algorithms ,and . In the following, we use the notation P=abc and S=a+b+c. The time required to calculate denumerants highly depends on the selected semigroup. This fact is reflected in the following subsections. All semigroups in this section will meet the property (a,b)=(a,c)=(b,c)=1. By Lemma <ref> of Brown, Chou and Shiue, this restrictions does not represent any loose of generality.Time costs of the involved algorithms are by Table <ref>. It is assumed that T=a,b,c and m≈ P=abc.According to Table <ref> there are some generic behaviours to be highlighted: (i) Algorithm has the best time cost.(ii) When a=1, algorithmsandare faster than Algorithm when b≪ c. However, when c≈ b, algorithms ,andrun at similar speed.(iii) When a≠1, there are two different behaviours, (iii.1) if ab<c, Algorithm is faster than algorithmsand ,(iii.2) otherwise, when ab>c, Algorithm winsand .(iv) When b≈ c, Algorithm is faster than algorithmsandprovided that a≫1.In the following subsections we take elements m of the semigroup that are closed to P-S. §.§ a>1, c∉a,b In this case, inequality c<ab always holds. Then, as it has been comment in Remark <ref>-(iii.2), Algorithm isfaster than Algorithm and Algorithm . Table <ref> shows how this assertion is kept for the semigroups T_1,k=7^k,11^k,(7^k,11^k) and m_k=P_k-S_k-k for k∈{1,2,3,4}. The non increasing sequence of times in the column of Algorithm is because the corresponding L-shapes. Their entries do not always increase as the value of k does.Table <ref>, for the semigroups T_2,k=7^k,11^k,11^k+1, shows an instance of the case c≈ b and, as it has been noticed in Remark <ref>-(iv), Algorithm is faster than algorithmsand . §.§ a>1, c∈a,b In this subsection we take the semigroups T_3,k=7^k,11^k,7^k+11^2k for the case ab<c and T_4,k=7^k,11^k,7^k+11^k for ab>c. Tables <ref> and <ref> show the influence of inequalities ab<c and ab>c in the resulting time cost. Here, item (iii) of Remark <ref> is also clear. §.§ a=1Let us take the semigroups T_5,k=1,7^k,11^k. Table <ref> confirms that Algorithm is slower than algorithmsand . This rule is not noticeable with respect to Algorithm for small values of k. However, it turns apparent from the value k=6.Consider now the semigroups T_6,k=1,7^k,7^k+1, where c≈ b. According to Remark <ref>-(ii), the algorithms ,andhave similar time cost. Table <ref> shows this behaviour in algorithmsand . Algorithm runs between three and four times slower.§.§ Almost medium and large input dataNow we take larger input values for the Algorithm . Usually, our algorithm can manage almost middle input values at acceptable time output. However, when the involved S^± sums take some proper parameters, the time cost can be almost constant. These cases allow the Algorithm to take large input values.We consider the same semigroups of the previous sections to see these behaviours. When the output values m_k and (m_k,T) turn to be large, tables will show ℓ(m_k) and ℓ((m_k,T)).Table <ref> and Table <ref> belong to the case a>1 with c∉a,b. The case a>1 with c∈a,b, is represented by Table <ref> when ab<c and Table<ref> when ab>c. Finally, the case a=1 is represented by tables <ref> and <ref>.Algorithm allows almost middle length inputs, above a hundred digits. Several instances of this inputs at reasonable time output are given in tables <ref>, <ref> and <ref>. The nature of the involved S^± sums has an interesting property. Some parameters taken by these sums make almost constant the time cost of the denumerant's calculation. In these cases, the algorithm can handle large inputs (million digits) at a small time cost. Tables <ref>, <ref> and <ref> show some instances of this good behaviour.Now, we briefly comment this almost constant time cost behaviour of Algorithm in tables <ref>, <ref> and <ref>. In fact, almost all the time is spent in the computation of the related L-shape, that is O(log c).The semigroup T_1,n=7^n,11^n,(7^n,11^n), from Theorem <ref>, has related the L-shape _1,n=(11^n-1,7^n-1,1,1) with δ=θ=1. Then, this is the case a>1 with c∉a,b. We have to calculate some of the sums ∑_k=0^A_m⌊z_0,1-kθ/δ⌋, ∑_k=0^A_m⌊z_0,2-kδ/θ⌋, ∑_k=0^k_1-1⌊y_0+kh/y⌋, ∑_k=0^A_m⌊y_0+kh/y⌋, ∑_k=0^k_0-1⌊x_0+kl/w⌋ and∑_k=0^k_0-1⌊x_0+kl/w⌋. From δ=θ=w=y=1, all these sums are calculated at constant time. Therefore, the fast computation of denumerants in Table <ref> is clear now.The semigroup T_4,n=7^n,11^n,7^n+11^n has related the L-shape _4,n=(11^n,7^n+1,11^n-1,7^n) with δ=0 and θ=1. This is the case a>1 with c∈a,b and parameters λ=μ=1 and λ<b=11^n. Thus, following this case at page abcd, we have two possibilities: * When k_0=0, it has to be computed the sum ∑_k=0^A_m⌊y_0+kh/y⌋ with h=7^n+1 and y=7^n. From h=y+1, it follows that h=1 and the sum can be computed at constant cost from Theorem <ref>.* Otherwise, when 1≤ k_0≤ A_m, the algorithm calculates the sum ∑_k=0^k_0-1⌊x_0+kl/w⌋. Then, l=w+1 holds and, by the previous argument, the sum can be computed at constant time cost.Therefore, the fast behaviour of the algorithm in Table <ref> is now clear.Finally, let us consider the semigroups T_6,n=1,7^n,7^n+1. A related L-shape is _6,n=(7^n,2,7^n-1,1) with δ=0 and θ=1. This is the case a=1 with parameters λ=μ=1 and c<2b. Here we also have two possible cases: * When k_0=0, there is only one sum to be computed, S_1^+=∑_k=0^A_m⌊y_0+kh/y⌋. Here we havey=1. Thus, this sum is calculated at constant time.* If 1≤ k_0≤ A_m, we have to compute S_1^+ of the previous case and S_2^+=∑_k=0^k_0-1⌊x_0+kl/w⌋. Again, from l=w+1 and Theorem <ref>, the sum S_2^+ can also be calculated at constant time cost.Thus, the speed of the algorithm in Table <ref> is now clear.Many semigroups have related an L-shape (l,h,w,y) with δ=1 and/or θ=1, w=1 and/or y=1. Additionally, many elements of the semigroup m∈ T have null coefficient multiplying k in the S^± sums. So, the fast behaviour of this algorithm eventually can be habitual.§ CONCLUSIONAlgorithm accepts almost medium input data to calculate denumerants of numerical 3-semigroups at acceptable speed using an ordinary computer (tables <ref>, <ref> and <ref>). As far as we know, this algorithm is faster than usual known implemented algorithms for embedding dimension three numerical semigroups. This is the behaviour in the worst case. Eventually, this algorithm accepts large input data (tables <ref>, <ref> and <ref>).The main tool of this algorithm is the hS-type set of ordered indices of intervals. As the computation techniques for obtaining these sets become faster, the time cost of this algorithm turns to be smaller.It is difficult to generalize the algorithm to larger embedding dimensions because of the related minimum distance diagrams. Less is known about these diagrams related to numerical n-semigroups for n≥4, mainly a generic geometrical description.12 AB:2008 F. Aguiló and J. 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Reine Angewandte Math. 226 (1967) 1–19.EhII:1967 E. Ehrhart, Sur un problème de géometrie diophantienne linéaire II, J. Reine Angewandte Math. 227 (1967) 25–49.numericalsgps M. Delgado, P.A. García-Sánchez, and J. Morais, “numericalsgps”: a gap package on numerical semigroups, (<http://www.gap-system.org/Packages/numericalsgps.html>).Gap The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.5; 2014, (<http://www.gap-system.org>).Li:1995 P. Lisoněǩ, Denumerants and their approximations, J. Combin. Math.& Combin. Comput. 18 (1995) 225–232.Math Wolfram Research, Inc., Mathematica, Version 8.0, Champaign, IL (2010).Po:1953 T. Popoviciu, Asupra unei probleme de patitie a numerelor, Acad. Republicii Populare Romane, Filiala Cluj, Studii si cercetari stiintifice 4 (1953) 7–58.Ra:2005 J.L. Ramírez Alfonsín, The Diophantine Frobenius Problem. Oxford Univ. Press (2005) Oxford. ISBN 0-19-856820-7 978-0-19-856820-9.RoGa-NS:2009 Rosales, J. C. and García-Sánchez, P. A., Numerical semigroups. Developments in Mathematics, 20. Springer (2009) New York, ISBN: 978-1-4419-0159-0.Sage The Sage Developers, SageMath, the Sage Mathematics Software System (Version 7.3), 2016, (http://www.sagemath.org).Sch:1926 I. J. Schur, Zur additiven zahlentheorie, Sitzungsberichte Preussische Akad. Wiss. Phys. Math. Kl. (1926) 488–495.SeOz:1991 S. Sertöz and A.E. Özlük, On the number of representations of an integer by a linear form, Istanbul Üniv. Fen Fak. Mat. Derg. 50 (1991) 67–77.Sy:1857 J. Sylvester, On the partition of numbers, Quart. J. Pure Appl. Math. 1 (1857) 141–152.Sy:1882 J. Sylvester, On subinvariants, i.e. semi–invariants to binary quantities of an unlimited order, Am. J. Math. 5 (1882) 119–136. | http://arxiv.org/abs/1706.08768v1 | {
"authors": [
"F. Aguiló-Gost",
"D. Llena"
],
"categories": [
"math.CO",
"05-04, 05A15, 20M99"
],
"primary_category": "math.CO",
"published": "20170627104256",
"title": "Computing denumerants in numerical 3-semigroups"
} |
Domain reduction techniques for global NLP and MINLP optimization [================================================================= The aim of this short note is to fill in a gap in our earlier paper <cit.> on 2BSDEs with reflections, and to explain how to correct the subsequent results in the second paper <cit.>. We also provide more insight on the properties of 2RBSDEs, in the light of the recent contributions <cit.> in the so-called G-framework.Key words: 2BSDEs, reflections, Skorokhod condition.AMS 2000 subject classifications: 60H10, 60H30. § INTRODUCTIONIn this short note, we fill in a gap in our earlier wellposedness result on so-called second-order reflected 2BSDEs (2RBSDEs for short). The issue stemmed from a wrongly defined minimality condition which ensures uniqueness of the solution, which we correct here. We also use this occasion to prove that an alternative minimality condition, taking the form of a Skorokhod-like condition, leads to the exact same solution, provided that an additional assumption linked to the oscillations of the lower obstacle is added (see Assumption <ref> below). This new condition appeared recently in the two contributions <cit.> on reflected G-BSDEs, and our result proves that the two notions do coincide, and that our formulation produces more general results. Since some of the results of <cit.> were used in our subsequent paper <cit.> considering doubly reflected 2BSDEs, we also explain how to change the minimality condition there, as well as which results are impacted by this change. Roughly speaking, all our previous results still hold true, except for the a priori estimates, where we no longer control the total variation of the bounded variation process appearing in the solution, but only its 𝔻^2,κ_H-norm.§ THE LOWER REFLECTED CASEThe notations in this section are the ones in <cit.>. Our only change is that to remain coherent with the notations in the next section, we will denote the lower obstacle in the 2RBSDE by L instead of S. §.§ The gapThe mistake in the paper <cit.> can be found right after Equation (3.3), when we try to prove that the process K^'-k^' is non-decreasing. Indeed, appealing to the minimality condition (2.6) in <cit.> does not imply the sub–martingality of this process, since it only gives the required inequality for any t∈[0,T] and the fixed time T. The end of Step (ii) of the proof of Theorem 3.1 in <cit.> therefore does not go through. The issue here is that the minimality condition (2.6) that we wrote is only adapted to the case where the generator F is 0, described in Remark 3.1 in <cit.>, and it has to be modified. One could argue there that K^'-k^' might still be non-decreasing, and that an appropriate minimality condition should reflect this fact. However, K^'-k^' is not non–decreasing in general. The issue was actually partially pointed out in Remark 3.6 of <cit.>. It is explained there (and the proof of this result is independent of the mistake in the minimality condition) that on the event {Y_t^-=L_t^-}, one has K^=k^, -a.s., for any ∈_H^κ, meaning that K^-k^ is constant (and thus non–decreasing) as long as Y_t^-=L_t^-. Similarly, the Skorokhod condition satisfied by k^ implies that K^-k^ is still non-decreasing on the event {y^_t^->L_t^-}. But there is nothing we can say on the event {Y_t>y^_t^-=L_t^-}. Also, the following counter–example, communicated to us by Jianfeng Zhang, proves that K^'-k^' is not non–decreasing in general.[Jianfeng Zhang]Fix T=2 and take as a lower obstacle a process L satisfying the required assumptions in <cit.> as well as L_t:=2(1-t),0≤ t≤ 1, andL_t≤ 2,1≤ t≤ 2. Furthermore, take the generator F of the 2RBSDE to be 0, and the terminal condition to be L_2. In this case, the solution to the 2RBSDE being necessarily the supremum of the solutions to the associated RBSDEs, we will have automatically the representations Y_t=ℙ'∈^κ_H(t^+,) essup^ τ∈_t,T essup^ 𝔼^ℙ'[L_τ|_T],y_t^=τ∈_t,T essup^ 𝔼^ℙ'[L_τ|_T]. Furthermore, in this case since F=0, K^'-k^' being a '-sub–martingale is equivalent to Y-y^' being a '-supermartingale, which would imply in particular that Y_0-y_0^'≥𝔼^ℙ'[Y_1-y_1^']. However, it is clear by definition of L that Y_0=y_0^'=2. However, there is absolutely no reason why in general one could not have, for some ', and for an appropriate choice of L, Y_1>y_1^' (recall that we always have Y_1≥ y_1^'), at least with strictly positive '-probability, which then contradicts (<ref>). §.§ The new minimality condition and uniqueness This being clarified, let us now explain what should be the appropriate minimality condition replacing (2.6) in <cit.>. Using the Lipschitz property of F (see Assumption 2.3(iii) in <cit.>), we can define bounded functions λ:[0,T]×Ω×××^d×^d× D_H⟶ R and η:[0,T]×Ω×××^d×^d× D_H⟶ R^d such that for any (t,ω,y,y',z,z',a) F_t(ω,y,z,a)-F_t(ω,y',z',a)=λ_t(ω,y,y',z,z',a)(y-y')+η_t(ω,y,y',z,z',a)· a^1/2(z-z'). Define then for any ∈^κ_H, and for any t∈[0,T] the processM_s^t,:=exp(∫_t^s(λ_u-1/2|η_u|^2)(Y_u,y_u^,Z_u,z_u^,a_u)du-∫_t^sη_u(Y_u,y_u^,Z_u,z_u^,a_u)·a_u^-1/2dB_u). Following the arguments in the beginning of the proof of Theorem 3.1 in <cit.>, we have then for any t∈[0,T], any ∈_H^κ and any '∈_H^κ(t^+,) Y_t-y_t^'=𝔼_t^'[∫_t^TM^t,'_sd(K^'_s-k^'_s)], -a.s. Therefore, the representation formulaY_t=ℙ'∈^κ_H(t^+,) essup^y_t^', -a.s., is equivalent to the new minimality conditionℙ'∈^κ_H(t^+,) essinf^ 𝔼_t^'[∫_t^TM^t,'_sd(K^'_s-k^'_s)]=0, -a.s.If one replaces the minimality condition (2.6) in <cit.> by (<ref>) above, as well as in the statement of Theorem 3.1 in <cit.> the representation (3.1) by simplyY_t=ℙ'∈^κ_H(t^+,) essup^y_t^', -a.s.,then the proof of (<ref>) is immediate as soon as one has proved (<ref>). This allows us to recover uniqueness of the solution (see Section <ref> below for details). The representation formula (3.1) in <cit.> does not only involve t and T, but any pair 0≤ t≤ s≤ T. If one only assumes the new minimality condition (<ref>), then it cannot be proved immediately that Y_t=ℙ'∈^κ_H(t^+,) essup^y_t^'(s,Y_s), -a.s.However, once we have proved that the solution Y of the 2RBSDE satisfies the dynamic programming principle, then the above is immediate. Furthermore, the case s=T is enough to obtain uniqueness, which is the purpose of Theorem 3.1 in <cit.>.Since in general K^'-k^' is not non–decreasing, we cannot reduce (<ref>) to a statement involving only K^' and k^', as is the case for non–reflected 2BSDEs, see for instance <cit.>. However, when L=-∞ and there is no reflection, k^' becomes identically 0, and (<ref>) is indeed equivalent to ℙ'∈^κ_H(t^+,) essinf^ 𝔼_t^'[K^'_T-K^'_t]=0, -a.s.,see the arguments in Step (ii) of the proof of <cit.>.The need to depart from the "standard" minimality condition has also been pointed out by Popier and Zhou <cit.>, when dealing with 2BSDEs under a monotonicity condition, with hypotheses relaxing the earlier work <cit.>. As a sanity check, let us verify here that the new minimality condition (<ref>) indeed allows to recover the classical RBSDE theory when ^κ_H is reduced to a singleton {}. In this case, (<ref>) says exactly that the bounded variation process ∫_0^· M^0,_sd(K^_s-k^_s) is a –martingale. Since the filtration ^ satisfies the predictable martingale representation property, it means that this process is identically 0. Now since M^0, is -a.s. positive, this implies that K^=k^, which is the desired property. §.§ Recovering existenceThe second instance of the use of the wrong conclusion that K^-k^ was non-decreasing in <cit.> is in the existence proof, during the discussion after Equation (4.6). At this point, the last thing to prove is that K satisfies the new minimality condition (<ref>). However, we already have the result of Proposition 4.2 in <cit.> which shows that the process V^+ satisfies the representation formula (<ref>). Therefore, the fact that (<ref>) is indeed satisfied is immediate, since both statements are equivalent (see Section <ref> below for details).To summarise, one should replace Definition 2.3 in <cit.> by the following, and use the corrections explained above in the proofs. For ξ∈𝕃^2,κ_H, we say (Y,Z)∈𝔻^2,κ_H×ℍ^2,κ_H is a solution to the 2RBSDE if ∙ Y_T=ξ, and Y_t≥ L_t, t∈[0,T], 𝒫_H^κ-q.s.∙ ∀ℙ∈𝒫_H^κ, the process K^ℙ defined below has non-decreasing paths ℙ-a.s.K_t^ℙ:=Y_0-Y_t - ∫_0^tF_s(Y_s,Z_s)ds+∫_0^tZ_sdB_s,0≤ t≤ T, ℙ-a.s.∙ We have the following minimality conditionℙ'∈^κ_H(t^+,) essinf^ 𝔼_t^'[∫_t^TM^t,'_sd(K^'_s-k^'_s)]=0 , ℙ-a.s.,0≤ t≤ T, ∀ℙ∈𝒫_H^κ.§.§ Detailed proofsFor the ease of the reader, we give the details of the proof for the uniqueness, which is a correction of Theorem 3.1 in <cit.>. Let Assumptions 2.1 and 2.2 in <cit.> hold. Assume ξ∈𝕃^2,κ_H and that (Y,Z) is a solution to 2RBSDE in Definition <ref>. Then, for any ℙ∈𝒫^κ_H and 0≤ t ≤ T, Y_t =ℙ^'∈𝒫^κ_H(t^+,ℙ)^ℙy_t^ℙ^', ℙ-a.s. Consequently, the 2RBSDE in Definition <ref> has at most one solution in 𝔻^2,κ_H×ℍ^2,κ_H.We start by proving representationref.(i) Fix 0≤ t≤ T and ℙ∈𝒫^κ_H. For any ℙ^'∈𝒫^κ_H(t^+,ℙ), we haveY_t = ξ + ∫_t^TF_s(Y_s,Z_s)ds - ∫_t^T Z_sdB_s + K_T^ℙ^'-K_t^ℙ^',ℙ^'-a.s. Now, it is clear that we can always decompose the non-decreasing process K^ℙ intoK^ℙ^'_t=A_t^ℙ^'+B_t^ℙ^',ℙ^'-a.s.,were A^ℙ^' and B^ℙ^' are two non-decreasing processes such that A^ℙ^' only increases when Y_t^-=S_t^- and B^ℙ^' only increases when Y_t^->S_t^-. With that decomposition, we can apply a generalisation of the usual comparison theorem proved by El Karoui et al. (see Theorem 8.3 in <cit.>), whose proof is given in the appendix of <cit.>, under ℙ^' to obtain Y_t≥ y_t^ℙ^' and A_T^ℙ^'-A_t^ℙ^'≤ k^^'_T-k^^'_t, ℙ^'-a.s. Since ℙ^'=ℙ on ℱ_t^+, we get Y_t≥ y_t^ℙ^', ℙ-a.s. and thusY_t≥ℙ^'∈𝒫^κ_H(t^+,ℙ)^ℙy_t^ℙ^', ℙ-a.s.(ii) We now prove the reverse inequality. Fix ℙ∈𝒫^κ_H. For every ℙ^'∈𝒫^κ_H(t^+,ℙ), denoteδ Y:=Y-y^ℙ^',δ Z:=Z-z^ℙ^' and δ K^ℙ^':=K^ℙ^'-k^ℙ^'.As in (<ref>), there exist two bounded processes λ and η such that for all t≤ Tδ Y_t=∫_t^T(λ_sδ Y_s+η_sa_s^1/2δ Z_s)ds-∫_t^Tδ Z_sdB_s+δ K_T^ℙ^'-δ K_t^ℙ^',ℙ^'-a.s. Then, by Itô's formula, we obtainδ Y_t=𝔼_t^ℙ^'[∫_t^TM^t,ℙ^'_s dδ K_s^ℙ^'],where M^t,ℙ^'_s is defined in (<ref>). Taking the essential infimum on both sides, then (<ref>) implies (<ref>). For the existence, there are only changes in the case when ξ is in UC_b(Ω). And we only give details for the part where the minimality condition is concerned.In Proposition 4.2 in <cit.>, we proved that the process V^+ satisfies the representation formula V_t^+=ℙ^'∈𝒫^κ_H(t^+,ℙ)^ℙy_t^ℙ^', ℙ-a.s., ∀ℙ∈𝒫_H^κ. We have to check that the minimality condition eq:new holds. Fix ℙ in 𝒫^κ_H and ℙ^'∈𝒫^κ_H(t^+,ℙ). By the Lipschitz property of F, we know that there exists bounded processes λ and η as in (<ref>) such thatV^+_t-y_t^ℙ^' =∫_t^T[λ_s(V^+_s-y_s^ℙ^')+η_sa_s^1/2(Z_s-z_s^ℙ^')]ds-∫_t^Ta_s^1/2(Z_s-z_s^ℙ^')a^-1/2_sdB_s+K_T^ℙ^'-K_t^ℙ^'-k_T^ℙ^'+k_t^ℙ^'. Then, by Itô's formula, we obtainV^+_t-y_t^ℙ^'=𝔼^ℙ^'_t[∫_t^TM_s^t,ℙ^'d(K_s^ℙ^'-k^ℙ^'_s)]. Taking the essential infimum on both sides, eq:repV implies that the minimality condition eq:new is satisfied.Now since Y_t=ℙ^'∈𝒫^κ_H(t^+,ℙ)^ℙy_t^ℙ^', ℙ-a.s.,t∈ [0,T],for all ℙ∈𝒫^κ_H,Y is thus unique. Then, since we have that d<Y,B>_t=Z_td<B>_t, 𝒫^κ_H-q.s., Z is also unique. Finally, the process K^ℙ is uniquely determined.§.§ An alternative: Skorokhod minimality conditionReaders familiar with the theory of standard reflected BSDEs should be wondering whether there is an equivalent, in the second–order setting, of the so–called Skorokhod condition. The latter states that the non-decreasing process appearing in the definition of a RBSDE acts in a minimal way, only when the solution actually reaches the obstacle, and implies uniqueness of the solution (see the seminal paper <cit.> for more details). There are actually two recent papers which treat the very related problem of reflected G-BSDEs, namely <cit.>, and which use a generalisation of this condition. The aim of this section is to show that this condition also implies wellposedness in our framework, under an additional assumption on the obstacle L, and that the two definitions are actually equivalent. We also provide a more detailed comparison between <cit.> and our work at the end of the section.§.§.§ Wellposedness under Skorokhod conditionUsing the same notations as before, the Skorokhod condition for 2RBSDEs readsℙ'∈^κ_H(t^+,) essinf^ 𝔼_t^'[∫_t^T(Y_s^--L_s^-)dK_s^']=0,t∈[0,T], -a.s., ∀∈^κ_H. For ease of reference, we provide the corresponding alternative definition of a solution to the 2RBSDE.For ξ∈𝕃^2,κ_H, we say (Y,Z)∈𝔻^2,κ_H×ℍ^2,κ_H is a Skorokhod–solution to the 2RBSDE if∙ Y_T=ξ, and Y_t≥ L_t, t∈[0,T], 𝒫_H^κ-q.s.∙ ∀ℙ∈𝒫_H^κ, the process K^ℙ defined below has non–decreasing paths ℙ-a.s.K_t^ℙ:=Y_0-Y_t - ∫_0^tF_s(Y_s,Z_s)ds+∫_0^tZ_sdB_s,0≤ t≤ T, ℙ-a.s.∙ The minimality condition (<ref>) holdsIn more mundane terms, this condition is saying that if there is a probability measuresuch that the supremum in the representation formula (<ref>) is attained, then on the support of , the classical Skorokhod condition is satisfied by the solution of the 2RBSDE. Let us now argue how (<ref>) can be used instead of (<ref>) to recover wellposedness, and that both conditions actually lead to the exact same solution. Notice however that the method of proof here requires the followingcondition on L, which basically asks that the variations of L are not too "extreme".We have for any m∈ℕ, for any sequence {(t_i^n)_1≤ i≤ n,n≥ 0} of partitions of [0,T] (allowing for stopping times) whose mesh goes to 0 as n goes to +∞, and for any >0n→+∞lim ∈^κ_Hsup[∑_i=0^n-1 1_{|L_t_i+1^n--L_t_i^n-|≥}≥ n-m]=0. The above assumption puts somehow restrictions on the oscillations or the variations of L. Before pursuing, let us give the following two sufficient conditions. Either of the following two conditions imply Assumption <ref>(i) For any non–decreasing sequence of stopping times (ρ_n)_n≥ 0 converging to T and for any >0, we haven→+∞lim ∈^κ_Hsup[|L_ρ_n+1^--L_ρ_n^-|≥]=0.(ii) Let Π_[0,T] be the set of all partitions of [0,T] (which allow for stopping times). We have for some p≥ 1ℓ:=(ρ_i)_i∈Π_[0,T]sup ∈_H^κsup𝔼^[∑_i=0^n-1L_ρ_i+1^--L_ρ_i^-^p]<+∞. For (i), It actually suffices to notice that for any ∈_H^κ, for any m≤ n and for any >0[∑_i=0^n-1 1_{|L_t_i+1^n--L_t_i^n-|≥}≥ n-m]≤∑_i=n-m^n[|L_t_i+1^n--L_t_i^n-|≥].As for (ii), a simple application of Markov inequality provides for any p≥ 1[∑_i=0^n-1 1_{|L_t_i+1^n--L_t_i^n-|≥}≥ n-m] ≤[∑_i=0^n-1|L_t_i+1^n--L_t_i^n-|^p≥ (n-m)^p]≤1/^p(n-m)^[∑_i=0^n-1|L_t_i+1^n--L_t_i^n-|^p]≤ℓ/^p(n-m). Obviously,conditions in Lemma <ref>, and thus Assumption<ref> are satisfiedfor Lbeinga semi–martingale of the formL_t=L_0+∫_0^tU_sds+∫_0^tV_sdB_s+C_t,𝒫_H^κ-q.s.where C is càdlàg process of integrable variation such that the measure dC_t is singular with respect to the Lebesgue measure dt and which admits the following decompositionC_t=C_t^+-C_t^-,where C^+ and C^- are non–decreasing processes. Besides, U and V are respectively ℝ- and ℝ^d-valued, and 𝔽-progressively measurable processes such that for somep ≥ 1, ∈^κ_Hsup𝔼^[sup_0 ≤t ≤ TU_t^p + (∫_0^T V_t^2dt )^p/2+ (C_T^+)^p +(C_T^-)^p]<+∞. Notice that in the proof of Theorem <ref>, we actually only use Assumption <ref> for a specific partition of [0,T] defined in terms of successive crossings of Y-L. We therefore could have formulated Assumption <ref> in terms of this partition only. We choose not to do so in order to make the assumption more natural.The main result is now as follows, and its proof borrows a lot from the seminal paper of Ekren, Touzi and Zhang <cit.>. Let Assumption <ref> hold, as well as the necessary assumptions for wellposedness in <cit.>. Then there is a unique Skorokhod–solution to the 2RBSDE which coincides with the unique solution to the 2RBSDE.We now argue in two steps.Step 1: uniquenessThis is the easiest part. Assume that there exists a Skorokhod–solution (Y,Z). We will argue that Y=Y, which implies immediately that Z=Z, since Z is uniquely defined by the quadratic co–variation between Y and B.Fix first some ∈_H^κ. By definition, Y is a super–solution underto the standard BSDE with terminal condition ξ, generator F. Since it is also always above L, and since solutions to reflected BSDEs are also the minimal super–solutions of the associated BSDEs, we deduce that necessarily we have Y≥ y^, -a.s., which implies by arbitrariness ofand by (<ref>) thatY≥ Y.For the converse inequality, fix some >0 and some ∈_H^κ, and define the following stopping timeτ̃_ :=inf{t≥ 0, Y_t^--L_t^-≤}∧ T.The Skorokhod condition implies that0=ℙ'∈^κ_H(t^+,) essinf^ 𝔼^'_t[∫_t^T(Y_s^--L_s^-)dK_s^']≥ℙ'∈^κ_H(t^+,) essinf^ 𝔼^'_t[K_τ̃_^'-K_t^'].Next, let (𝒴^'(τ̃_, L_τ̃_),𝒵^'(τ̃_, L_τ̃_)) be the solution, under ' and on [0,τ̃_], of the BSDE with generator F and terminal condition L_τ̃. We have '-a.s.Y_t =Y_τ̃_ + ∫_t^τ̃_F_s(Y_s,Z_s)ds-∫_t^τ̃_Z_sdB_s+∫_t^τ̃dK^'_s,0≤ t≤τ̃_, ^'_t(τ̃_, L_τ̃_) =L_τ̃_1_{τ̃<T}+ξ1_{τ̃_=T} + ∫_t^τ̃_F_s(^'_s(τ̃_, L_τ̃_),^'_s(τ̃_, L_τ̃_))ds-∫_t^τ̃_^'_s(τ̃_, L_τ̃_)dB_s,0≤ t≤τ̃_.We can now use classical linearisation arguments as above to show the result. First, we define M^t,ℙ^'_s in the same way as in (<ref>). Notice that as in <cit.>, for any p≥ 1, there exists C_p>0 such that𝔼^^'_t[t≤ s≤ Tsup (M^t,ℙ^'_s)^p]≤ C_p,^'-a.s.Then, by Itō's formula, we obtainY_t-^'_t(τ̃_, L_τ̃_) = 𝔼^'_t[M^t,ℙ^'_τ̃_(Y_τ̃_- ^'_τ̃_(τ̃_, L_τ̃_)) +∫_t^τ̃_M^t,ℙ^'_s dK^'_s]≤𝔼^'_t[t≤ s≤ Tsup M^t,ℙ^'_s(Y_τ̃_- ^'_τ̃_(τ̃_, L_τ̃_) + K^'_τ̃_-K^'_t)]≤𝔼^'_t[t≤ s≤ Tsup M^t,ℙ^'_s] +𝔼^'_t[t≤ s≤ Tsup M^t,ℙ^'_s(K^'_τ̃_-K^'_t)]≤𝔼^'_t[t≤ s≤ Tsup M^t,ℙ^'_s] +(𝔼^'_t[t≤ s≤ Tsup(M^t,ℙ^'_s)^3])^1/3 (𝔼^'_t [(K^'_τ̃_-K^'_t)^3/2])^2/3≤𝔼^'_t[t≤ s≤ Tsup M^t,ℙ^'_s] +(𝔼^'_t[t≤ s≤ Tsup(M^t,ℙ^'_s)^3])^1/3 (𝔼^'_t [K^'_τ̃_-K^'_t]𝔼^'_t [(K^'_τ̃_-K^'_t)^2])^1/3≤ C_1 +(C_3)^1/3(𝔼^'_t[K^'_τ̃_-K^'_t])^1/3(𝔼^'_t[(K^'_τ̃_-K^'_t)^2])^1/3.Define for simplicityC_t^:= ℙ'∈^κ_H(t^+,) esssup^ 𝔼^'_t[(K^'_τ̃_-K^'_t)^2]. Taking the essential infimum on both sides of the previous inequality, we deduceY_t-ℙ'∈^κ_H(t^+,) essup^^'_t(τ̃_, L_τ̃_) ≤ C_1+ (C_3)^1/3(C_t^)^1/3( ℙ'∈^κ_H(t^+,) essinf^ 𝔼^'_t [K^'_τ̃_-K^'_t])^1/3.Now, as in <cit.>, since the set of probabilities is upward directed, we obtain that𝔼^[ℙ'∈^κ_H(t^+,) esssup^ 𝔼^'_t[(K^'_τ̃_-K^'_t)^2]] ≤ℙ'∈^κ_H(t^+,) sup 𝔼^'[(K^'_T-K^'_t)^2]<∞.As such, we can use the Skorokhod condition to deduce nowY_t-ℙ'∈^κ_H(t^+,) essup^^'_t(τ̃_, L_τ̃_)≤ C_1.By the classical comparison theorem, we have that ^'_t(τ̃_, L_τ̃_) ≤y^'_t, so that we deduceY_t≤ℙ'∈^κ_H(t^+,) essup^ y^'_t +C_1≤ Y_t+C_1,which implies the required result by arbitrariness of .Step 2: existenceThe only thing that needs to be done here is to prove that the solution we constructed in the sense of Definition <ref> is also a Skorokhod–solution. In other words, we simply have to prove that Y satisfies the Skorokhod minimality conditionℙ'∈^κ_H(t^+,) essinf^ 𝔼_t^'[∫_t^T(Y_s^--L_s^-)dK_s^']=0,t∈[0,T], -a.s., ∀∈^κ_H. Without loss of generality, we prove that this holds for t=0, which is equivalent to proving thatℙ∈^κ_H inf 𝔼^[∫_0^T(Y_s^--L_s^-)dK_s^]=0.Let us start by fixing some >0, and define the following sequence of stopping times (τ_n)_n≥ 1 byτ_0:=0, τ_1:=inf{t≥ 0,Y_t^--L_t^-≤}∧ T, τ_2n:=inf{t>τ_2n-1,Y_t^--L_t^-≥ 2}∧ T, τ_2n+1:=inf{t>τ_2n,Y_t^--L_t^-≤}∧ T,n≥ 1.We start by proving that for any n≥ 1 and any ∈_H^κ∈_H^κinf^[K^_τ_1]='∈_H^κ(τ_2n^+,) essinf^^'_τ_2n[K^'_τ_2n+1-K_τ_2n^']=0.By the dynamic programming principle (see <cit.>), we know that for any n≥ 0, and using the link between reflected BSDEs and optimal stopping problems, where for any 0≤ s≤ t≤ T, 𝒯_s,t denotes the set of stopping times taking values in [s,t]Y_τ_2n='∈_H^κ(τ_2n^+,) essup^ τ∈𝒯_τ_2n,τ_2n+1 essup^^'_τ_2n(τ,L_τ 1_{τ<τ_2n+1}+Y_τ_2n+1 1_{τ=τ_2n+1}).We claim that the family {^'_τ_2n(τ,L_τ 1_{τ<τ_2n+1}+Y_τ_2n+1 1_{τ=τ_2n+1}):(',τ)∈_H^κ(τ_2n^+,)×𝒯_τ_2n,τ_2n+1},is upward directed. Indeed take for i∈{1,2}, (^i,τ^i)∈_H^κ(τ_2n^+,)×𝒯_τ_2n,τ_2n+1. We need to find a pair (',τ)∈_H^κ(τ_2n^+,)×𝒯_τ_2n,τ_2n+1 such that^'_τ_2n(τ,L_τ 1_{τ<τ_2n+1}+Y_τ_2n+1 1_{τ=τ_2n+1}) =max{^^1_τ_2n(τ^1,L_τ^1 1_{τ^1<τ_2n+1}+Y_τ_2n+1 1_{τ^1=τ_2n+1}),^^2_τ_2n(τ^2,L_τ^2 1_{τ^2<τ_2n+1}+Y_τ_2n+1 1_{τ^2=τ_2n+1})}.Define the following sets in _τ_2n^+^A:={^^1_τ_2n(τ^1,L_τ^1 1_{τ^1<τ_2n+1}+Y_τ_2n+1 1_{τ^1=τ_2n+1})≥^^2_τ_2n(τ^2,L_τ^2 1_{τ^2<τ_2n+1}+Y_τ_2n+1 1_{τ^2=τ_2n+1})},B:=Ω∖ A.Define then τ:=τ^11_A+τ^21_B, as well as ^'[C]:=^1[A∩ C]+^2[B∩ C], for any C∈_T. It can be checked directly that τ∈𝒯_τ_2n,τ_2n+1, and that ∈_H^κ (see similar arguments in <cit.>), and we have by definition that (<ref>) holds for this choice of (^',τ). Therefore, we know that the essential supremum in Y_τ_2n is attained along a subsequence, meaning that for any δ>0, there exists some some '_δ∈_H^κ(τ_2n^+,) and some τ^δ∈_τ_2n,τ_2n+1 such thatY_τ_2n≤_τ_2n^'_δ(τ^δ,L_τ^δ 1_{τ^δ<τ_2n+1}+Y_τ_2n+1 1_{τ^δ=τ_2n+1})+δ.Recall η and λ from (<ref>), and define then (notice that this process is slightly different from M defined in (<ref>))_s^t,'_δ:= exp(∫_t^s(λ_u-1/2|η_u|^2)(Y_u,_u^'_δ,Z_u,_u^'_δ,a_u)du-∫_t^sη_u(Y_u,_u^'_δ,Z_u,_u^'_δ,a_u)·a_u^-1/2dB_u),where we denoted for simplicity^'_δ:=^'_δ(τ^δ,L_τ^δ 1_{τ^δ<τ_2n+1}+Y_τ_2n+1 1_{τ^δ=τ_2n+1}), ^'_δ:=^'_δ(τ^δ,L_τ^δ 1_{τ^δ<τ_2n+1}+Y_τ_2n+1 1_{τ^δ=τ_2n+1}).Notice that for any p∈ℝ, the boundedness of λ and η imply that for some constant C_p>0∈_H^κsup𝔼^[(t≤ s≤ Tsup_s^t,)^p+(t≤ s≤ Tinf_s^t,)^p]≤ C_p.Then, linearization arguments similar to the ones used before in this note imply thatY_t-^'_δ_t=𝔼^'_δ_t[_τ^δ^t,'_δ(Y_τ^δ-L_τ^δ) 1_{τ^δ<τ_2n+1}+∫_t^τ^δ_s^t,'_δdK^'_δ_s], τ_2n≤ t≤τ_2n+1, -a.s.By definition of the (τ_n)_n≥ 0, we deduce thatδ≥ Y_τ_2n-^'_δ_τ_2n≥𝔼^'_δ_τ_2n[_τ^δ^τ_2n,'_δ 1_{τ^δ<τ_2n+1}]+𝔼^'_δ_τ_2n[τ_2n≤ s≤τ^δinf_s^τ_2n,'_δ(K^'_δ_τ^δ-K^'_δ_τ_2n)].We then estimate that'_δ[τ^δ< τ_2n+1] =^'_δ[(_τ^δ^τ_2n,'_δ)^-1/2(_τ^δ^τ_2n,'_δ)^1/2 1_{τ^δ< τ_2n+1}]≤(^'_δ[(_τ^δ^τ_2n,'_δ)^-1]^'_δ[(_τ^δ^τ_2n,'_δ)^1/2 1_{τ^δ< τ_2n+1}])^1/2≤ C_-1^1/2√(δ/).Recall as well that by Step (iii) of the proof of <cit.> that for some C̅>0'∈_H^κ(τ_2n^+,) essup^^'_τ_2n[(K^'_τ_2n+1-K_τ_2n^')^2]≤C̅.Therefore, we have^'_δ_τ_2n[K^'_δ_τ_2n+1-K_τ_2n^'_δ]≤^'_δ_τ_2n[K^'_δ_τ^δ-K_τ_2n^'_δ]+ ^'_δ_τ_2n[(K^'_δ_τ_2n+1-K_τ_2n^'_δ) 1_{τ^δ<τ_2n+1}] = ^'_δ_τ_2n[(τ_2n≤ s≤τ^δinf_s^t,(K^'_δ_τ^δ-K_τ_2n^'_δ))^1/3(K^'_δ_τ^δ-K_τ_2n^'_δ)^2/3(τ_2n≤ s≤τ^δinf_s^t,)^-1/3]+^'_δ_τ_2n[(K^'_δ_τ_2n+1-K_τ_2n^'_δ) 1_{τ^δ<τ_2n+1}]≤(^'_δ_τ_2n[τ_2n≤ s≤τ^δinf_s^t,(K^'_δ_τ^δ-K_τ_2n^'_δ)]^'_δ_τ_2n[(K^'_δ_τ^δ-K_τ_2n^'_δ)^2]^'_δ_τ_2n[(τ_2n≤ s≤τ^δinf_s^t,)^-1])^1/3+ (^'_δ_τ_2n[(K^'_δ_τ^δ-K_τ_2n^'_δ)^2]ℙ'_δ[τ^δ<τ_2n+1])^1/2≤(C̅C_-1)^1/3δ^1/3+C̅^1/2C_-1^1/4(δ/)^1/4.This implies immediately that '∈_H^κ(τ_2n^+,) essinf^^'_τ_2n[K^'_τ_2n+1-K_τ_2n^']≤(C̅C_-1)^1/3δ^1/3+C̅^1/2C_-1^1/4(δ/)^1/4,which proves the second equality in (<ref>) by letting δ go to 0.Now, in order to prove that ∈_H^κinf^[K^_τ_1]=0, notice that we just obtained'∈_H^κ(0^+,) essinf^^'_0[K^'_τ_1]=0.Taking expectations, and using the fact that since the family of measure sis upward directed, the essential infimum is attained along some sequence (_n)_n∈⊂_H^κ(0^+,), we deduce by the monotone convergence theorem under , and the fact that all the (_n)_n∈ coincide withon _0+0=^['∈_H^κ(0^+,) essinf^^'_0[K^'_τ_1]]=^[lim_n→∞↓^_n_0[K^_n_τ_1]]=lim_n→∞↓^[^_n_0[K^_n_τ_1]]= lim_n→∞↓^_n[K^_n_τ_1]≥∈_H^κinf^[K^_τ_1],which proves the desired equality.Therefore, for any n≥ 0, we can find some _1∈_H^κ, and some _n+1∈_H^κ(τ_2n^+,_n) such that for some C̃>0^_n+1_τ_2n[K^_n+1_τ_2n+1-K_τ_2n^_n+1]≤/2^n.By definition, we have Y_t-L_t≤ 2 for t∈[τ_2n-1,τ_2n], so that𝔼^_n[∫_0^τ_2n(Y_s^--L_s^-)dK_s^_n] = ∑_i=0^n-1𝔼^_n[∫_τ_2i+1^τ_2(i+1)(Y_s^--L_s^-)dK_s^_n+∫_τ_2i^τ_2i+1(Y_s^--L_s^-)dK_s^_n]=∑_i=0^n-1𝔼^_i+1[∫_τ_2i+1^τ_2(i+1)(Y_s^--L_s^-)dK_s^_i+1+∫_τ_2i^τ_2i+1(Y_s^--L_s^-)dK_s^_i+1]≤∑_i=0^n-12𝔼^_n[K_τ_2(i+1)^_i+1-K_τ_2i+1^_i+1]+/2^i𝔼^_n[τ_2i≤ s≤τ_2i+1sup(Y_s-L_s)]≤C̃,where we used the a priori estimates satisfied by the solution of the 2RBSDE, see <cit.> and the definition of _n.Next, notice that Y-L is right–continuous, and therefore uniformly continuous from the right (see <cit.>). Besides, by definition, we have for any n≥ 0, that on {τ_n+1<T}(Y_τ_n+1^--L_τ_n+1^-)-(Y_τ_n^--L_τ_n^-)≥.Therefore the τ_n cannot accumulate and for n large enough we necessarily have τ_n=T. We now assume that the n we have chosen satisfies this property.Finally, fix some m≤ n. We have the following estimate for any ∈_H^κ[τ_2n<T] ≤[⋂_i=0^n-1{|Y_τ_2(i+1)^--Y_τ_2i+1^-|+|L_τ_2(i+1)^--L_τ_2i+1^-|≥}]≤[⋂_i=0^n-1({|Y_τ_2(i+1)^--Y_τ_2i+1^-|≥/2}⋃{|L_τ_2(i+1)^--L_τ_2i+1^-|≥/2})]≤[{_i=0^n-1|Y_τ_2(i+1)^--Y_τ_2i+1^-|^2≥m^2/4}⋃{_i=0^n-1 1_{|L_τ_2(i+1)^--L_τ_2i+1^-|≥/2}≥ (n-m)}]≤4/m^2^[_i=0^n-1|Y_τ_2(i+1)^--Y_τ_2i+1^-|^2]+[∑_i=0^n-1 1_{|L_t_i+1^n--L_t_i^n-|≥/2}≥ n-m].Now notice that we have for some constant C which may change value from line to line, by definition and using Doob's inequality as well as the elementary inequality ∑_i a_i^2≤(∑_i |a_i|)^2 and the estimates of <cit.>^[_i=0^n-1|Y_τ_i+1^--Y_τ_i^-|^2]≤ C𝔼^[∫_0^T|F_s(Y_s,Z_s)|^2ds+∫_0^T|a_s^1/2Z_s|^2ds+(K_T^)^2]≤ C.Consequently, using Assumption <ref>ℙ∈^κ_H inf 𝔼^[∫_0^T(Y_s^--L_s^-)dK_s^] ≤𝔼^_n[∫_0^T(Y_s^--L_s^-)dK_s^_n]≤C̃+(𝔼^_n[0≤ s≤ Tsup(Y_s-L_s)^2]𝔼^_n[(K_T^_n)^2])^1/2^n[τ_2n<T]≤C̃ +4C/m^2+[∑_i=0^n-1 1_{|L_t_i+1^n--L_t_i^n-|≥/2}≥ n-m].It thus suffices to let n go to +∞ first, then m to +∞ and finallyto 0. §.§.§ Comparison with the literatureIn the recent months, two independent studies of the so-called reflected G-BSDEs have appeared, the first by Li and Peng <cit.>, and the second in the PhD thesis of Soumana Hima <cit.>. Both these papers obtain wellposedness, in the G-framework of Peng of solutions to reflected G-BSDEs with a lower obstacle. Unlike our first paper <cit.>, they ensure uniqueness by using the Skorokhod minimality condition (<ref>). However, as shown by the result of the previous section, under Assumption <ref>, both minimality conditions actually lead to the exact same solution. Let us now detail a bit more the other differences between the two different approaches.(i) First of all, concerning the assumptions made, the main difference is on the obstacle. In <cit.>, in addition to our own assumptions, it is assumed to either be bounded from above or that it is a semimartingale under every measure considered (see their Assumptions (H4) and (H4')). Similarly, <cit.> requires the obstacle to be a semimartingale (see the equation just after (5.4) in <cit.>). In our framework, if one is satisfied with Definition <ref>, then we only require classical square integrability on L. If one also wants to recover the Skorokhod condition, then we need more in the form of Assumption <ref>. In any case, this does not imply that L has to be a semimartingale nor bounded from above, and as shown in Lemma <ref>, it would be enough for L to have finite p-variation for some p≥ 1, which is obviously satisfied if L is a semimartingale, making our assumption weaker in general. (ii) Concerning the method of proof, both <cit.> and <cit.> use the classical penalisation method introduced by <cit.> to prove existence, while uniqueness is obtained through a priori estimates. Our proof is more constructive and in the spirit of the original paper <cit.>. We expect that the penalisation approach should be applicable in our setting as well, but we leave this interesting question to future research.(iii) Maybe more important than the above point, one has to keep in mind that the very essence of the G-BSDE theory requires that the data of the equation, meaning here the generator F, the terminal condition ξ and the obstacle L, have to have some degree of regularity with respect to the ω variable. More precisely, they have to be quasi-continuous in ω, which loosely speaking means that they must be uniformly continuous (for the uniform convergence topology) outside a "small" set (see the references for more details). This is inherent to the construction itself, as soon as the set _H^κ is non–dominated, and cannot be avoided with this approach. Granted, it is also the case in our paper <cit.>. However, since then, many progresses have been achieved in the 2BSDE theory, and the recent paper <cit.> has proved that the (non–reflected) 2BSDE theory worked perfectly without any regularity assumption. Furthermore, a general modus operandi is given in <cit.> to extend those results to many type of 2BSDEs, including the reflected ones. This program has actually been carried out in the recent PhD thesis Noubiagain <cit.>(see also <cit.> and <cit.>). Combined with the results and discussions of the present note, the 2RBSDEs can therefore be defined in a much more general framework than the reflected G-BSDEs.§ A SUPER–HEDGING DUALITY FOR AMERICAN OPTIONS IN UNCERTAIN, INCOMPLETE AND NONLINEAR MARKETS This short section is devoted to obtain some clarifications concerning the link made in <cit.> between solutions to 2RBSDEs and super–hedging prices for American options under volatility uncertainty. The recent years have seen a flourishing of papers treating the above super–replication problem of American options in discrete time financial markets under uncertainty, allowing or not for static trading of European options, see among others Dolinsky <cit.>, Neuberger <cit.>, Hobson and Neuberger <cit.>, Bayraktar et al. <cit.>, Bayraktar and Zhou <cit.>, or Deng and Tan <cit.>, and Aksamit, Deng, Obłój and Tan <cit.>. In a continuous–time setting, non–linear markets were considered by Dumitrescu, Quenez and Sulem <cit.>, with some level of ambiguity, in the sense that both the non–linear driver of the wealth process and the default intensity could be not perfectly known. This however corresponds to families of probabilities which are absolutely continuous with respect to each other, and thus does not require to consider second–order BSDEs. As far as we know, beyond the results given in <cit.>, there are no other results allowing to tackle volatility uncertainty in continuous–time (see however the recent contribution <cit.> for partial hedging issues). Given that this is a corrigendum, we do not wish to go into too many details, and simply want to point out that <cit.> only provided an upper bound for the super–hedging price of an American options as the initial value of a 2RBSDE, and that using techniques similar to the ones in <cit.> for instance, it can readily be checked that this is actually the super–hedging price itself. § GENERAL REFLECTIONS§.§ UniquenessLet us now consider our second paper <cit.>. First of all, the definition of a solution should be replaced by the following.We say (Y,Z)∈𝔻^2,κ_H×ℍ^2,κ_H is a solution to a 2DRBSDE if∙ Y_T=ξ, 𝒫_H^κ-q.s. ∙ ∀ℙ∈𝒫_H^κ, the process V^ℙ defined below has paths of bounded variation ℙ-a.s.V_t^ℙ:=Y_0-Y_t - ∫_0^tF_s(Y_s,Z_s)ds+∫_0^tZ_sdB_s,0≤ t≤ T, ℙ-a.s.,and admits the following decompositionV_t^ℙ=K_t^ℙ-𝒦_t^ℙ,+, t∈[0,T], -a.s.,where the two processes K^ and 𝒦^,+ are non-decreasing, and where 𝒦^,+ satisfies the following Skorokhod condition∫_0^T(S_s^--Y_s^-)d𝒦^,+_s=0, -a.s. ∙ We have the following minimality condition for 0≤ t≤ Tℙ'∈^κ_H(t^+,) essinf^ 𝔼_t^'[∫_t^TM^t,'_sd(V^'_s+k_s^',+-k^',-_s)]=0, ℙ-a.s., 0≤ t≤ T, ∀ℙ∈𝒫_H^κ,where M^t, is defined as in (<ref>) but using the solution (y^,z^) of the doubly reflected BSDE under .∙ L_t≤ Y_t≤ S_t, 𝒫_H^κ-q.s. There are two main differences with the earlier definition in our paper <cit.>. The first one is obviously the new minimality condition (<ref>), which is simply the version with two obstacles of (<ref>). The second main difference is the decomposition (<ref>) of the bounded variation process V^. It is not really new, per se, as it was already implicit in the existence proof we provided in <cit.>, see in particular the lignes between the statements of Lemma 4.3 and Proposition 4.4. In particular, it does not require any additional argument in the existence proof. Under this new definition, the proof of uniqueness of a solution follows exactly the same lignes as in the lower obstacle case described above, it suffices to use the new minimality condition (<ref>), which is equivalent to the representation formula of the solution to the 2DRBSDE as an essential supremum of solutions of the associated DRBSDEs. §.§ A priori estimatesThe main change in <cit.> with the introduction of the new minimality condition (<ref>) above concerns the a priori estimates for 2DRBSDEs. Let us start with Proposition 3.5 in <cit.>, which has to be corrected as follows. Notice that the references (2.5) and (2,6) are the ones from <cit.>, and not the present paper. Let Assumption 2.3 hold. Assume ξ∈𝕃^2,κ_H and (Y,Z)∈𝔻^2,κ_H×ℍ^2,κ_H is a solution to the 2DRBSDE (2.5). Let {(y^,z^,k^,+,k^,-)}_∈^κ_H be the solutions of the corresponding DRBSDEs (2.6). Then we have the following results for all t∈[0,T] and for all ∈^κ_H (i) V_t^,+:=∫_0^t 1_Y_s^-=L_s^-dV_s^=∫_0^t 1_Y_s^-=L_s^-dk_s^,-, -a.s., and is therefore a non–decreasing process.(ii) V_t^,-:=∫_0^t 1_y^_s^-=S_s^-dV_s^=-∫_0^t 1_y^_s^-=S_s^-dk_s^,+, -a.s., and is therefore a non–increasing process. The proof of (ii) above is given in <cit.> and does not use the minimality condition and is thus correct. (i) can be proved similarly. The issue now is that we no longer have a nice Jordan decomposition of V^, which changes a lot how we can prove and obtain a priori estimates for the solution. Actually, the main point here is to rely on the decomposition (<ref>), which is almost a Jordan decomposition. In the proof of Theorem 3.7 in <cit.>, the proof of the estimates for Y, y^, z^, k^,+ and k^,- does not change and is still correct. In the estimate for Z, corresponding to the calculations in (3.15) in <cit.>, one has to use the decomposition (<ref>) for V^, and the fact that we know that for some constant C independent of 𝔼^[|K_T^|^2+|_T^|^2]≤ C.Indeed, this is a consequence of <cit.> and the fact that the unique solution to the 2DRBSDE is constructed through the Doob–Meyer decomposition of a doubly reflected g-supermartingale. The rest of the proof is then the same, still using the decomposition (<ref>). Thus Theorem 3.7 in <cit.> should be replaced by Let Assumptions 2.3, 2.5 and 2.8 hold. Assume ξ∈𝕃^2,κ_H and (Y,Z)∈𝔻^2,κ_H×ℍ^2,κ_H is a solution to the 2DRBSDE (2.5). Let {(y^ℙ,z^ℙ,k^ℙ,+,k^ℙ,-)}_ℙ∈𝒫^κ_H be the solutions of the corresponding DRBSDEs (2.6). Then, there exists a constant C_κ depending only on κ, T and the Lipschitz constant of F such thatY^2_𝔻^2,κ_H+Z^2_ℍ^2,κ_H+ℙ∈𝒫^κ_Hsup{y^ℙ^2_𝔻^2(ℙ)+z^ℙ^2_ℍ^2(ℙ)}+ℙ∈𝒫^κ_Hsup𝔼^ℙ[ Var_0,T(V^)^2+(K_T^)^2+(_T^,+)^2+(k_T^ℙ,+)^2+(k_T^ℙ,-)^2]≤ C_κ(ξ^2_𝕃^2,κ_H+ϕ^2,κ_H+ψ^2,κ_H+φ^2,κ_H+ζ^2,κ_H).Next, concerning the estimates for the difference between two solutions, the proof of Theorem 3.8 in <cit.> also has to be modified. More precisely, the three lignes after (3.19) should be erased. Then the proof of the estimate for δ Z is still correct. However, we only have control over the difference between V^,1 and V^,2, not individually for K^,1 and K^,2 on the one hand, and ^,1 and ^,2 on the other hand. Theorem 3.8 of <cit.> should therefore be replaced byLet Assumptions 2.3, 2.5 and 2.8 hold. For i=1,2, let (Y^i,Z^i) be the solutions to the 2DRBSDE (2.5) with terminal condition ξ^i, upper obstacle S and lower obstacle L. Then, there exists a constant C_κ depending only on κ, T and the Lipschitz constant of F such thatY^1-Y^2_𝔻^2,κ_H≤ Cξ^1-ξ^2_𝕃^2,κ_HZ^1-Z^2^2_ℍ^2,κ_H+ℙ∈𝒫^κ_Hsup𝔼^ℙ[0≤ t≤ TsupV_t^ℙ,1-V_t^ℙ,2^2]≤ Cξ^1-ξ^2_𝕃^2,κ_H(ξ^1_𝕃^2,κ_H+ξ^1_𝕃^2,κ_H+(ϕ^2,κ_H)^1/2+(ψ^2,κ_H)^1/2+(φ^2,κ_H)^1/2+(ζ^2,κ_H)^1/2).Notice also that Remark 3.9 in <cit.> no longer holds. Similarly, Remark 3.12 should be deleted. As a consequence, in Proposition 3.10 in <cit.>, the constant γ should always be taken as equal to 0. Finally, direct computations using the decomposition (<ref>) prove that Proposition 3.14 in <cit.> should be replaced byLet Assumptions 2.3, 2.5, 2.8 and 3.13 hold. Let (Y,Z) be the solution to the 2DRBSDE, then for all ℙ∈𝒫^κ_HZ_t=P_t, dt×ℙ-a.s. on the set {Y_t^-=S_t^-},and there exists a progressively measurable process (α_t^ℙ)_0≤ t≤ T such that 0≤α≤ 1 andd_t^ℙ,+=α_t^ℙ 1_Y_t^-=S_t^-([F_t(S_t,P_t)+U_t]^+dt+dC_t^++dK^_t). §.§ ExistenceBecause we no longer control the total variation of V^ in Theorem <ref>, the proof of existence we gave in <cit.> only holds for ξ∈ UC_b(Ω). However, this is not an issue at all, since the only reason we had to restrict to uniformly continuous terminal condition was to obtain the measurability result in <cit.> and the dynamic programming principle of <cit.>. Using the results of <cit.>, in particular Proposition 2.1, these two results were obtained in <cit.> (see also <cit.>) for doubly reflected BSDEs, and allow to extend the construction carried out in <cit.> to any ξ∈𝕃^2,κ_H.Finally, notice that similar arguments as in the lower reflected case should in principle allow to prove that wellposedness can be recovered for 2DRBSDEs when the minimality condition (<ref>) is replaced by asking that the process K^ in the decomposition (<ref>) satisfies some sort of Skorokhod condition similar to (<ref>), and provided that an conditions similar to Assumption <ref> hold. In such a situation, both processes K^ and ^,+ would then satisfy some Skorokhod type conditions. Such a program has been carried out recently for G–RBSDEs with two obstacles in <cit.>, and <cit.>. §.§ Game optionsIn Section 5.1 of <cit.>, we introduced game options and claimed that the second order doubly reflected BSDEs (2DRBSDEs for short) allow us to obtain super– and sub–hedging prices for game options in financial markets with volatility uncertainty. Actually,it is proved that the amount Y_t, where Y is the solution of the 2DRBSDE in Definition 3.1 of <cit.>, allows the seller of the game option to build a super–hedging strategy under any probability measure '.We emphasise however that we are not able to guarantee that this amount is optimal in the sense that it is the lowest value for which we can find a super–hedging strategy, though, as explained in the case of American options above in Section <ref>, we strongly expect this result to hold. For related problems in nonlinear market with default, but without volatility uncertainty, we refer the reader to the recent paper <cit.> and the references therein. Note that <cit.> studies an associated (non-linear) robust Dynkin game problem, in particular, in the case when there is default intensity ambiguity on the model.Moreover, we have claimedinSection 5.1 of <cit.>, that the whole interval of prices, given by [Y_t , Y_t ] with Y_t :=ℙ'∈^κ_H(t^+,) essinf^y^'_t (Page 2309 in <cit.>), can be formally considered as arbitrage free. This also requires a proper justification. Actually, we may define the super–hedging price for a game option as in Section 6.1 of <cit.>. Using this definition, the link between the super–hedging price and the solution of a 2DRBSDE will be considered in the forthcoming working paper <cit.>. plain | http://arxiv.org/abs/1706.08588v3 | {
"authors": [
"Anis Matoussi",
"Dylan Possamaï",
"Chao Zhou"
],
"categories": [
"math.PR",
"math.OC",
"q-fin.MF"
],
"primary_category": "math.PR",
"published": "20170626204836",
"title": "Corrigendum for \"Second-order reflected backward stochastic differential equations\" and \"Second-order BSDEs with general reflection and game options under uncertainty\""
} |
Department of Astronomy, Keio University, Hiyoshi, Yokohama 223-8521, Japan; [email protected] Institute, Graduate School of Science, Tohoku University, Sendai, 980-8578, Japan Department of Earth Science and Astronomy, College of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan We present 1500 cycles of hydrogen shell flashes on a 1.38 M_ whitedwarf (WD) for a mass accretion rate of1.6 × 10^-7 M_ yr^-1, the mass ejection of which is calculated consistently with the optically thick winds. This model mimics the one-year-recurrence-period nova M31N 2008-12a.Through these hydrogen flashes a helium ash layer grows in massand eventually triggers a helium nova outburst. Each hydrogen flash is almost identical and there is no precursor for the forthcoming He flash either in the outburst or in the quiescentuntil the next He flash suddenly occurs.Thus, M31N 2008-12a is a promising candidate of He novae, outbursting in any time within a millennium years.The prompt X-ray flash of He nova lasts as short as 15 min with the X-ray luminosity being about a half of the Eddington luminosity,making the observation difficult.In the very early phase of a He flash, the uppermost H-rich layer isconvectively mixed into the deep interior and most of hydrogen is consumedby nuclear burning. In comparison with hydrogen shell flashes of M31N 2008-12a, we expect the forthcoming He nova with a very short prompt X-ray flash (15 min), a very bright optical/NIR peak(∼3.5 mag brighter than M31N 2008-12a), a much longer nova duration(>2 years), and a longer supersoft X-ray sourcephase (40-50 days or more).§ INTRODUCTIONA nova is a hydrogen flash on a mass-accreting white dwarf (WD) <cit.>. Multicycle nova outbursts have been calculated with Henyey-type evolution codes.Those codes, however, meet numerical difficulties when the nova envelope expands to a giant size. To continue the numerical calculation beyond this stage, various authors have adopted various mass-loss schemes and approximations <cit.>. In the previous paper <cit.>, we establishedan iteration method for calculating the extended stage of novae withtime-dependent mass-loss rates of optically thick winds, and presented a model for one full cycle of a nova outburstfor the recurrent nova M31N 2008-12a. M31N 2008-12a has exploded almost every year which makes thisobject as the shortest record of the recurrence period ofP_ rec∼ 1 yr <cit.> orP_ rec∼ 0.5 yr.[<cit.> proposed a 0.5 yr recurrence period to explain the discrepancy betweenthe early (around 2000) and recent trends of the outburst cycles.Even if an outburst occurred in the middle of the 1 yr cycle,we could not observe it due to Sun constraint.]<cit.> presented the outburst modelof a 1.38 M_ WD with a mass-accretion rate ofṀ_ acc = 1.6 × 10^-7 M_⊙ yr^-1.During the outburst of the model, a part of the envelope mass is blown in the optically thick wind, and the rest is processedto helium and accumulates on the WD.Thus, we expect that, after many flashes,the helium mass gradually increases and eventually reaches a critical value for ignition, leading to a He flash.In other words, M31N 2008-12a is a promising candidate of He novae. Helium novae were theoretically predicted by <cit.> as a nova explosion caused by a helium shell flash on a WD.Binary systems of He nova progenitors were categorized into three types<cit.>.(1) WDs accreting helium matter from a helium star companion. (2) WDs accreting hydrogen with rates high enough to keep steadyhydrogen burning, i.e., the accretion rate is higher than the stability line<cit.>.Such objects correspond to persistent supersoft X-ray sources. (3) WDs accreting hydrogen with rates lower than the stability line,but high enough to increase the helium layer mass.Such objects correspond to recurrent novae. Kato et al.'s (1989) prediction was realized as a type (1) when V445 Pup was discovered on UT 2000 December 30 by K. Kanatsu <cit.>.V445 Pup is the first and only identified helium nova that underwenta helium shell flash <cit.>. The other two types of helium nova systems were not detected yet.As mentioned above, the recurrent nova M31N 2008-12ais a promising candidate of type (3) He novae. As M31N 2008-12a is the shortest recurrence period nova,we expect that the He layer is now growing in mass at high rates thatresult in He ignition in the near future. Thus, the theoretical description for He nova outbursts could be useful for making observational plans. A successive shell flash calculation is not easy, because we need substantial computer resources. Onlyfew groups have ever presented such calculations. <cit.> calculated >1000 successive cycles of nova outburstson a 0.65 M_ WD for the mass-accretion rate ofṀ_ acc= 1× 10^-9M_ yr^-1 and on a1.0 M_ WD for Ṁ_ acc= 1× 10^-11M_ yr^-1.In these models, the WDs are eroded inevery outburst and the WD masses secularly decrease.As the He ash is lost, no He flashes occur. <cit.> calculated a few thousand successive hydrogen flashesfor 1.0, 1.25, 1.35, and 1.4 M_ WDs withṀ_ acc=1 × 10^-6M_ yr^-1,and showed that the He layers grow in mass to result in He flashes.This accretion rate is much higher than that of the stability line,i.e., hydrogen should steadily burn without flashes,unless the accretion is stopped and restarted artificially.We call such novae “the forced novae”<cit.>. This accretion rate is too high to be applicable to M31N 2008-12a.<cit.> also presented successive hydrogen flashesstarting from a hot 1.34 M_ WD withṀ_ acc∼ 1 × 10^-7M_ yr^-1,which result in a He flash after 2573 cycles of hydrogen flashes. The recurrence period of the H flashes is about 2 years, close to1 yr of M31N 2008-12a.These authors, however, paid little attention todescribing the flash properties in detail.Moreover, the adopted parameters are not appropriate forM31N 2008-12a, which makes difficult drawing practical information for observation. The aim of this work is to calculate a number of hydrogen flashes untila He flash occurs for the appropriate parameters of M31N 2008-12a.This paper is organized as follows. Section <ref> introduces our numerical method andvarious parameters for the model.Section <ref> describes physical properties inour thousands of nova outbursts.Section <ref> shows the physical properties ofthe early phase of helium burning.Discussion and conclusions follow in Sections <ref> and <ref>. llllInitial Model Parameters0ptSubjectunitsquantityM_ WD M_⊙ 1.380log R_ WD R_⊙ -2.572 log L_ WD L_⊙ 2.00 log g cm s^-2 9.72 log V_ esc cm s^-1 9.15log T_ ca K8.03Ṁ_ acc 10^-7 M_⊙ yr^-1 1.6M_ H env 10^-8 M_⊙ 6.47log L_ ph L_⊙ 4.08 log L_ nuc L_⊙ 4.05 log T_ ph K 6.05 log R_ ph R_⊙ -2.54 aThe temperature at the WD center. § NUMERICAL METHODWe have calculated 1500 consecutive nova outburstsleading to a helium flash.We adopt the WD mass of 1.38 M_ and the mass-accretion rate of 1.6 × 10^-7 M_ yr^-1, taken from the model for the one-year-recurrence-period nova M31N 2008-12a <cit.>.We use the same Henyey-type code as in the previous work<cit.>.The chemical compositions of the accreting matter and initial hydrogen-rich envelope of the WD are assumed to be X=0.7, Y=0.28, and Z=0.02. To save computer time, we use a small nuclear reaction networkup to magnesium.Convective mixing is treated diffusively adopting the effective diffusioncoefficient derived by <cit.>. Although the coefficient was derived for semiconvective mixing(corresponding to the Schwarzschild type), it leads to a uniform chemical composition distribution in the fully convective zone.Therefore, we use the coefficient whenever radiative temperaturegradient exceeds adiabatic temperature gradient <cit.>.Neither convective overshooting nor diffusion processes of nuclei are included.We neglect the effects of rotation for simplicity.Including rotation into the evolution code is difficult and beyond thescope of the present work.Also, accretion energy outside the photosphere is not included.For technical simplicity, we assume that gas is accreted with the same temperature as the stellar surface.These assumptions are discussed in Section <ref>.It should be noted that the quiescent luminosity, L_ ph, whichappears in Figures <ref>, <ref>,<ref>, and <ref>, cannot be directly compared with observation because it does not include the accretion energy outside the photosphere that is reprocessed with the accretion disk surface<cit.>.During the extended stages of nova outbursts,the optically-thick wind mass-loss occurs.<cit.> presented two series of time-dependentwind mass-loss rates for a nova outburst model of M31N 2008-12a.To avoid time consuming process of iterative numerical fitting withoptically thick wind solutions in each stage,we here simply assumed wind mass-loss rates in our Henyey-codecalculation and followed a few thousand flashes.The adopted mass-loss rates<cit.> reasonably mimic those of the optically thick winds,but are slightly overestimated. We adopt > 5000 mass zones to cover the entire configuration including a carbon-oxygen (CO) WD core, a He layer, and a H-rich envelope.Such a large number of (i.e., very fine) mass zones are necessary to guaranteenumerical accuracy especially in rapidly changingphysical variables, such asthe temperature, density, and chemical composition of nuclear burningregion and also expanding region.Rezoning is adopted when it is necessary in a way to conservemass, energy, and chemical composition.The time step is chosen to be short enough (< 4× 10^4 seconds)in calculation of the 1500 cycles of hydrogen shell flashes,but much shorter (1 second) for the He ignition.It took about a week of CPU time on a PC (Xeon E5-1660, 3.70 GHz) for entire sequence of 1500 hydrogen shell flashes followed by the He flash until we stop calculation. We adopted an initial WD modelin which an energy balance is already establishedbetween heating (by the mass accretion and nuclear energy generation) and cooling (by the radiative transfer and neutrino energy loss) <cit.>.This is a good approximation of the long time-averaged evolutionof a mass accreting WD. Starting fromsuch an equilibrium state, the nova cycle approaches a limit cycle<cit.> in a short time.We will discuss in more detail on the initial modelin Section <ref>. Parameters of our initial WD model is summarized in Table <ref>. § 1500 HYDROGEN FLASHES §.§ Nearly Identical Flashes with No Indicationof Forthcoming Helium Flash Figure <ref> shows the change of the recurrence period,P_ rec, throughout our calculation.After 1543 hydrogen flashes, a He flash occurs passing half a periodfrom the last H flash (open blue star).The recurrence period increases by ∼ 8 %just after the start of calculation and then decreases during the first 70 cycles.After that, it staysat ∼ 0.91 yr.This early period change is caused by our choice of the initial WD modelwhich is slightly different from the thermal equilibriumstructure (see Section <ref> for the effectsowing to the choice of initial models).Figure <ref> shows the first 12 years of our calculation.There are 12 outbursts as shown by the change in the photosphericluminosity L_ ph.The mass of the hydrogen-rich envelope M_ H,env(defined as the mass above X > 0.01)increases during the inter-pulse phase owing to accretion, and decreasesduring the outburst, owing partly to wind mass-lossand partly to hydrogen nuclear burning. The He envelope (defined asthe region between the CO WD boundary and the bottom of aH-rich envelope) increases its massduring the outburst phase.The envelope mass, M_ env, which is the summation of the H-rich and He envelopes, increases in the quiescent phases, and sharply decreases due to wind mass-loss during the outburst phases. Figure <ref> shows a close-up view of three flashes atthree epochs of t ∼10 yr, ∼210 yr, and ∼1420 yr.As the recurrence period becomes constant after 70 years (see Figure <ref>), the middle and bottom panels showthe same P_ rec=0.91 yr. Through these flashesthe He envelope steadily increases its mass with time.The bottom panel showsthe last three outbursts before the He flash occurs at t=1421.13 yr.Note that the photospheric luminosity L_ ph and temperature T_ ph change almost identically in panels (b) and (c),until the He nova outburst occurs.Thus, we have no precursors for the coming He nova outburst.§.§ Toward Helium IgnitionAlthough there are no apparent diagnostics of approaching a He novain L_ ph and T_ ph until the He ignition,there is a gradual change in deep interior of the envelope.Figure <ref> compares the energy budgets att∼ 210 years and 1420 years. The photospheric luminosity,L_ ph, is the summation of the integrated nuclear burningrate, L_ nuc, and integrated gravitational energy release,L_ G.In the very early phase of the outburst, L_ nuc reachesas large as > 10^6 L_, which is mostly absorbedin the burning region as indicated as L_ G < 0.As a result the photospheric luminosity is as small asL_ ph∼ 5.8 × 10^4 L_ at most in both the epochs. Figure <ref> also shows that L_ G turnsfrom negative to positive after the peak of L_ nuc.The absorbed energy (L_ G < 0) is released in the later phase of the outburst.This means that the burning region is slightly sinking back towardthe original, geometrically thin,plane parallel configuration.This energy release continues until the end of the H flash.The hydrogen-rich envelope mass has decreased owing to nuclear burning,being unable to support enough high temperature for hydrogen burning.Thus, nuclear burning extinguishes where L_ nuc quickly decreases as shown in Figure <ref>.In the interpulse phase, L_ G owing to mass accretion is the main source of radiation, L_ ph. Hydrogen nuclear burning is only the source of L_ nuc atepoch t∼ 210 yr, while both H and He burning contributeto L_ nuc at t∼ 1420 yr.We plot the contribution of hydrogen burning (dotted black line) andHe burning (dotted red line) separately in Figure <ref>(b). The hydrogen burning rate varies just in the same way asin t∼ 210 yr, but additional energy release owing to He burning continuously increaseswith time, which is absorbed in the inner envelope andnot transferred upward. As a result, L_ ph behaves just like the epoch of t∼ 210 yr. In this way,every hydrogen flash is almost identical until just before the He flash,in the recurrence period, flash duration, and quiescent luminosity, even though the He envelope is growing in massand its nuclear energy generation rate is increasing.In other words, there is no observational precursorof the forthcoming He flash. Note that helium ignited in the last interpulse phase of our H flash cyclecalculation, i.e., a hydrogen flash itself does not trigger directly the He ignition.As shown later in Figure <ref>, the temperature at the bottom of the He layer gradually increases as its mass increases with time.A small temperature peak appears at the epoch of the last hydrogen flash,at logρ ( g cm^-3)∼ 5.1, i.e., just above the CO core, where He burning already occurs with low rates.This small temperature peak eventually triggers the He flashduring the interpulse phase. § HELIUM SHELL FLASH§.§ Onset of Helium Shell Flash Figure <ref> shows close-up views of very early phases ofH and He shell flashes.The He flash is significantly different from the H flash in many points. One of the differences is the nuclear energy release rate.At maximum, the integratednuclear energy release rate reachesL_ nuc^ max= 3.4× 10^6 L_⊙ in the H flash,whereas it reaches as large asL_ nuc^ max= 3.7× 10^11 L_⊙ in the He flash,10^5 times larger than that of H burning.The timescale is also very different.The H flashundergoes explosive nuclear burning in ∼0.1 days,whereas the He flash proceeds in as short as ∼ 0.5 minutes. In both the cases most of the nuclear energy is absorbedin the lower part of the burning region asshown by a large negative value of L_ G (< 0). Thus, only a very small part of the nuclear energyL_ nuc is transported outwardand emitted at the photosphere as L_ ph.As a result, the photospheric luminosity L_ ph does not exceed the Eddington luminosity. These properties were already reported for the H flashmodel of 1.38 M_ WD <cit.>.The present calculation demonstrates thatthe He nova has similar propertieseven for much larger nuclear energy release rates and much shorter timescales.§.§ H-R Diagram and X-ray FlashFigure <ref> shows the track in the H-R diagram forthe final H flash (solid black line)followed by the He shell flash (solid red line).Now we define the time t_ He starting atthe onset of the He flash. We set t_ He=0when the total nuclear energy generation ratereaches its maximum, L_ nuc=L_ nuc^ max. After the final hydrogen flash,the star becomes faint keeping the photospheric radiusalmost constant, from point H (maximum T_ ph) toward the pointof t_ He=0.0.It takes 0.43 yr.After that the starbrightens up within a minute,along with a constant but a bit larger than the photospheric radius in the case of H flashes.A similar track in the H-R diagram was already shown by <cit.>as an evolution passing the phase of a planetary nebula nucleus ofan 0.6 M_post-AGB star. As the star evolves down in the H-R diagram,a final He flash occurs and the star brightens up again.The decay timescale of the final H shell flash is as long as 4000 yr,while the rising timescale of the He shell flash is 20 yr.These values should not be directly compared with our case because the WD mass is different,but we see similar characteristic properties of a He shell flash. Figure <ref> demonstrates the difference in the rising timescalesbetween H and He flashes.The figure shows the evolutions of the photospheric temperature,total luminosity, and (0.3 – 1.0) keV X-ray luminosity for the Hand He flashes.The X-ray luminosities are calculated assuming the black body spectrumof the photospheric temperature. This assumption may not beaccurate for observational X-ray fluxes but enough to estimatethe duration of the X-ray flash, becausethe rising and decay timescales are very short compared with the duration. The X-ray flash is a brief X-ray bright phasein the very early phase of the outburst before the optical maximum<cit.>.As the He flash is much more violent, the duration of the X-ray flash isas short as 15 min and much shorter than that of the H flash(∼ 1 day). Nevertheless, the X-ray luminosity is a little bit smaller than that of the H flash because of a lower photospheric temperature (red line in Figure <ref>).Thus, detection of X-ray flash in the He flash would be difficult evenin high cadence satellite observations as planned for the X-ray flashin the 2015 outburst of M31N 2008-12a <cit.>. §.§ Internal Structures before/after Helium IgnitionFigure <ref> shows the temporal changesof the internal structurein the ρ-T plane during the course of H and He shell flashes.The rightmost point corresponds to the center of the WD.Figure <ref>(a) shows the structure changein the later phase of the final H flash, starting from stage Gin Figure <ref>,which roughly corresponds to the beginning of a late supersoft X-ray source (SSS) phase.Hydrogen ignites at the bottom of the H-rich envelope, i.e.,logρ ( g cm^-3)∼ 2, but until this stage, heat was transferred both outward and inward to form a large hot region(0< logρ ( g cm^-3) < 4).The internal structure hardly changesin the SSS phase (from stage G to stage H: dotted line). In the following cooling phase toward t_ He=-25 days,the star moves down in the H-R diagram (Figure <ref>)from stage H to stage A along with a constant radius.The temperature profile (logρ ( g cm^-3) < 4) changes up and down, in the flash and interpulse phases, in every cycle.On the other hand, the temperature in the deep interior of the envelopehardly changes,except that a tiny peak appears at the base ofthe He zone (logρ ( g cm^-3)= 5.2) in the last three H flashes.The middle and bottom panels show the temperature change immediatelybefore and after the He ignition (t_ He = 0), respectively.The middle panel demonstrates that thetiny peak at logρ ( g cm^-3)= 5.2 in the top panelextends toward lower-density region and the maximum temperature increasesto log T (K) ∼ 8.8.A convection zone develops outward from the temperature maximum.Lower density layersbecome hotter and hotter, whereasthe temperature at the He burning zone gradually drops because of adiabatic expansion (negative L_ G).§.§ Mixing of Hydrogen into Helium Burning ZoneFigures <ref>(a) and (b) show the temporal changeof the H/He profile, while Figures <ref>(c) and (d) showthe entropy distribution in the corresponding stages. After the final H-flash, freshly accreted matter(X=0.7, Y=0.28, and Z=0.02) accumulateson top of the H-rich/He layer.Figure <ref>(a) shows that, at t_ He=-3.4 min, the freshly accreted layer of X=0.7 lies on top ofthe leftover of the final H-flash, where X is decreasing inward. Convection occurs, before t_ He=0, at the He nuclearburning region.The convection spreads almost all over the envelope.The inner edge of the convective region is shown as the originof the black arrow in Figure <ref>(d).The outer edge of the convective regionis indicated by the small open circles. Note that the convection does not reach the photosphere,thus, the surface hydrogen content is always X=0.7 until the wind occursat t_ He=16.1 min. The convective region extends all over the He layer andpenetrates into the upper H-rich envelope(see Figures <ref>(c) and (d)). The H-rich matter is carried inward and mixed into deep interior of He-rich zone where the temperature is very high,and hydrogen is burned into helium.Therefore, the H mass fraction rapidly decreases with time.As shown in panel (a), most hydrogen disappearsuntil t_ He=16 min. Figures <ref> and <ref> show the changes ofthe energy budget, nuclear-burning energy-generation rate, and chemicalcomposition, for the selected stages of (a) t_ He=-34 min,(b) -0.75 min, (c) 0.0 min, and (d) 16.1 min.The top panels showthe nuclear luminosity integrated from the center of the WD up to theradius r, L_ nuc(r) (solid blue lines),integrated gravitational energy release rate, L_ G(r)(solid black lines), and local luminosity at radiusr, L_r (solid orange lines), which is the sum of radiative and convective luminosities.The temperature profile (red lines) is added. The middle panels show the energy generation rates per unit massowing to pp-chain, CNO-cycle, and He burning,and the neutrino energy loss rate. The bottom panels show the mass fractions of selected elements. The convective region is also indicated by the short horizontal barsin the bottom panels. The convection started from the He burning region and spreadsoutward all over the He layer and penetrates into the H-rich envelopeuntil t_ He=0. Thus, the surface hydrogen is mixed into the inner He layerand explosively burns with very high temperatures.As shown in the middle panels, the specific energy generation rate owing to CNO-cycle becomes comparableto that of He burning even in the region where the H mass fractionis very small.Therefore, hydrogen burning contributes as much as two thirdsto the total nuclear burning rate L_ nuc as in Figure <ref>(b).§.§ Occurrence of Optically Thick Wind Figure <ref> showsenvelope structures just before and after the optically thick winds occur.The solid/dashed lines denote the structuresjust before/after the optically thick winds occur. We also indicate two places corresponding to the inner edge of the opacity peak owing to highly ionized Fe, C, O, and Ne (labeled “C/O”)and inner edge of the peak owing to low/mid-degree ionized iron (labeled “Fe”) by the arrows.The optically thick winds are driven by the Fe opacity peak,not by the C/O peak, because the sonic point <cit.> of the optically thick winds is located at the inner edge of the Fe opacity peak, not at the C/O peak. The structure changes little at the onset of optically thick winds.This property is the same as that of acceleration in hydrogen flashes<cit.>.Our calculation of the He flash stopped at t_ He=1.6 years before it evolves to a SSS phase because of numerical difficulties.As well known Henyey-type code calculations do not work (do not converge)when the envelope extends to a giant size and surface region becomesradiation-pressure dominant. One way to continue numerical calculationis to assume very large mass-loss rates, but such large rates are ofteninconsistent with realistic wind acceleration such as optically-thick winds.In our previous paper, we developed an iteration method tocalculate a complete cycle of hydrogen shell flashes withmass-loss rates consistent with the optically-thick winds<cit.>. However, the He shell flash is so violent that we did not succeed in calculating the wind mass-loss phase with the iteration method. Thus,in this work, we did not adopt the iteration cycle, instead, we assumed relatively large trial mass-loss ratesduring the He flash.Although we assumed mass-loss rates as small as possible, they aremuch larger than those of the optically-thick winds. This makesthe outburst evolution faster, so we sickly underestimatethe flash duration. We suppose the duration of the He flash possibly longerthan 1.6 years, namely about 2 years or more. § DISCUSSION§.§ Initial WD Model and Its Central Temperature<cit.> and <cit.> presented nova calculationsfor a wide range of three parameters, the WD mass, central temperature of the WD, and mass-accretion rate.These three parameters are not independent of each otherbut linked through long-term evolution of the binary system.<cit.> calculated 1000 cycles of H flashes on a 1.0 M_ WDwith a mass-accretion rate of 1× 10^-11M_ yr^-1.The recurrence period quickly increases by a factor of 10 from the initial1.8× 10^6 years in the first 400 cycles and then turn to a gradual increase to 2.18× 10^7 years in thefinal 100 cycles (see their Figure 2).They adopted an initial WD temperature of T_ c=3 × 10^7 K, and during the calculation, the central temperature decreases with time bya factor of 5.Such a large change occurs because they assumeda much hotter initial WD model than that ofan equilibrium model with the corresponding mass-accretion rate.The above authors took a 0.6 M_ WD withṀ_ acc=1.0 × 10^-9M_ yr^-1 andshowed that an initially hotter WD of5 × 10^7 K cools down but an initially cooler WD of5 × 10^6 K becomes hotter and the WD temperatures approach a common equilibrium value after 3000 cycles.This means that if they adopted an initial WD model close to that ofthe equilibrium model, nova cycle would approach a steady-statemuch earlier. <cit.> also presented a similar phenomena,in successive helium shell flashes in a He accreting WDwith a mass-accretion rate of 2.0 × 10^-7M_ yr^-1.The central temperature increasedby a factor of 4 through 400 helium shell flashesduring which the WD mass increases from 1.105 M_ to 1.247 M_.In the present paper we adopted an initial WD model very close to thethermal equilibrium with the mass-accretion rate(see Section <ref>).Thus, the WD interior is already hot and the central temperature is as high as log T (K)=8.0299 at the start ofcalculation (t=0.0 yr) that is close to the final valueof log T (K)=8.0304 at the onset of He shell flash (t=1421.13 yr). Therefore, in our calculation, the recurrence period soon (after 70 cycles)approaches the final value with a small amplitude (8 %) variation (see Figure <ref>).<cit.> suggested thatthe recurrent nova M31N 2008-12a is consistent withthe 1.38 M_ WD model because of its short recurrenceperiod and rapid decline.Such an extremely massive WD is unlikely born as it is,but likely has grown up through long-term mass-accretion from the companion star <cit.>.Thus, the WD is likely as hot as expected in an equilibrium modelwith the mass-accretion rate of ∼ 10^-7M_ yr^-1. Therefore, our assumption of the initially hot WD is reasonable.§.§ Accretion EnergyWe suppose that gas is accreted through an accretion disk, releasinga part of gravitational energy from its surface which isemitted perpendicularly to the disk.Still, remaining energy is expected to be released in the boundarylayer. We have neglected the energy released above the photosphereas in other previous nova calculations<cit.>, whereas<cit.> included the energy from boundary layer of whichamounts 15 % of the gravitational energy based on the work by <cit.>.The 15 % of the gravitational energy release rate corresponds to1.5 ×10^36 erg s^-1 (390 L_) in our case. Thus, the quiescent luminosity increases to log L (erg s^-1) =36.37.The additional energy, however, hardly causes appreciable effectsin the nova calculations as discussed below.The heat flux from the boundary layer amounts 1.5 times the quiescentphase luminosity of our model.If this additional heat source changes thermal structure deep interior,the flash properties, such as the ignition mass, maximum temperature, and recurrence period may change.<cit.> included 15 % of the gravitational energy, and their1.0 M_ WD model with the mass accretion rate of1.0 × 10^-6 M_⊙ yr^-1 (T_ WD=5 × 10^7 K)has the accreted mass of 2.15 × 10^-6 M_⊙ andthe maximum temperature T_ max=1.03 × 10^8 K.For the same WD mass and accretion rate, <cit.> obtained 2.06 × 10^-6 M_⊙ andT_ max=1.06 × 10^8 K.Considering the differences in the input parameters, these two models are in good agreement.<cit.> also obtained a 1.35 M_ model withṀ_ acc=5 × 10^-7M_ yr^-1 that shows2.0 × 10^-7M_ and T_ max=1.47 × 10^8 K,also being consistent with the grid models of1.25 M_ and 1.4 M_ withṀ_ acc=1 × 10^-7M_ yr^-1andṀ_ acc=1 × 10^-6M_ yr^-1 in <cit.>.Thus, we may conclude that the inclusion of the additional15 % of the gravitational energy releasedoes not make much difference on the flash properties.§.§ Mass Accumulation EfficiencyIn our long-term evolution model, a part of the accreted H-rich matteris lost during the wind phase.In the present work, we adopted slightly overestimated mass-loss ratesas described in Section <ref>, to simplify our calculation.The adopted wind mass-loss rates result in the 64% lost of mass during one cycle of H flash.Thus, the mass accumulation efficiency, η≡ 1- (the ratio of lost mass to accreted mass),is η=0.36.This ratio increases to η=0.40 if we use the self-consistent wind mass-loss rates <cit.>.This accumulation efficiency is, however, highly uncertainbecause there are many unsolved problems associated with nova light curves.In classical novae, theoretical free-free emission light curveswell reproduce the decay phase of light curves in opticaland NIR bands <cit.>.We expect that the optical peak corresponds to the peak ofthe wind mass-loss rate, because the free-free emission optical flux isin proportion to the square of the wind mass-loss rate. In the rising phase, on the other hand, no reliable light curveshave been calculated neither for classical novae nor for recurrent novae.<cit.> presented an idea for the pre-maximum evolution,based on color-color evolution of fast novae, thatthe mass ejection begins just before the optical maximum.The quick start of the mass ejection in our model may correspond tothe fast expansion of the photosphere rising toward optical/NIR maximum.If this is the case, the real mass-ejection begins shortly before theoptical peak. In M31N 2008-12a outbursts, optical/NIR magnitudesrose in a short time toward the peak (< 1 day)<cit.>. This timescale is muchshorter than the slow pre-maximum evolutionof theoretical model <cit.>, in which thewind phase lasts 2 weeks before the peak. Moreover, the wind velocity does not reach the escape velocityV_ esc=√(2GM_ WD/r_ ph) in the beginning of the wind phase <cit.>. One possible idea to solve the inconsistency is that,in the premaximum phase, the wind solutions should be treated asa theoretical representative of fast-expanding surface of the hydrogen-rich envelope when the static solutions do not existas the luminosity approaches the Eddington luminosity,and the wind mass-loss rate is just a parameter to characterizethe expanding envelope solutions.In this case, wind solutions give the photospheric temperature,radius, and luminosity, but the wind mass-loss ratesshould not be taken as real mass-outflow rates.If we adopt this idea, the mass lost from the system becomesroughly a half, i.e., ∼ 30 %of the accreted mass and the mass accumulationefficiency is η∼ 0.7. Another important problem is the effect of rotation.<cit.> calculated He flashes including rotation and showed that rotation generally makes flashes milder because of a decrease in effective gravity and contamination of C and O by rotational mixing at the base of He layer. On the other hand, the ignition mass seems to be hardlyaffected by rotation (we judged it from their Figure 1).If we simply apply these results to our model,we may say that H flashes will be milderbecause of a decrease in effective gravity and contamination of He(not WD material, because heavy element enrichment is not observed),while the timescale of inter-flash phases would be unchanged. Thus, the mass lost from the system could become further small(i.e., η≳ 0.7).§.§ Observational Properties of Helium Flash Figure <ref> shows the wind mass-loss rates against the photospheric temperature.The filled red circles areof the evolution model of a H flash on a 1.38 M_ WD withṀ_ acc=1.6 × 10^-7M_ yr^-1,calculated for M31N 2008-12a outbursts <cit.>, in whichthe wind mass-loss rates are obtained from iteration process andare consistent with the optically thick wind acceleration.The large filled red circle represents the stage of the maximumwind mass-loss rate of the H flash, which may correspond to the stage ofthe optical/NIR maximumbecause in free-free emission, the optical/NIR magnitudes arein proportion tothe square of the wind mass-loss rate <cit.>.The red line depicts the wind mass-loss rateof the steady-state sequence of optically thick wind solutions. The envelope mass at the large filled red circle is2.2 × 10^-7 M_.This figure also shows a steady-state sequence for a helium flashon a 1.38 M_ WD with the chemical composition, Y=0.68, X_ C+O=0.2, X_ Ne=0.1 and Z=0.02 <cit.>. The upper end of this line (large filled black circle) represents the steady-state solutionof the envelope mass, 7.2 × 10^-5 M_,consistent with the ignition mass of our He flash,7.5 × 10^-5 M_. The corresponding mass-loss rate is ∼ 7.2 × 10^-5 M_ yr^-1,about 20 times larger than ∼ 3.5 × 10^-6 M_ yr^-1 of the H flash.If we further assume that the proportionality constant of Equation (9)in <cit.> is common among He-rich and H-rich matter, thefree-free emission flux of He nova isroughly 20^2 /2/8 ∼ 25 times higher than those of the H flash,where the factors 2 and 8 are the difference ofelectron and nuclei number densities between the H-rich and He-rich envelopes.Thus, the peak magnitude in optical/NIR emissionof He nova is 2.5 ×log (f_ He nova/f_ H nova)=2.5 ×log 25=3.5 mag brighter than that of the M31N 2008-12a outburst.We can expect a bright He nova outburst.Thus, we encourage search for a He flash in archival plates.Figure <ref> shows the sequence of the steady/static solutions(solid green line) that approximately represents the decay phase of He nova if the gravitational energy release rate L_ G is negligible <cit.>. The SSS phase of this sequence lasts 37 days,from the end of the wind mass-loss (small open circle) to the left-end of the line, where He nuclear burning ends.The effect of the gravitational energy release would beslowdown of the evolution <cit.>.If this effect amounts a few ten percent, the SSS phase may last ∼ 40-50 days.However, He burning produces substantial amount of carbon, which wouldtrigger thick dust-shell formation. In the outburst of the He nova V445 Pupdust blackout occurs 210 days after the optical discovery,which prevents direct observation of the WD until now<cit.>. In the same way, thick dust-shell formation would possibly hinder optical observation of a later phase of the He nova outburst of M31N 2008-12a. It is interesting that, in the He flash, the prompt X-ray flash lasts much shorter (15 min) but the late SSS phase lasts longer (40 – 50 days)than those in the H flash (a day and a week, respectively).The He flash is much strongerbecause of high temperature of He ignition, which results in a shorterrising time, i.e., shorter X-ray flash.In the decay phase, on the other hand,the He envelope mass is 100 times larger than that of H envelope but nuclear burning energy release of He is 10 times smaller.Thus, the late SSS phase lasts almost one order of magnitude longerthan that in the H flash.§ CONCLUSIONS Our main results are summarized as follows.1. We present 1500 consecutive hydrogen shell flasheson a 1.38 M_ WDwith a mass accretion rate of 1.6 × 10^-7M_ yr^-1. These parameters are taken from a model of M31N 2008-12a.The shell flash soon reaches a steady-state only after∼ 70 cycles with a small period variation of 8%. Until the He ignition, each H shell flash is almost identicalin the photospheric luminosity and there are no noticeableprecursors of theforthcoming He flash even in the epoch of the last hydrogen flash.2. The helium thermonuclear runaway occurs in an extremely short timescalecompared with that of hydrogen.The nuclear burning rate reaches L_ nuc=4 × 10^11 L_ atits peak, 10^5 times larger than that of H burning.Even such large energy production rates, most of the nuclear energy is absorbedby the inner part of the burning region. As a resultthe photospheric luminosity L_ ph is almost equal to the Eddington luminosity. 3. We present a prompt X-ray flash light-curve of the He nova. The duration of the X-ray flash of He nova is as short as 15 min,which makes the detection very difficulteven in high cadence observations as donein the X-ray flash of the 2015 outburstof M31N 2008-12a <cit.>. 4. During the early phase of the He outburst, most of the surface hydrogenis convectively mixed into the deep interior and is burned into heliumbefore the optically thick wind mass-loss occurs.Thus, the ejecta would contain much less hydrogen(i.e., ≪ 1 × 10^-7 M_) than we expect fromthe amount of accreted hydrogen-rich matter before the He flash occurs. 5. The optically thick winds begin at the end of theX-ray flash (t_ He=16.1 min), when the photospheric temperature decreases to log T_ ph (K) = 5.55, owing to acceleration by the Fe opacity peak. Characteristic properties, such as the occurrence of the wind and interior structure of the envelope,are essentially the same as those in hydrogen shell flashes<cit.>.6. M31N 2008-12a is a promising candidate of He novae. We expect a He nova outburst having a very short X-ray flash (15 min), very bright optical/NIR peak (∼3.5 mag brighter than M31N 2008-12a),much longer nova duration (>2 years), and longer SSS phase(40 – 50 days or more).Thus, we encourage search for a He flash in archival plates. M.K. and I.H. thank M. Henze for fruitful discussion on M31N 2008-12aand He novae.We thank the anonymous referee for useful comments that improved the manuscript.This research has been supported in part by Grants-in-Aid forScientific Research (15K05026, 16K05289)of the Japan Society for the Promotion of Science.[Ashok & Banerjee (2003)]ash03 Ashok, N. M., & Banerjee, D. P. K. 2003, ,409, 1007[Darnley et al. (2015)]dar15 Darnley, M. J., Henze, M., Steele, I. A. et al. 2015, , 580, 45 [Darnley et al. (2016)]dar16 Darnley, M. J., Henze, M., Bode, M.F. et al. 2016, , 833, 149 [Darnley et al. (2016)]dar16errata Darnley, M. J., Henze, M., Steele, I. A. et al. 2015, , 593, 3 (Erratum) [Darnley et al. 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(2005)]yar05 Yaron, O., Prialnik, D., Shara, M.M., & Kovetz, A. 2005, , 623, 398 [Yoon et al. (2004)]yoo04 Yoon, S.-C., Langer, N., & Scheithauer, S. 2004, , 425, 217 | http://arxiv.org/abs/1706.08654v1 | {
"authors": [
"Mariko Kato",
"Hideyuki Saio",
"Izumi Hachisu"
],
"categories": [
"astro-ph.SR",
"astro-ph.HE"
],
"primary_category": "astro-ph.SR",
"published": "20170627030106",
"title": "A Millennium-Long Evolution of One-Year-Recurrence-Period Nova -- Search for Any Indication of the Forthcoming He Flash"
} |
http://arxiv.org/abs/1706.08456v2 | {
"authors": [
"F. Aprile",
"J. M. Drummond",
"P. Heslop",
"H. Paul"
],
"categories": [
"hep-th"
],
"primary_category": "hep-th",
"published": "20170626161134",
"title": "Unmixing Supergravity"
} |
|
http://arxiv.org/abs/1706.08984v1 | {
"authors": [
"Greg Ver Steeg"
],
"categories": [
"stat.ML"
],
"primary_category": "stat.ML",
"published": "20170627180234",
"title": "Unsupervised Learning via Total Correlation Explanation"
} |
|
Correspondence should be addressed to [email protected] Department of Systems Biology, Columbia UniversityDepartment of Chemical and Biomolecular Engineering, Rice UniversityDepartment of Chemical and Biomolecular Engineering, Rice UniversityDepartment of Chemical and Biomolecular Engineering, Rice University Quantifying the statistics of occupancy of solvent molecules in the vicinity of solutes is central to our understanding of solvation phenomena.Number fluctuations in small `solvation shells' around solutes cannot be described within the macroscopic grand canonical framework using a single chemical potential that represents the solvent `bath'. In this communication, we hypothesize thatmolecular-sized observation volumes such as solvation shells are best described bycoupling the solvation shell with a mixture of particle bathseach with its own chemical potential. We confirm our hypotheses by studying the enhanced fluctuations in the occupancy statistics of hard sphere solvent particles around a distinguished hard sphere solute particle.Connections with established theories of solvation are also discussed.Mini-grand canonical ensemble: chemical potential in the solvation shell Dilip Asthagiri========================================================================A quantitative description of thermodynamics at the nano-scale is of crucial importance in many biological as well as nano-technological systems <cit.>. While traditional statistical mecahnical ensembles can describe macroscopic systems, they are inadequate in describing `small systems'. For example, the probability distribution p(r̅) of degrees of freedom (dof)r̅ of a small system exchanging energy with a surrounding bath isnot solely determined by its Hamiltonian H_ sys(r) and a unique temperature that describes system-bath interactions <cit.>,p(r̅) ∝̸exp ( -β H_ sys(r̅)). Todescribe the distribution of dof accurately, the system Hamiltonian must be augmented by a temperature dependent potential of mean force ϕ(r̅; β) that also depends on the molecular details of the interactions between the system and the bath <cit.>,p(r̅) ∝exp ( -β H_ sys(r̅) - βϕ(r̅;β)).We note that Eq. <ref> is formally correct but the functional form of ϕ in Eq. <ref> andits explicit dependence on the bath temperature is seldom known a priori. The molecular field becomes irrelevant for macroscopically large systems (with short range interactions) <cit.>; we expect ϕ(r̅;β) → 0. The failure of the canonical ensemble in describing small systems can be understood by noting that the magnitude of system-bath interactions, H_ sys-bath, is comparable to the magnitude of system-system interactions H_ sys. As a result, small systems cannot weakly couple with a realistic bath <cit.>. The inability of small systems to couple weakly to their surroundings is likely to hold true for all statistical mechanical ensembles. In other words, if the system-bath exchanges are comparable to the corresponding property of the system,statistical mechanics based on average extensive quantities is expected to fail. For example,the grand canonical framework with a unique chemical potential is likely to be inadequate if the number fluctuations are comparable to the average number of particles in a system. Recently, we hypothesized that the notion of unique intensive bath parameterscan be relaxed when studying small systems <cit.>.We showed, using all-atom molecular dynamics (MD) simulations, that the equilibrium properties and dynamics of a small system exchanging energy with its surrounding can be accurately described by a super-statistical generalization of the canonical ensemble wherein the small system is coupled to multiple heat baths each with a different temperature <cit.>.In this work, we focus our attention on the grand canonical ensemble at the microscopic scale. We study the statistics of number fluctuations in solvation shells of solute molecules. Understanding the number statistics at small length scales is of particular interest in biochemistry;typically small molecules bind to biological macromolecules such as proteins and nucleic acid polymers (RNA and DNA) in small `binding sites' whose chemical composition can fluctuate. Indeed, the thermodynamics of preference of small molecules over their competitors in such binding sitesdirectly depends on the statistics of`ligands'in the binding site <cit.>.We work with a hard sphere system to avoid confounding effects due to energetic interactions. For concreteness, we consider abath of N≫ 1 hard sphere particles with one solute particle fixed at the origin. Let the radius of each hard sphere particle, including the solute, be r_p.Within the bath,imagine a `solvation shell'of radius R around the solute. The number n of solvent particles inside the shell fluctuates as the bath samples configurations according to the microcanonical ensemble (see Fig. <ref>). The probability p(n|R) of observing n solvent particles in the solvation shell is a centrally important quantity in the study of hydration phenomena, such as ion solvation and the hydrophobic effect <cit.>.We note that if R ≫ r_p, the grand canonical ensemble predicts that the probability of observing n particles in the solvation shell is p_μ(n|R) ∼ e^- F(n) +μ nwhere F(n) is the free energy of assembling n solvent particles around the solute in a shell of radius R in the absence of the rest of the solvent. Without loss of generality we have assumed that β = 1. Here,μ is the chemical potential that dictates system-bath coupling. Note that F(n) only depends on the configurations of the system and does not depend on the nature of exchange of particles between the system and the bath and on the chemical potential of the bath. A key feature of the grand canonical description is that a single bath parameter μ describes all moments of the number distribution p_μ(n|R) as derivatives of the grand canonical partition function <cit.>. Is the grand canonical prescription accurate when the size of the solvation shell is comparable to the size of the particle (R ∼ r_p)?In Fig. <ref>, we show the distribution p(n|R) of observing n hard sphere solvent particles in a `solvation shell' of radius R=2.2× r_p around a distinguished hard sphere solute particle of the same size (black circles). The reduced density of the solvent is 8ρ r_p^3 = 0.9. In order to ensure efficient sampling of rarely occupied states, we employ the expanded ensemble technique developed by Merchant et al. <cit.>. The average number of solvent particles in the solvation shell is ⟨ n ⟩≈ 4.5.Next, in order to find the best grand-canonical description of the solvation shell, we conducted grand canonical Monte Carlo simulations in a solvation shell of radius R = 2.2× r_ p for μ∈ [-10, 5] with an interval of δμ = 0.05 (see inset for ⟨ n ⟩ and √(⟨ n^2 ⟩ - ⟨ n ⟩^2) as a function of μ). Then, we chose the value of μ^*≈ -0.6 that reproduced the average number of particles ⟨ n ⟩≈ 4.5 in the solvation shell.The dashed blue line shows the grand canonical estimate of the number distribution p_μ = μ^*(n|R).Notably,the grand canonical ensemble predicts a distribution with alower variance ⟨ n^2 ⟩ - ⟨ n ⟩^2 in the occupancy statistics compared to the explicit simulation of hard sphere particles. What factors lead to these enhanced number fluctuations in the solvation shell compared to the grand-canonical ensemble? For molecular-sized solvation shells, local density fluctuations in the solvent are of the same magnitude as the occupancy of the solvation shell itself. Moreover, the density fluctuations outside the solvation shell will depend on the occupancy of the solvation shell. This implies that the work required to transfera solvent particle across the boundary of the solvation shell will depend on both the local density of solvent particles just outside the solvation shell as well as the density of solvent particles in the solvation shell <cit.>. As a result, a single chemical potential cannot represent the exchange of solvent particles between the solvation shell (`system') and the rest of the solvent (`bath'). From the point of view of statistical inference, the grand canonical distribution (Eq. <ref>)specifies all moments of n with a single parameter μ. Consequently the mean ⟨ n ⟩ and the variance ⟨ n^2 ⟩ - ⟨ n ⟩^2 are coupled to each other through their dependence on μ. In contrast, the mean and variance of p(n) in Fig. <ref> can vary independent of each other. How do we capture these enhanced number fluctuations in the solvation shell? Based on our previous work with the canonical ensemble <cit.>, we propose a superstatistical generalization. We hypothesize that a small system that exchanges particles with a surrounding medium can be represented by a system that is in contact with multiple baths, each characterized by a unique chemical potential μ. Let P(μ) be the probability distribution over the baths. We can obtain the distribution over the number of solvent particles by marginalizing the variation over bath chemical potentials.p_ ss(n|R)= ∫ P(μ) × p_μ(n|R) dμIn Eq. <ref>, p_μ(n|R) is given by Eq. <ref>.What is the functional form of P(μ)? One numerical approach, inspired by research in image processing <cit.>, is to constrain the L_2 error between the observed distribution p(n|R) and the predicted distribution p_ ss(n|R) while maximizing the entropy of P(μ). While this numerical approach can lead to accurate predictions <cit.>, the numerically inferred distribution P(μ) offers little physical clarity. Another, more conceptual approach is to motivate the functional form of P(μ) using first principles. Previously, we have shown using maximum entropy arguments that in a superstatistical generalization of the canonical ensemble <cit.>, the distribution of inverse temperatures P(β) can be described as an inverse gamma distribution.Unfortunately, this functional form is not suitable for P(μ). This is because while inverse temperature β for classical systems is always positive, chemical potential can take both positive and negative values. Notably, the inverse gamma distribution does not support negative arguments. However, the activity z = exp (μ) only takes positive values. In this work, as a first guess,we assume that bath chemical activities z = exp (μ )are distributed as an inverse gamma distribution.Thus, we assume that the bath chemical potential μ is distributed as P(μ) = e^ (-λ_1e^-μ- λ_2μ )/Γ(λ_2)λ_1^-λ_2. As we see below, this particular functional form accurately describes the solvent number fluctuations around a solute molecule. In the future, we would like to explore the relationship between P(μ) and the nature of system-bath interactions. Before we investigate whether P(μ) in Eq. <ref> can capture the solvent number fluctuations around the solute molecule, let us inspect its behavior. In Fig. <ref>, we plot different cases of P(μ). All shown distributions are constrained to have the same mean (⟨μ⟩ = 0) and increasing standard deviation from black (σ = 0.5) to red (σ = 1.5) in steps of δσ = 0.25. Next, we test whether Eq. <ref> accurately capture the number statistics in the solvation shell. In Fig. <ref>, we show p(n|R) at various reduced densities 8ρ r_p^3 = 0.7, 0.8, and 0.9. At each reduced density we find the chemical potential μ^* of the grand canonical ensemble that matches the mean occupancy ⟨ n ⟩. From Fig. <ref>, it is clear that for molecular-sized solvation shells, the occupancy statistics predicted by the grand canonical ensemble (Eq. <ref>) cannot capture the distribution of the number of solvent particles in the solvation shell.Using the probabilities p_μ(n|R) of observing n solvent particles in the solvation shell in grand canonical simulations, at each reduced density 8ρ r_p^3, we numerically determined (using a simulated annealing scheme) the parameters λ_1 and λ_2 that lead to the lowest error when comparing log p(n|R)and log p_ ss(n|R). The dashed red lines show the predicted probability p_ ss(n|R) using Eq. <ref>. Remarkably, the superstatistical distribution p_ ss(n|R) can capture the entire distribution of solvent occupancy numbers very well for multiple solvent densities. In contrast, the grand canonical distribution of Eq. <ref> under-predicts the fluctuations in solvent occupancy numbers. Notably, a maximum entropy approach using a Gibbs prior or a flat prior on the occupancy distribution constrained by the mean occupancy and the variance in occupancyalso fails to capture the p(n|R) distribution <cit.>. We next investigated the approach to the eventual `macroscopic'grand canonical description.Using the expanded ensemble technique, we estimated the number distribution p(n|R) for solvation shells of size R = 2.2× r_p, 2.4× r_p, 2.6 × r_p, and 2.8 × r_p.The reduced density of the solvent was fixed at 8ρ r_p^3 = 0.8.Next, for each solvation shell radius R, we performed grand canonical Monte Carlo simulations and estimated theμ^* that reproduced the mean occupancy ⟨ n ⟩. In Fig. <ref> we show p(n|R)in black dots and the corresponding grand canonical estimate p_μ=μ^*(n|R) in dashed blue lines. As expected, we observe thatthe discrepancies between p(n|R) and the grand canonical estimate p_μ=μ^*(n|R) decrease as R increases. Discussion:The statistics of occupancy of the solvation shell of a solute molecule is of central importance in understanding hydration phenomena <cit.> as well as in understanding preference of `binding sites' in proteins and nucleic acids towards small molecules <cit.>. The grand canonical framework, if applicable at the microscopic scale, would be a versatile framework to understand solvation phenomena. This is because it would allow us to understand the occupancy statistics of a solvation shell as a combination of free energy F(n) of formation of solute-solvent clusters that is independent of the bulk solvent and the effect of the bulk solvent represented by a single parameter; the chemical potential. Unfortunately, our recent work <cit.> and the current study shows that the grand canonical framework is inadequate in describing solvation shells. However,our superstatistical approach allows us to interpret the effect of the bulk medium not as a single number (the chemical potential) but as a distribution. How do we reconcile our development with previous work on understanding solvation phenomena? The non-constant nature of the bath chemical potential is well-recognized in the quasi-chemical approach <cit.>. We describe connections with two previous works. First is the work of Merchant and Asthagiri that describes the molecular aufbau principle of the thermodynamic reorganization of the solvation shell upon addition of subsequent solvent molecules <cit.>. Second is the work by Bansal et al. that developes explicit solvent-related corrections to the grand canonical picture <cit.>. To be concrete, let p^0(n|R) denote the probability of observing n solvent particles in the solvation shell in the absence of the solute and G(n) denote the free energy of assembling n solute molecules in the solvation shell in the presence of the solvent (note that G(n) is different from F(n) in Eq. <ref>).Merchant and Asthagiri showed that <cit.>G - G(0) =G(n) + log p(n|R) - log p^0(n|R)where G is the excess free energy of introducing the solute particle in the solution. From Eq. <ref>, we can calculate the chemical potential G(n) - G(n-1) of the n^ th solvent particleG(n) -G(n-1) = logp(n-1|R)× p^0(n|R)/p(n|R) × p^0(n-1|R)From Eq. <ref>, it is clear that the work required to insert a solvent particle in the solvation shell is not constant. The work depends on the current occupancy of the solvation shell. In contrast, for a macroscopic system, the work in Eq. <ref> will be independent of the occupancy of the solvation shell and equal to the chemical potential. Thus, on the one hand, the quasi-chemical approach allows a detailed calculation of the dependence of chemical potential on the occupancy. On the other hand, the presented work complementarily allows us to rationalize enhanced density fluctuations in the solvation shell in terms of fluctuations in the bath chemical potential.An approach more closely related to the current work is that of Bansal et al. <cit.>. In that work, we analyzed the solvation shell for hard sphere solutes explicitly in terms of the interactions within the cluster around the solute, interactions of the cluster with the rest of the medium and the interactions for the bulk solvent molecules which are not in the cluster. Based on the partition function in the canonical ensemble for the system which has n molecules within the spherical shell around thesolute, we obtained the probabilities p(n|R) as (in the notation of the current work)p(n|R)∼ p_μ_p(n|R)× e^Ωσ_nwhere μ_p is the excess chemical potential of the solvent particleand p_μ_p(n|R) is given by Eq. <ref> with the μ equal to the chemical potential of the solvent. Ωσ_n approximately represents the field imposed by the bulk solvent on the solute-solvent cluster. The field was explicitly recognized as a surface interaction term that depends on the occupancy of the surface sites in the cluster around the solute and represented as followsΩσ_n = ζ_1 ·n^2+ζ_2· n.In other words, Bansal et al. <cit.>apply a correction to the grand canonical ensemble explicitly for each coordination state (see also Reiss and Merry <cit.>). Interestingly, Eq. <ref> can also be derived using maximum entropy arguments with the grand canonical distribution p_μ_p(n|R) as the prior (with the μ equal to the chemical potential of the solvent) with mean occupancy and second moment of occupancy as constraints. Note that absence of the correction term in Bansal et al. is equivalent to having P(μ) as a Dirac Delta function centered around the solvent chemical potential in the current work. In summary, the grand canonical ensemble is inadequate in describing fluctuations in number statistics of molecular-sized solvation shells. In this work, we modeled the correction to the grand canonical ensemble in the form of a super-statistical ensemble where the solvation shell is coupled with multiple solvent baths, each with its own chemical potential. We found that the superstatistical description accurately captures the statistics p(n|R) of the number n of solvent molecules in the solvation shell of radius R. As the size of the solvation shell increases compared to the size of the solute particle, the agreement with the grand canonical description is restored. In this work, we approximated the distribution P(z) of the bath chemical activitieswith an inverse gamma distribution (and correspondingly P(μ) by Eq. <ref>). However, relevant experimental constraints can inform the choice of P(μ) as well. For example, in case of the canonical ensemble, previously, we have used maximum entropy arguments to motivate the form of the distribution over bath temperatures <cit.>. Acknowledgments AB, WGC, and DA gratefully acknowledge support from Robert A. Welch Foundation (Grant # C-1241) to Professor Walter Chapman. WGC and DA additionally acknowledge support from the Abu Dhabi National Oil Company (ADNOC). 10dill2010molecular K. Dill and S. Bromberg, Molecular driving forces: statistical thermodynamics in biology, chemistry, physics, and nanoscience (Garland Science, 2010).dixit2011elastic P. 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Cox, and W. G. Chapman, The Journal of Chemical Physics 145, 074904 (2016).bansal_thermodynamics_2017 A. Bansal, A. Valiya Parambathu, D. Asthagiri, K. R. Cox, and W. G. Chapman, 146, 164904 (2017).merchant2011water S. Merchant, J. K. Shah, and D. Asthagiri, The Journal of chemical physics 134, 124514 (2011).gull1984maximum S. Gull and J. Skilling, Communications, Radar and Signal Processing, IEE Proceedings F 131, 646 (1984).hummer1996information G. Hummer, S. Garde, A. E. Garcia, A. Pohorille, and L. R. Pratt, Proceedings of the National Academy of Sciences 93, 8951 (1996).paliwal2006analysis A. Paliwal, D. Asthagiri, L. Pratt, H. Ashbaugh, and M. Paulaitis, The Journal of chemical physics 124, 224502 (2006).merchant2011ion S. Merchant, P. Dixit, K. Dean, and D. Asthagiri, J. Chem. Phys. 135, 054505 (2011).beck2006potential T. L. Beck, M. E. Paulaitis, and L. R. Pratt, The potential distribution theorem and models of molecular solutions (Cambridge University Press, 2006).reiss1981upper H. Reiss and G. A. Merry, The Journal of Physical Chemistry 85, 3313 (1981). | http://arxiv.org/abs/1706.08998v1 | {
"authors": [
"Purushottam D. Dixit",
"Artee Bansal",
"Walter G. Chapman",
"Dilip Asthagiri"
],
"categories": [
"cond-mat.stat-mech"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170627182738",
"title": "Mini-grand canonical ensemble: chemical potential in the solvation shell"
} |
Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel [Current address: ]Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel ] Spectroscopic Measurement of the Softness of Ultra-Cold Atomic CollisionsThe softness of elastic atomic collisions, defined as the average number of collisions each atom undergoes until its energy decorrelates significantly, can have a considerable effect on the decay dynamics of atomic coherence. In this paper we combine two spectroscopic methods to measure these dynamics and obtain the collisional softness of ultra-cold atoms in an optical trap: Ramsey spectroscopy to measure the energy decorrelation rate and echo spectroscopy to measure the collision rate. We obtain a value of 2.5 (3) for the collisional softness, in good agreement with previously reported numerical molecular dynamics simulations. This fundamental quantity was used to determine the s-wave scattering lengths of different atoms but has not been directly measured. We further show that the decay dynamics of the revival amplitudes in the echo experiment has a transition in its functional decay. The transition time is related to the softness of the collisions and provides yet another way to approximate it. These conclusions are supported by Monte Carlo simulations of the full echo dynamics. The methods presented here can allow measurements of a generalized softness parameter for other two-level quantum systems with discrete spectral fluctuations.[ Nir Davidson December 30, 2023 =====================Elastic collisions are of great importance in atomic physics, both from a theoretical and a practical point of view. They are relevant for atomic clocks, metrology, quantum information, evaporative cooling, atom-ion hybrid systems and more <cit.>. Collisions may also have a significant effect on the coherence properties of an ensemble of atoms, providing either elongation <cit.> or shortening <cit.> of the atomic coherence time.Considering a rapid collisional process compared to other dynamical timescales [the collision time can be estimated by the range of atomic interaction divided by the relative velocity of the colliding atoms. For dilute ultra-cold atoms and excluding mean-field interactions the collision time is on the nanosecond scale, much shorter than the mean time between collisions and the oscillation time in the trap (≳10 msec and ≳1 msec, respectively for our experiment).], there exist two extremities for a colliding atom in the center-of-mass frame of the interacting ensemble: “hard collisions”, in which the energy of the atom is completely randomized after a single collision, and “soft collisions” in which the atomic energy remains almost unchanged after each collision <cit.>. We therefore define the “collisional softness” parameter, s, as the number of times an atom has to collide in order for the correlation between its initial and final energies to drop to 1/e [This definition of s is related to the strength parameter of velocity changing collisions, α, defined in <cit.>, by s = 1/1-α^2]. The collisional softness of hard collisions is one, since the energy correlation drops to zero after a single collision. Collisions are considered “soft” if their softness parameter is much larger than unity. Even though the s-wave collisional process considered here is itself is of universal nature, the softness of the collisions can be affected by the confining potential. This can be intuitively understood by considering that only the kinetic energy changes due to a collision whereas the potential energy does not, carrying a “memory” of the energy prior to the collision.More formally, an ensemble of colliding trapped thermal atoms has two relevant characteristic rates. First, the atomic collisions, treated as a Poisson process energy-randomization events, occur at an average collision rate Γ_coll. Second, the single atom temporal energy autocorrelation function, averaged over the atomic ensemble, decays exponentially with an energy decorrelation rate Γ. The collisional softness is then defined as: [eq:softness_definition]s = Γ_coll/Γ. Collisions with s=1 are hard, and collisions with s ≫ 1 are soft.The softness of s-wave elastic collisions of ultra-cold bosonic atoms trapped in a harmonic potential and far from a Feshbach resonance was evaluated using molecular dynamics simulations and found to be 2.5 <cit.>. This value of the softness has been used to determine the elastic collision cross sections of different atoms <cit.>, but has not been measured directly. For a given number of collisions the softness is the number of collisions required for thermalization in a perturbed trap, having immediate repercussions on the physics of evaporative cooling <cit.>.In this paper we present a direct spectroscopic measurement of the softness of ultra-cold atomic collisions. We do so using a combination of two spectroscopic methods (Ramsey and echo <cit.>), in two opposite regimes of low and high collision rates. First we show that the coherence of an atomic ensemble in an echo experiment at low density asymptotically depends only on Γ_coll. We then show that in a high density Ramsey experiment the decay is independent of the collision rate and can be fitted to reveal the energy decorrelation rate Γ. By combining these two measurements, we are able to quantitatively extract the s-wave collisional softness of cold ^87Rb atoms in an optical dipole trap. We obtain good agreement with previously reported theoretical results from molecular dynamics simulations, validating our method and laying the foundations for its application in measuring the softness of other collisional processes.We further show that the coherence decay in an echo experiment is qualitatively different for short and long times <cit.>. This can be used to approximate the softness by combining a Ramsey measurement and an echo measurement in a single, intermediate density regime. These methods may allow measurements of a softness parameter for other two-level quantum systems that have discrete energy fluctuations <cit.>. §.§ Spectroscopic signatures of collisional softness atoms trapped in an optical dipole trap experience a differential AC-Stark shift imposed by the different detuning of the trapping laser from their two ground state hyperfine levels. If the mean time between atomic collisions is larger than the oscillation period in the trap, the fast oscillations can be averaged. The rate of the phase accumulated by the wavefunction, determined by the detuning, then depends on the average energy <cit.>. Effectively this creates a stationary inhomogeneous broadening of the spectrum, decreasing the coherence time of the ensemble. Due to this effect, the dynamics of the hyperfine coherence in a Ramsey (π/2-π/2) experiment with no collisions is given by [eq:Meschede]C_R(t) = [1+0.95(t/τ)^2]^-3/2 for an ensemble of two-level atoms in thermal equilibrium in an optical harmonic potential <cit.>. The bare Ramsey time is given by τ≈2ħ/η k_BT. Here T is the temperature of the cloud and η is the ratio between the hyperfine splitting and the detuning of the trapping laser [For ^87Rb and a YAG 1064 nm trapping laser η≈7×10^-5]. In an echo experiment (π/2-π-π/2), where the echo-pulse is given at time t_π, the echo coherence C_E(2t_π)≡ C(t=2t_π) fully revives in the absence of elastic atomic collisions due to the stationarity of the trap perturbation [For strong trap perturbations and for chaotic traps this may not be the case <cit.>].Factoring in the effect of elastic atomic collisions, the Ramsey and echo coherences have a complicated behaviour <cit.>. However, they both have some useful, simple limits. For high density n_high, the spectrum is collisionally-narrowed, resulting in an elongated Ramsey coherence, that can be approximated by the generalized Gumbel function <cit.> C_R(t) ∼exp[-2.86/Γ^2τ^2(e^-Γt+Γt-1)], dependent only on τ and the energy decorrelation rate Γ, and not on the collision rate Γ_coll.In the opposite regime, of low density n_low, the asymptotic long-time echo coherence behaves as[eq:low_density_corr]C_E(2t_π) ∼exp(-2Γ_collt_π).In this regime the coherence depends solely on the collision rate Γ_coll and not on the energy decorrelation rate Γ. This is due to the fact that every collision, no matter how soft, causes a finite deflection in the atomic trajectory. As t→∞ this deflection will fully decohere the atom. This implies that the coherence in this long-time regime is nothing but the fraction of atoms that did not collide.Fig. <ref> illustrates the role of collisions in an echo experiment. It presents the normalized upper hyperfine state population as a function of the time between the Ramsey pulses. In (a) the measurement is performed with very low collision rate (Γ_collτ≈ 0.15) [Low/high collision rates are defined, throughout this paper, with respect to the bare Ramsey time τ. In both regimes the collision rate is much smaller than the trapping frequencies.]. The coherence decays fast, and the echo revival amplitude is significant. On the other hand, in (b) the collision rate is much higher (Γ_collτ≈ 10). The Ramsey decay is slower (partially due to collisional narrowing) and the echo revival is negligible, manifesting the failure of the echo due to atomic collisions. Both the collision rate and the energy decorrelation rate are proportional to the multiplication of the atomic density n with the average velocity v_rel∼√(T): Γ_coll = n σ v_rel. Therefore, knowing the ratios n_high/n_low and T_high/T_low at the two different experimental conditions allows for the normalization of the collision rate, measured at low density, and the energy decorrelation rate, measured at high density, and extraction of the collisional softness s = (n_high/n_low√(T_high/T_low))Γ_coll^low/Γ^high. We note that in the intermediate regime Γτ≈ 1 it was theoretically shown that the spectrum of an ensemble of colliding atoms depends weakly on the softness <cit.>. It was further suggested that it may be possible to distinguish between the two extreme cases of hard and soft collisions, by measuring a Dicke narrowed spectrum in a Ramsey experiment. Practically, this task turns out to be challenging. Small uncertainties in the experimental conditions, such as the collision rate and the inhomogeneous broadening of the spectrum, may cause an incorrect model to fit well to the experimental data <cit.>. §.§ Measuring the collisional softness Our apparatus is described in detail in <cit.>. Briefly, the experiment consists of ^87Rb atoms trapped in a 1064 nm far-detuned crossed-beam optical dipole trap. The atoms are evaporatively-cooled down to two distinct regimes: high density with n_high = 3.5 (2)×10^12 cm^-3 and T_high = 0.56 (2) μK, and low density with n_low = 3.6 (3)×10^11 cm^-3 and T_low = 6.8 (3) μK [The temperature is measured using time-of-flight and the peak atomic density using n=ω_xω_yω_zN(m/2π k_BT)^3/2, where ω_i are the trap frequencies, N is the total number of atoms, m is the atomic mass, k_B is the Boltzmann constant and T the measured temperature]. Measurement of the total number of atoms, and hence the peak density, is susceptible to common systematic errors and obtaining an exact value for it is challenging <cit.>. However, as our method relies only on the knowledge of the ratio between densities [Eq. (<ref>)], systematic errors are common-mode rejected. All errors stated in throughout the paper represent a 1σ confidence level.The coherence is measured between the first-order Zeeman insensitive hyperfine |1⟩≡|F=1,m_F=0⟩ and|2⟩≡|F=2,m_F=0⟩ states of the 5^2S_1/2 manifold. The atoms are prepared by optical pumping and microwave transitions in state |1⟩. We then use a microwave ∼ 6.8 GHz control to perform Ramsey (π/2-π/2) or phase-scanned echo (π/2-π-π/2) manipulations on the atoms. At the end of each experiment we use a state-selective fluorescence-detection scheme to evaluate N_2/(N_1+N_2), the fraction of atoms at state |2⟩. The coherence is defined as the normalized amplitude of the fringes of the Ramsey and echo data.We measure the collisional softness using Eq. (<ref>), by first obtaining Γ_coll^low from an echo measurement in the low density regime by fitting the asymptotic decay described by Eq. (<ref>) and then obtaining Γ^high, given by Eq. (<ref>) from a Ramsey measurement in the opposite regime.Focusing first on the low density regime, the resulting echo coherence and an additional Ramsey measurement at the same experimental conditions are presented in Fig. <ref>(a). The echo decay time is indeed much longer than that of the Ramsey experiment (by about a factor of four). The extracted low-density bare Ramsey time is τ_low = 37 (1) ms, compared to 32 (1) ms obtained directly from the measured temperature. The echo measurement exhibits a long-time linear decay in a semi-logarithmic scale [Fig. <ref>(b)], confirming the expected exponential decay of Eq. (<ref>). The slope, excluding short times, gives a collision rate of Γ_coll^low = 9.4 (3) s^-1. Next we obtain the energy decorrelation rate, Γ^high, from the collisional narrowing of a high-density Ramsey measurement. The atomic coherence is shown in Fig. <ref>(c). From the measured temperature, we expect τ_high = 390 (15) ms. We use this value as a fixed parameter and fit the coherence data to Eq. (<ref>), extracting Γ^high = 10.6 (1) s^-1. The softness is then calculated using Eq. (<ref>) to be s = 2.5 (3) [This value is correct for harmonic trapping potential that describes well our crossed Gaussian beam optical potential. For other trap shapes and energy distributions it may vary, for a flat box potential s=1.5 <cit.>], in excellent agreement with molecular dynamics simulations <cit.>.We perform Monte Carlo simulations to study the effect of the softness of the collisions on the full dynamics of the echo decay. The simulation calculates the ensemble coherence of 2×10^4 two-level atoms with the energy distribution corresponding to a 3D harmonic potential <cit.> as a function of time. The collision rate Γ_coll is drawn from a Poisson distribution and the collisional softness s is generated by introducing controlled correlations between the energy jumps of successive collisional events using the Cholesky decomposition method of the required correlation matrix <cit.>. Typical energy trajectories for s=1 and s=10 are illustrated in the insets of Fig. <ref>. A comparison between the experimental data and simulation results is presented in Fig. <ref> for the echo experiment with hard (s=1), soft (s=10) and s=2.5 collisions, using the measured energy decorrelation rate, rescaled using Eq. (<ref>), Γ^low = (n_high/n_low√(T_high/T_low))^-1Γ^high = 3.76 s^-1. The simulation of the s=1 and s=10 clearly disagrees with the data, whereas the s=2.5 collisions agrees best with the experimental data for all times and with no fit parameters. The echo decays of the hard and of the soft collisions, with the same energy decorrelation rate, clearly do not agree with the experimental results. §.§ Measuring the softness using a transition in the functional decay of the echo dynamics In the low density regime of Γτ≪ 1, valuable information can be extracted by observing the entire dynamics of the decay of the echo coherence. Eq. (<ref>) gives the long-time limit of the coherence C_E(2t_π) ∼exp(-2Γ_collt_π), depending solely on the collision rate Γ_coll. The short-time limit, however, depends on the energy decorrelation rate Γ and the bare Ramsey time τ and is given by <cit.>- [eq:low_density_short_time]C_E(2t_π) ∼exp[-Γ(2t_π)^3/6τ^2]. In both limits the echo coherence decays as C_E(2t_π)∼exp(-β(2t_π)^α), with (α,β)=(1,Γ_coll) for long times or (α,β)=(3,Γ/6τ^2) for short times. Defining t_tr≡τ/√(s), a transition time between the two regimes, an interpolating function can be written to describe the entire dynamics, which is the accurate solution in the limit of soft collisions <cit.>: C_E(2t_π) ∼exp{-2Γ_collt_π[1-√(π/2)t_tr/2t_πerf(√(2)t_π/t_tr)]}.We measure the echo coherence decay with a moderate collision rate (Γ_coll t_tr≈ 1). The resultant coherence is summarized in Fig. <ref>. The transition of α is evident from the fit to Eq. (<ref>). An indication for the α = 3 decay is shown in Fig. <ref>(b), where we plot the echo revival amplitudes in a logarithmic scale against (2t_π)^3. Fig. <ref>(c) shows the same data in a semi-logarithmic scale as a function of 2t_π. Here the linear dependence at long times indicates the α = 1 decay. A similar transition was observed for warm molecular gases in the limit of soft collisions <cit.>. More quantitatively, extracting t_tr from the echo decay using Eq. (<ref>), and τ from the Ramsey decay under the same experimental conditions, we can evaluate the softness by the definition of the transition time s = (t_tr/τ)^2. Fitting the echo decay of Eq. (<ref>) to the data of Fig. <ref>, using Γ_coll=5.1 (3) s^-1 obtained from fitting the long-time decay of Fig. <ref>(c), we get 2t_tr=205 (17)ms [To evaluate the error 2Δ t_tr we use the value Γ_coll±ΔΓ_coll as a fitting parameter. The upper 1σ confidence bound on t_tr is that which is obtained for Γ_coll+ΔΓ_coll and the lower bound is that which is obtained for Γ_coll-ΔΓ_coll]. This, in addition with a Ramsey experiment under the same experimental conditions [inset of Fig. <ref>(a)] that gives τ=76 (3) ms, yields s = (t_tr/τ)^2=1.8 (3). The agreement with the theoretical value of 2.5 is not as good as for the measurement described in the previous section.We investigate the use of the interpolation function of Eq. (<ref>) as a fitting function for extracting the softness using the Monte Carlo simulations described previously. We find that for a Gaussian energy distribution the method is quite precise. Fig <ref> presents a summary of the softness obtained for a Gaussian energy distribution (full symbols) by fitting Eq. (<ref>) to the numerically simulated decay of coherence as a function of the input softness for different values of Γτ. For most cases, the output softness is very close to the input softness. The relative error is <10% as long as Γ_collt_tr is within the range 0.1<Γ_collt_tr<10. Outside this range, the value of the coherence at the transition time C_E(2t_tr)≈exp(-2Γ_collt_tr) is either too high or too low for the fitting procedure to accurately extract the transition time. For the case of an energy distribution of a 3D harmonic trap similar to the one we have in the experiment the situation is different [Fig. <ref>, empty symbols]. The approximate solutions for the Ramsey [Eq. (<ref>)] and echo [Eq. (<ref>)] fit only qualitatively, yielding significant errors similar to the ones observed in the experiment.§.§ Summary and outlook We have measured the softness of ultracold elastic atomic collisions using a combination of two spectroscopic methods: a measurement of the energy decorrelation rate obtained from the collisional narrowing of a Ramsey experiment with a high collision rate, and a direct measurement of the collision rate obtained from an echo experiment with a low collision rate. The obtained collisional softness is 2.5 (3) - in excellent agreement with the value obtained by molecular dynamics simulations.We have also demonstrated a transition in the functional decay of the echo coherence, from exp(-t^3) at short times, to exp(-t), at long times <cit.>. This transition occurs at a softness-dependent time t_tr=τ/√(s). We show that this transition in the functional decay can be used to approximate the softness at a single, intermediate density regime. Our results validate the spectroscopic method, allowing for its use in measuring the softness of other, non-trivial, collisional processes such as extensions to higher temperature involving the inclusion of more partial waves into the scattering process, fermionic collisions and thermalization <cit.>, inter-species hybrid collisions <cit.> and atom-ion collisions <cit.>. It can also be useful as a tool in simulating the efficiency of evaporative cooling <cit.>, the investigation of high-density atom interferometers <cit.> and slow and stored light <cit.>. Our methods may allow measurements of a generalized softness parameter for other two-level quantum systems with discrete spectral jumps <cit.>.Our work can be extended to warm vapor systems where the bare Ramsey time is dominated by Doppler broadening and the presence of a buffer gas induces collisional narrowing <cit.>. As in our system, the effect of collisional softness on the Ramsey signal is in itself very small and challenging to detect <cit.>. Combining Raman Rabi, Ramsey and echo spectroscopy at high and low collisional-rate regimes can provide an accurate measure of the collisional softness for different buffer gases. In this case, it is possible to change Γ_collτ by simply altering the angle between the Raman beams without actively changing the density <cit.>. The authors would like to thank Yoav Sagi, Hagai Edri and Noam Matzliah for discussions. This work was supported in part by the ICore Israeli excellence center Circle of light and the Weizmann Institute Texas A&M collaboration program. apsrev4-1 | http://arxiv.org/abs/1706.09004v2 | {
"authors": [
"Jonathan Coslovsky",
"Gadi Afek",
"Alexander Mil",
"Ido Almog",
"Nir Davidson"
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"categories": [
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"published": "20170627183935",
"title": "Spectroscopic Measurement of the Softness of Ultra-Cold Atomic Collisions"
} |
Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371 Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34051, Republic of Korea Instituto de Energías Renovables, Universidad Nacional Autónoma de México, Temixco, Morelos 62580, Mexico We demonstrate the potential of quantum operation using lattices of exciton-polaritons in patterned semiconductor microcavities. By introducing an inverse four-wave mixing scheme acting on localized modes, we show that it is possible to develop non-classical correlations between individual condensates. This allows a concept of quantum exciton-polariton networks, characterized by the appearance of multimode entanglement even in the presence of realistic levels of dissipation. Quantum Exciton-Polariton Networks through Inverse Four-Wave Mixing Y. G. Rubo December 30, 2023 ===================================================================Recently, there has been a significant attention devoted to the study of exciton-polaritons in lattices <cit.>. As systems of nonlinear interacting bosons, they have often been suggested as potential candidates of quantum simulators <cit.> and indeed the minimization of the energy of a particular Hamiltonian on a graph was a problem considered recently <cit.>. While the majority of studies of exciton-polaritons have been restricted to the classical regime <cit.>, the quantum nature of polaritons has received revived attention recently <cit.>. Therefore, it is natural to question whether exciton-polaritons can be used to form lattices of entangled modes. Here we must be aware that a lattice or graph of polaritons does not behave as a system of qubits. Instead each node of a polariton network could be described by the quantum field amplitude â_n or the continuous amplitude and phase variables associated with the operators: q̂_n=â_n+â_n^†/√(2), p̂_n=â_n-â_n^†/i√(2). Since continuous variable modes can be entangled, networks of continuous variable modes are highly relevant for quantum applications. As an example, cluster state computation <cit.> based on continuous variables <cit.> is a potential route towards universal computation. It relies on producing a highly entangled state from an arbitrary lattice or graph of modes coupled by two-mode squeezing type interactions, with a Hamiltonian of the form: ℋ_𝒮=∑_nmw_nm(â_nâ_m+â^†_nâ^†_m), where w_nm describes the weights of different connections in the graph. Arranging such a Hamiltonian is already a problem and it must be done making use of some interaction process that is stronger than any detrimental processes in the system (dissipation, dephasing, etc.). While evidence of strongly interacting polaritons <cit.> was reported recently, it is not clear if any nonlinear interaction process in microcavities is sufficiently strong for the generation of quantum resources. In the absence of strong interactions, exciton-polaritons tend to only demonstrate nonlinear effects at high densities, when they are well described by the classical physics corresponding to the mean-field approximation. For this reason only a handful of experimental reports of quantum exciton-polariton effects have appeared in the literature <cit.>.Two-Mode Squeezing.—Before considering how to build a polariton network, it is instructive to consider the effect of the two-mode squeezing type Hamiltonian: Ĥ=-iα/2(12-12). Such a Hamiltonian generates entanglement, which can be characterized by the violation of the inequality <cit.> 1≤ S_12=1/2[V(q̂_1-q̂_2)+V(p̂_1+p̂_2)], where the variances are defined by V(Ô)=⟨Ô^2⟩-⟨Ô⟩^2.The Heisenberg equations of motion give the evolution of the quantum field operators 1,2(t)≡ e^iĤt1,2e^-iĤt (we set ħ=1) 1,2(t)=cosh(α t/2)1,2+sinh(α t/2)2,1.To calculate the second order correlators, we consider operators K̂=1+11+22, L̂=12+12, M̂=i(12-12) and use the Lie algebra [M̂,K̂]=2iL̂, [M̂,L̂]=2iK̂ to obtainK̂(t)= cosh(α t)K̂ + sinh(α t)L̂,L̂(t)= cosh(α t)L̂ + sinh(α t)K̂.Using these relations and taking the vacuum state as an initial condition one arrives at S_12=<K̂(t)-L̂(t)>=e^-α t<1, indicating evolution of the system towards the entangled co-eigenstate of the EPR pair of operators q̂_1-q̂_2 and p̂_1+p̂_2. This result, S_12=e^-α t, remains unchanged for initial coherent states of the fields. We note that the EPR pair of operators <cit.> needed to demostrate the nonclassical correlations depends on the Hamiltonian. Other possible pairs can be obtained with the gauge transformation of operators in (<ref>), 1,2→1,2e^iϕ_1,2, with subsequent optimization over the phases ϕ_1,2.Inverse Four-Wave Mixing.—Let us now consider a single cavity with a four-wave mixing (parametric) type resonance, described with the Hamiltonian ℋ_0=α_0/2(â^†â^†â_Lâ_U+â_L^†â_U^†ââ), where α_0 describes the strength of the four-wave mixing process. Physical realizations of the above Hamiltonian could be made in exciton-polariton micropillars <cit.> or a Kerr nonlinear photonic crystal cavity <cit.>. Finding a parametric resonance would however require careful tuning <cit.>, which suggests that systems compatible with post-growth tuning would be the most realistic choices. For example, dipolariton based setups allow electrical control of mode energies <cit.>. Regardless of the mechanism of introducing Hamiltonian (<ref>), it is typically the case that α_0 will be weak compared to the system losses Γ, that is, typical optical systems are only weakly nonlinear (α_0≪Γ).The Hamiltonian (<ref>) is usually considered for generating the fields â_L and â_U from initial excitation of the field â, however, we can also consider the inverse process illustrated in Fig. <ref>a.Namely, if the modes â_L and â_U are driven by coherent laser fields then particles scatter in pairs from â_L and â_U to the mode â. It is true that under such conditions the modes â_L and â_U should behave only classically, such that their physics can not go beyond what is expected from making the mean-field approximation on these modes, but doing so leaves still a reduced quantum Hamiltonian acting on the mode â: ℋ_0=α/2(â^†â^†+ââ), where α=α_0⟨ a_U⟩⟨ a_L⟩. While this is just the Hamiltonian of two particle creation, by considering its introduction via the aforementioned inverse four-wave mixing process we have a way to make this a strong effect: since ⟨ a_L⟩ and ⟨ a_U⟩ can be increased by the resonant driving intensity, one can reach the regime α≫Γ.Considering exciton-polariton systems the regime α≫Γ has essentially been realized previously under different conditions, where the blueshift due to polariton-polariton interactions may exceed the linewidth and cause bistability <cit.>. It is worth mentioning that four-wave mixing experiments also revealed an interesting polarization dependence <cit.>, which allow the signal mode â to have a different linear polarization to that of the others (â_U and â_L), useful for better resolution and limiting other scattering processes.Coupled cavities.—If we now consider a pair of coupled cavities, which could be made with the techniques of Ref. <cit.>, the model Hamiltonian becomes (see Fig. <ref>b): ℋ=α/2(1+2+1+2)-J(â_1^†â_2+â_2^†â_1), where J is the coupling constant between the cavities and we can set α>0 without loss of generality. We show below that this Hamiltonian results in entanglement between modes â_1 and â_2.It is convenient to define new operators, representing a symmetric-antisymmetric basisâ_1=â_+ + â_-/√(2), â_2=â_+ - â_-/√(2), decoupling the Hamiltonian into two parts: Ĥ=1/2∑_σ=±[ α(σ+σ) - 2σ Jσσ]. We can then consider the Bogoliubov transform â_σ=cosh(x/2)b̂_σ+σsinh(x/2)b̂^†_σ, which in the case |J|>α and tanh(x)=α/J reduces the Hamiltonian into the simple form ℋ=ω(++---), where ω=√(J^2-α^2).If we take the vacuum state as the initial condition then only the second-order correlators of a-fields contribute to the inequality (<ref>). It is easy to show that ⟨σσ(t)⟩ = sinh^2(x/2), ⟨σ(t)⟩ = -σ/2sinh(x)e^-2iσω t,which results in⟨σσ(t)⟩=sinh^2(x)sin^2(ω t), ⟨σ(t)⟩=sinh(x)sin(ω t)[σcosh(x)sin(ω t)+icos(ω t)].In the symmetric-antisymmetric basis we have S_12=1+⟨+++---(+--)⟩, wheredenotes the real part and the first order correlators vanish in our case. Substituting the explicit form of the correlators, we obtain: S_12 =1+2sinh^2(x)sin^2(ω t)-sinh(2x)sin^2(ω t) =J+αcos(2ω t)/J+α.While this expression can never reach the value of zero, for the case J>α, one can reach the value (J-α)/(J+α) for the specific time when the cosine function evaluates to -1. Since J-α can be tuned to be small, one can then in principle reach arbitrarily small values of S_12. To give a visualization, Fig. <ref>a shows the variation in S_12 at some fixed time as a function of J. Figure <ref>b then shows the minimal value of S_12 obtainable for increasing values of α. The formation of entanglement in the above scheme might be seen as a round about way to create entanglement from four-wave mixing, which could be obtained already from Hamiltonian (<ref>). Indeed the usual method of exciting the central mode â and looking at correlations between â_L and â_U has been considered before, in different contexts <cit.>. It should be stressed however that the conventional method requires α_0 to be significant compared to the dissipation rate and also α_0 should be stronger than other scattering processes (e.g., scattering with acoustic phonons) that may resonantly couple the modes to be entangled. In the scheme that we consider here α can become the dominant interaction in the system as it is enhanced by the density of modes a_L and a_U. Furthermore, local interactions, such as scattering with phonons and sample disorder are not able to couple spatially separated modes â_1 and â_2.Dissipation.—We have shown so far that the system of coupled cavities driven by parametric resonance can generate entangled states, which become asymptotically close to the level of entanglement expected from a two-mode squeezing type operation, as measured by the violation of inequality (<ref>). As we have noted in the previous section α can be controlled by the intensity of external lasers. In principle, J can also be controlled by external fields, for example, by using external electric <cit.> or optical fields <cit.> to modify the potential between lattice points.While we expect the regime α≫Γ to be experimentally accessible, given the parametric driving scheme, it is still instructive to consider the influence of dissipation in the system. This is readily introduced by modification of the Heisenberg equations: d⟨Ô⟩/dt=i⟨[ℋ̂,Ô]⟩+Γ/2∑_n⟨2nÔn-nnÔ-Ônn⟩. This introduces additional dissipation terms in our equations of motion, which are solved in the Supplementary Material <cit.>. The resulting effect of dissipation is illustrated in Figs. <ref>(a,b).As one would expect, too much dissipation results in a loss of entanglement. However, given the parametric pumping scheme it is in principle possible to work in the limit where Γ≪α. At some short time such that τ≪ 1/Γ one then obtains a high degree of entanglement despite the presence of dissipation.Multimode Entanglement.—In comparison to conventional methods of entanglement generation with respect to four-wave mixing, a further advantage of our scheme that entangles polariton modes separated in real space is that it is in principle scalable; by coupling more cavities in space, arbitrary networks could be considered such as the one illustrated in Fig. <ref>c.As an example let us consider a system of four identical cavity, which are subjected to the Hamiltonian: ℋ_4 =∑_n=1^4α/2(â_n^†â_n^†+â_nâ_n)-J_A(t)(â_1^†â_2+â_2^†â_1+â_3^†â_4+â_4^†â_3)-J_B(t)(â_1^†â_3+â_3^†â_1+â_2^†â_4+â_4^†â_2). This Hamiltonian is a generalization of Hamiltonian (<ref>), where we assume that it is possible to control the linear coupling in time. For simplicity, we will consider (J_A(t)=J, J_B(t)=0) for the time 0<t<τ and (J_A(t)=0, J_B(t)=J) for time τ<t<2τ. It is possible to write Heisenberg equations of motion and their time dependent solution can be obtained analytically <cit.>. Alternatively, in the absence of dissipation, it is more efficient to solve for the operator evolution in the Heisenberg picture <cit.>.While violation of inequality (<ref>) is a sufficient condition for entanglement, the definition given of S_12 is not ideal for all states. In particular, varying the phases of modes â_1 and â_2 changes the value of S_12 and thus to demonstrate the entanglement we should minimize S_12 over all choices of local phases. As we mentioned above, this is equivalent to finding the best EPR pair of operators. The procedure is detailed in <cit.>, where we define S̃_12 as the value of S_12 minimized over phase rotations. The result is shown in Fig. <ref>a, where, in addition to characterizing the entanglement between modes â_1 and â_2, we find also entanglement between other pairs of modes, using similar definitions for S_13 and S_14 (other combinations of modes display identical entanglement characteristics due to symmetry). In addition to entanglement between pairs of modes, multimode entanglement, simultaneously between all four modes of the system can be evidenced by the violation of the inequality <cit.>: 1/2(V(q̂_1-q̂_2)+V(p̂_1+p̂_2+gp̂_3+gp̂_4))=I≥1, where g is an arbitrary real parameter that should be chosen so as to optimize the violation of the inequality. In the general case of four modes, one should also break two other inqualitites to evidence an entangled state, obtained by permuting the modes <cit.>. However, given the symmetry of our four mode example in a ring these inequalities are equivalent and the violation of inequality <ref> is a sufficient condition. Following correct choice of the parameter g and optimization over the phases of the modes <cit.>, we indeed find that the quantity I can drop below one and even reach zero, as shown in Fig. <ref> for different values of α and Γ.Conclusion.—The evolution of polariton networks from the classical to quantum regime implies finding a mechanism of generating quantum correlations that can overcome the dissipation of the system. Nonlinearity, in the form of polariton-polariton interactions is traditionally weak, however, here we have shown theoretically that an inverse four-wave mixing geometry allows enhancement to an effective strongly nonlinear regime. Local nonlinearity and standard Josephson coupling between spatially separated modes is then sufficient to generate quantum entanglement both between pairs of modes and multiple modes. 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"authors": [
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"Y. G. Rubo"
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"published": "20170626160020",
"title": "Quantum Exciton-Polariton Networks through Inverse Four-Wave Mixing"
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Department of Physics, Stockholm University, AlbaNova University Center, SE-106 91 Stockholm, SwedenDepartment of Physics and Technology, University of Bergen, N-5007 Bergen, NorwayDepartment of Physics and Technology, University of Bergen, N-5007 Bergen, NorwayFaculty of Technology, Art and Design, Oslo and Akershus University College of Applied Sciences, NO-0130 Oslo, NorwayDepartment of Physics, Stockholm University, AlbaNova University Center, SE-106 91 Stockholm, SwedenWe present a generalized velocity gauge form of the relativistic laser-matter interaction. In comparison with the (equivalent) regular minimal coupling description, this new form of the light-matter interactionresults in superior convergence properties for the numerical solution of the time-dependent Dirac equation. This applies both to the numerical treatment and, more importantly, to the multipole expansion of the laser field. The advantages of the alternative gauge is demonstrated in hydrogen by studies of the dynamics following the impact of superintense laser pulses of extreme ultraviolet wavelengths and sub-femtosecond duration.Alternative gauge for the description of the light-matter interaction in a relativistic framework Eva Lindroth December 30, 2023 ================================================================================================= § INTRODUCTION Withhigh laser intensities, available alreadynow or in the near future <cit.>, and the interesting possibilities then opening, discussed for example in Ref. <cit.>, the description of the light-matter interaction in a relativistic framework is of growing importance. The ionization dynamics initiated with few-cycle laser pulses calls further for a time-dependent treatment. Several attempts have consequently been made to solve the time-dependent Dirac equation (TDDE), see, e.g., Refs. <cit.>, but it has been proven a hard task to explore the truly relativistic region while simultaneously accounting for the spatial dependence of the electromagnetic field and the full dimensionality of the problem. Recently, however,a numerical study wasmade where high orders of multipole interaction terms were successfully accounted for <cit.>. Field intensities up to the strength where electrons are expected to reach quiver velocities, v_quiv≈ e E_0/ mω, of around 20% of the speed of light were treated, and emerging relativistic effects could be detected. Still, the study alsounderlined some severe problems appearing when one is tackling the TDDE, concerning in particular the inclusion of magnetic effects.When electrons are driven to high velocities by laser fields the magnetic part of the electromagnetic field inevitably becomes increasingly important. A qualitatively new effectemerging is then the force imposed on the particle in the propagation direction of the light. Simulationsin the low- or medium-intensity regimes are usually made within the dipole approximation, where the spatial dependence of the vector potential of the pulse is neglected completely. Since this approximation implies a neglect of all magnetic effects it is rather pointless in the high intensity regime <cit.>. To understand the importance of different types of effects beyond the dipole approximation it is illustrative to look at the studies within this regime that have been done with the non-relativistic time-dependent Schrödinger equation (TDSE). In that case the spatial dependence of the vector potential may conveniently be treated through a Taylor expansion <cit.> and the lowest order contribution has been shown to dominate the dynamics beyond the dipole completely – at least up to intensities that drive the electron to velocities just above ten percent of the speed of light <cit.>.Surprisingly enough, as shown in Ref. <cit.>, when the same approach is used with the TDDE, the lowest order spatial contribution from the Taylor expansion gives results that deviates significantly from the non-relativistic results already at modest intensities, far below the relativistic regime. This can be corrected by adding the next term in the expansion, but when the intensity is increased further the situation is repeated and one is forced to include also the following term and so on. This behavior can be analyzed and understood in the non-relativistic limit, as shown in Ref. <cit.> and also discussed in Sec. <ref> below.The problem stems from contributions that are known to cancel (approximately), but which enterin differentformal orders with respect to the Taylor expansion when it is applied to the Dirac equation. Wherever the expansion is truncated, there will be unbalanced contributions which at some intensity will play a significant role.This imbalance is inherent to the four-component Dirac equation, and if the TDDE is to be solved for strong relativistic pulses the Taylor expansion approach in the regular minimal coupling Hamiltonian quickly leads to an intractable problem.Recently a generalized velocity gauge form of the non-relativistic light-matter interaction was presented <cit.>. Within this gauge the dipole contribution is given as in velocity gauge, while the so called diamagnetic term disappears and instead new terms appear. Of these, the leading order ones depend explicitly on the momentum in the direction of the laser propagation and the new gauge was consequently coined the propagation gauge. It was further shown that it is possible to use a series of gauge transformations tosuccessively remove all field-dependent terms that do not depend explicitly on the momentum.The lowest order interaction within this gauge was further tested <cit.> and compared to simulations performed with the traditional minimal coupling Hamiltonian.Impressive numerical advantages were then demonstrated. One manifestation of this was the evolution of the momentum expectation value along the direction of propagation of the light. In the minimal coupling description it showed a strong oscillating behavior, but in the new gauge it was replaced by a smooth curve that could be sampled with much larger time steps. Moreover, for a wave function expanded in spherical harmonics, the ionized wave packet could be described with considerably fewer angular momenta. In the following we will show that a corresponding gauge choice for the TDDE is even more advantageous. It requires just a single gauge transformation, it takes a simpler form, and it is a promising candidate for studying strong relativistic multipole interactions.The paper is structured as follows: In the next section we outline the theoretical framework. Brief details on the implementation are provided in Sec. <ref>, while the results are presented and discussed in Sec. <ref>. Finally, we present our conclusions in Sec. <ref>. Atomic units are used throughout the text unless explicitly stated otherwise.§ THEORYIn a non-relativistic framework, the evolution of a wave packet representing a particle of mass m and charge -e in the scalar field φ and vector potential A is governed by the TDSE,i ħd/d tΨ_NR = H_NR(t) Ψ_NR,with the HamiltonianH_NR(t)= [ 𝐩^2/2m -eφ + e/m p· A + e^2 A^2/2m].Here the potential A has been taken to fulfill the Coulomb gauge condition, ∇· A = 0. Letting the electromagnetic pulse be defined in terms of the vector potential A, and assuming the field to be linearly polarized along the z axis and propagating along the x axis,the pulse may be writtenA(η) =A(η) ẑ = E_0/ω f(η) sin(η + ϕ)ẑ ,where η = ω t - k x and k = ω/c. The envelope function is chosen to be sine squared:f(η) = {[ sin^2 ( πη/ω T),0 < η < ω T; 0,otherwise ]. . In the dipole approximation, when the spatial dependence of the vector potential is neglected, theA^2 term can be removed by a gauge transformation and consequently does not affect the dynamics. On the other hand, this diamagnetic term is known to give the leading contribution beyond the dipole approximation in the high-intensity limit. This has, e.g., been shown in Ref. <cit.>,where the spatial dependence of the vector potential was examined with the help of a Taylor series expansion:A(η) ≈∑_n=0^n_trunc1/n!d^n A (η)/d η^n|_η=ω t(- ω x/c)^n .In Ref. <cit.> n_trunc≤ 2 was considered. A point worth noticing here is that the diamagnetic term is second order in 𝐀, and thus an expansion of A to a particular order in (ω x/c)^n does not imply an expansion of the Hamiltonian to the same order. The contributions to the Hamiltonian with, for example, n=2 come from the square of the n=1 term in Eq. (<ref>) and from the cross term between the n=0 and n=2 terms. Furthermore, there should be considerable cancellations between these terms; their sum oscillates with twice the frequency of the light while each of them have a constant sign. Large cancellations were indeed found in Ref. <cit.>, and it was concluded that it is decisive to include all terms that contribute to the Hamiltonian to a given order. When the corresponding time-dependent Dirac equation is solved using the minimal coupling Hamiltonian <cit.>, the diamagnetic term, which is then only implicitly included, causes severe convergence problems in terms of the multipoles of the external field. This is connected to an effective blocking of the aforementioned cancellations as will be clear in the following.Turning now to the TDDE, the first step will be to consider the minimal coupling Dirac Hamiltonian;H(t) = c α·[p +e A(η) ] -e φ(r) 1_4 + m c^2 β,andα = ( [ 0 σ; σ 0 ]). As usual,σ is given by the Pauli matrices, andβ = ( [1_20;0 -1_2 ]).We set out to solve the TDDE:i ħd/d tΨ̃ = H(t)Ψ̃,where the four-component wave function Ψ̃ can be written asΨ̃( r,t) = ( [ Ψ̃_F( r,t); Ψ̃_G( r,t) ]),with Ψ̃_F and Ψ̃_G being two-component spinors, often called the large and small component, respectively. The potential φ(r) is for the present purposes simply the Coulomb potential from a point nucleus, i.e., we neglect retardation effects in the electron-nucleus interaction and take the nuclear mass to be infinite.§.§ The non-relativistic limit of the relativistic minimal coupling Hamiltonian In order to understand the origin of the problems encountered with the TDDE expressed in terms of the orignal minimal coupling Hamiltonian, it is important to study its non-relativistic limit. Since we are aiming for a solution to the TDDE which describes a positive energy state, we may writeΨ̃( r,t) = Ψ( r,t) e^-i mc^2 t.Eq. (<ref>) can then be rewritten as:i ħd/d tΨ( r,t) = ( H(t) -mc^2 )Ψ( r,t).Using the form of the wave function given in Eq. (<ref>) we can write Eq. (<ref>) as two coupled differential equations:-e φΨ_F +c σ·( 𝐩 +e 𝐀)Ψ_G = iħdΨ_F/d tc σ·( 𝐩 +e 𝐀) Ψ_F + ( -e φ - 2mc^2 )Ψ_G = iħdΨ_G/d t.If only the dominating terms on the second line is retained (i.e. assuming that the mass-energy term is large both compared to the potential energy, 2mc^2 ≫ eφ, and to the time variation of the small component) it is possible to write the small component as:Ψ_G ≈1/2mcσ·( 𝐩 + e 𝐀) Ψ_F.When inserting this into the first line of Eq. (<ref>) we get-e φΨ_F +1/2m( σ·( 𝐩 + e 𝐀))^2Ψ_F = iħdΨ_F/d t,and with some operator algebra, detailed in Ref. <cit.>, this expression can be rewritten as( 𝐩^2/2m -e φ + e/m𝐩·𝐀 +e^2 A^2/2m + e ħ/2 mσ·𝐁) Ψ_F = iħdΨ_F/d t,where 𝐁 = ∇×𝐀. Apart from the spin-dependent term, the operators on the left-hand side are the same as those that appeared in Eq. (<ref>). In particular, we note the diamagnetic contribution, which apparently is implicitly included in the TDDE through the coupling between the small and large component of the wave function. Hence the advice from Ref. <cit.> regarding the consistent inclusion of the x^n terms from the Taylor expansion of 𝐀 is not easy to follow for the TDDE. Since the Dirac equation is linear in the vector potential, truncation after a particular n in Eq. (<ref>) will result in an implicitly included diamagnetic term that contains the x^2 n contributions from the square of the x^n term, but not the cross terms between higher- andlower-order terms that are also x^2 n contributions. As a consequence, the solution of the TDDE with only the lowest order spatial correction (n=1) to the vector potential generally gives meaningless results, as demonstrated in Ref. <cit.>. Furthermore, the convergence of the dynamics with respect to n_trunc in Eq. (<ref>) was shown to be very slow once the laser pulse parameters started to enter the relativistic regime. In the case where the electron was accelerated to a quiver velocity of v_quiv∼ 0.2c, a fifth order expansion was necessary for converged results. It is natural to assume that an even higher order expansion would be necessary further into the relativistic regime, and with each additional term x^n in Eq. (<ref>), the computational demand quickly turns this into an intractable problem. It is thus highly relevant to instead seek an alternative route less prone to grow so complex when the dynamics become increasingly relativistic. §.§ The relativistic propagation gaugeSince it is the actual way the diamagnetic contribution resurfaces in the TDDE that causes the convergence problems, it might be possible to find an alternative form where it is easier to balance the terms included in Eq. (<ref>). We are for instance free to make a gauge transformation to change the scalar field and vector potential as;𝐀→𝐀 + ∇ζ φ→φ - ∂ζ/∂ t,which will yield a transformed Hamiltonian:H = c α·[p +e A(η)+ e∇ζ]+ [ e∂ζ/∂ t - e φ(r) ] 1_4+ m c^2 β.In Ref. <cit.> it was shown that by choosing a gauge that followed the classical electron momentum in the direction of the light propagation, p_k, the diamagnetic term in the Schrödinger equation could be removed and replaced by operators that showed superior convergence properties. In the relativistic case, as shown in Refs. <cit.>, a free classical particle that is initially at rest will acquire the momentump_k(η) =mc/2( e A(η) /mc)^2,when exposed to the electromagnetic field given by A(η).It is natural to assume that a suitable gauge can be found if ζ is defined using Eq. (<ref>), but we start by defining it with an additional operator, ℵ(η), that remains to be determined:ζ( η) =- mc^2/e ω∫_-∞^η dη'1/2( e A(η') /mc)^2ℵ( η' ).The introduction of ℵ is related to the distinction between the relativistic and the non-relativistic version of the gauge transformation leading to the propagation gauge formulation <cit.>.We will return to its specific form in the following.With the vector potential polarized along the z axis and the field propagating along the x axis we obtaine∇ζ = - mc^2/ωx̂ ∂η/∂ xd/d η∫_-∞^η dη' 1/2( e A(η') /mc)^2ℵ( η')=+ x̂k mc^2/ω1/2( eA(η)/mc)^2ℵ( η) =x̂mc 1/2(e A(η)/mc)^2ℵ( η) ,where k=ω/c has been used in the last step. Equation (<ref>) is a vector operator in the propagation direction of the field. Further, withe∂ζ/∂ t =- m c^2 /2( e A(η) /mc)^2ℵ( η),we may now write down the propagation gauge Dirac Hamiltonian, H_ PG:H_ PG = c α·[p +e 𝐀(η) ]-e φ(r) 1_4+ m c^2 β + e^2 A^2(η)/2 mℵ( η)(α_x -1_4 ),where the first line is just the minimal coupling Dirac Hamiltonian from Eq. (<ref>). The second line, on the other hand, displays one operator proportional to α_x, the relativistic velocity operator in the direction of the propagation of the light, and one counter term. As we will see, this counter term cancels the implicit diamagnetic term contributed by the first line when the equation is examined in the non-relativistic limit.§.§ The non-relativistic limit of the relativistic propagation gaugeStarting again from Eq. (<ref>) but now adding the new terms from the second line in Eq. (<ref>), we will instead of Eq. (<ref>) find- ( φ + e^2 A^2 /2mℵ)Ψ_F + ( c σ·( 𝐩 + e 𝐀) + σ_x e^2 A^2 /2mℵ)Ψ_G = iħdΨ_F/d t ( c σ·(𝐩 + e 𝐀) + σ_x e^2 A^2 /2mℵ)Ψ_F - ( e φ +2mc^2 + e^2 A^2 /2mℵ)Ψ_G = iħdΨ_G/d t .Following the derivation preceding Eq. (<ref>), and assuming in addition that 2mc^2 dominates also over (e^2 A^2/2m) ℵ, we obtain a new approximate relation between the large and small component:Ψ_G ≈1/2mc( σ·( 𝐩 + e 𝐀)+ σ_x e^2 A^2 /2mcℵ) Ψ_F. Inserting this expression for Ψ_G into Eq. (<ref>) we find the propagation gauge Hamiltonian in the non-relativistic limit (cf. the expression for the minimal coupling Hamiltonian on the left-hand side of Eq. (<ref>));H_ PG^ NR = 𝐩^2/2m+ e/m𝐩·𝐀+e^2 A^2/2m-e ϕ+e ħ/2 mσ·𝐁+ 1/2 m c{e^2 A^2 /2mℵ,p_x} - e^2 A^2 /2m(ℵ- ℵ^2 e^2 A^2 /4 m^2 c^2),where {a,b} denotes an anticommutator. In addition to the original terms in Eq. (<ref>), two new terms have appeared on the last line of Eq. (<ref>). It is evident that if we put ℵ=1 the diamagnetic term is cancelled.However, another possibility is to requiree^2 A^2/2m - e^2 A^2 /2m(ℵ- ℵ^2 e^2 A^2 /4 m^2 c^2) = 0and thus get rid also of the term proportional to A^4. If Eq. (<ref>) is regarded as the defining equation for ℵ, we can readily write down its expression asℵ = 1 - √(1-(e A/mc)^2)/1/2(e A/mc)^2 .It is clear from Eq. (<ref>) that its range of validity is restricted to the region where(e A/mc)^2 < 1 ,which is consistent with the approximation made to obtain Eq. (<ref>). In this case we may also expand Eq. (<ref>) and findℵ = 1 +1/4(e𝐀/mc)^2 + 1/8(e𝐀/mc)^4 + 5/64(e𝐀/mc)^6 + … ,which in fact is the series that was found in Refs. <cit.>, i.e.,ℵ =∑_j=0^∞ 2 a_j+1(e𝐀/mc)^2jwitha_j=(2j)!/4^j (2j-1) (j!)^2 . With ℵ defined this way we may write Eq. (<ref>) asH_ PG^ NR= 𝐩^2/2m+ e 𝐩·𝐀/m -e ϕ+e ħ/2 mσ·𝐁+ 1/2 m c{e^2 A^2 /2mℵ,p_x},which, apart from the spin-dependent term, is identical to the propagation gauge Hamiltonian obtained directly from the TDSE in Refs. <cit.>.§.§ The long-wavelength approximation While a vector potential without spatial dependence does not introduce any magnetic interaction in the ordinary minimal coupling Dirac Hamiltonian, Eq. (<ref>), a purely time-dependent A does provide an additional dynamical term in Eq. (<ref>); the term proportional to α_x. Again this is in agreement with the findings in Refs. <cit.>; in the propagation gauge the radiation pressure is accounted for through a velocity gauge-like operator acting along the propagation direction of the laser in spite of a spatially independent 𝐀. The effective Dirac Hamiltonian in this long-wavelength approximation (LWA) is given byH_ LWA = c α·[p +e 𝐀(ω t) ]-e φ(r) 1_4+ m c^2 β+ e^2 A^2(ω t)/2 mℵ(ω t) α_x,where the terms that lack spatial dependence altogether have been removed since they do not affect the dynamics. In Sec. <ref> we will show that the Hamiltonian Eq. (<ref>) can account fully for the dominating effects beyond the dipole approximation for a wide range of electromagnetic pulses. It gives in fact excellent agreement with the much more demanding fifth order expansionof the Hamiltonian Eq. (<ref>), as applied in Ref. <cit.>.In principle we are free to choose either ℵ =1, to follow the relativistic momentum in the direction of the propagation of the laser light, or as given in Eq. (<ref>) to allow for a more straight forward comparison with the non-relativistic treatment. For high enough fields there will of course be differences for any non-exact implementation, as will be demonstrated in Sec. <ref>. Lastly, although the properties of the LWA-Hamiltonian Eq. (<ref>) are very promising, we want to emphasize that a practical implementation of Eq. (<ref>) is by no means restricted to the LWA-approximation. It is indeed possible to go further and introduce spatial dependence in 𝐀, which should become important for large enough field strengths E_0 and/or in the limit of very high laser frequencies. For illustrative purposes we present a first order beyond the LWA Hamiltonian in the next section and later demonstrate that it gives negligible contributions for the laser pulses considered in this article, which accelerate the electron to quiver velocities v_ quiv up to about 0.2c.§.§ Beyond the long-wavelength approximationWe introduce a spatial dependence in 𝐀 by using n_trunc=1 in Eq. (<ref>). Then, using ℵ=1, the first order beyond the long-wavelength approximation (BYLWA1) Hamiltonian can be written asH_ BYLWA1 = c α·[p +e 𝐀(ω t) ] -e α_z x ω A'(ω t)-e φ(r) 1_4+ m c^2 β+e^2/2m(x ω/c 2 A(ω t) A'(ω t) - x^2 ω^2/c^2 (A'(ω t))^2 )1_4 +e^2/2m(A^2(ω t)-x ω/c 2 A(ω t)A'(ω t))α_x .It may seem odd that the two terms proportional to A^2 in Eq. (<ref>) have been expanded differently.However, according to the discussion in Sec. <ref>, this is indeed the proper way of expandingthe field-dependent terms as this minimizes the problem with inconsistent terms appearing in the corresponding non-relativistic Hamiltonian. Note also that terms lacking spatial dependence altogether have been removed from Eq. (<ref>). In the continuation, we will demonstrate that the Hamiltonian H_ LWA in Eq. (<ref>) provides practically all dynamics for fields penetrating into the relativistic region, simply by comparing its results with the corresponding results obtained with H_ BYLWA1 as defined above. For clarity we emphasize that ℵ=1 has been used in both Hamiltonians for a just comparison. Before presenting our results we briefly describe our numerical implementation. § IMPLEMENTATION We expand the wave function in eigenstates of the time-independent Hamiltonian, i.e., Eq. (<ref>) without 𝐀, givingΨ(t) = ∑_n,j,m,κ c_n,j,m,κ( t ) ψ_n,j,m,κ( r),withψ_n,j,m,κ( r) = ( [ F_n,j,m,κ( r); G_n,j,m,κ( r) ]),where( [ F_n,j,m,κ( r); G_n,j,m,κ( r) ])= 1/r( [P_n,κ(r) X_κ,j,m(Ω); i Q_n,κ(r) X_-κ,j,m(Ω) ]).Here κ=l for j=l-1/2 and κ=-(l+1) for j=l+1/2. X_κ,j,m represents the spin-angular part which has the analytical form𝑋_κ, j, m = ∑_m_s,m_l⟨ l_κ,m_l ;s,m_s | j,m ⟩Y^l_κ_m_l (θ, ϕ) χ_m_s,where Y^l_κ_m_l (θ, ϕ) is a spherical harmonic and χ_m_s is an eigenspinor. The radial components P_n,κ(r) and Q_n,κ(r) are expanded in B-spline functions <cit.>;P_n,κ(r) = ∑_i a_i B_i^k_1(r),Q_n,κ(r) = ∑_j b_j B_j^k_2(r). Just as in Ref. <cit.> we use B-spline functions of orders k_1=7 and k_2 = 8 for the small and large components, respectively. As has been shown by Froese Fischer and Zatsarinny <cit.>, the use of different k for the two components effectively removes the so called spurious states which are known to appear when the Dirac equation is solved within a finite basis set. We use a linear knot sequence with 500 B-spline functions for the large component and 501 for the small component up to R_max = 150 a.u.. To avoid unphysical reflections at the box boundary we have used a complex absorbing potential starting from r = 110 a.u.. We include all spin-orbitals with angular momenta up to a certain l_max (as defined for the large component) and keep all the associated magnetic quantum numbers m, cf. Eqs. (<ref>, <ref>). To speed up the propagation without compromising the results, high energy components have been filtered out from the basis.In Sec. <ref> we present converged data for the energy distribution, the expectation value of the momentum operator along the pulse propagation direction and, finally, the total ionization yieldfrom the hydrogen ground state exposed to a 15 cycle, 95 eV (ω=3.5 a.u.) laser field of intensity 7× 10^19 W/cm^2 (E_0=45 eV). For the two former quantities, converged data were obtained with l_ max=30 for the propagation gauge LWA, cf. Eq (<ref>), corresponding to 1,902,594 states and about 2.32 × 10^10 non-zero matrix elements. In order to arrive at the same result with the minimal coupling Hamiltonian, Eq. (<ref>), it was necessary to include l_ max=50 for the case with n_trunc=5 in Eq. (<ref>). From now on we will refer to this level of approximationas fifth order beyond dipole (BYD5). The BYD5 simulation required 5,125,954 states and about 1.64 × 10^12 non-zero matrix elements, i.e., roughly 70 times more than our converged propagation gauge simulations.For the ionization yield, which was systematically investigated for both lower and higher values of E_0, convergence was always achieved with l_ max = 40 for both H_ LWA, Eq. (<ref>), and H_ BYLWA1, Eq. (<ref>). For further details on the implementation, such as how interaction matrix elements are computed and which numerical schemes that are applied, readers are referred to Ref. <cit.>.§ RESULTS We will first present results for the following scenario: A hydrogen atom with the electron initially prepared in the ground state is exposed to a laser pulse, defined in Eqs. (<ref> - <ref>), with the parameters[ E_0 = 45.0 ,ω = 3.5 ,ϕ = 0,; andT = N_c 2 πω withN_c = 15. ]The pulse parameters are such that the electron's quiver velocity is expected to reach about v_quiv∼ 0.1c and have been chosen primarily to demonstrate the convergence property of the relativistic propagation gauge – not to reveal relativistic effects per se. To show the convergence properties, the lowest order interaction in the propagation gauge, LWA, cf. Eq. (<ref>), has been compared to the minimal coupling Hamiltonians ranging from BYD1 to BYD5, that is with n_trunc=1-5 in Eq. (<ref>). Figure <ref> shows a comparison of the energy distribution of the ionized electron after interaction with the pulse. A somewhat typical convergence pattern for the minimal coupling simulations can be seen, where each successive interaction type pushes the distribution to either side of the fully converged result.Figure <ref> also shows the energy distribution but now only for the minimal coupling BYD5 resultand the propagation gauge LWA result. The energy grid has been extended to include the three first ionization peaks and a logarithmic scale is used to better resolve the data. The coincidence between the LWA result in the propagation gauge, which only involves purely time-dependent fields, and the result using a fifth order Taylor expansion of the vector field within the minimal coupling formulation Eq. (<ref>), is evident. In the non-relativistic version of the propagation gauge <cit.> an important demonstration of its computational advantages was the smooth evolution of the expectation value of the momentum in the propagation direction of the laser field. The same behaviour is found also for the corresponding relativistic results in Fig. <ref>, where violent oscillations seen using the minimal coupling Hamiltonian are transformed into a smooth development in the propagation gauge. Interestingly, to the naked eye, ⟨ p_x ⟩ seems to be converged already at BYD3 while the probability distribution clearly requires at least BYD5, as seen in Fig. <ref>.A comparison with non-relativistic simulations is also in order. As mentioned, the chosen pulse parameters result in an expected quiver velocity of v_quiv∼ 0.1c and only small relativistic corrections, if any, are expected. Figure <ref> shows a comparison of the relativistic and non-relativistic probability distributions, as obtained by solving both the TDSE in the propagation gauge LWA, cf. Ref. <cit.>, and the TDDE with the propagation gauge LWA Hamiltonain, Eq. (<ref>),using an equivalent basis set in both cases. Indeed, there are no relativistic effects displayed in Fig. <ref>, and one only expects these to appear at even higher intensities. In order to search for possible relativistic effects we now systematically increase the field strength up to about E_0 = 100 a.u., corresponding to I ∼ 3.5 × 10^20^2 and v_quiv∼ 0.2c. Figure <ref> shows the resulting ionization yield as a function of the electric field strength. The minimal coupling results from Ref. <cit.> (TDDE BYD5), therelativistic propagation gauge results obtained both within the LWA (Eq. (<ref>)) and beyond (Eq. (<ref>)), as well as the corresponding TDSE result, are shown for comparison. Again, the relativistic corrections seem to be very small. Nevertheless, a tiny relativistic shift manifested as a decrease in the ionization yield, is displayed as the quiver velocity approaches v_ quiv∼ 0.2c. Furthermore, Fig. <ref> shows that the favorable behavior of the LWA propagation gauge Hamiltonian persists over a wide range of intensities, up to the onset of the relativistic regime. Finally, going even further into the relativistic regime the important question of how to properly incorporate the full spatial dependence of the field in the propagation gauge Dirac Hamiltonian needs to be addressed. In an exact calculation, the choice of ℵ, be it simply ℵ=1 or as in Eq. (<ref>) or, equivalently, Eq. (<ref>), does of course not matter. However, with a truncated representation of the field, cf. Eq. (<ref>), this choice may be ofcrucial importance – and increasingly so for increasing field strengths. This dependence has been investigated and the first results are shown in Fig. <ref>. As in Fig. <ref>, ionization probabilities are shownfor both the TDSE and the TDDE, but here for field strengths E_0 in the interval 80-105 a.u. and only within the LWA. In going fromℵ as defined in Eq. (<ref>) to simply choosing ℵ=1, i.e., truncation at the first term in Eq. (<ref>), a small shift downwards is introduced – consistently both in the relativistic and non-relativistic treatments, respectively. Based on the present results, it is still unclear which of the choices for ℵ represent the best approximation, nor where the LWA approximation breaks down. However, the correspondence between the gauge transformation and the relativistic momentum in the propagation direction for a corresponding free, classical electron moving in the field, cf. Eq. (<ref>), suggests that simply ℵ=1 should be the best choice. Although we here leave these open questions for future research, simply due to the computational complexity of the problem, it should be noted that they could all be studied within the current computational framework.§ CONCLUSION We have presented a generalized velocity gauge form of the relativistic light-matter interaction and demonstrated its superior convergence properties compared to the regular minimum coupling Hamiltonian. As in the non-relativistic case, the alternative relativistic gauge relaxes the requirement on the maximum angular momentum needed during the time propagation. However, the major advantage goes even beyond that. While the usual minimal coupling formulation is numerically tough for high-intensity fields treated in a non-relativistic framework, it constitutes an intractable problem in the relativistic case due to inherent imbalance in the Dirac equation. The propagation gauge to a large extent removes this imbalance and opens up for calculations on atoms subjected to electromagnetic pulses in the truly relativistic regime.§ ACKNOWLEDGMENTSSimulations have been performed partly on resources provided by the Swedish National Infrastructure for Computing (SNIC) at TRIOLITH and KEBNEKAISE and partly on resources provided by the Norwegian Metacenter for Computational Science (UNINETT Sigma2) at HEXAGON (Account number NN9417K). Financial support by the Swedish Research Council (VR), Grant No. 2016-03789, is gratefully acknowledged. We also acknowledge support for our collaboration through the Nordic Institute for Theoretical Physics (Nordita) and from the research group Mathematical Modelling at Oslo and Akershus University College of Applied Sciences. 19 natexlab#1#1bibnamefont#1#1bibfnamefont#1#1citenamefont#1#1url<#>1urlprefixURL[Moore et al.(1999)Moore, Ting, McNaught, Qiu, Burris, and Sprangle]Moore1999 authorC. I. Moore, authorA. 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"authors": [
"Tor Kjellsson",
"Morten Førre",
"Aleksander Skjerlie Simonsen",
"Sølve Selstø",
"Eva Lindroth"
],
"categories": [
"physics.atom-ph"
],
"primary_category": "physics.atom-ph",
"published": "20170627210732",
"title": "Alternative gauge for the description of the light-matter interaction in a relativistic framework"
} |
On branching-point selection for trilinear monomials in spatial branch-and-bound: the hull relaxation[This work extends and presents parts of the first author's doctoral dissertation<cit.>, and itcorrects results first announced in the short abstract <cit.>.]Emily Speakman Jon Lee============================================================================================================================================================================================================================================================================empty Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. The worst-case hardness of SAT lies at the core ofcomputational complexity theory. The average-case analysis of SAT has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures.Despite a long line of research and substantial progress, nearly all theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a scale-free distribution of the variables, which results in distributions closer to industrial SAT instances.We study random k-SAT on n variables, m=Θ(n) clauses, and a power law distribution on the variable occurrences with exponent β. We observe a satisfiability threshold at β=(2k-1)/(k-1). This threshold is tight in the sense that instances with β≤(2k-1)/(k-1)-ε for any constant >0are unsatisfiable with high probability (). For β≥(2k-1)/(k-1)+ε, the picture is reminiscent of the uniform case: instances are satisfiable for sufficiently small constant clause-variable ratios m/n; they are unsatisfiable above a ratio m/n that depends on β.§ INTRODUCTIONSatisfiability of propositional formulas (SAT) is one of the most researched problems in theoretical computer science. SAT is widely used to model practical problems such as bounded model checking, hardware and software verification, automated planning and scheduling, and circuit design. Even large industrial instances with millions of variables can often be solved very efficiently by modern SAT solvers. The structure of these industrial SAT instances appears to allow a much faster processing than the theoretical worst-case of this NP-complete problem. It is an open and widely discussed question which structural properties make a SAT instance easy to solve for modern SAT solvers.Random SAT: For modeling typical inputs, we study random propositional formulas. In random satisfiability, we have a distribution over Boolean formulas in conjunctive normal form (CNF). The degree of a variable in a CNF formula is the number of disjunctive clauses in which that variable appears either positively or negatively. Two interesting properties of random models are its degree distribution and its satisfiability threshold. The degree distribution F(x) of a formula Φ is the fraction of variables that occur more than x times (negated or unnegated). A satisfiability threshold is a critical value around which the probability that a formula is satisfiable changes from 0 to 1.Uniform random SAT: In the classical uniform random model, the degree distribution is binomial. On uniform random k-SAT, the satisfiability threshold conjecture <cit.> asserts if Φ is a formula drawn uniformly at random from the set of all k-CNF formulas with n variables and m clauses, there exists a real number r_k such thatlim_n →∞{Φ is satisfiable} =1 m/n < r_k;0 m/n > r_k.A well-known result of Friedgut <cit.> establishes that the transition is sharp, even though its location is not known exactly for all values of k (and may also depend on n). For k=2, the critical threshold is r_2 = 1 <cit.>. Recently, Coja-Oghlan and Panagiotou <cit.> gave a sharp bound (up to lower order terms) withr_k = 2^k log 2 - 12 (1 + log 2) ± o_k(1). Ding, Sly, and Sun <cit.> derive an exact representation of the threshold for all k≥ k_0, where k_0 is a large enough constant. Explicit bounds also exist for low values of k, e.g., 3.52 ≤ r_3 ≤4.4898 <cit.>, and numerical estimates using the cavity method from statistical mechanics <cit.> suggest that r_3 ≈ 4.26.Other random SAT models: In the regular random model <cit.>, formulas are constructed at random, but the degree distribution is fixed: each literal appears exactly ⌊km/2n⌋ or ⌊km/2n⌋ + 1 times in the formula. Similarly, Bradonjic and Perkins <cit.> considered a random geometric k-SAT model in which 2n points are placed at random in [0,1]^d. Each point corresponds to a unique literal, and clauses are formed by all k-sets of literals that lie together within a ball of diameter Θ(n^-1/d). Again, this model has a binomial variable distribution.Power law random SAT: Recently, there has been a paradigm shift when modeling real-world data. In many applications, it has been found that certain quantities do not cluster around a specific scale as suggested by a uniform distribution, but are rather inhomogeneous <cit.>. In particular, the degree distribution in complex networks often follows a power law <cit.>. This means that the fraction of vertices of degree k is proportional to k^-β, where the constant β depends on the network. To mathematically study the behavior of such networks, random graph models that generate a power law degree distribution have been proposed <cit.>. While there has been a large amount of research on power law random graphs in the past few years <cit.>, there is little previous work on power law SAT formulas. Nevertheless, the observation that quantities follow a power law in real-world data has also emerged in the context of SAT <cit.>. As all aforementioned random SAT models assume strongly concentrated degree distributions, it was conjectured that this property might be modeled well by random formulas with a power law degree distribution.To address this conjecture, and to help close the gap between the structure of uniform random and industrial instances, Ansótegui, Bonet, and Levy <cit.> recently proposed a power-law random SAT model. This model has been studied experimentally <cit.>, and empirical investigations found that (1) indeed the constraint graphs of many families of industrial instances obey a power-law and (2) SAT solvers that are constructed to specialize on industrial instances perform better on power-law formulas than on uniform random formulas.To complement these experimental findings, we contribute with this paper the first theoretical results on this model.Our results: We study random k-SAT on n variables and m=Θ(n) clauses. Each clause contains k= Θ(1) different, independently sampled variables.Each variable x_i is chosen with non-uniform probability p_i and negated with probability 1/2. A formal definition can be found in Section <ref>. We first study sufficient conditions under which the resulting k-SAT instances are unsatisfiable.Assume a probability distribution p⃗ on the variables where p_i is non-decreasing in i∈{1,…, n}. If the k most frequent variables are sufficiently common, we prove in unsat the following statement: theoremstatethmunsatLet Φ be a random k-SAT formula with probability distribution p⃗ on the variables (c.f. randomSAT), with k≥2 and mn = Ω(1). If p_n-k+1 = Ω( (log nn)^1/k), then Φ isunsatisfiable.Our focus are power law distributions with some exponent β. unsat implies that power law random k-SAT formulas with β=2k-1k-1 - for an arbitrary constant ε > 0 are unsatisfiable with high probability[We say that an event E holds , if there exists a δ > 0 such that [E] ≥ 1- (n^-δ).], cf. unsat.In Section <ref> we show that something similar holds for the clause-variable ratio mn, i.e. power law random k-SAT formulas with mn bigger than some constant are unsatisfiable with high probability. Although this already follows from basic observations, we derive a better bound on the value of the constant. theoremstatethmflip Let Φ be a random k-SAT formula with probability distribution p⃗ on the variables (c.f. randomSAT), with k≥2 and r= mn. With high probability, Φ is unsatisfiable if(1-12^k)^r[∏_i=1^n[2-(1-k· p_i/2^k-11/(1-12 k^2 p⃗^2_2))^m]]^1/n<1. In sat we prove the following positive result, which complements our picture of the satisfiability landscape: theoremstatethmsatLet Φ be a random k-SAT formula whose variable probabilities follow a power law distribution (c.f. general). If the power law exponent isfor an arbitrary >0, Φ is satisfiable with high probability if mn is a small enough constant.Together our main theorems prove that random k-SAT instances whose variables follow power law distributions do not only exhibit a phase transition for some clause-variable ratio r= mn, but also around the power law exponent β=2k-1k-1. illustration contains an overview of our results. To prove these statements, we borrow tools developed for the uniform random SAT model. Note, however, that many of their common techniques like the differential equation method seem difficult to applyto non-uniform distributions; as removing a variable results in a more complex rescaling of the rest of the distribution. It is therefore crucial to perform careful operations on the formulas that leave the distribution of variables intact. To this end, we use techniques known from the analysis of power law random graphs.Clause length: We focus on power law variable distributions but fix the length of every clause to k ≥ 2. Power law models have also been proposed in which clause length is distributed by a power law as well <cit.>. As long as there is a constant minimum clause length k_min≥ 2, our results can be extended to this case in the following way.If the clause lengths are distributed as a power law, there will appear Θ(n) clauses of length k_min, and all other clauses are of larger size. In that case, unsatsat are directly applicable to the linear number of clauses with size k_min (obtaining different hidden constants); and we have that the formula is satisfiable with high probability if β≥2k_min - 1k_min -1 + and m/n is a small enough constant. On the other hand, the formula is unsatisfiable with high probability, ifβ≤2k_min - 1k_min -1 -. Consequently, the satisfiability of the formula does (asymptotically) not depend on the second power law. § DEFINITION OF THE MODEL AND PRELIMINARIES We analyze random k-SAT on n variables and m=Θ(n) clauses, where k ≥ 2. The constant r :=mn is called clause-variable ratio or constraint density. We denote by x_1, …, x_n the Boolean variables. A clause is a disjunction of k literals ℓ_1 …ℓ_k, where each literal assumes a (possibly negated) variable. Finally, a formula Φ in conjunctive normal form is a conjunction of clauses c_1 … c_m. We conveniently interpret a clause c both as a Boolean formula and as a set of literals. Following standard notation, we write |ℓ| to refer to the indicator of the variable corresponding to literal ℓ. We say that Φ is satisfiable if there exists an assignment of variables x_1, …, x_n such that the formula evaluates to 1. Let m,n be given, and consider any probability distribution p⃗ on n variables with ∑_i=1^n p_i = 1. To construct a random SAT formula Φ, we sample m clauses independently at random. Each clause is sampled as follows: * Select k variables independently at random from the distribution p⃗. Repeat until no variables coincide. * Negate each of the k variables independently at random with probability 1/2.Observe that by setting p_i =1n for all i, we obtain again the uniform random SAT model.The probability to draw a specific clause c is∏_ℓ∈ c p_|ℓ|/2^k∑_J∈𝒫_k({1,2,…,n})∏_j∈Jp_j,where 𝒫_k(·) denotes the set of cardinality-k elements of the power set. The factor 2^k in the denominator comes from the different possibilities to negate variables. Note that k!∑_J∈𝒫_k({1,2,…,n})∏_j∈Jp_j is the probability of choosing a k-clause that contains no variable more than once.To see that this probability is almost 1 for most distributions, we apply the following result from <cit.>. Let p⃗ = (p_1, …, p_n) be any probability distribution on n items. Assume you sample t items from p⃗. Let ℰ(t) be the event that there is a collision, i.e. that at least 2 of t items are equal. Then,[ℰ(t)] ≤12 t^2 p⃗^2_2 = 12 t^2 ∑_i=1^n p_i^2. The probability that a sampled k-clause thereby has collisions is at most 12 k^2 p⃗_2^2; so for p⃗_2^2 = o(1) and constant k we obtain that the probability to draw a specific clause c is(1+o(1))k!/2^k∏_ℓ∈ c p_|ℓ|. Power law Distributions. In this paper, we are mostly concerned with distributions p_i that follow a power law. To this end, we define two models: A general model to capture most power law distributions (which is harder to analyze), and a concrete model that gives us one instance of p⃗ depending only on n that can be used to compute precise leading constants. We use the general model to derive some asymptotic results; and the concrete model to compare with the uniform random SAT model and for the experiments.Before we define these two models, let us establish the concept of a weight w_i of a variable x_i. The weight gives us (roughly) the expected number of times the variable appears in the formula. That is,p_i := w_i/∑_j w_j.Thus, fixing the weights w⃗ = (w_1, …, w_n) also fixes the probability distribution p⃗.It is important to distinguish between the initial distribution of variables p⃗ and modified distributions that may arise as a result of stochastic considerations. For instance, the smallest-weight variable in a clause is clearly not distributed according to p⃗ (except in 1-SAT). To avoid confusion, we identify a variable with its weight, as the weights stay fixed throughout the analysis. For convenience, we further assume that the variables are ordered increasingly by weight, for i ≤ j we have w_i ≤ w_j. Note that our definition of power law ensures that for β>2, we have ∑_j w_j = Θ(n).We are now ready to define the two models. Let the weights w⃗ := w_1, …, w_n be given, and let W be a weight selected uniformly at random. We say that w⃗ follows a power law with exponent β, if w_1 = Θ(1), w_n = Θ(n^1/β-1), and for all w ∈ [w_1, w_n] it holdsF(w) := [W ≥ w] = Θ(w^1-β) Whenever we need the explicit constants bounding the distribution function, we refer to them by α_1, α_2 as inα_1 w^1-β≤ F(w) ≤α_2 w^1-β.We point out that general assumes a deterministic weight sequence; but it can be easily generalized to also support randomly generated weights.For the concrete model, we define the weights as follows. Given a power law exponent β, we call w⃗ the concrete power law sequence, if w_n-i+1 := ( ni)^1/β-1. One can check that for these concrete weights, it holds n · F(w) = ⌊ nw^1 -β⌋, so in a sense, they are a canonical choice for producing a power law weight distribution.To analyze power law distributions, we often make use of the following result ofBringmann, Keusch, and Lengler <cit.>, which allows replacing sums by integrals. Let fℝ→ℝ be a continuously differentiable function, and let F^>(w) := [W > w]. Then, for any 0 ≤w≤w̅,∑_i∈ [n], w≤ w_i ≤w̅1n f(w_i) = f(w) · F(w)- f(w̅) ·F^>(w̅) + ∫_w^w̅ f'(w) · F(w)w. Using this theorem, the following corollary can be shown: Let the variables w_i be power law distributed with exponent β > 2, and define W_≥ w := ∑_i∈[n] w_i ≥ w w_i.Then, W_≥ w = Θ( nw^2-β). Observe that ∑_w' ≥ w w' = ∑_i ∈ [n], w_i ≥ w w_i.We apply karl to obtain1n ∑_i ∈ [n], w_i ≥ w w_i= w · F(w) + ∫_w^w_n F(v)v ≤α_2 w^2-β + [α_22-β v^2-β]_w^w_n≤α_2β-1β-2 w^2-β.In a similar fashion, one may show that 1n ∑_i ∈ [n], w_i ≥ w w_i ≥α_1β-1β-2(1-o(1)) w^2-β. Hence, ∑_j w_j = W_≥ w_1 = Θ(n) and therefore p_i = Θ(w_in). Finally, we denote by V the random variable describing the weight of a SAT variable chosen according to a power law distribution p_i, that is, [V = w] = ∑_i p_i ·1[w_i = w], where 1 denotes the indicator variable of the event. Note that this is not equivalent to W, since there is a subtle difference in the two random processes: W is a random variable drawn uniformly at random from w_1, …, w_n, whereas V is a random variable drawn from the same set, but with the non-uniform distribution p_1, …, p_n. Hence, by sumweights,[V ≥ w] = Θ(w^2-β) . Using karl, we can show that the probability to draw a certain clause c is as given by clausesample for general with exponent β>2, sincep⃗^2_2 = ∑_i=1^n p_i^2 = Θ(n^-2) ∑_i=1^n w_i^2 = Θ(n^-1) · n^3-β/β-1 = o(1). It remains to show that using a power law distribution in randomSAT indeed results in a power law distribution of variable occurrences. Ansótegui et al. <cit.> provide a proof sketch for this fact, we prove it rigorously. Let Φ be a random k-SAT formula that follows an arbitrary power law distribution with exponent β (c.f. general) and m=Θ(n). Then, there areand d_max=Θ(w_max), such that for all d_min≤ d≤ d_max it holds thatN_≥ d=Θ(n· d^1-β), where N_≥ d is the number of variables that appear at least d times in Φ. Let f_x be the number of appearances of x or x̅ in Φ. Observe that f_x≤ k· m · p_x, since each variable can appear at most once in a clause. On the other hand, it holds f_x≥ m · p_x, since this is the expected number of appearances of x in a 1-SAT formula. Thus, since m = Θ(n) and k=Θ(1) by assumption, it holds thatf_x=Θ(w_x). We first prove the statement for d≥ 2cln n, where c>0 is some suitable constant. By applying Chernoff bounds, we can derive that all variables x with f_x<d 2 appear fewer than d times; and all variables x with f_x≥ 2d appear at least d times. The requirement d_min≤ d≤ d_max is needed so that the Chernoff bounds work, which might not be the case if d is too close to w_min or w_max. Due to general and exp-freq this impliesN_≥ d = Θ(n· d^1-β).Now let us consider the case d< 2cln n and let Y_i be random variables indicating if f_x_i≥ d for i=1,2,…,n. To show a lower bound on N_≥ d, we again look at variables x with f_x≥2d.For those it holds that(Y_i=0)≤(f_x_i < 1/2f_x_i)≤ e^-f_x_i/8≤ e^-d/4,again due to Chernoff Bounds. Also, by exp-freq, it holds for variables x with f_x≥2d that w_x=Ω(2d). By the requirements on the weight distribution from general there are Θ(n· d^1-β) such variables. Therefore, it holds thatN_≥ d≥∑_i∈[n] f_x_i≥2dY_i≥ c'(1-e^-d/4)· n · d^1-βfor a suitable constant c'>0. Observe that if we condition on Y_i = 1, that x_i appears at least d times, this lowers the probability of all other variables to appear d times, and vice versa. Thus, the random variables Y_1,…,Y_n are negatively correlated and we may apply a Chernoff bound <cit.>. Since 1 - e^-d/4 = Ω(1) and d = (log n), we obtain thatN_≥ d≥1/2 c'·(1-e^-d/4)· n · d^1-β = Ω(n· d^1-β). To show an upper bound on N_≥ d we consider variables x with f_x≤ d 2e of which there are n· (1 -Θ(d^1-β)) due to general. For these variables, by a Chernoff bound <cit.> it holds that(Y_i=1)≤(f_x_i > 2e·f_x_i)≤ 2^-d.Now let N'_≥ d be the number of variables with f_x≤d/2e and f_x≥ d. Thus, there exists a constant c” > 0 such thatN'_≥ d≤ n· 2^-d(1- c”· d^1-β).Due to negative association of the Y_i's we can again use a Chernoff bound, yielding that ,N'_≥ d≤ n·2^1-d(1- c”· d^1-β).If N'_≥ d is very small, for example N'_≥ d=(log n), then we can use negative association to apply the Chernoff bound with t>2e·N'_≥ d to achieve N'-chernoff with high probability, since t=n· d^1-β=Ω(n/polylog(n)). Observe that 1 - c” d^1-β = (1). Furthermore, for variables x with f_x> d/2e, we pessimistically assume f_x≥ d. This gives usN_≥ d = (n· (d^1-β+2^-d)) = (n· d^1-β),since 2^-d=(d^1-β) for constant β. § SMALL POWER LAW EXPONENTS ARE UNSATISFIABLE For small power law exponents, one can show that they result in formulas that are unsatisfiable (for large n) for all constant clause-variable ratios. The rationale behind this is that large variables with weight Θ(w_n) appear polynomially often together in a clause. For constant k, they thus appear in all 2^k configurations (negated and non-negated), making the formula trivially unsatisfiable. Theorem <ref>, already stated in the introduction, gives a sufficient condition on the variable distribution to make a random k-SAT formula unsatisfiable.Recall that p_i is without loss of generality increasing in i. Consider the k largest variables n-k+1, …, n. We call ℰ_i the event that clause i consists of these variables. Then,[ℰ_i] = Ω(p_n-k+1^k) = Ω(log nn).Since each clause is drawn independently at random, we obtain by a Chernoff bound (see for example Theorem 1.1 in <cit.>) that with high probability, the total number of clauses consisting of these variables is|ℰ| := ∑_i=1^m1[ℰ_i] = Ω(log n).In other words, the number of clauses in which the k largest variables appear together increases as a logarithm in n. Since in each of these clauses, the literals appear negated or non-negated with constant probability 1/2, we have that all 2^k possible combinations of negated and non-negated literals appear in the formula with probability at least1 - 2^k · (2^k-12^k)^|ℰ| = 1 - n^-Ω(1)by the union bound. Since all 2^k combinations cannot be satisfied at once, the resulting formula is unsatisfiable.By applying unsat to a power law distribution on the variables, we obtain the following power law threshold for unsatisfiability. Let Φ be a random k-SAT formula that follows an arbitrary power law distribution fulfilling general. If the power law exponent is β≤2k-1k-1 - for an arbitrary >0, Φ is unsatisfiable with high probability. Observe that from β = 2k-1k-1 - it follows k = β-1β-2 - ' for some constant '. By setting nF(w) ≤ k we obtain that the largest k variables all have weightΘ(w_n) = Θ(n^1/β-1).Consequently, when β > 2,(p_n-k)^k = Θ(n^-kβ-2/β-1) = Θ(n^-1+ε'β-2/β-1) = ω(log nn),and the statement follows from unsat. For the case where β≤ 2, one can show using karl that ∑_i w_i = Θ(n^1/β-1), and therefore p_n-k = Ω(1). Again, the statement follows from unsat. § LARGE CLAUSE-VARIABLE RATIOS ARE UNSATISFIABLEIt is a well-known result that random SAT on any probability distribution will result in unsatisfiable formulas if the clause-variable ratio is high. This follows from the probabilistic method: The expected number of assignments that satisfy a formula is 2^n (1-2^-k)^m. This is independent from the variable distribution as long as each variable is negated with probability 1/2. Hence, if the clause-variable ratio exceeds ln(2)/ln(2^k2^k-1), the resulting formula will be unsatisfiable with high probability. This constant is rather large, however: In the case of k=3 this yields an upper bound on the clause-variable ratio of ≈ 5.191. For the concrete power law distribution in concrete, the true threshold is much smaller. In fact, it appears to be below the satisfiability threshold for uniform random SAT.Let us restate the main result, which will be proven with the Single Flip Method <cit.>. The following is a corollary from this theorem:Let Φ be a random k-SAT formula that follows randomSAT with k≥2, r= mn and p⃗^2_2=o(1). With high probability, Φ is unsatisfiable if(1-12^k)^r(2-exp(-(k/2^k-1r)(1+o(1))))<1.We can upper-bound the left-hand side of the inequality as follows(1-12^k)^r[∏_i=1^n[2-(1-k· p_i/2^k-11/(1-12 k^2 p⃗^2_2))^m]]^1/n ≤(1-12^k)^r[1/n∑_i=1^n[2-(1-k· p_i/2^k-11/(1-12 k^2 p⃗^2_2))^m]] =(1-12^k)^r[2-1/n∑_i=1^n(1-k· p_i/2^k-11/(1-12 k^2 p⃗^2_2))^m]by applying the inequality of arithmetic and geometric means. Since k· p_i/2^k-11/(1-12 k^2 p⃗^2_2) is upper-bounding a probability, we can assume it to be at most 1. It now holds that(1-k· p_i/2^k-11/(1-12 k^2 p⃗^2_2))^m > exp(-m/(2^k-1)(1-12 k^2 p⃗^2_2)/k· p_i-1) = exp(-k· p_i/2^k-1m/(1-12 k^2 p⃗^2_2)(1+o(1))),since p⃗^2_2=o(1) implies max_i(p_i)=o(1). By plugging this into the inequality from before and applying the inequality of arithmetic and geometric means again, we get(1-12^k)^r[∏_i=1^n[2-(1-k· p_i/2^k-11/(1-12 k^2 p⃗^2_2))^m]]^1/n ≤(1-12^k)^r[2-1/n∑_i=1^nexp(-(k· p_i/2^k-1m/(1-12 k^2 p⃗^2_2))(1+o(1)))]≤(1-12^k)^r[2-(∏_i=1^nexp(-(k· p_i/2^k-1m/(1-12 k^2 p⃗^2_2))(1+o(1))))^1/n] = (1-12^k)^r[2-exp(-(k/2^k-1r/(1-12 k^2 p⃗^2_2))(1+o(1)))].For p⃗^2_2=o(1) this is roughly(1-12^k)^r(2-exp(-(k/2^k-1r)(1+o(1)))).Interestingly, the above corollary gives the same inequality as the Single-Flip Method for uniform random SAT <cit.>. This shows that the uniform distribution resembles a worst-case for this method; and all other distributions can only improve this bound.If p⃗ follows a power law distribution as in concrete, we can derive the following theorem, which gives an upper bound independent of n.Let Φ be a random k-SAT formula with k≥2 and r= mn that follows a power law distribution fulfilling concrete. Let further N∈ℕ^+ be any constant. If the power law exponent is β>2, then Φ isunsatisfiable if ((1-12^k)^r 2^1/N∏_l=1^N-1[2-exp(-(1+o(1))rk/2^k-1β-2/β-1(N/l)^1/β-1)]^1/N)<1.We apply singleflip-general. If p⃗ follows a power law distribution as in concrete, we can further simplify N_SF toN_SF ≤(1-12^k)^m∏_i=1^n[2-exp(-k· p_i/2^k-1m/(1-12 k^2 p⃗^2_2)-k· p_i/2^k-1)]using the inequality 1-x≥ e^-x/1-x which holds for all x<1. We upper bound the probabilities p_i by choosing an integer N≥2 and dividing the set of variables into N buckets of equal size. For i∈[⌈l-1/N n⌉+ 1, ⌈l/Nn⌉] and 1≤ l≤ N-1 we can estimatep_i ≤(n/n-⌈l/Nn⌉+1)^1/β-1/∑_i=1^n(n/i)^1/β-1 = (N/N-l)^1/β-1/∑_i=1^n(n/i)^1/β-1.The last bucket i∈[⌈N-1/N n⌉+ 1, n] is simply bounded by 2^n/N in the overall product. W.l.o.g. we assume that there are exactly n/N variables in each bucket, aswe could split the factor for ⌈l/Nn⌉ into appropriate parts which both obey the upper bound on p_i for bucket l and l+1 respectively. We can now upper-bound N_SF by(1-12^k)^m 2^n/N∏_l=1^N-1[2-exp(-k/2^k-1· m·(N/l)^1/β-1/(∑_i=1^n(n/i)^1/β-1)(1-12 k^2 p⃗^2_2)-k/2^k-1(N/l)^1/β-1)]^n/NWe are now interested in what happens to the expression in the exponent when n tends to infinity. First, ∑_i=1^n(ni)^1/β-1→β-1β-2n for β>2. Second, by karl we have that p⃗^j_j = Θ(n^-jβ-2/β-1) → 0 whenever j > β-1 and p⃗^j_j = Θ(n^-j+1) → 0 whenever j≤β-1. Finally, for every constant N we have that k/2^k-1(Nl)^1/β-1 is also constant. Using m=r· n we can thus simplifyk/2^k-1· m·(N/l)^1/β-1/(∑_i=1^n(n/i)^1/β-1)(1-12 k^2 p⃗^2_2)-k/2^k-1(N/l)^1/β-1 = (1+o(1))rk/2^k-1β-2/β-1(N/l)^1/β-1.Plugging this into our inequality we getN_SF ≤(1-12^k)^m 2^n/N∏_l=1^N-1[2-exp(-(1+o(1))rk/2^k-1β-2/β-1(N/l)^1/β-1)]^n/N = ((1-12^k)^r 2^1/N∏_l=1^N-1[2-exp(-(1+o(1))rk/2^k-1β-2/β-1(N/l)^1/β-1)]^1/N)^nThis establishes singleflip-powerlaw. The bound from this Theorem improves as N→∞. As this expression is rather terse, we also numerically determine in flip the smallest constant r such that the formula is unsatisfiable.We compare these values to the upper bounds for uniform random SAT obtained from the Single-Flip Method.In the remainder of this section, we show singleflip-general and singleflip-powerlaw.For a random formula Φ a truth assignment A has the single-flip property iff A satisfies Φ and every assignment A' obtained from A by flipping exactly one zero to one does not satisfy Φ. Let N_SF be the number of truth assignments with the single-flip property for Φ. As argued in <cit.>, such an assignment exists if Φ is satisfiable. From Markov's Inequality, we thus know[Φ satisfiable] ≤N_SF. In the following, we derive a bound on N_SF. By birthday, the probability of choosing a clause c is at mostk!/2^k·∏_ℓ∈ cp_|ℓ|/1-12 k^2 p⃗^2_2.To bound the number of assignments with the single-flip property, we use the following result. The expected number of assignments with the single-flip property isN_SF=(1-12^k)^m∑_assignment AA single-flip| A satisfying. Note that for a certain truth assignment A, the probability of choosing a clause which is not satisfied by A is 1/2^k. Therefore, the probability that A is a satisfying assignment for Φ is exactly (1-12^k)^m.We next bound the probability that a satisfying assignment A has the single-flip property. For a satisfying assignment A=(a_1, a_2, …, a_n)∈{0,1}^n it holds that A single-flip | A satisfying≤∏_i: a_i=01-(1-k· p_i/2^k-11/(1-12 k^2 p⃗^2_2))^m. For a satisfying assignment A to have the single-flip property, all assignments A^i obtained by flipping a bit a_i=0 of A must not satisfy Φ. To fulfill this property for A^i, we have to choose at least one clause which contains X̅_̅i̅ and k-1 other variables with appropriate signs so that A^i does not satisfy the clause. Let S^i (c) denote the event that a clause c is satisfied by A, but not by A^i. Then,[S^i(c)]= k!· p_i∑_J∈𝒫_k-1([n]∖{i})∏_j∈ Jp_j/2^k(1-12 k^2 p⃗^2_2)≤k· p_i/2^k(1-12 k^2 p⃗^2_2)since ∑_J∈𝒫_k-1([n]∖{i})∏_j∈ Jp_j≤p⃗^k-1_1/(k-1)!. The probability of choosing a clause not satisfied by A^i under the condition that A is satisfying is then[S^i(c) | Asat] = [S^i(c) | Asatisfiesc]≤k· p_i/2^k-11/(1-12 k^2 p⃗^2_2)as the probability of choosing a clause which is satisfied by any assignment is exactly 2^k-1/2^k. For a fixed assignment A^i we concludeA^i unsat | A sat = 1 -( 1 - [S^i(c) | Asat] )^m ≤ 1-(1-k· p_i/2^k-11/(1-12 k^2 p⃗^2_2))^m.It remains to find the joint probability that all single-flipped assignments A^i for 1≤ i ≤ n with a_i=0 are not satisfying. We show this using a correlation inequality by Farr <cit.>. The sets of clauses which are not satisfied by the A^i's are pairwise disjoint as each clause in the set for A^i has to contain X̅_̅i̅, whereas each clause in the set for A^j (j≠ i) can not contain X̅_̅i̅. In the context of the correlation inequality from <cit.> we set V={1,2,…,m}, I={i∈{1,2,…,n} | a_i=0}, X_v=i iff the v-th clause is satisfied by A, but not by A^i, and ℱ_i the “increasing” collection of non-empty subsets of V. The application of the Theorem then directly yields[Asingle-flip| Asat]= ⋂_i: a_i=0 A^i unsat | A sat≤∏_i: a_i=0[1-(1-k· p_i/2^k-11/(1-12 k^2 p⃗^2_2))^m].Combining expectation1conditional we get that the expected number of assignments with single-flip property is at mostN_SF ≤ (1-12^k)^m∑_I⊆{1,2,…,n}∏_i∈ I[1-(1-k· p_i/2^k-11/(1-12 k^2 p⃗^2_2))^m] =(1-12^k)^m∏_i=1^n[2-(1-k· p_i/2^k-11/(1-12 k^2 p⃗^2_2))^m].This establishes singleflip-general.§ CONDITIONS FOR SATISFIABILITYIn this section, we provide a complementary result to unsatsingleflip-powerlaw proving that ifand the clause-variable ratio r =mn does not exceed some small constant, then a random k-SAT formula with exponent β is satisfiable with high probability. Let us first restate the main result: We show this statement by constructing an algorithm that satisfies Φif the clause-variable ratio is small. simple contains a formal description. The main idea is to shrink all clauses to size 2 by selecting the literals with smallest weight in each clause; and then running any well-known (polynomial time) 2-SAT algorithm ( <cit.>). In the following, we seek to establish that simple will find a satisfying assignment (for small constraint densities) with high probability. To this end, we first analyze the probability distribution of a clause c after it has been shrunk.Let ℓ_1, ℓ_2 be the selected literals of an arbitrary clause c ∈Φ in simple. Then, [|ℓ_1| = i, |ℓ_2|=j] + [|ℓ_1| = j, |ℓ_2|=i] ≤(1n^2 (w_iw_j)^1-1/2(k-2)(β-2))., we assume that w_i ≤ w_j. Then, [|ℓ_1| = j, |ℓ_2| = i] = 0 by the definition of simple. For the event |ℓ_1| = i, |ℓ_2|=j to happen, all other k-2 literals in the clause must be of larger weight. By clausesamplevarsample,[|ℓ_1| = i, |ℓ_2| = j]= 1/2·k2· (1 + o(1))· p_i · p_j ·[V ≥ w_j]^k-2= Θ(1n^2) · w_iw_j^1- (k-2)(β-2)≤(1n^2) ·(w_iw_j)^1-1/2(k-2)(β-2).The last statement holds since w_i ≤ w_j.Having derived a bound on the probability distribution of a shrunk clause, it is possible to compute the probability that the resulting 2-SAT formula is satisfiable. We use that the clauses are sampled independently. To avoid confusion, we write Φ' and c', whenever we talk about the shrunk formula and clauses. To upper bound the probability of Φ not being satisfiable, we look at so-called bi-cycles in Φ'. A bi-cycle of length l is a sequence of l+1 clauses of the form (u,ℓ_1),(ℓ̅_1,ℓ_2),…,(ℓ̅_l-1,ℓ_l),(ℓ̅_l,v),where ℓ_1,…,ℓ_l are literals of distinct variables and u,v∈{ℓ_1,…,ℓ_l,ℓ̅_1,…,ℓ̅_l}.Chvatal and Reed <cit.> show that if the formula Φ' is unsatisfiable, it must contain a bi-cycle. Consequently, by upper bounding the probability that a bi-cycle appears, we immediately obtain an upper bound on the probability that Φ' and henceforth Φ is unsatisfiable. Let Φ' be any 2-SAT formula. If Φ' contains no bi-cycle, it is satisfiable.Before we are able to prove the main Theorem, we need the following auxiliary Lemma.Let β = δ + 1 + for some > 0. For all 1 ≤ l ≤ n, there is a constant c with ∑_S⊆[n] |S|=l∏_i∈ Sw_i^δ≤ n^l· c^l1l!.We begin by observing that the term on the left side of the equation is obviously monotone in w_i: If δ≥ 0 (δ < 0), then increasing (decreasing) w_i increases the sum. Thus, instead of considering the true distribution function F(w), we may consider the upper (lower) bound on F(w), see sandwhich. For the sake of brevity, we consider the distribution F(w) = α w^1-β, where α is chosen to be either α_1 if δ≥ 0, or α_2 otherwise.To estimate this sum, we arrange the elements of S increasingly by weight, such that w_s_1 < w_s_2 < … < w_s_l. This gives us∑_S⊆[n]|S|=l∏_i∈ Sw_i^δ = ∑_s_1=1^n-l+1(w_s_1^δ∑_s_2=s_1+1^n-l+2(w_s_2^δ…∑_s_l=s_l-1+1^nw_s_l^δ)).We are now inductively estimating these sums, beginning with the innermost. Recall that F(w) = α w^1-β. Let d be a large enough constant. We establish the following induction hypothesis:∑_s_l-i = s_l-i-1 + 1 ^ n-i (w_s_l-i^δ∑_s_l-i+1= s_l-i + 1^n-i+1(w_s_l-i+1^δ…∑_s_l=s_l-1+1^nw_s_l^δ)) ≤n^i+1· w_s_l-i-1^(i+1) · (δ+1-β)· d^i+11(i+1)!Now we apply karl to prove the induction basis. For i=0, we have∑_s_l=s_l-1+1^nw_s_l^δ ≤ n ·α w_s_l-1^δ +1-β + n ∫_w_s_l-1^w_nαδ w^δ-β w≤ n ·α w_s_l-1^δ +1-β + n ·αδβ-δ-1 w_s_l-1^δ+1-β= n ·α(β-1)β-δ-1 w_s_l-1^δ +1-βas desired, since β>δ +1.Now suppose the induction hypothesis holds for i-1. For i we get∑_s_l-i = s_l-i-1 + 1 ^ n-i (w_s_l-i^δ∑_s_l-i+1= s_l-i + 1^n-i+1(w_s_l-i+1^δ…∑_s_l=s_l-1+1^nw_s_l^δ)) ≤n^i· d^i1i!∑_s_l-i = s_l-i-1 + 1 ^ n-iw_s_l-i^i · (δ+1-β) + δTo bound the sum, we distinguish two cases. If i(δ + 1 - β) + δ≥ 0, then1n ∑_s_l-i = s_l-i-1 + 1 ^ n-iw_s_l-i^i · (δ+1-β) + δ ≤ α w_s_l-i-1^(i+1) · (δ+1-β) + ∫_w_s_l-i-1^w_nα (i · (δ+1-β) + δ) w^i · (δ+1-β) + δ - β w= α w_s_l-i-1^(i+1) · (δ+1-β) + [α (i · (δ+1-β) + δ)(i+1) · (δ+1-β) w^(i+1) · (δ+1-β)]_w_s_l-i-1^w_n= ( α + α (i · (δ+1-β) + δ)(i+1)(β-δ-1)) w_s_l-i-1^(i+1) · (δ+1-β) - α (i · (δ+1-β) + δ)(i+1)(β-δ-1) w_n^(i+1) · (δ+1-β)= α (β-1)(i+1)(β-δ-1) w_s_l-i-1^(i+1) · (δ+1-β)·( 1 - i · (δ + 1 - β) + δβ-1 (w_nw_s_l-i-1)^(i+1)· (δ + 1 - β)).For the integration, we need to make sure that the special case -1 = i · (δ+1-β) + δ - β does not occur. By rearranging, one can see that this is only true for i = -1, however, our weights begin at i=1. We now bound the error term that appears from the integration limit w_n. Note that we only need to consider the case where i (δ + 1 - β) < - δ, otherwise the error term is smaller than 1 and may simply be omitted.Observe from induction that w_s_l-i-1≤ w_n-i. Further, by sandwhich we have that w_n = Θ(n^1/β-1). Similarly,in = F(w_n-i) = α w_n-i^1-β,therefore w_n-i = Θ(1) · (ni)^1/β-1. Therefore, we havew_n/w_s_l-i-1≤w_n/w_n-i = Θ(i^1/β-1).Recall that we are in the case where the error term is positive; and that the exponent (δ + 1 -β) is negative. Substituting the above inequality, we obtain1 - i · (δ + 1 - β) + δβ-1 (w_nw_s_l-i-1)^(i+1)· (δ + 1 - β) ≤ 1 - i · (δ + 1 - β)β-1 i^i+1/β-1· (δ + 1 - β)= 1 - δ+1-ββ-1 i^1 + i+1/β-1· (δ+1-β)By inspecting the exponent 1 + i+1β-1· (δ + 1 - β), we observe that it is of order (1). In particular, once i is a large enough constant, the exponent becomes negative. Therefore, we may conclude that1 - δ+1-ββ-1 i^1 + i+1/β-1· (δ+1-β) = (1),where the constant is not dependent on the iteration i. Thus, as d was chosen large enough, 1n ∑_s_l-i = s_l-i-1 + 1 ^ n-iw_s_l-i^i · (δ+1-β) + δ≤di+1 w_s_l-i-1^(i+1) · (δ + 1 - β),Plugging this into inequality (<ref>) proves the induction step.Choosing i=l-1 and setting s_0=0 yields∑_S⊆[n] |S|=l∏_i∈ Sw_i^δ ≤ n^l· w_1^l · (δ+1-β)· d^l1l!.Since w_1^δ + 1 - β = Θ(1), we can choose an appropriate constant c such that the statement holds.We are now able to show sat. As discussed above, we do this by upper bounding the probability that a bi-cycle appears in Φ'. To this end, we calculate the expected number of bi-cycles in Φ', observe that it is poly(n)^-1, and apply Markov's inequality. This yields that , Φ' and thus Φ are satisfiable. We calculate the expected number of bi-cycles in Φ'. First, we fix a set S⊆[n] of l≥ 2 variables to appear in a bi-cycle. Let X_B denote the random variable counting how many times a specific bi-cycle B with the variables from S appears in F. Then𝔼[X_B] ≤ ml+1(l+1)!·[ux_1] [x̅_̅l̅ v] ·∏_i=1^l-1[x̅_̅i̅ x_i+1 ].The factor ml+1(l+1)! counts the possible positions of B in F. By shrinked,𝔼[X_B] ≤ m^l+1·(c_1n^2)^l+1·( w_|u| w_|v|∏_i∈ Sw^2_i)^1-1/2(k-2)(β-2)for some suitable constant c_1. Now let X_S denote the random variable counting how many times any bi-cycle with the variables from S appears in F. There are l! permutations of the l variables; and 2^l combinations of literals on l variables.Similarly, literals u and v have 4 possible sign combinations. Thus,𝔼[X_S]≤ m^l+1· l!· 2^l·(c_1n^2)^l+1·4(∑_i∈ Sw_i^1-1/2(k-2)(β-2))^2∏_i∈ Sw_i^2-(k-2)(β-2).To estimate the sum, we upper bound w_i ≤ w_n for all sets up to a certain size l_0, which we will determine later. We set δ := 2 - (k-2)(β-2) and define α(l) as(∑_i∈ Sw_i^δ/2)^2 ≤α(l):= (l^2), if δ≤ 0,l_0^2· w_n^δ, if δ > 0andl≤ l_0,(n^2), otherwise.Now let X denote the random variable counting the number of bi-cycles that appear in F.X ≤∑_l=2^n2^l+2· m^l+1· l!· (c_1n^2)^l+1·α(l)∑_S⊆[n]|S|=l∏_i∈ Sw_i^δ.Since δ+1 = 2-(k-2)(β-2) + 1 < β by our assumption , we can apply Lemma <ref>. Using r:=m/n, we obtain that the right-hand side is at mostX≤∑_l=2^n2^l+2· m^l+1· l!· (c_1n^2)^l+1·α(l) · n^l· c^l1l!≤ 1n ∑_l=2^n c_2^l · r^l·α(l), for some suitable constant c_2. Since r is a small enough constant we thus have c_2 · r < 1. If δ≤ 0, we are finished, since then1n ∑_l=2^n c_2^l · r^l·α(l) ≤ 1n ∑_l=2^n (c_2· r)^l·l^2 ≤( 1n).Otherwise, if δ>0, we choose l_0:=-4·ln^-1(c_2 r)ln(n), which ensures (r· c_2)^l = ( n^-4) for all l> l_0. For l=2,…,l_0, equation (<ref>) sums up to at most 1n ∑_l=2^l_0 (c_2r)^l · l_0^2 · w_n^δ = (log^3(n) · n^1 - kβ-2/β-1),where we substituted w_n = Θ(n^1/β-1) and δ = 2 - (k-2)(β-2). Since , the exponent 1 - kβ-2/β-1 < -' is negative, and we thus have X ≤ 1n ∑_l=2^n c_2^l r^lα(l) ≤(log^3(n) · n^-') + (1n),which proves the Theorem by Markov's inequality. § DISCUSSION OF THE RESULTSIn this work, we have shown that with high probability, a power law random k-SAT formula is satisfiable, if β≥2k-1k-1 + and the clause-variable ratio is not too large; and that it is unsatisfiable if β≤2k-1k-1 -, or if the clause-variable ratio is too large. Here, we give a few observations following these results.First, as explained in Section 1 our results translate directly to the model where clause lengths are power law distributed. This observation might help to explain a phenomenon that arose in <cit.>: The authors experimentally observed that a random-sat formula with double power law distribution (both variables and clause lengths are drawn from a power law) can be solved extremely fast by MiniSAT. Although the formula was of length 5 · 10^5, MiniSAT already gave an answer after 4 seconds! Using our results, we are now able to provide a potential explanation for this phenomenon: Disregarding the double power law distribution, the smallest clause length k_min occurring in their generated formulas is one. Thus, there will be Θ(n) clauses of length one and by unsat the formula is likely unsatisfiable. Second, we observe a sharp threshold in the sense of Friedgut <cit.> (for small constraint densities r) for β at the point 2k-1k-1. In contrast, it is unclear whether such a sharp threshold exists (and can be analytically derived) for fixed β but variable r. Considering however, that decades of research were dedicated to the same question in the uniform case—an arguably simpler model—it is unlikely that we obtain a satisfying answer any time soon; at least for all k. As in the uniform model, however, it might be more tractable to get sharp thresholds for k →∞. | http://arxiv.org/abs/1706.08431v1 | {
"authors": [
"Tobias Friedrich",
"Anton Krohmer",
"Ralf Rothenberger",
"Thomas Sauerwald",
"Andrew M. Sutton"
],
"categories": [
"cs.DM",
"cs.CC"
],
"primary_category": "cs.DM",
"published": "20170626151225",
"title": "Bounds on the Satisfiability Threshold for Power Law Distributed Random SAT"
} |
Bit-Reversible Version of Milne's Fourth-Order Time-Reversible Integrator for Molecular Dynamics William Graham Hoover and Carol Griswold Hoover Ruby Valley Research InstituteHighway Contract 60, Box 601Ruby Valley, Nevada 89833December 30, 2023 ============================================================================================================================================================================================================= In this paper, we present two main results. First, by only one conjecture (<Ref>) for recognizing a vertex symmetric graph, which is the hardest task for our problem, we construct an algorithm for finding an isomorphism between two graphs in polynomial time O(n^3). Second, without that conjecture, we prove the algorithm to be of quasi-polynomial time O(n^1.5log n). The conjectures in this paper are correct for all graphs of size no larger than 5 and all graphs we have encountered. At least the conjecture for determining if a graph is vertex symmetric is quite true intuitively. We are not able to prove them by hand, so we have planned to find possible counterexamples by a computer. We also introduce new concepts like collapse pattern and collapse tomography, which play important roles in our algorithms.§ INTRODUCTIONCurrently, the best general algorithm for graph isomorphism problem is due to Babai <cit.>, who shows that the graph isomorphism is of quasi-polynomial time exp((log n)^O(1)). We give a constructive proof of this result in the current paper. And by only one conjecture, which is quite true intuitively, we give a polynomial time algorithm for the problem. This also means that we have reduced the graph isomorphism problem in polynomial time to the problem of determining whether a graph is vertex symmetric or not.A detailed review is needed on the applications and related problems such as group isomorphism. The author has not seen Babai's <cit.> work in very detail. There may be some common techniques between this paper and previous papers not pointed out, which is another reason for the need of a review paper. Currently, if you want to know more about the origin and research history of the graph isomorphism problem, please refer to <cit.>. In this paper, G=(V,E) means an undirected graph G with a vertex set V and an edge set E and without self-loop or multiple edges connecting two vertexes. The case that graphs containing self-loop and multi-edges is discussed after we have presented the main results. Let V={v_1, v_2, ⋯, v_n}. If there is a direct connecting between v_i and v_j for v_i, v_j∈ V and v_j≠ v_i, we denote it by (v_i, v_j). Let T be the set of all (i,j) pairs with (v_i, v_j). E={(v_i,v_j):(i,j)∈ T }. We denote the size of V as |V|. Let V_j={v_k∈ V: (v_j, v_k)∈ E }. The degree of a vertex v_j∈ V is |V_j|.Given two graphs G_1=(V_1, E_1) and G_2=(V_2, E_2), if there is a one-to-one correspondence π between V_1 and V_2, s.t., for all v_i, v_j∈ V_1, (v_i,v_j)∈ E_1 iff (π(v_i), π(v_j))∈ E_2, then we say G_1 is isomorphic to G_2 and π is a graph isomorphism between G_1 and G_2. Any isomorphism from a graph G to itself is called the automorphism of G. Note that, as the identity permutation is always an automorphism for any graph, we are not interested in this trivial automorphism. For the graph isomorphism and automorphism, we have an intuitive understanding, i.e., all directly connected vertexes must be also directly connected after an isomorphic or automorphic mapping, which is a permutation of vertex names.Now we define several problems.The graph isomorphism problem, denoted as GI(G_1, G_2): determine whether G_1 and G_2 are isomorphic.The graph automorphism problem, denoted as GA(G): determine if there exists a non-trivial automorphism of the undirected graph G.The graph automorphism counting problem, denoted as # GA(G): find the total number of automorphisms of the undirected graph G.If we can solve a problem P by using polynomially many times of the procedure for solving another problem Q, we say P is polynomially reducible to Q. If P and Q are polynomially reducible to each other, we say they are polynomially equivalent.It is shown GI(G_1, G_2) is polynomially equivalent with # GA(G) <cit.>, and GA(G) is polynomially reducible to GI(G_1, G_2) <cit.>.It seems # GA(G) is much harder than GA(G). Just take the complete graph as an example, the total number of automorphisms is n!-1, which is hard to find one by one. With an oracle for GI(G_1, G_2), it will be easy to do # GA(G).In this paper, we introduce two more problems. One is the graph automorphism with constraint problem, denoted as GA(G, C): determine if there exists a non-trivial automorphism of the undirected graph G, with the constraint C. For v_1, v_2∈ G, (v_1, v_2)∈ C if v_1 and v_2 cannot replace each other under any permutation or ⟨ v_1,v_2⟩∈ C if v_1 and v_2 can only correspond to each other in any automorphic mapping. The other is the graph isomorphism with constraint problem, denoted as GI(G_1,G_2, C): determine whether G_1 and G_2 are isomorphic, with the constraint C. For v_1∈ G_1, v_2∈ G_2, (v_1, v_2)∈ C if v_1 and v_2 cannot replace each other under any mapping or ⟨ v_1,v_2⟩∈ C if v_1 and v_2 can only correspond to each other in any isomorphic mapping.It is easy to see that (1) GI(G_1,G_2) is a special case of GI(G_1,G_2,C); (2) GA(G) is a special case of GA(G, C); (3) GI(G_1,G_2) is equivalent to GA(G_1∪ G_2, C), with C={⟨ v_1,v_2⟩: v_1, v_2∈ G_1 or v_1, v_2∈ G_2 }; (4) GA(G, C) seems easier than GA(G), as some permutations are ruled out by the constraint C. In our polynomial algorithm for GI(G_1,G_2), we make use of GI(G_1,G_2,C) as a subroutine. In <cit.>, it is said that the checking and counting of graph isomorphism are polynomially equivalent, which is an evidence to the conjecture that the graph isomorphism is not 𝐍𝐏-complete.In <cit.>, they define a complexity class 𝐆𝐈 of all problems polynomially reducible to the graph isomorphism problem, and claim that 𝐆𝐈=𝐏 if the graph isomorphism is in class 𝐏. We introduce concepts and theoretic work, including our conjectures, in Section <ref>. The algorithm for graph isomorphism is described in Section <ref>. The final section is the conclusion. The reader may skip <Ref> if not interested in too much theoretical work.§ PREPARATIONS Given two graphs G=(V,E) and G'=(V',E'), with |V|=|V'|, our target is to find a one-to-one correspondence π: V→ V', s.t.,for all v_i, v_j∈ V, (v_i,v_j)∈ E iff (π(v_i),π(v_j))∈ E'. In this section, V={v_1,v_2,⋯, v_n }, V'={v'_1,v'_2,⋯,v'_n }; V_i={v_j∈ V: (v_i, v_j)∈ E }, V'_i={v'_j∈ V': (v'_i, v'_j)∈ E' }, for i=1, 2, ⋯, n. The first information we can use is that |V_i|=|V'_j| if π is an isomorphism and π(v_i)=v'_j, i.e., in any isomorphism between V and V', a vertex of V can only be mapped to a vertex of V' with the same degree. For this reason, we introduce a concept called base subgraph G^(w) for those vertexes of the same degree w in a graph G: G^(w)=(V^(w),E^(w)), V^(w)={v_i∈ V: |V_i|=w }, E^(w)={(v_i,v_j)∈ E: v_i, v_j∈ V^(w)}.Note that it is only possible to map a v_i∈ V^(w) to some v'_j∈V'^(w).Given G^(w) and v_i∈ V^(w), we define the extension based on v_i,G^ex(v_i,w)=(V^ex(v_i,w),E^ex(v_i,w)),as follows: * v_i∈ V^ex(v_i,w); * V_i-V^(w)⊆ V^ex(v_i,w); * For any v_j∈ V^ex(v_i,w)-V^(w), V_j⊆ V^ex(v_i,w); * E^ex(v_i,w)={(v_j,v_k)∈ E: v_j∈ V^ex(v_i,w)-V^(w), v_k∈ V^ex(v_i,w) }. We call v_i the base point of the extension G^ex(v_i,w).The base subgraph G^(w) is a separation of the the graph G, which means vertexes inside the base subgraph are different from those outside. As such separation is not limited to the degree argument, we can generalize the concepts, base subgraph and extension, to a given set of vertexes β⊊ V. The base subgraph of β is G^β=(V^β,E^β), where V^β=β. The extension of G^β based on v_i is G^ex(v_i,β). We just replace V^(w) by β in the definition of G^(w) and G^ex(v_i,w).As the definitions are not so intuitive, we explain them by an example. Let's consider the graph in Figure <ref>. The base subgraph of degree 3 is depicted in Figure <ref>. All extensions with respect to this base subgraph are depicted in Figure <ref>, <ref>. Note we have colored all base points in black. Now we introduce some sets of labels L_k={l_k1, l_k2, l_k3, ⋯}, for k=1,2,⋯. The total number of labels used in an algorithm will be finite. We use labels to replace extensions with respect to a base subgraph. The labels serve as carrying on the information of whether two extensions are isomorphic or not. If two extensions are isomorphic, we replace them with the same label, otherwise different labels. After pair-wise comparing Figure <ref>, <ref>, we find that they are not isomorphic with the constraint that a base point (black point) can only be mapped to a base point, so we label them differently. In Figure <ref>, we focus on the base subgraph of degree 3, the base point v_2 now has a label l_11 and the base point v_4 has a label l_12. It is better to write labels of a base point in a predefined order and combine identical labels, e.g., write l_23 l_11 l_12 l_12 as l_11 l_12^2 l_23. We will learn more about the labeling procedure in the algorithm for graph isomorphism.We say a graph is vertex regular of w if every vertex in the graph is of the same degree w. Given a graph G=(V,E), with v_i and V_i for i= 1, 2, ⋯, n, we define the collapse of G with the trigger v_k, G^col(v_k), as follows: * Layer 0: G^col(v_k,0), V^col(v_k,0)={v_k}, E^col(v_k,0)=∅; * Layer 1: G^col(v_k,1), V^col(v_k,1)=V_k, E^col(v_k,1)={(v_i,v_j)∈ E: v_i,v_j∈ V_k }; * Layer i+2: Given Layer i+1 (G^col(v_k,i+1)) and Layer i (G^col(v_k,i)), Layer i+2 is G^col(v_k,i+2), with V^col(v_k,i+2)=⋃_v_x∈ V^col(v_k,i+1) {v_y∈ V: v_y∈ V_x-V^col(v_k,i+1)-V^col(v_k,i) }, E^col(v_k,i+2)={(v_x,v_y)∈ E: v_x,v_y∈ V^col(v_k,i+2) }.See Figure <ref> for example. It is a collapse trigged by vertex v_1 for the vertex regular graph G_b, which is Figure <ref>. There are n collapses in a graph of n vertexes.Let deg(v_i,v_j)=|V_i|+|V_j|-|V_i∩ V_j| be the degree (edge degree) of an edge (v_i,v_j). Intuitively, the degree of an edge is the total number of collapses in which the edge is located before Layer 2. Likewise, we say a graph is edge regular of w if every edge in the graph is of the same degree w. Figure <ref> and Figure <ref> are both edge regular and vertex regular, and they are actually isomorphic. Figure <ref> is vertex regular but not edge regular.Let's define here an important class of graphs, the vertex symmetric graph. For a graph G=(V,E), by designating a nailed vertex (imagine that you nail the graph to the wall with the nail on the designated vertex), we distinguish |V| possible nailed graphs of G, G(v_i), i=1,2,⋯,|V|. Those independent parts of the graph will fall down to the floor. Thus, G(v_i) is equal to G^col(v_i) with the edges between its layers added back. Any nailed graph is connected. G is vertex symmetric if and only if all of its nailed graphs are isomorphic to each other. More precisely, if for every i from 1 to |V|, we have for every j from i to |V|, GI(G(v_i),G(v_j),{⟨ v_i,v_j ⟩})=True, then we say the graph G is vertex symmetric. Here, {⟨ v_i,v_j⟩} means v_i can only correspond to v_j in any isomorphic mapping and vice versa. Complete graph is vertex symmetric. A graph consisting of one circle or multiple identical circles is also vertex symmetric. Figure <ref> and Figure <ref> are further examples of vertex symmetric graphs. Later we will see that the only obstacle of the graph isomorphism problem is how to recognize a vertex symmetric graph. Given G=(V,E), if G is vertex symmetric, then it is vertex regular. We can generalize the concepts, collapse and nailed graph, to multi-collapse and multi-nailed graph, i.e., using multiple triggers or nailed vertexes at the same time. Given a graph G, a multi-collapse (sometimes just called collapse if it is clear from the content) with a set of triggers T is G^col(T), in which layer 0 contains vertexes in the set T, and layer k+1 contains all vertexes connecting those vertexes in layer k. A multi-nailed graph with a set of nailed vertexes T is G(T), which looks the same as G^col(T) with those edges between layers added back. The normal (not nailed) graph and the nailed graph are special cases of the multi-nailed graph.A graph G is edge symmetric if and only if all of its multi-nailed graphs G((v_i,v_j)), with (v_i, v_j)∈ E, are isomorphic to each other under the constraint {⟨ (v_i,v_j), (v_i', v_j')⟩}, which means that the edge (v_i,v_j) can only be mapped to (v_i', v_j') in any isomorphism between G((v_i,v_j)) and G((v_i', v_j')). Given G=(V,E), if G is edge symmetric, then it is edge regular. Actually, there exist graphs that is both vertex regular and edge regular but not vertex symmetric, see Figure <ref>. There exist graphs that is edge symmetric but not vertex symmetric, e.g., Figure <ref>, the complete bipartite graph in Figure <ref> and Figure 3.2 in <cit.>. There exist graphs that is vertex symmetric but not edge symmetric, see Figure <ref>. There exist graphs edge symmetric but not vertex regular, see Figure <ref>. There exists graphs that is both vertex regular and edge regular but not vertex symmetric or edge symmetric, e.g., a graph consisting of two independent circles of different sizes. An arc is an edge treated as directed. A graph G is arc symmetric if and only if all of its arc nailed graphs G(v_i; v_j), with (v_i, v_j)∈ E, are isomorphic to each other under the constraint {⟨ v_i,v_i'⟩, ⟨ v_j, v_j'⟩}, which means that the vertex v_i of the edge (v_i,v_j) can only be mapped to v_i' of the edge (v_i', v_j') and v_j can only be mapped to v_j' in any isomorphism between G(v_i;v_j) and G(v_i'; v_j'). The difference between arc symmetry and edge symmetry is that we treat the trigger edge as directed in the arc symmetry. There exists graphs that is both vertex symmetric and edge symmetric but not arc symmetric, e.g., the Doyle-Holt graph, see Figure 3.3 in <cit.>. Given G=(V,E), if G is arc symmetric, then it is both vertex symmetric and edge symmetric. In the literature, vertex symmetric is equivalent to vertex transitive and 0-arc transitive, edge symmetric is equivalent to edge transitive, and arc symmetric is equivalent to arc transitive and 1-arc transitive. We prefer the word `symmetric' more than `transitive', as the former is shorter and more intuitive.Let's point out the levels of symmetry of a graph:No pair of isomorphic vertex nailed subgraphs and no pair of isomorphic edge nailed subgraphs ⟹ Some pairs of isomorphic vertex nailed subgraphs or some pairs of isomorphic edge nailed subgraphs ⟹ Vertex symmetric or edge symmetric ⟹ Arc symmetric. Given two multi-sets A={a_1,a_2,⋯,a_s } and B={b_1,b_2,⋯,b_t }, we say A matches B if the sort of a_1,a_2,⋯,a_s is equal to the sort of b_1,b_2,⋯,b_t. For instance, {3,7,3,2,1 } matches {1,2,3,3,7 } but does not match {3,7,2,1,4 }. More generally, {{5,3,3},{5,2,2,8},{1,4,2},{1,3,3}} matches {{1,2,4},{1,3, 3},{2,2,5,8},{3,3,5}} but does not match {{5,3,3},{1,3,4},{5,2,2,2},{1,4, 2}}.The vertex property of a graph G=(V,E) is the multi-set P_V(G)={|V_i|:i=1,2,⋯,n }, where n=|V|. We say two graphs G and G' are of the same vertex property if their vertex properties match. Similarly, the edge property of a graph G=(V,E) is the multi-set P_E(G)={deg(v_i,v_j):(v_i,v_j)∈ E }. We say two graphs G and G' are of the same edge property if their edge properties match.Given one collapse of G, say G^col(v_k), which has l+1 layers, G^col(v_k,i) for i=0,1,⋯, l, the collapse tomography of G^col(v_k) is an ordered list of l ordered pairs of the vertex property and the edge property, i.e.,C^tom(G,v_k)=[[P_V(G^col(v_k,i));P_E(G^col(v_k,i))]:i=1,2,⋯,l ].Two collapse tomographies match if all properties in the ordered list match at the corresponding position. For instance,[ [{5,3,3};{5,2,2,8} ], [{1,4,2}; {1,3, 3} ], [{7,5}; {3,3} ] ]matches[ [ {3,3,5};{2,2,5,8} ],[{1,2,4}; {1,3,3}], [{5,7};{3,3} ] ]but not[ [ {3,3,5};{2,2,5,8} ],[{1,3,3};{1,2,4} ], [{5,7};{3,3} ] ]. The collapse tomography of Figure <ref> is[ [{0,2,2,2};{3,3,3} ],[{2,2,2};{3,3,3}] ]. Given a graph G, the collapse pattern of G is a multi-set of collapse tomographies of G, i.e.,C^pat(G)={C^tom(G,v_i):i=1,2,⋯,|V| }. Given a nailed graph G(v_a), let G^col(v_a,j), j=0,1,2,⋯,l, be l+1 layers of its collapse. The collapse pattern of G(v_a) is an ordered list of multi-sets of collapse tomographies of G, i.e.,C^pat(G(v_a))=[{C^tom(G,v_i): v_i∈ V^col(v_a,j) }: j=0,1,2,⋯ l ]. Given a multi-nailed graph G(T) and its collapse G^col(T), which has l+1 layers, the corresponding collapse pattern isC^pat(G(T))=[{C^tom(G,v_i): v_i∈ V^col(T,j) }: j=0,1,2,⋯ l ]. If each vertex v of the graph G is labeled by L(v), then the collapse tomography under labeling L isC^tom(G,L,v_i)=[[L(v_i);P_V(G^col(v_k,i));P_E(G^col(v_k,i))]:i=1,2,⋯,l ],and the collapse pattern under labeling L for a nailed graph G(v_a) isC^pat(G(v_a),L)=[{C^tom(G,L,v_i): v_i∈ V^col(v_a,j) }: j=0,1,2,⋯ l ]. We leave the study of properties of the collapse pattern of normal graph, nailed graph and multi-nailed graphs as a future work. Given G=(V,E) and G'=(V',E'), if G is isomorphic to G', then the collapse pattern of G matches that of G'. Given G=(V,E) and G'=(V',E'), G is isomorphic to G' if the collapse pattern of G matches that of G'. Given G, G', and their nailed graphs, G(v_a) and G'(v_b), if G(v_a) is isomorphic to G'(v_b) with the constraint {⟨ v_a,v_b ⟩}, then the collapse pattern of G(v_a) matches that of G'(v_b). Given G, G', and their nailed graphs, G(v_a) and G'(v_b), G(v_a) is isomorphic to G'(v_b) with the constraint {⟨ v_a,v_b ⟩} if the collapse pattern of G(v_a) matches that of G'(v_b). Given G=(V,E), if G is vertex symmetric, then all |V| collapse tomographies of G match each other, i.e., the collapse pattern of a vertex symmetric graph consists of |V| equal collapse tomographies. Given G=(V,E), G is vertex symmetric if all |V| collapse tomographies of G match each other, i.e., the collapse pattern of a vertex symmetric graph consists of |V| equal collapse tomographies.Conjecture <ref> is a corollary of Conjecture <ref>. In the algorithm for graph isomorphism, we only use Conjecture <ref>, as it is quite true intuitively. Let's call a graph G vertex indistinguishable if all of its |V| collapse patterns C^pat(G(v_i)), with v_i∈ V, are equivalent. The question corresponding to Conjecture <ref> is “Is there any graph that is vertex indistinguishable but not vertex symmetric?”. Given G=(V,E), if G is edge symmetric, then all |E| collapse patterns of edge nailed graphs G(v_i,v_j), with (v_i,v_j)∈ E, match each other. Given G=(V,E), G is edge symmetric if all |E| collapse patterns of edge nailed graphs G(v_i,v_j), with (v_i,v_j)∈ E, match each other.In order to have a similar conjecture for arc symmetry, we redefine the collapse pattern for the arc nailed graph G(v_a;v_b) as an ordered list of collapse patterns:C^pat(G(v_a;v_b))=[C^pat(G(v_a,v_b)), C^pat(G[v_a,v_b](v_a)), C^pat(G[v_a,v_b](v_b)) ],where C^pat(G(v_a,v_b)) is the collapse pattern of the edge nailed graph G(v_a,v_b), C^pat(G[v_a,v_b](v_a)) is the collapse pattern of the nailed graph at v_a obtained by removing the edge (v_a,v_b) in G and nail the vertex v_a, and similarly for C^pat(G[v_a,v_b](v_b)). Given G=(V,E), if G is arc symmetric, then all 2|E| collapse patterns of arc (an edge treated as directed) nailed graphs G(v_i;v_j), with (v_i,v_j)∈ E and the direction is from v_i to v_j, match each other. Given G=(V,E), G is arc symmetric if all 2|E| collapse patterns of arc nailed graphs G(v_i;v_j) match each other.The quantity, collapse pattern, contains a lot of information of a graph, so it may be a good argument to distinguish graphs. Up to now, our conjectures have never failed. Although we have tried our best to prove them, we cannot. Thus, we hope of finding a counterexample by a computer. Now let's introduce the dual graph. Given a graph G=(V,E), its dual graph is G=(V,E), where (v_i,v_j)∈E if and only if (v_i,v_j)∉ E. G is isomorphic to G' if and only if G is isomorphic to G'. For any isomorphism π between G and G', (v_i,v_j) in G if and only if (π(v_i), π(v_j)) in G'. So it is also true that (v_i,v_j) not in G if and only if (π(v_i), π(v_j)) not in G'. When a graph of size n is vertex regular of a large degree w>n/2, it seems easier to consider its dual graph, which is vertex regular of a smaller degree n-1-w<n/2. Given two complete graphs G and G', if G has n vertexes with vertex v_i labeled by L(v_i) and G' has n vertexes with vertex v'_j labeled by L(v'_j), then G is isomorphic to G' if and only if {L(v_i):i=1,2,⋯,n } matches {L(v'_j):j=1,2,⋯,n }. The dual graph of a complete graph of size n is n independent vertexes. §.§ Variations of Collapse PatternIn the remaining part of Section <ref>, we provide more discussions on our conjectures, which is not important for the next section. The reader may skip it if not interested.In case that the above conjectures fail, we can vary the definitions of our collapse pattern to add more details of the graph.Given a nailed graph G(v_a), let G^col(v_a,j), j=0,1,2,⋯,l, be l+1 layers of its collapse, and let G^ex(v_i;v_a,j) be the extension with a base point v_i∈ V^col(v_a,j) and the base subgraph G^col(v_a,j). The varied collapse pattern of G(v_a) is an ordered list of multi-sets of ordered collapse tomographies, i.e., C^pat(G(v_a))=[{[ C^tom(G,v_i);C^tom(G^col(v_a,j),v_i);C^tom(G^ex(v_i;v_a,j),v_i) ]: v_i∈ V^col(v_a,j) }: j=0,1,⋯ l ]. This varied definition encodes more information of the nailed graph.For an ordinary graph, we also change the definition asC^pat(G)={C^pat(G(v_i)): v_i∈ V }. Let G(T) be a multi-nailed graph with nailed vertexes T and its collapse G^col(T), which has l+1 layers. The corresponding varied collapse pattern of G(T) is C^pat(G(T))=[{[ C^tom(G,v_i);C^tom(G^col(T,j),v_i);C^tom(G^ex(v_i;T,j),v_i) ]: v_i∈ V^col(T,j) }: j=0,1,⋯ l ]. All of our definitions of collapse tomography and collapse pattern are well-defined and computable in polynomial time. If G of size n is a graph constructed by linking n-1 vertexes to one vertex v, we call it a diverging graph and v is called the source of the graph.. If G is a diverging graph with the source v and n-1 vertexes labeled by L(v_i), and G' is another diverging graph with source v' and m-1 vertexes labeled by L(v'_j), then G(v) is isomorphic to G'(v') if and only if {L(v_i):v_i∈ V } matches {L(v'_j):v'_j∈ V' }. Suppose we want to define for a nailed graph G(v_a) a quantity that is useful to distinguish graphs and easy to be proved by induction, let such quantity be C^q(G(v_a)). We hope that two graphs are isomorphic if and only if their values of the quantity are equivalent. Given G(v_a) and G'(v_b), G(v_a) is isomorphic to G'(v_b) with the constraint {⟨ v_a,v_b ⟩} if and only if C^q(G(v_a)) matches C^q(G'(v_b)), if we define the quantity asC^q(G(v_a))={C^q(G^ex(v_i,v_a)):v_i∈ V_a },where G^ex(v_i,v_a) is the extension based on v_i linking to v_a and the base subgraph consisting only of the edge (v_i,v_a). Let L(v_i)=C^q(G^ex(v_i,v_a)), then by induction and Lemma <ref>, we can prove this theorem. If C^q(G(v_a)) is a quantity for nailed graphs, then C^q(G)={C^q(G(v_i)):v_i∈ V } is a quantity for normal graphs. If C^q(G)={C^q(G(v_i)):v_i∈ V } is a quantity for normal graphs, then C^q(G,L)={[C^q(G(v_i)),L(v_i)]:v_i∈ V } is a quantity for the case that each vertex is labeled by L(v_i). If G(v_a) and G'(v_b) are two nailed graphs with their collapses G^col(v_a) and G'^col(v_b) having only Layer 0 and Layer 1, then G(v_a) is isomorphic to G'(v_b) if and only if G^col(v_a,1) is isomorphic to G'^col(v_b,1). Given G(v_a) a nailed graph, if we define the quantity asC^q(G(v_a))={[C^q(G(v_i,V_a)),C^q(G^ex(v_i,V_a∪{v_a}))]:v_i∈ V^col(v_a,1) },where G(v_i,V_a) is the graph G^col(v_a,1) nailed at v_i, and G^ex(v_i,V_a∪{v_a}) is the extension based on v_i with the base subgraph consisting of vertexes V_a and v_a, then this quantity can distinguish graphs according to isomorphism. Let L(v_i)=C^q(G^ex(v_i,V_a∪{v_a})), then by induction and <Ref> and <Ref>, we can prove this theorem.As the lack of details of a graph, we cannot accelerate the graph isomorphism problem in the abstract approach. In the next section, we apply a bunch of methods for the acceleration of our algorithm.§ ALGORITHMS Now, we are ready to present the algorithm for determining whether two graphs are isomorphic.Note that our algorithm is only based on <Ref> and no other assumption not proved. If we remove this conjecture, we can show its quasi-polynomial time efficiency. Our algorithm for graph isomorphism is GI(G,G')=GI(G,G',L_0), where L_0=∅.Algorithm: Graph Isomorphism GI(G,G',L_k) with Labels Input: Two graphs G=(V,E) and G'=(V',E'), without self-loop or multi-edge. V={v_1,v_2,⋯,v_n }, V'={v'_1,v'_2,⋯,v'_n }. Every vertex v∈ V∪ V' is labeled by L_k(v). Output: If G and G' are isomorphic under the labeling function L_k, output `Yes', otherwise `No'. Runtime: O(n^3) with <Ref>; O(n^1.5log n) without <Ref>. The worst case is when the input is two vertex indistinguishable but not vertex symmetric graphs.Procedure: * Compute C^tom(G,L_k,v_i) for all v_i∈ V, and C^tom(G',L_k,v'_j) for all v'_j∈ V'. If they all match each other, then by <Ref>, G and G' are vertex symmetric. Then if G and G' are regular of degree w≤n/2, we call the sub-algorithm GI(G,G',C,L_k), with C={⟨ v_1,v'_1⟩}. If G and G' are regular of degree w> n/2, we call the sub-algorithm GI(G,G',C,L_k), with C={⟨ v_1,v'_1⟩}. Output `Yes' iff the sub-algorithm returns `Yes'. Here, our target is to find an isomorphism. Note that, if we do not appeal to <Ref>, then we have to check the isomorphism for all 2n nailed graphs of G and G' in the worst case, instead of only one pair. We may use the method `try-and-error' to further accelerate the algorithm, as we believe that there is a non-negligible probability for finding a pair of isomorphic nailed graphs if two graphs are vertex indistinguishable but not vertex symmetric. Because of the use of dual graph, the size of the next occurrence of vertex indistinguishable graph is no larger than n/2. * Compute C(G)={C^pat(G(v_i),L_k): v_i∈ V }, and C(G')={C^pat(G'(v'_j), L_k): v'_j∈ V' }.Sort these two multi-sets of collapse patterns. If they do not match, output `No'. Otherwise, suppose the rarest collapse pattern in C(G) is C^pat(G(v_x),L_k) and the corresponding one in C(G') is C^pat(G'(v'_y),L_k), let β={v_i:C^pat(G(v_i),L_k)=C^pat(G(v_x),L_k) } and β'={v'_j:C^pat(G'(v'_j),L_k)=C^pat(G'(v'_y),L_k) }. The base subgraph of G is G^β and that of G' is G'^β'. The extensions of G^β are Ex(β)={G^ex(v_i,β):v_i∈ V^β} and those of G'^β' are Ex(β')={G'^ex(v'_j,β'):v'_j∈V'^β'}. Call the sub-algorithm GI(G_1^ex(v_1,β_1), G_2^ex(v_2,β_2), C,L_k) for all pairs of extensions (G_1^ex(v_1,β_1), G_2^ex(v_2,β_2)) in the union set Ex(β)∪ Ex(β') with C={⟨ v_1,v_2⟩}. Next we assign two extensions with the same label iff they are isomorphic. The labeling function L_k+1 assigns two base points with the same label iff L_k assigns them with the same label and the labels for their extensions are of the same. Then we call the sub-algorithm GI(G^β,G'^β', L_k+1 ), with vertex v∈β∪β' labeled by L_k+1(v). Output `Yes' iff this sub-algorithm returns `Yes'. Algorithm: Graph Isomorphism GI(G, G',C,L_k) with Constraint and Labels Input: Two graphs G=(V,E) and G'=(V',E'), without self-loop or multi-edge. V={v_1,v_2,⋯,v_n }, V'={v'_1,v'_2,⋯,v'_n }. The constraint is C={⟨ v_a,v'_b⟩}, where v_a is the nailed vertex of G(v_a) and v'_b is the nailed vertex of G'(v'_b). Every vertex v∈ V∪ V' is labeled by L_k(v). Output: If G(v_a) and G'(v'_b) are isomorphic under the constraint C and the labeling function L_k, then output `Yes'; otherwise `No'. Procedure: * Compute C^pat(G(v_a),L_k) and C^pat(G'(v'_b), L_k). If they do not match, then output `No'. Otherwise, let the collapse G^col(v_a) have l+1 layers. * If l=1, call the sub-algorithm GI(G^col(v_a,1),G'^col(v'_b,1),L_k). Output `Yes' iff this sub-algorithm returns `Yes'. Here, the input graphs may be again vertex indistinguishable but not vertex symmetric. * For l≥ 1, from x=l-1 to x=0, we update the labels of vertexes in layer x with the layer x+1 by the following procedure. Let β(x) be the set of vertexes in layers from 0 to x of G^col(v_a) and β'(x) be the set of vertexes in layers from 0 to x of G'^col(v'_b). Let Ex(β(x))={G^ex(v_i,β(x)):v_i∈ V^col(v_a,x)} and Ex(β'(x))={G'^ex(v'_j,β'(x)):v'_j∈V'^col(v'_b,x) }. Suppose the labeling function for layer x is L_k,x with L_k,l=L_k. Call the sub-algorithm GI(G_1^ex(v_1,β_1(x)), G_2^ex(v_2,β_2(x)), C, L_k,x+1) for all pairs of extensions (G_1^ex(v_1,β_1(x)), G_2^ex(v_2,β_2(x))) in the union set Ex(β(x))∪ Ex(β'(x)) with C={⟨ v_1,v_2⟩}. Usually, we omit the labels of base points when we are determining isomorphism of their extensions. Next we assign two extensions with the same label iff they are isomorphic. The labeling function L_k,x assigns two base points with the same label iff L_k assigns them with the same label and the labels for their extensions are of the same. On the whole, we distinguish two nailed graphs, according to isomorphism, layer-by-layer. The algorithm GI(G,G',L_k) can correctly distinguish graphs according to isomorphism and is of polynomial time if <Ref> is true. If <Ref> is not true, then the algorithm GI(G,G',L_k) is of quasi-polynomial time O(n^1.5log n). The recursion is T(n)=n n^2 T(n/2) for the worst case, which has the most chance of encounter of graphs that is vertex indistinguishable but not vertex symmetric. Suppose now we allow graphs with self-loop and multi-edge. First, we remove all self-loop and multi-edge, and by the algorithm GI(G,G'), we can find an isomorphism. Then we can check this isomorphism for graphs with self-loop and multi-edge. Another method is to use the labeling procedure to remove self-loop and multi-edge before the algorithm for graphs without those. With the algorithm for isomorphism, we can construct an algorithm for automorphism without too much effort.§ CONCLUSIONIn this paper, we provide an algorithm for the graph isomorphism problem. We have shown that the graph isomorphism problem is based on the recognition problem of a vertex symmetric graph. <Ref> can solve the latter problem very efficiently. Without this conjecture, our algorithm is of quasi-polynomial time. It is possible to further improve our algorithm. Although some techniques in this article might have been already known before, we apologize for the possibility of not pointing out, for our limited knowledge. We plan to write a detailed and also interesting review paper to make it clear and talk all aspects of the graph isomorphism problem, including the research history and its applications. Left works: What is the algorithm for directed graphs? Experimental benchmark of algorithms for graph isomorphism. Problems relating to the graph isomorphism. Properties of collapse tomography and collapse pattern. A more refined analysis of our algorithm. Find a counter-example to <Ref> and other conjectures. | http://arxiv.org/abs/1706.09230v1 | {
"authors": [
"Caishi Fang"
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"categories": [
"cs.DS",
"cs.CC"
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"primary_category": "cs.DS",
"published": "20170627060316",
"title": "Accelerations for Graph Isomorphism"
} |
^1 Department of Electrical Engineering and Computer Science, University of California - Berkeley, Soda Hall, Berkeley, CA - 94720, USA ^2 Engineering Directorate, Lawrence Livermore National Laboratory, 7000 East Avenue L-795, Livermore, CA - 94550, USA ^3* Physical and Life Science Directorate, Lawrence Livermore National Laboratory, 7000 East Avenue L-367, Livermore, CA - 94550, USA [email protected] demonstrate that the Maximum Lyapunov Exponent for computable dynamical systems is isomorphic to the maximum capacity of a noiseless, memoryless channel in a Shannon communication model. The isomorphism allows the understanding of Lyapunov exponents in the simplified terms of Information Theory, rather than the traditional definitions in Chaos Theory. This work provides a bridge between fundamental physics and Information Theory to the mutual benefit of both fields. The result suggests, among other implications, that machine learning and other information theory methods can be successfully employed at the core of physics simulations.Isomorphism between Maximum Lyapunov Exponent and Shannon's Channel Capacity Gerald Friedland^1,2, Alfredo Metere^3* December 30, 2023 ============================================================================ Information Theory is a relatively new field of study introduced by Shannon in 1948, providing mathematical understanding of how information can be measured, stored and transmitted between senders and receivers, meant in the most generic way possible. Today, Information Theory represents the foundation of telecommunications, signal processing and machine learning, e.g., Artificial Neural Networks are defined in Information Theory as universal encoders <cit.>.Shannon's formulation <cit.> defines information as the message, and its flow as a channel-mediated communication model. As Fig. <ref> depicts, information is the content of a message that is sent by a sender to a receiver, passing through a channel. In order for the channel to pass the message, it needs to be encoded, and in order for the receiver to understand the message, it must be decoded. All these elements, known as the Shannon Communication Model, can be meant either very generally or very specifically without a change in the paradigm. For example, consider a speech system consisting of a person A, the sender, who wants to communicate his/her idea, a spoken message, to person B, the receiver. For the success of the communication, person A, using the vocal cords and the mouth-nose complex, must encode the message into codified sound waves: the verbal language. The air bulk between person A and person B represents the channel through which the encoded message propagate as sound waves, and the ear of person B is the decoder able to transform the codified sound waves back into electrical signals that the brain can use to infer person A's ideas contained in the message. The communication happened successfully if person B's idea is an accurate enough inference of person A's idea. The channel in this scenario is considered to be noisy, which means that the encoded message gets distorted by the channel, and therefore the chances that the message cannot be correctly transmitted are not negligible. In this example, the noise can either come from the channel itself dampening the amplitude of the signal.Noise can also be introduced in the channel by polluting it with additional undesired background noise consisting of other sound waves. In contrast, an ideal, noiseless channel will not be affected by any noise. However, noiseless channels can only be ideal, and they are studied to understand the theoretical limits of how real channels function. Channels could also have memory or be memoryless.A memory channel's behavior is influenced by the input, while memoryless channels exhibit input-independent behavior. The most important measure to establish the quality of communication is the channel capacity <cit.>, which measures how much information can be successfully communicated per unit of time by a certain channel, given a certain message. A channel is characterized by its channel capacity C:C = sup_p_X(x) I (X;Y),where X and Y are, respectively, the encoded input and output of the channel; p_X(x), is the marginal distribution <cit.>, which is the probability distribution of the variables contained in the subset x for a given set X; and I(X;Y) is the mutual information <cit.>, which measures the mutual dependence between the X and Y. Information Theory models rely on probability theory, and they are generic. Such a stochastic formulation stems from certain fundamental assumptions, e.g., that information can be treated statistically and that the Central Limit Theorem (CLT), which can be summarized by saying that when the size of a set of independent distributions tends to infinity, the convolution of all the distributions in the set converges to a Gaussian distribution. In the stochastic definition X and Y are true random variables that are not necessarily correlated, hence the statistical mutual dependence formulation in Eq. <ref>.However, in reality the message always needs time to transition from input to output through a channel, hence we can define the channel as a time-dependent function N(t) that maps the input X to the output Y as follows <cit.>:Y/X = N(t).Such relationship already suggests that a channel communication could in principle be treated as the time-evolution of a dynamical system in physics, where X and Y would represent respectively the initial and final state, and N(t) the equation of motion for such equivalent dynamical system.Thanks mainly to the work of Shannon and Hartley, several models have been reformulated in a more practical form to describe the behavior of real-world communication systems.For example, the channel capacity (Eq. <ref>) can be reformulated using the Shannon-Hartley theorem <cit.>, for a noiseless, memoryless channel as:C = lim_t →∞1/tlog_2( Y/X ),where t is the sampling period. The log_2(x) is an encoding operator for x and it returns the real-valued number of bits necessary to represent the message, and if t is expressed in seconds, then the dimensions of C are:C = [s^-1] [bits] .This formulation exposes that C necessarily depends on the sampling frequency t^-1 and on the input of the channel X, while the ratio Y/X is determined by the nature of the channel (see Eq. <ref>). The amount of information passing through the channel is called signal <cit.>, and it can be defined as:S = log_2(Y) - log_2(X) = log_2( Y/X ) ,which is the bit-wise difference between output and input. The sampling frequency t^-1 is commonly called bandwidth. Among the main contributors to the foundation of modern Chaos Theory, which is a branch of mathematics stemming from the study of non-linear dynamical systems in Physics, Kolmogorov and Sinai thoroughly investigated the link between Information Theory and Physics. Their work resulted into the stochastic interpretability of physical phenomena, especially at a quantum level, commonly referred to as Stochastic Physics or Stochastic Mechanics <cit.>. However, because of the generality and the abstraction of the used probabilistic formulations, their theoretical work needs non-trivial interpretation and adaptation to be used for real-world physical systems.Another important contributor to Chaos Theory is Lyapunov, who developed several, more practical tools for characterizing the behavior of non-linear dynamical systems. More specifically, the chaotic behavior of computable Hamiltonian dynamical systems can be characterized by the Lyapunov Exponents (LE), a set of quantities that estimate the rate of separation between infinitesimally close trajectories <cit.>. For example, let a 3D Hamiltonian dynamical system of arbitrary form consisting of N particles, defining a 6N-dimensional phase space. For each particle, we can calculate two trajectories, starting from two distinct points in phase space initially separated by a distance |δ𝐙(t_0)| at time t_0.The separation distance |δ𝐙(t)| at any time t > t_0, t →∞ can be calculated as follows:lim_t →∞| δ𝐙(t) | = |δ𝐙(t_0)| lim_t →∞ e^λ t,where λ is the Lyapunov exponent for each particle. We can reformulate Eq. <ref> to highlight λ as follows <cit.>:λ = lim_t →∞1/tln(|δ𝐙(t)| / |δ𝐙(t_0)| ).Therefore, for our system there will be a set of 6N Lyapunov exponents, known as Lyapunov spectrum. Because our system is symplectic, the volume of the phase space is preserved, thus resulting in a total of 3N negative and 3N positive Lyapunov exponents, the sum of them being zero.The presence of positive Lyapunov exponents is a necessary but not sufficient condition for a dynamical system to be defined as chaotic. However, in chaotic systems, the larger the LEs, the faster the chaotic system will become unpredictable. For this reason, the most important Lyapunov exponent is the largest, commonly referred in literature as Maximum Lyapunov Exponent (MLE), and it is obtained by assuming that the initial separation is infinitesimally small, as follows <cit.>:λ_M= lim_t →∞ lim_| δ𝐙(t_0) | → 0 1/tln (|δ𝐙(t)|/| δ𝐙(t_0) | ),where λ_M is the MLE. Lyapunov exponents can represent the exponential separation between trajectories of the same system for two infinitely close initial states, but could also represent the separation between the real trajectory of a dynamical system and its discrete, finite-state, computable surrogate <cit.>. The similarity between Eq. <ref> and Eq. <ref> is already evident. In fact, the two formulations are isomorphic.In this article we want to demonstrate such isomorphism, list and briefly discuss what we believe to be the most important implications of this relationship both in Information Theory and in Physics. § ISOMORPHISMLet Ψ be an noiseless, memoryless, channel conveying an initial separation distance X = | δ𝐙(t_0) | at time t_0 to a final separation distanceY = | δ𝐙(t) | at time t, with t →∞ in a Hamiltonian dynamical system:Ψ : X → Y . The signal S sent by the channel can be quantified, according to Eq. <ref>, as follows:S = log_2( Y/X ) = log_2( | δ𝐙(t) |/| δ𝐙(t_0) | ) .Hence, the channel capacity λ_Ψ of the channel Ψ can be calculated as:λ_Ψ= lim_t →∞1/tlog_2( Y/X )= lim_t →∞1/tlog_2( | δ𝐙(t) |/| δ𝐙(t_0) | ) .The forms of the Eq. <ref> and <ref> are similar. Since an isomorphism <cit.> exists between logarithms:log_x(a) ≅log_y(a)/log_y(x) ∀{ a, x ∈ℝ^1 | a, x > 0} ,by induction, we identify the following isomorphism:λ≅λ_Ψ .□§ DISCUSSIONTo the best of our knowledge, the isomorphism presented above has never been formalized before. Each implication is deep enough to deserve the writing of a separate manuscript. Therefore, we will only list and briefly comment on each of the most important consequences of this isomorphism, both from Chaos Theory and Information Theory perspectives.We would like to refresh the reader about Artificial Neural Networks (ANNs) being defined as universal encoders in Information Theory <cit.>. The reported isomorphism implies that ANNs can be thought of and can be used as non-linear, chaotic-approximant memory storage units, implementable in hardware, able to learn with enough accuracy any dynamical system trajectory. ANNs can therefore be used to predict states of the system that were not part of the input dataset, without explicitly calculating the system's equations of motion.The Lyapunov Exponent (or channel capacity) of a dynamical system can be seen as the amount of bits necessary to keep track of the time evolution of the system's states with satisfactory accuracy. Because of that, this isomorphism represents the first step towards the creation of a new field of investigation based on the adoption of Deep Learning and other Machine Learning methods to accurately simulate dynamical systems trajectories without explicitly solving the equations of motion. Such approach represents a revolution in the field of computational physics. It is well known <cit.> that Information Theory models are comparable to 1D recursive maps in Chaos Theory. This isomorphism implies that the computation of any N-dimensional dynamical system can always be reduced to compute 1D recursive maps without loss of generality: we can call this new definition asprinciple of computer-representability invariance of a physical system. From such principle it immediately follows that the complexity reduction of data representation translates into increased encoding complexity. This suggests that lossless compression could be seen as a complexity-preserving transformation.Another important implication of this isomorphism is that a new metric, the Maximum Channel Capacity C_M, can be defined for a noiseless, memoryless channel as follows:C_M = lim_t →∞ lim_X → 01/tlog_2( Y/X ) ,which we proved hereby to be isomorphic to Eq. <ref>. Differently from what was already reported by Shannon (Eq. <ref>), we introduced the limit lim_X → 0, which has the physical meaning of an infinitesimally small message to be sent through a channel.In this form, we actually measure the maximum channel capacity needed for a particular channel independently on the size of the input, rather than a specific, input-dependent channel capacity, thus defining the upper limit of possible capacities for the range of accepted inputs.Kolmogorov-Sinai (KS) Entropy, a very important and well-known metric for the study of dynamical systems, can be defined as the sum of the positive Lyapunov exponents of a dynamical system <cit.>. This isomorphism reveals that analogously, KS Entropy can also be measured as the total bit rate corresponding to the sum of the positive channel capacities of a communication system. Hence, this result extends the validity of Lyapunov analysis to all known systems treatable by Information Theory, although we should remark that Lyapunov exponents have been previously used, not surprisingly, to determine the channel capacity of finite-state Markov channels and memory channels <cit.>.We are also aware of the work done by M. Ebeid about the stochastic relationship between LEs and Information Theory limits, using Control Theory models <cit.>. Our work validates and significantly extends the results previously reported by M. Ebeid. We prove that the LE is not simply equal to the communication rate in some broad family of channels, as M. Ebeid reported, but the LE is in fact isomorphic to the communication rate. We additionally propose a fully deterministic formulation, in net contrast with the exclusively stochastic approach adopted by M. Ebeid, suggesting that systems studied by information theory can also be legitimately described deterministically and viewed in physics as dynamical systems.§ CONCLUSIONSIn conclusion, this isomorphism constitutes a decisive proof that Information Theory models can be adopted for the analysis and characterization of dynamical systems simulation data, including the possibility of rationally using Machine Learning to accelerate the sampling of the phase space of Hamiltonian systems. § ACKNOWLEDGEMENTSThis work has been supported in part by the Joint Design of Advanced Computing Solutions for Cancer (JDACS4C) program established by the U.S. Department of Energy (DOE) and the National Cancer Institute (NCI) of the National Institutes of Health and was performed under the auspices of the U.S. Department of Energy byLawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. IM release number LLNL-TR-733786. It was also partially supported by a Lawrence Livermore Laboratory Directed Research & Development grants (17-ERD-096, 17-SI-004, and 18-ERD-021). Any findings and conclusions are those of the authors, and do not necessarily reflect the views of the funders. We want to warmly thank Prof. Angelo Vulpiani, Dr. Aaron Wilson, and Dr. Sachin S. Talathi for suggestions, and Dr. Jeffrey Hittinger for illuminating suggestions and funding support. 1mackay2003 MacKay DJC (2003) Information theory, inference and learning algorithms. (Cambridge university press).shannon2001mathematical Shannon CE (2001) A mathematical theory of communication. ACM SIGMOBILE Mobile Computing and Communications Review 5(1):3–55.vulpiani2009 Cencini M, Cecconi F., Vulpiani A. (2009) Chaos: From Simple Models to Complex Systems. (World Scientific).weisstein Weisstein EW (2017) Isomorphism. MathWorld.caiani1997 Caiani L, Casetti L, Clementi C, Pettini M (1997) Geometry of dynamics, lyapunov exponents, and phase transitions. Phys. Rev. Lett. 79(22):4361.misha1998 Dzugutov M, Aurell E, Vulpiani A (1998) Universal relation between the kolmogorov-sinai entropy and the thermodynamical entropy in simple liquids. Phys. Rev. Lett. 81(9):1762.vulpiani1 Boffetta G, Cencini M, Falcioni M, Vulpiani A (2002) Predictability: a way to characterize complexity Phys. Reports 356(6):367-547.vulpiani2 Cecconi F, Cencini M, Falcioni M, Vulpiani A (2012) Predicting the future from the past: An old problem from a modern perspective Am. J. of Phys. 80(111):1001-1008.holliday2006 Holliday T, Goldsmith A, Glynn P (2006) Capacity of finite state channels based on lyapunov exponents of random matrices. IEEE Trans. on Inf. Th. 52(8):3509.pfister2003 Pfister HD (2003) On the Capacity of Finite State Channels and the Analysis of Convolutional Accumulate-m Codes. (UC San Diego).hani M Hebeid HJ (2010) Relating Information-Theoretic Limits to the Lyapunov Exponent of a Dynamical System (University of Illinois at Urbana-Champaign) | http://arxiv.org/abs/1706.08638v5 | {
"authors": [
"Gerald Friedland",
"Alfredo Metere"
],
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"primary_category": "cond-mat.stat-mech",
"published": "20170627013411",
"title": "Isomorphism between Maximum Lyapunov Exponent and Shannon's Channel Capacity"
} |
/ #1#2#1#2 | http://arxiv.org/abs/1706.08510v3 | {
"authors": [
"James M. Cline",
"Jorge Martin Camalich"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170626175950",
"title": "$B$ decay anomalies from nonabelian local horizontal symmetry"
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Saturable nonlinearity-induced mode-locked laser self-starting]Effect of saturable nonlinearity on cw stability in passively mode-locked lasers with fast saturable absorbers^1 Laboratory of Research on Advanced Materials and Nonlinear Sciences (LaRAMaNS), Department of Physics, Faculty of Science, University of Buea P.O. Box 63, Buea Cameroon The self-starting dynamics of a model for passively mode-locked lasers with saturable absorber, in which the optical amplifier has a saturable nonlinearity, is examined. The basic assumption is that the laser will operate in the mode-locked regime when the continuous-wave regime becomes unstable. Within the framework of the modulational-instability analysis, a global map for the laser self-starting criteria is constructed. According to this map, the saturable nonlinearity enhances the input-field intensity required for laser self-starting. Analytical expression of the zero modulation-frequency regime of this threshold input field is derived, which turns out to be valid in the normal as well as in the anomalous dispersion regime, where evidence of self-starting is also given.Keywords: Operation regimes of laser optical systems, modulation and mode locking, optical systems with saturable nonlinearities§ INTRODUCTIONMode-locked fiber lasers with saturable absorbers have become a particularly attractive source of high-intensity short pulses <cit.>, for use in a vast area of modern communication technology. In these optical devices, an intensity fluctuation acts in conjunction with the fiber nonlinearity to modulate the cavity loss without need of some external control. Although several mode-locking techniques have been reported, for applications involving ultrashort pulses produced at finite repetition rates passive mode-locking remains the most preferred.Passive mode-locking rests essentially on an appropriate choice of the gain medium, in this regard rare-earth-doped fiber amplifiers have demonstrated high efficiency in femtosecond pulse multiplexing and soliton-train laser applications <cit.>. Indeed this specific class of fiber-based gains is characterized by long upper-state lifetimes, so long that the gain changes only slowly within the cavity roundtrip. In general, because of the slow gain change, a fast saturable absorber will be required to clean up both the leading and the trailing edges of the pulse. Erbium-doped fibers <cit.> are the most appreciated choice among rare-earth-doped fiber amplifiers, they consist of silica optical fibers doped with rare earth ions (Er^3+), where the core of the amplifier is smaller than in typical fibers so as to increase the available density of erbium ions thus decreasing the optical pump threshold. These optical fibers can be transformed into laser amplifiers by simply adding positive feedback mechanisms, in this way erbium-doped fiber amplify optical signals by means of stimulated emission inducing a population inversion, where erbium ions are raised from their natural ground state to a higher energy level <cit.>. Doping also provides means for controlling nonlinearity of the amplifier, which can be tuned from weak to strong <cit.>. Though passively mode-locked fiber lasers can display a wealth of operation regimes including continuous-wave (cw), plane-wave, period-doubling cascade, soliton and chaotic regimes <cit.>, in most applications it is desired that the device operates in the pulse regime. In this respect, a fundamental issue in recent studies of passively mode-locked lasers with saturable absorber has been their operation regime. As pioneer on this issue Haus proposed <cit.> two possible schemes, the first involves direct simulations of the entire evolution of the optical light starting from noise, while the second focuses on the evolution of light intensity from cw. The second scheme rests on a master equation describing the propagation of the optical field in the laser cavity over roundtrips, thus allowing one follow its evolution from cw to pulse. If numerical simulations provide details about the evolution of the optical field from noise to either stable pulse or chaotic structures <cit.>, it does not permit a global view of the field evolution over a relatively broad range of values of characteristic parameters of the laser system. On the contrary the second scenario, which is more analytical, puts into play the modulation of an input field of arbitrary intensity over the cavity roundtrips. Most importantly it has the merit to involve the cw as one of the transient regimes preceeding the fully pulsed regime, a feature reminiscent of the phenomenon of modulational instability suggesting that the cw will grow over roundtrips in the laser cavity, and should become unstable above some threshold intensity.The self-starting mechanism of passively mode-locked lasers has been discussed previously <cit.> by considering a complex Ginzburg-Landau equation (CGLE) with cubic nonlinearity, coupled to a two-level type dynamic gain. However, for applications involving pulse multiplexes, higher-order nonlinear terms and saturable nonlinearities in general <cit.> are required to favor multi-periodic solitons and bound-soliton states<cit.>. Therefore in this study we shall examine the self-starting dynamics of a mode-locked laser with saturable absorber as well as a saturable nonlinearity of the active medium. We follow the modulational instability analysis which, as stressed above, provides a global map of the dynamics of the laser system in different (i.e. cw and pulse) regimes of operation. In this approach, the laser is assumed to self-start when the cw regime is unstable and the laser operates in the mode-locked regime. Therefore, the main objective of this work is to investigate the effect of saturable nonlinearity on the self-starting feature of the family of mode-locked lasers considered.§ STEADY-STATE CW SOLUTION We are interested in the self-starting dynamics of a passively mode-locked laser with fast saturable absorber, for which the optical amplifier has a saturable nonlinearity. Typical examples of optical amplifiers with saturable nonlinearity are semiconductor-doped glass fibers, indeed doping in these materials leads to non-cubic and sometimes extremely high optical nonlinearities <cit.> as compared with conventional optical amplifiers with Kerr (i.e. cubic) nonlinearity <cit.>. The propagation of the laser field is governed by the following saturable-nonlinearity CGLE: ∂ U/∂ z= (g - ℓ + iθ)U + (B + iD)∂^2 U/∂ t^2 + Γ +iK/1+γ| U|^2| U|^2 U,where U(z,t) is the optical field, z is the cavity roundtrip number, t is time, g is the gain, ℓ is the constant loss and θ is the phase change over each roundtrip. The characteristic parameters C and D account respectively for the spectral filter and group-delay dispersion, Γ and K are the fast saturable absorber and nonlinearity coefficients respectively, and γ accounts for nonlinearity saturation in the fiber amplifier. The gain dynamics will be described by the following equation <cit.>:dg/dt=-(g-g_0)/T_0-|U(z,t)|^2/T_0 P_sg,where g_0 is the homogeneous gain, P_s is the saturation power of the saturable absorber and T_0 is the gain relaxation time. When γ=0, the above model reduces to the case discussed by Chen et al. in ref. <cit.>. For small γ, the last term in eq. (<ref>) can be expanded. This leads among others to the CGLE with a cubic-quintic nonlinearity <cit.>. The self-starting dynamics in the case when the fiber amplifier has a cubic nonlinearity has been investigated by Chen et al. <cit.>, within the framework of the modulational-instability approach. In the present study we shall examine the effect of nonlinearity saturation on self-starting, starting with the analysis of characteristic features of plane-wave solutions in the steady state. In this context solutions to the coupled set (<ref>)-(<ref>) are given by:U(z)= √(P_c) e^i q_s z,0.5truecm g(t)= g_s,where P_c=U_c^2, q_s is the plane-wave wave-number and g_s is the steady-state gain. Replacing these in eqs. (<ref>) and (<ref>) and separating real from imaginary parts, we obtain:g_s= ℓ - Γ P_c/1+γ P_c = g_0/1 + P_c/P_s, q_s= θ + K P_c/1+γ P_c.Eqs. (<ref>) and (<ref>) determine the steady-state gain g_s and the wave-number q_s for which the laser has a plane-wave form. Eq. (<ref>) is particularly relevant since according to eq. (<ref>), the existence of plane wave will depend on the balance between the gain g and the loss ℓ. It follows from eq. (<ref>) that this balance should be determined by P_c and γ, for a given saturation power P_s of the saturable absorber. Fig. <ref> summarizes the laser self-starting in the steady state regime, where the small signal power gain is defined as in ref. <cit.> by exp(2g_0) and where we considered three different values of γ, namely γ=0, γ=0.1 and γ=0.5.§ CONTINUOUS-WAVE STABILITY To investigate the stability of the cw field U(z) given in formula (<ref>), we carry out a modulational-instability analysis by following the evolution of the cw as it co-propagates with a plane-wave noise. Thus, consider a small perturbation ũ(z,t) to the cw and a small deviation g̃ from the steady-state gain g_s, such that solutions to eqs. (<ref>) and (<ref>) now read: U(t) = [√(P_c) + ũ(z,t)]e^(iq z),g(t) =g_s + g̃.Replacing these in eqs. (<ref>) and (<ref>) and linearizing, we find:ũ_z= αũ_tt + (ũ + ũ^*)F_u + √(P_c)g̃,g̃_t =-g̃/T_e + F_g(ũ, ũ^*),with α =B+iD,0.5truecm F_u=P_cΓ + iK/(1+γ P_c)^2, F_g =-ϵ_c/T_e(ũ+ũ^*),0.2truecm ϵ_c= g_0√(P_c)/P_s(1+P_c/P_s)^2,and the effective relaxation time T_e is defined as:T_e= T_0/1+P_c/P_s.The linear inhomogeneous first-order ordinary differential equation eq. (<ref>) can be solved by means of the Green-function method yielding:g̃(t)=∫_-∞^tG(t,t')F_g[ũ(t'), ũ^*(t')]dt',where G(t,t')= e^-(t-t')/T_eH(t-t') is the Green function with H(t-t') the step function. Now assuming:[ũ(z,t), ũ^*(z,t)]= [A_1, A_2] e^(κ z + iω t),where κ is the rate of spatial amplification of the noise and ω the associate time modulation frequency, eqs. (<ref>) together with its complex conjugate lead to the following secular equation in matrix form:κ( [ A_1; A_2 ])=[( [ m_1 m_2; m_2^* m_1^* ])-m_0 ( [ 1 1; 1 1 ])] ( [ A_1; A_2 ]),with:m_1=-α + F_u,0.5truecm m_2= F_u,0.5truecm m_0= √(P_c)ϵ_c/1+iω T_e.The determinant of the above 2× 2 matrix gives rise to a quadratic polynomial in the associate eigenvalue κ, the two possible roots of which read:κ_1,2 = Γ P_c/(1+γ P_c)^2 - Bω^2 - m_0 ± √([m_0 - Γ P_c/(1+γ P_c)^2]^2 - (Dω^2)^2 + 2D K P_cω^2/(1+γ P_c)^2),where the subscripts 1, 2 refer to the plus and minus signs, respectively. According to formula (<ref>), the cw will be unstable (and hence the laser will self-start) if the real part of κ is positive. It turns out that at zero modulation frequency, when κ_1= 0 and κ_2=2Γ P_c/(1+γ P_c)^2 - 2√(P_c)ϵ_c, we need ϵ_c < Γ√(P_c)/(1+γ P_c)^2 for the laser to self-start. Quantitatively, this condition implies two characteristic values of P_c above which self-starting can occur. One of them is negative and hence unstable, while the positive one,P_c^(γ)=P_c^(0)/1 - γ√(P_s g_0/Γ),sets a threshold input-field intensity above which the laser will self-start. Note that in the case of mode-locked lasers with a cubic-nonlinearity optical amplifier this threshold value should be:P_c^(0)=P_s(√(g_0/P_sΓ) - 1).For a best understanding of laser self-starting when the two eigenvalues vary with the modulation frequency ω, we resorted to a global mapping of the κ_1,2 in the complex plane. Thus, fig. <ref> are parametric plots of the two eigenvalues for the modulation frequency in the range -5≤ω≤ 5, where the imaginary part Im(κ) is plotted as a function of the real part Re(κ). Values of characteristic parameters of the model are indicated in the caption.According to the graphs, self-starting is stronger for the cubic nonlinearity and is enhanced by an increase of P_c. However, when we increase γ for a fixed value of the input intensity P_c, the cw field bedomes highly unstable. Actually this later observation is consistent with the dependence of the threshold value P_c^(γ) of P_c, on the nonlinearity saturation coefficient γ obtained in formula (<ref>) and suggesting a higher input field for laser self-starting when the nonlinearity is of a saturable type. It is worthwile recalling that the above analysis assumes the laser will self-start (automatically in the pulse regime) when the cw regime is unstable. On the other hand in our discussions we assumed that the optical system is in a normal dispersion regime (i.e. B and D are positive), where the CGLE is equivalent to the NLSE such that the system admits quasi-Schrödinger sech-type pulses in the mode-locked regime. Still, although negative group-delay dispersion and spectral filter act against NLS sech-type pulses, experiments have demonstrated that pulses can still form in this case. The most interesting experimental evidences have been reported in ref. <cit.>, where multi-periodic and bound pulse states have been shown to form when B and D cross zero from the positive branch. As the two parameters decrease in the negative branch, multiple-pulse structures sharpen while pulse durations get shorter and shorter. So to say self-starting is also possible in the anomalous dispersion regime, and in the present particular context this is evidenced by the parametric plots of κ_1 and κ_2 shown in fig. <ref>.§ CONCLUSION The present work is to be seen as an extension of the study carried out in ref. <cit.>, to the general context of mode-locked lasers with highly nonlinear optical amplifiers. In this respect, while we assume, as in this previous work, that the laser will self-start (i.e. operate instantaneously) once the cw regime is unstable, the modulational-instability analysis can actually only provide relevant information about the stability of the cw regime. It strictly cannot help identiy the exact structures of the pulse and/or multiple-pulse fields that would be stabilized in the mode-locked regime. To determine the exact structures of the laser field in the mode-locked regime we must solve the CGLE. In this task, numerical simulations will enable us explore all the possible operation regimes inherent to the dynamics of the above model. This is expected to reveal not just stable pulse or multipulse regimes, but also transient states characterized by period-doubling cascades and chaotic phases as established for the cubic-quintic CGLEs <cit.>. The work of A. M. Dikandé is supported by the Alexander von Humboldt foundation.§ REFERENCES99 haus1 Haus H. A. 1975 J. Appl. Phys. 46, 3049.haus2 Haus H. A. and Silberberg Y. 1986 IEEE J. Quantum Electron. 22, 325. haus3 IppenE. P., Haus H. A. and Liu L. Y. 1989 J. Opt. Soc. Am. B6, 1736. gordon1 Martinez O. E., Fork R. L. and Gordon J. P. 1984 Opt. Lett. 9, 156. gordon2 Well R., Vodonos R. 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B., Borges N. M., Gouveia E. A. and Gouveia-Neto A. S. 1993 Opt. Lett. 18, 182. hick1Lyra M. L. and Gouveia-Neto A. S. 1994 Opt. Commun. 108, 117. zhao2Zhao M. L., Tang D. L. and Zhao B. 2005 Opt. Commun. 252, 167. kart Kärtner F. X., Jung I. D. and Keller U 1996 IEEE J. Selec. Topics Quantum Electron. 2, 540. haus4 Haus H. A., Ippen E. P. and Tamura K. 1994 IEEE J. Quantum Electron. 30, 200. akhmed4Soto-Crespo J. M. and Akhmediev N. N. 1999 J. Opt. Soc. Am. B16, 674. akhmed2Soto-Crespo J. M., Akhmediev N. N.and Town G. 2002 J. Opt. Soc. Am. B19, 234. akhmed3Soto-Crespo J. M., Grapinet M., Grelu P. and Akhmediev N. N. 2004 Phys. Rev. E70, 066612. akhmed5 Akhmediev N. N., Afanasjev V. V. and Soto-Crespo J. M. 1996 Phys. Rev. E53, 1190. | http://arxiv.org/abs/1706.09028v1 | {
"authors": [
"Alain M. Dikande",
"P. Achankeng Leke"
],
"categories": [
"physics.optics",
"nlin.PS"
],
"primary_category": "physics.optics",
"published": "20170627195252",
"title": "Effect of saturable nonlinearity on cw stability in passively mode-locked lasers with fast saturable absorbers"
} |
http://arxiv.org/abs/1706.08633v2 | {
"authors": [
"Fan Yang",
"Ranjith Nair",
"Mankei Tsang",
"Christoph Simon",
"Alexander I. Lvovsky"
],
"categories": [
"physics.optics",
"quant-ph"
],
"primary_category": "physics.optics",
"published": "20170627005633",
"title": "Fisher information for far-field linear optical superresolution via homodyne or heterodyne detection in a higher-order local oscillator mode"
} |
|
[email protected] de Física, CCE, Universidade Federal do Espírito Santo, Av. Fernando Ferrari 514, Vitória, ES, 29075-910 [email protected] de Física, CCE, Universidade Federal do Espírito Santo, Av. Fernando Ferrari 514, Vitória, ES, 29075-910 Brazil [email protected] de Física, CCE, Universidade Federal do Espírito Santo, Av. Fernando Ferrari 514, Vitória, ES, 29075-910 BrazilInstitute for Theoretical Physics, University of Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany [email protected] de Física, CCE, Universidade Federal do Espírito Santo, Av. Fernando Ferrari 514, Vitória, ES, 29075-910 Brazil We apply the full Will-Nordtvedt version of the Parameterized Post-Newtonian (PPN) formalism to a class of General Relativity extensions that are based on nontrivial renormalization group (RG) effects at large scales. We focus on a class of models in which the gravitational coupling constant G is correlated with the Newtonian potential. A previous PPN analysis considered a specific realization of the RG effects, and only within theEddington-Robertson-Schiff version of the PPN formalism, which is a less complete and robust PPN formulation. Here we find stronger, more precise bounds, and with less assumptions.We also consider the External Potential Effect (EPE), which is an effect that is intrinsic to thisframework and depends on the system environment (it has some qualitative similarities to the screening mechanisms of modified gravity theories). We find a single particular RG realization that is not affected by the EPE. Some physical systems have been pointed out as candidates for measuring the possible RG effects in gravity at large scales, for any of them the Solar System bounds need to be considered. Will-Nordtvedt PPN formalism applied to renormalization group extensions of general relativity Nicolas Bertini December 30, 2023 ============================================================================================== § INTRODUCTIONThe use of general relativity (GR) onthe very small scales of the universe, or onthe very large ones, leads to inconsistencies or to the need of unexpected new features in the universe. These issues related with GR include quantum-gravity issues <cit.>, the cosmological dark sector <cit.>, inflation <cit.>, and perhaps smaller details like the small scale issues of the standard cosmological model <cit.>. On the other hand, GR has achieved great success through several tests, specially at the Solar System scale <cit.> (see however <cit.> for some anomalies). Here we analyse a class of GR extensions that is based on Renormalization Group (RG) expectations considering gravity at large distances. There are different approaches for extendingGR using RG effects, both in the high and the low energy limits <cit.>, these also include the asymptotic safety approach to quantum gravity <cit.>. We consider the RGGR (Renormalization Group extended General Relativity) approach <cit.>, which extends and generalizes the proposals of Refs. <cit.>. One of the characteristic features of this approach is the use of a correlation between the RG scale μ and the Newtonian potential in the context of stationary and weak field systems (some other proposals use μ∝ 1/r, which coincides with the RGGR proposal for point particles, see also Ref. <cit.>). AnotherRGGR feature is the use of a constant infrared β-function[A β-function of a coupling X is defined by β_X ≡μ∂ X/ ∂μ, where μ is the RG scale <cit.>. From the integration of the β-function one finds X(μ).] for the gravitational coupling constant G (as explained in Ref. <cit.>, for instance). The existence of a correlation between μ and the Newtonian potential is here assumed, but this work is not restricted to a specificform of this correlation, or to a single specific β-function. Some different systems have been considered for evaluating large scale RG effects in gravity, from galaxies to cosmology. Considering galaxies, previous tests of RGGR in galaxies have found that the non-Newtonian effects can act as a kind of effective dark matter if ν̅≳ 10^-9 <cit.>, otherwise the effect could still be true but it would have negligible impact as a kind of dark matter, even for the smallest galaxies. The dimensionless parameter ν̅ sets the strength of the RG effects in a given system, and it is such that ν̅=0 corresponds to classical GR.The Solar System data have always to be considered, since it provides some of the clearest and precisest results on gravity. The first work on RGGR and the Solar Syetem used the Laplace-Runge-Lenz (LRL) vector and found|ν̅_⊙| ≲ 10^-17 <cit.>. A second work evaluated a number of different observations in the Solar System, to concludethat |ν̅_⊙| ≲ 10^-21 <cit.>. A third work used up to date data on LRL vector and the Eddington-Robertson-Schiff PPN formulation to find |ν̅_⊙| ≲ 10^-16 <cit.>. The third work expressthe best bound on ν̅ at the Solar System up to this paper. This bound is softer than the others since the corresponding paper was the first to notice andconsider an effect that depends on the system environment and it is part of RGGR: the external potential effect (EPE). Qualitatively, this effect can be explained as follows, for a non-relativistic system: the higher is the value of the Newtonian potential due to matter outside the system, the lower are the non-Newtonian effects inside the system. This effect was not imposed as an additional feature, it was already present in the theory, but it was neglected in previous Solar System analyses. It has superficial similarities withthe screening mechanisms of modified gravity <cit.>. It is also important to stress that all these bounds consider a particular form for the β-function of G, and a particular form for the scale setting. Although they are natural options, they constitute nonetheless additional hypothesis that will not be necessary to the main results of this paper.Here we apply the full Will-Nordtvedt version of the Parameterized Post-Newtonian (PPN) formalism (see <cit.> for a review) to a class of RG extensions of GR that includes RGGR as a particular case. The development of the PPN formalism relied and still relies on many researches, took many years and has currently some different bifurcations (e.g., <cit.>). Probably the most well known and simplest version is the Eddington-Robertson-Schiff PPN formulation, which essentially depends on two parameters, γ and β. Theseare derived from theory solutions considering a massive point within a static and spherically symmetric space-time (for a review, see e.g. <cit.>). This massive particle would represent the Sun and test particles would be in place of the planets or photons. The first parameter can be found from the measurement of light bending, while the second fromMercury's orbit precession. This simpler PPN formulation was applied to RGGR in Ref. <cit.>. Nonetheless, a theory whose values of γ and β are compatible with observations may be incompatible with other experiments. Also, the use of static spherically symmetric space-time and “point” particles are just rough approximations which in general are not irrelevant for post-Newtonian dynamics <cit.>. The Will-Nordtvedt version considers more tests, it depends on 10 parameters (nine of them are observationally constrained) and it is based on fluids, not on particles.This work is organized as follows: in the next section we present a review on RGGR that focuses on its main features that are important for the PPN evaluation. The review includes the original noncovariant formulation and the newer covariant version. Section <ref> briefly reviews a few essential PPN features and apply them to RG extensions of GR. In Sec. <ref> the observational bounds are determined for any case in which G is an analytical function of the Newtonian potential. The latter section also considers the EPE and two particular classes of β-functions, one of them being the RGGR case. Our conclusions are presented in Sec. <ref>. In the appendices <ref> and <ref> considerations on the covariant formulation are presented, and it is shown that the standard Will-Nordtvedt PPN formalism cannot be applied to the full covariant formulation.§ LARGE SCALE RG EXTENSIONS OF CLASSICAL GR §.§ A brief review on RGGR and a larger class of theoriesNot all RG extensions of GR at large scales have an effective action that captures all the dynamical information, without the need of imposing field equations that are outside the action. For instance, as classified in Ref. <cit.>, some consider the RG improved equations (e.g., <cit.>), in which the coupling constants are promoted to running ones at the level of the field equations. In the case of RGGR, this promotion is done at the level of the action, and hence it is a case of RG improved action (see e.g., <cit.>). Moreover, it has an action that leads to all the field equations <cit.>. At the action level, RGGR depends explicitly on the dimensionless constant ν, which is such that ν=0 corresponds to classical GR (i.e., the β-function of the gravitational coupling becomes zero). According to the RGGR action proposal of Ref. <cit.>, which extends and generalizes the proposals in Refs. <cit.>, large scale RG effects can bedescribed by an effective action which reads (using c=1),S =∫[ R - 2 Λ{μ}/16 π G(μ)+ λ( μ - f(g, γ,Ψ))] √(-g) d^4x + S_m.In the above,S = S[g,γ, μ,λ,Ψ], S_m = S_m[g,Ψ], Ψ stands for any matter fields of any nature, and μ is the RG scale, whose relation to all the other fields is stated in the action in a constraint-like way, as imposed by the Lagrange multiplier λ. The field γ_αβ is called the reference metric, it only appears inside f, without derivatives, and its variation at the action level ensures energy-momentum conservation <cit.>.The scalars G and Λ depend on the RG scale μ, but in different ways. Namely, G is a standard function of μ, which is fixed at the action level. The relation between Λ and μ is not fixed at the action level, but it can and must be derived from the field equations; equivalently, this means that the corresponding β-function of Λ is not universal, and it depends on the matter fields. Examples on how to derive Λ for different systemscan be found in Ref. <cit.>. This system-dependent relation between Λ and μ is stressed by the use of a different notation, namely Λ{μ} instead of Λ(μ). In essence this implies that a local analysis of Λ cannot determine its global behaviour, andthat Λ is not in general an analytical function ofμ. Before proceeding, a comment on the nature of γ_αβ and background independence is in order. The splitting of the spacetime metric into a background plus quantum corrections is a convenient procedure that is largely used in the context of unveiling RG effects ingravity. Nonetheless, it is expected that the physical phenomena uncovered from this splitting does not depend on the chosen background, that is, the RG effectsshould be background independent <cit.>. Considering the f function as proposed in Ref. <cit.>, see also Appendix <ref>, the dependence on the reference metric γ_αβ is such that, when the metrics coincide, the RG effects become null and one recovers classical GR.[More precisely,if, in a given neighborhood of a spacetime pointthe metrics satisfy g_αβ = γ_αβ, then in that neighborhood there are no RG effects.] Thus, γ_αβ sets a background for the RG effects. Formally, the action (<ref>) is background independent in the sense that there is no particular geometry that is preferred. On the other hand, different coices ofγ_αβ at the level of the field equations lead to different solutions for the spacetime metric. Therefore, in the sense of the split symmetry, as discussed for instance in Ref. <cit.>, action (<ref>) is not explicitly background independent. This does not imply that γ_αβ is a physically independent quantity, only that action (<ref>) does not handle background changes (i.e., changes of γ_αβ within a fixed coordinate system). In classical GR, sometimes the boundary conditions are not obvious and one has to use physical intuition (or large computational efforts) to discover the proper boundary conditions. In the case of the RG extended action (<ref>), this problem includes finding the proper reference metric.[Actually the problem is much simpler, since for the proposed f function one only has to specify a scalar quantity, u^αu^βγ_αβ (see Appendix <ref>).] For systems that are close to Newtonian, the most natural assumption for γ_αβ is the Minkowski metric, in this case the RG effects depend on the Newtonian potential, and this is the choice assumed in this work. This choice is relevant for the passage from the covariant action (<ref>) to the noncovariant RGGR formulation (further details are in Appendix <ref> and in Refs. <cit.>). Considering the field equations, fromaction (<ref>), the variation with respect to λ yields the scale setting μ = f, the variation with respect to μ yields a condition between G, Λ, λ which ensures that the matter energy-momentum tensor satisfies[Due to the constraint term that depends on both μ and the matter fields, the diffeomorphism invariance of S_m is not sufficient to assure that T_μν is conserved <cit.>.]∇_α T^α_β∝λ. The variation with respect to γ_αβ sets λ = 0 at the level of the field equations (whenever ∂ f / ∂γ^αβ≠0), thus ensuring energy-momentum conservation (see Ref. <cit.>for further details). At last, the variation withrespect to the metric yields G_αβ + Λ g_αβ = 8 π G T_αβ,where G_αβ≡ G_αβ +g_αβ GG^-1- G ∇_α∇_β G^-1,≡ g^αβ∇_α∇_β, and ∇_α is thecovariant derivative.If it is possible to neglect the contribution from Λin the Solar System up to first post-Newtonian order, which seems natural since Λ should be a correction to the cosmological constant Λ_0 that appears in classical GR, it is useful to write eq. (<ref>) asR_αβ= G[8π (T_αβ-1/2g_αβ T)++∇_α∇_β(G^-1) + 1/2g_αβ(G^-1)] . The Λ term was not considered in the previous Solar System analysis of RGGR <cit.>. In Appendix <ref> we comment on the possible effects of Λ and show that the Will-Nordtvedt PPN formalism in its standard form cannot handle the Λ term, in particular because Λ cannot be both an analytical function and be compatible with asymptotic flatness.§.§ Running gravitational constant and scale settingFor concreteness, it helpful to present an example for the G(μ) function. We present below a simple expression for G(μ) that some of us used in previous publications, and which was also derived from different RG approaches, namely <cit.>,G(μ) = G_0/1 + 2 νln (μ/μ_0),where G_0 and μ_0 areconstants such that G(μ_0) = G_0, and ν is a small dimensionless constant. GR is recovered withν=0. We present further details on the consequence of this expression in Sec. <ref>, but our main results in this work are not limited to this expression. From the action (<ref>), the relation of the scale μ to other physical quantities (i.e., the scale setting <cit.>) is a field equation that comes from the variation of the action with respect to the Lagrange multiplier λ. Contrary to some other approaches, the scale setting is not an additional equation outside the action, it is derived from the action. In Ref. <cit.>, considering RG expectations within GR at large scales, some of us proposed that, within stationary weak field gravitational systems, there should be a function f such that, in a given reference frame,μ = f(U) .Our results do not depend on specifying a particular f function. In the above, U is the negative of the Newtonian potential[For conciseness, commonly we will call U the Newtonian potential, without writing “negative” in front of it. We use U since we are following the notation of Ref. <cit.> on the PPN parameters and the potentials.] <cit.>, and it is given by,U(t, x⃗) = G_N∫ρ(t, x⃗')/| x⃗ - x⃗'|d^3x' ,where G_N is the Newtonian gravitational constant (which may be different from G_0). The scale setting (<ref>) is a development over some previous RG application to gravity at large scales, in which it was used the qualitative relation μ∼ 1/r <cit.>, that is, a large value for the RG scale μ should correspond to small distances r. The use of the Newtonian potential is in qualitative agreement with the previous assumption, but it includes a dependence on the mass distribution and uses the most relevant potential for systems close to the Newtonian regime, that is, U.Since U is not a spacetime scalar, the scale setting (<ref>) is not covariant. In Refs. <cit.> we proposed a covariant generalization of the above scale setting, but we leave further details on the covariant version to the Appendix <ref>.As shown in the latter appendix, the covariant version leads to the appearance of potentials that are not part of the Will-Nordtvedt PPN formalism.In Ref. <cit.> we used eq. (<ref>) and presented a natural f(U) function which lead to a metric solution with spherical symmetry that could be handled through the Eddington-Robertson-Schiff PPN formalism. Here we will proceed with more generality,namely we will simply demand that G can be expanded as a function of U as follows,G^-1(μ) =G^-1(U)= G^-1_e + 2∑_n=1^∞ν_n U^n.With this parametrization, GR is recovered with ν_n =0. It will be shown that all the ν_n terms with n ≥ 3 are not relevant to the Solar System dynamics up to the first post-Newtonian order. In eq. (<ref>), ν_n are real constants and G_e is the value of G(U) when U=0. We use the index e in reference to the external value of G. That is, far from the Sun, the Newtonian potential of the Solar System should become close to zero, but the value of G at such distance may depend on the environment of the Solar System. This will be further developed in Sec. <ref> and the relation between G_e and G_N is shown in the next section.§ THE POST-NEWTONIAN APPROXIMATION In this section we apply the Will-Nordtvedt PPN formalism<cit.> to a RG extension of GR whose RG scale μ is correlated with the Newtonian potential. This formalism uses a perfect fluid as the gravitational source and describes the metric of a gravitational theory in terms of ten observable PPN parameters in a theory-independent way. The main small parameter of the formalism is the velocity field |v⃗| = v < 1. The metric is expanded about Minkowski spacetime,g_αβ=η_αβ+h_αβ ,whereη_αβ is the Minkowski metric, which is of zeroth-order on v, and h_αβ∼ O(v^2), at least. We use the signature (-,+,+,+).Up to the first post-Newtonian order, the metric must be known as follows: g_00 to order v^4, g_0i to order v^3 and g_ij to order v^2 (Latin indices run from 1 to 3). Thus, up to the required order, the Ricci tensor components can be expressed as R_00 =-1/2∇^2h_00 - 1/2(h^k_ k,00- 2 h^k_ 0,k0) - 1/4 |∇⃗h_00|^2++1/2 h_00,l(h^lk_ ,k- 1/2 h^k_ k,jδ^j_l)+ 1/2 h^klh_00,lk ,R_0i=-1/2(∇^2h_0i - h^k_ 0,ik + h^k_ k,0i - h^k_ i,k0) ,R_ij=-1/2(∇^2h_ij - h_00,ij + h^k_ k,ij- h^k_ i,kj-h^k_ j,ki).The comas refer to simple derivatives, ∇^2 ≡η^ij_i_j , and it was used that time derivatives effectively count as one order increase. Thus, if a quantity X is of order v^n then X,_k∼ O(v^n) and X,_0∼ O(v^n+1).Using eq. (<ref>) and thatU ∼ O(v^2) for systems not far from equilibrium, then∇_α∇_β(G^-1)=(G^-1)_, αβ - Γ_αβ^λ (G^-1)_, λ, =2ν_1( U,_αβ-Γ^λ_αβU,_λ) + +4ν_2 (U,_αU,_β + UU,_αβ) + O(v^6) . Since the gravitational source isa perfect fluid, T^μν=(ρ+ρΠ+p)u^μ u^ν + pg^μν ,where Π is the specific energy density, p is the pressure and u^μ=(u^0,v^i) is the four velocity of the fluid element, withu^0= √(1+v^2/1-h_00) ,such that u_μ u^μ=-1. The mass density ρ, Π and p/ρ are of order v^2 <cit.>.With the expressions above, we compute the metric components order by order on powers of v. h_00 up to order v^2 (Newtonian limit): Up to the required order,R_00=-1/2 ∇^2h_00 T_00=-T=ρ .Therefore,∇^2h_00=-8πρ + 2ν_1 ∇^2Uand, from eq. (<ref>),h_00=2/G_N(1+G_Nν_1)U . In order to be in agreement with the Newtonian physics,h_00=2U ,thuswe must set(1+G_Nν_1)=G_N .The equation above sets the relation betweenand G_N. Since this relation is now clear, henceforth weuse G_N =1 .Thus,=1/1+ν_1 .h_ij up to order v^2:Imposing the three gauge conditions,h^μ_ i,μ-1/2h^μ_ μ,i=2ν_1 U_,i ,the spatial part of eq. (<ref>) reduces to,∇^2h_ij= -8πρδ_ij - 2 ν_1∇^2Uδ_ij .The above equation is easily integrated,h_ij=2(1-2 ν_1/1+ν_1)U δ_ij ,where eq. (<ref>) was used.h_0i up to order v^3: With a fourth gauge condition,h^μ_ 0,μ-1/2h^μ_ μ,0=-1/2h_00,0 +3ν_1 U_,0and from eq. (<ref>),∇^2h_0i +U_,0i= 16πρ v_i .To integrate the above equation, we will use the super-potential χ(t,x⃗) <cit.>, which is given byχ(t,x⃗)≡∫ρ(t,x⃗')|x⃗-x⃗'| d^3x' .From the above definition,∇^2χ=-2U χ_,0i=V_i-W_i ,whereV_i=∫ρ(t,x⃗') v_i'/|x⃗-x⃗'| d^3x', ∇^2V_i=-4πρ v_i ,andW_i=∫ρ(t,x⃗')v⃗· (x⃗-x⃗')(x-x')_i/|x⃗-x⃗'|^3 d^3x' .Therefore, from eq. (<ref>) it resultsh_0i= -7V_i/2(1+ν_1) - W_i/2(1+ν_1) .h_00 up to order v^4: To develop the right hand side of the dynamical equation, we need the explicit expression of some components of the connection. To the required order, that terms areΓ^i_00=-U_,i ,Γ^k_ij= (1-2 ν_1/1+ν_1)(U_,iδ^k_j+ U_,jδ^k_i- U^,kδ_ij).For the energy-momentum tensor, up to order v^4, one findsT_00-1/2 g_00T=1/2 ρ[1+2(v^2-U+Π/2 + 3p/2ρ) ] ,where the expansion of eq. (<ref>) was used. By considering the gauge fixing conditions (<ref>), (<ref>), introducing the potentials below <cit.>,∇^2Φ_1=-4πρ v^2, ∇^2Φ_2=-4πρ U, ∇^2Φ_3= -4πρΠ , ∇^2Φ_4=-4π p,and using the relation,|∇⃗U|^2=∇^2(U^2/2-Φ_2) ,the dynamical equation can be integrated, leading toh_00=2U -2[1+ ν_1^2-ν_2(1+ν_1)/(1+ν_1)^2]U^2+ + 4Φ_1/1+ν_1 +2Φ_3/1+ν_1+ 6Φ_4/1+ν_1 + + [4(1-ν_1+ν^2_1)/(1+ν_1)^2-4ν_1/1+ν_1]Φ_2+O(v^6) . With the above, we conclude the expansion of the RGGR perturbations as a function of the PPN potentials. In the next section, we infer the values of the PPN parameters and compare with the observational values. § THE PPN PARAMETERS AND THEIR INTERPRETATION §.§ General analysis With the results obtained in the previous section, themetric up to the first post-Newtonian (1PN) order can be written asg_00=-1 +2U -2[1+ ν^2_1-ν_2(1+ν_1)/(1+ν_1)^2]U^2++(1-ν_1/1+ν_1)(4Φ_1 + 2Φ_3 + 6Φ_4)++ 4[1-4ν_1+ν^2_1/(1+ν_1)^2]Φ_2 , g_0i= (1-ν_1/1+ν_1)( -7V_i/2 - W_i/2) ,g_ij=δ_ij+ 2(1-2ν_1/1+ν_1)U δ_ij .To extract the PPN parameters from the above geometric structure we compare it to the Will-Nordtvedt generic post-Newtonian metric <cit.>, namelyg_00 =-1 + 2U - 2β U^2 + (2 γ +2+α_3 +ζ _1-2 ξ ) Φ_1++ 2(3 γ -2β+1+ζ _2+ ξ ) Φ_2 +2(1+ζ _3 ) Φ_3++ 2(3 γ +3ζ _4-2 ξ ) Φ_4 - (ζ _1-2 ξ )A-2ξΦ_W, g_0i =- 1/2(4 γ +3+α_1-α_2+ ζ_1-2ξ) V_i- - 1/2(1+α_2- ζ_1+2ξ) W_i , g_ij =(1+2γ U) δ_ij . From the coefficients of U in g_ij and U^2 in g_00, one infers the parameters γ and β as functions of ν_1 and ν_2. Using the data from Table <ref>, one finds|ν_1| < 1.2 × 10^-5, |ν_2| < 8 × 10^-5.We stress that the above considers only the observational constraints from γ and β, which are not all the observational constraints. Since ν_1 and ν_2 need to be much smaller than one, their relations to the PPN parameters can be expressed from linear expansions on ν_1 and ν_2, which readsγ=1- 2ν_1, β=1-ν_2, α_2=-ν_1 , ζ_2=-2(ν_1+ν_2) ,ζ_3=-ζ_4=-ν_1, α_1=α_3=ξ=ζ_1=0 . Using the relations above and the observational constraints of all the the PPN parameters, listed in Table <ref>, the resulting strongest constraints on the parameters ν_1 and ν_2 are displayed in Table <ref>. One sees that they do not come from β or γ, but from α_2 and ζ_2. There are well known examples of theories that come from an action and have α_1 and α_2 different from zero, which are related with special frame effects <cit.>, but theories with an action are not expected to yield non-zero values for any of the ζ's and α_3 if ξ = 0 <cit.>. On the other hand, we are not using the full covariant action, which demands energy-momentum conservation, but the noncovariant approximation. The derived bound from ζ_2 changes the bound found from the β parameter by a factor 4 (from eq. <ref>). That is, the noncovariant approximation works as an order of magnitude approximation, at the 1PN order, to the covariant version <cit.>. The situation would bedifferent in caseα_3 would depend on ν_1 or ν_2. Further considerations on the effects from Λ and the full covariant action are in appendices <ref> and <ref>. §.§ A constant infrared β-function and the External Potential EffectThere is a particular expression for G(U) that is well motivated and particularly simple. This expression was proposed in Ref. <cit.> and it readsG^-1 = G^-1_0 [ 1 + 2ν̅ln( U/U_0)],where ν̅ is a constant and U_0 is a reference potential (the one that satisfies G(U_0) = G_0). The above expression uses the following infrared β-function of G <cit.> (with c = ħ = 1),β_G^-1≡μ∂ G^-1(μ)/∂μ = 2 ν M^2_ = 2 ν G_0^-1,whose integration leads to G^-1(μ) = G_0^-1(1 + 2 νlnμ/μ_0). The latter expression is combined with the scale setting <cit.>μ = ( U/U_0)^α. In eq. (<ref>) we used ν̅≡να. The G(U) expression from eq. (<ref>) is notin general compatible with the expansion (<ref>), but it becomes compatible once the external potential effect (EPE) is considered <cit.>.Since in this picture G depends on the potential U, G will in general depend on both the matter distribution inside the system under investigation and also the matter outside it. Following Ref. <cit.>, we write,ρ = ρ_s + ρ_e , U = U_s + U_e ,where ρ_s refers to the matter density contribution that is inside the system under consideration, while ρ_e refers to the external mass density. The quantities U_s and U_e are computed from eq. (<ref>), but withρ replaced by ρ_s and ρ_e respectively.We consider that the scale of the system is much smaller than the typical scale of the exterior contributions (e.g., the Solar System inside the Galaxy), such that inside the system U_e behaves as a constant. Hence, instead of using the arbitrary U_0 scale, it is convenient to use U_e as the reference potential, as follows,[The change on the scale from U_0 to U_e actually changesG_0 to G_e and also changes ν̅, such that the product G^-1ν is constant. The relevant change is on the reference potential, the changes on G and ν are second order on ν. For clarity, we opted not to introduce an index on ν to label this small change. The exact expressions can be found in Ref. <cit.>.]G^-1 =G^-1_e [ 1 + 2ν̅ln( 1 + U_s/U_e)], =G^-1_e (1 + 2 ν̅U_s/U_e -ν̅U_s^2/U_e^2) +...with G(U_e) = G_e (or, equivalently, G|_U_s = 0 = G_e) and U_s < U_e. The expression above is compatible with eq. (<ref>), with U_s in place of U. Hence, we identify,ν_1= ν̅/G_e U_e,ν_2= - ν̅/2 G_e U_e^2.It should be remembered that the expressions above assume U_e>U_s, hence the limit U_e → 0 is meaningless. The PPN bound on ν̅ depends on the value of U_e, and it is such that the larger is U_e, the softer is the bound on ν̅. Since U_e is a gravitational potential, U_e < 1, and hence the most conservative bound on ν̅ comes from using U_e ∼ 1, which reads|ν̅| < 10^-9. The minimum structure outside the Solar System that it should be considered is the Milky Way, whose Newtonian potential at the Solar System position can be estimated to be about U_e ∼ 10^-6 <cit.>, thus,|ν̅| ≲ 10^-17.Beyond the Milky Way, one should consider the Local Group contribution to U. Since the Milky Way is already one of the two most massive galaxies of the Local Group, the other being Andromeda, the bound will not change appreciably. Beyond the Local Group there is the Virgo super-cluster, but the Local Group is not gravitationally bound to it, thus one starts to enter a domain in which cosmology becomes important, and hencebeyond the validity of the scale setting (<ref>). Therefore, unless there is some nontrivial cosmological contribution, eq. (<ref>) is the most reasonable bound on ν̅ that can be inferred at the Solar System. The bound that appears in eq. (<ref>) is slightly stronger than the bound from Ref. <cit.>, where it was found |ν̅| ≲ 10^-16 for the same value of the external potential. The reason for the disagreement comes from that here we use all the Will-Nordtvedt parameters, and the strongest bound on ν_2 is not the one from β, but from ζ_2. These two bounds only differ by a factor 4, but since 8 × 10^-5∼ 10^-4 and 2 × 10^-5∼ 10^-5, the final answer has an order of magnitude of difference. §.§ Infrared β-function proportional to μ^n and the External Potential EffectAlthough the case of a constant infrared β-function is a natural one, here we consider another simple possibility that also appears frequently in diverse contexts,β_G^-1≡μ∂ G^-1(μ)/∂μ = νμ^n,where n is a dimensionless real constant different from zero and ν is a constant. Again ν is used to set the strength of the RG effects, but for the β-function above,ν is a dimensionful quantity.After integrating eq. (<ref>) and using the scale setting[One could consider μ=f(U), but for clarity we consider this simpler case.] μ = U,one findsG^-1(U) = G_0^-1 + ν/n U^n.In the above, G_0 is an integration constant. Upon considering the presence of matter outside the system, U is divided into U_s and U_e (the latter being a constant) and the G expression can be stated as a function of U_s as follows,G^-1 =G_0^-1 + ν/n (U_s + U_e)^n =G_e^-1 - ν/nU_e^n + ν/n (U_s + U_e)^n = G_e^-1 + νU_e^n-1 U_s + νn-1/2 U_e^n-2 U_s^2 + O( U_s^3/U_e^3),where G_e is defined from G(U_s=0) = G_e. From the expansion above and eq. (<ref>), one identifiesν_1= 1/2ν U_e^n-1, ν_2= νn-1/4U_e^n-2.As in the previous subsection, for U_e ∼ 1, the bound comes from α_2 and reads (using c=ħ = G_N =1),|ν| < 10^-9.If U_e ≪ 1 and n=1, then the bound above is also valid. For the case U_e ∼ 10^-6 (which corresponds to the contribution from the Milky Way at the Solar System), and if n is not close to one, the bound becomes,|(n-1)ν | ≲ 10^-16 + 6 n .The above inequality shows that the larger is the external potential U_e, the softer is the bound on ν, as expected. This example with G given by eq. (<ref>) shows that, for some cases, the EPE does notimprove concordance with GR. Namely, for n=1 the bound on ν is given by eq. (<ref>), which is independent of U_e.§ CONCLUSIONSIn this work we used, for the first time, the Will-Nordtvedt PPN formalism to address Solar System bounds on a class of RG-based proposals that extend GR. This class is such that the RG scale is a function of the Newtonian potential, hence in particular it includes the RGGR proposal <cit.>. We also consider the External Potential Effect (EPE), which is an intrinsic effect of these proposals and which depends on the environment of the system <cit.>.In Ref. <cit.>, using a more heuristic approach within the less rigorous and simpler Eddington-Robertson-Schiff PPN version, it was found the bound |ν̅_⊙| ≲ 10^-16 for RGGR. Here we find a slightly stronger bound for RGGR,[Indeed, as argued in Ref. <cit.>, although the used approach was not as rigorous as the one employed here, the bounds derived on <cit.> should be an order of magnitude estimation.] |ν̅_⊙| ≲ 10^-17 (both of these bounds consider the Solar System as part of the Milky Way, see Sec. <ref>). Moreover, the present work also addressbounds for a more general class of theories, whose relation between G and the Newtonian potential is given by the expansion (<ref>). The bounds for such class are stated in Table <ref>. These bounds should be seen with care since they, for technical convenience, do not consider the EPE. Implementations of the EPE, for different RG extensions, are presented in Secs. <ref> and <ref>. In Sec. <ref>, we explore relations between G and the Newtonian potential that are simple considering the RG motivation, and that do not follow the original RGGR proposal <cit.>. In particular, we find a single peculiar case in which the EPE is irrelevant to the observational bound (the case n=1 in eq. <ref>).Renormalization group extensions of GR at the large scales, as presented in several works (some of them are cited in the introduction), constitute a theoretical possibility which demands to be analysed. We add that it is in connection with QFT in curved spacetime and quantum gravity from the asymptotic safety approach. Also, it leads to results and a framework that cannot be naturally achieved by other means.Among the possible phenomenological consequences, some works have developed on the possibility that perhaps such RG modifications of classical GR may be related to dark matter-like effects <cit.>. The latter line of research,has achievedinteresting nontrivial consequences, but there is not yet an approach sufficiently developed and tested to be clearly better than the standard dark matter approach. Apart from such uncertainties, and on whether one should look for RG effects associated to dark matter or to other effects, the constraints from the Solar System commonly depend on less hypothesis than larger scale phenomena and are commonly of higher precision, hence they should always be taken into consideration.We thank Felipe T. Falciano and Sebastião Mauro forremarks on the PPN formalism application, and to Ilya Shapiro for discussions on the RG application to gravity. DCR thanks CNPq and FAPES (Brazil) for partial financial support, AOFA and NB thank CAPES (Brazil) for financial support.§ COVARIANT SCALE SETTING AND NEW PN POTENTIALSHere we consider the covariantextension as proposed in Refs. <cit.>. Considering the latter references, the scale setting (<ref>) has a covariant extension given byμ= f(Ψ),withΨ≡ h_αβu^α u^β .In the above, u^α is the fluid four-velocity defined in (<ref>),h_αβ≡ g_αβ - γ_αβ, andγ_αβ is the reference metric, which we use the Minkowski metric. Hence, the h_αβ that appears in eq. (<ref>) is the same that appears in eq. (<ref>).We expand G^-1 as a power series on Ψ, similarly to eq. (<ref>),G^-1=G_f^-1+ ∑_n=1^∞σ_nΨ^n .In the above,G_f is the value of G when Ψ=0 (it needs not to coincide with G_e from eq. <ref>). We use σ_n in place of ν_n to avoid confusion, since these quantities are in general different. Rewriting the field equation (<ref>) up to the first post-Newtonian order, it resultsR_αβ=G_f (1-G_fσ_1Ψ) [8π(T_αβ-T/2g_αβ)+ + σ_1 ∇_α∇_βΨ + σ_2 ∇_α∇_βΨ^2 ++1/2 g_αβσ_1Ψ + 1/2 g_αβσ_2Ψ^2] . Using eq. (<ref>), Ψ is expanded as follows,Ψ= h_00 + h_00^2 + h_00v^2 + 2h_0iv^i + h_ijv^i v^j + O(v^6) . The relation between G_f and G_N is found from the Poisson equation ∇^2h_00 = - 8 π G_Nρ at the Newtonian order, which impliesG_f = G_N/1 + σ_1 G_N.In the following, we use G_N=1. The relation above is similar to eq. (<ref>), but we stress that ν_n and σ_n are associated to different expansions, threfore their values willin general be different as well.Before expressing the metric solution up to the 1PN order, first we solve eq. (<ref>) for h_00 and h_ij up to order v^2, and h_0i to order v^3. In this case, it is sufficient to considerΨ≈ h_00. The procedure is the same one of Sec. <ref>, and it yields,h_00=2U+O(v^4) ,h_ij=2(1-2σ_1/1+σ_1)Uδ_ij+O(v^4) ,h_0i=- 1/1+σ_1(7/2 V_i+1/2 W_i)+O(v^5) .Now we proceed to obtain h_00 up to v^4 order. In this case, the fourth-order terms that appear in eq. (<ref>) do contribute. The resulting expression for h_00 reads,h_00=2U-2[1-σ_1+2σ_2(1+σ_1)/1+σ_1] U^2+ +4Φ_1+ 4(1-3σ_1/1+σ_1) Φ_2+2Φ_3+6Φ_4+ +2σ_1Uv^2- 7σ_1V_iv^i-σ_1W_iv^i + O(v^6) .The standard Will-Nordtvedt PPN formalism <cit.> does not include the three last terms in eq. (<ref>).The above is not the only field equation of the covariant formulation, and neither it is complete, since the Λ term was not considered (see Appendix <ref>). Nonetheless, it is sufficient to show that new potentials will appear. In conclusion, the PPN analysis of the covariant extension of the scale setting (<ref>), as proposed in <cit.>, demands an extension of the formalism, including the potentials above, which is beyond the purpose of this work. Theories that are not covered by the PPN formalism are not rare in the literature <cit.>.§ Λ AND VIOLATION OFASYMPTOTIC FLATNESS OR ANALYTICITYIn this appendix it is shown that the Λ term either violates asymptotic flatness or it cannot be expressed as an analytical function, which are necessary conditions for the application of the PPN formalism. We also comment on the possible physical impact of Λ in the Solar System.The Λ term includes a Λ_0 constant, which reduces to the cosmological constant of GR if ν=0, and RG corrections that depend on the RG scale μ and on powers of ν. The constant Λ_0 in GR necessarily leads to non-asymptotically flat spacetimes, hence it is not considered in standard PPN Solar System analysis. This is also physically reasonable since, up to first Post-Newtonian order (1PN), considering its valueas inferred from the cosmological observations, it has negligible impact on the Solar System dynamics <cit.>. Therefore, as a starting point on the Λ contribution analysis up to 1PN, we consider Λ_0=0 .According to Ref. <cit.>, in any region without matter (i.e., T_μν=0), writing Λ and G as Λ = Λ_0 + O(ν) and G = G_0 + O(ν), then,Λ = Λ_0 G_0 G^-1 + O(ν^2).Consequently, in vacuum and using Λ_0=0, one finds Λ=0 + O(ν^2). From the above, one concludes that, within the approximation that the Solar System is composed by point particles representing the Sun and the planets, Λ should not have a relevant role up to the 1PN order. This is in accordance in particular with the Laplace-Runge-Lenz vector approach of Refs. <cit.>.On the other hand, the Will-Nordtvedt PPN approach uses a fluid instead of point particles. This change from particles to fluid may lead to different answers depending on the theory <cit.>, for instance it may change the value of β appreciably. As commented in Sec. <ref>, the Λ expression as a function of μ should be derived from the field equations and hence it is not universal (say, in vacuum, inside a star or in cosmology Λ may have different dependences on μ). Nonetheless, for a fixed system, the Solar System, Λ should be a fixed function of μ. Using the scale setting (<ref>) and expanding Λ similarly to what was done for G in eq. (<ref>), letΛ = Λ_0+ ∑_n=1^∞Λ_n U^n .The hypothesis in the above is that, although Λ is not in general an analytical function, perhaps it can be approximated by one in the Solar System and up to 1PN order. We will show that this hypothesis cannot be true in an asymptotically flat spacetime. With Λ,the field equations (<ref>) becomeR_μν= G[8π (T_μν-1/2g_μν T)+ +∇_μ∇_ν(G^-1) + 1/2g_μν(G^-1)]+ Λ g_μν .To proceed with the PPN analysis, one needs to find the metric solution up to order v^4. As a first step,the equation for the zeroth order on v contribution leads to eq. (<ref>), as expected. The next step is to compute the Newtonian limit which means evaluate h_00 up to order v^2. Thus, using G_N=1, h_00= 2U-Λ_1χ ,which extends eq. (<ref>). The potential χ is defined in eq. (<ref>). According the PPN formalism, the weak field expansion is about Minkowski metric, but χ is a potential that diverges at infinity and there is no gauge freedom to remove it, therefore,Λ_1=0.With the above result, any contribution from Λ to the metric may appear only at the v^4 order or higher.Since the Λ contribution to the field equations is simply an additional term that depends on no derivates, its contribution to the metric can be easily obtained following the same steps used to derive eq. (<ref>), leading to, up to the terms of order v^4,h_00=2U -2[1+ ν_1^2-ν_2(1+ν_1)/(1+ν_1)^2]U^2+ + 4Φ_1/1+ν_1 +2Φ_3/1+ν_1+ 6Φ_4/1+ν_1 + + [4(1-ν_1+ν^2_1)/(1+ν_1)^2-4ν_1/1+ν_1] Φ_2 + 2Λ_2ℵ ,where ℵ is a new post-Newtonian potential defined asℵ=-1/4π∫U'^2/|x-x'|d^3x' .The other metric componentsare the same as in eq. (<ref>). For large distances from the system, U should decay linearly with the distance, and therefore ℵ diverges logarithmically, implying that Λ_2 = 0,to preserve asymptotic flatness. With above, the contribution from Λ is completely eliminated up to the 1PN order.In conclusion, the Λ term cannot be considered within the standard form of the Will-Nordtvedt PPN formalism. We have not proved that its contribution is dynamically negligible, and hence by not considering it one may be inserting violations of energy-momentum conservation that are relevant at 1PN order within the fluid description. However, considering the point particle case, in which Λ becomes zero everywhere, it is unlikely that its inclusion can change the derived bounds by orders os magnitude. For instance, in case a full inclusion of Λ in the dynamics can lead to ζ_2 = 0, the bound on ν_2 in table <ref> will change, but hardly by an order of magnitude, in particular since the constraint on β is rather close to the constraint that comes from ζ_2. apsrev4-1 | http://arxiv.org/abs/1706.09032v2 | {
"authors": [
"Júnior D. Toniato",
"Davi C. Rodrigues",
"Álefe O. F. de Almeida",
"Nicolas Bertini"
],
"categories": [
"gr-qc"
],
"primary_category": "gr-qc",
"published": "20170627200323",
"title": "Will-Nordtvedt PPN formalism applied to renormalization group extensions of general relativity"
} |
] Global center stable manifold for the defocusing energy criticalwave equationwith potentialIn this paper we consider the defocusing energy critical wave equation with a trapping potential in dimension 3. We prove that the set of initial data for which solutions scatter to an unstable excited state (ϕ, 0) forms a finite co-dimensional path connected C^1 manifold in the energy space. This manifold is a global and unique center-stable manifold associated with (ϕ,0). It is constructed in a first step locally around anysolution scattering to ϕ, which might be very far away from ϕ in the Ḣ^1× L^2(^3) norm. In a second crucial step a no-return property is proved for any solution which starts near, but not on the local manifolds. This ensures that the local manifolds form a global one.Scattering to an unstable steady state is therefore a non-generic behavior, in a strong topological sense in the energy space.This extends our previous result <cit.> to the nonradial case.The new ingredients hereare (i) application ofthe reversed Strichartz estimate from <cit.> to construct a local center stable manifold near any solution that scatters to (ϕ, 0). This is needed since the endpoint of the standard Strichartz estimates fails nonradially.(ii) The nonradial channel of energy estimate introduced by Duyckaerts-Kenig-Merle <cit.>,which is used toshow that solutions that start off but near the local manifolds associated with ϕemit some amount of energy into the far field in excess of the amount of energy beyond that of the steady state ϕ.Hao Jia:School of Mathematics, Institute of Advanced study, 1 Einstein Drive, Princeton, New Jersey 08540, [email protected] Baoping Liu: Beijing International Center for Mathematical Research, Peking University, Beijing, China [email protected] Wilhelm Schlag: University of Chicago, Department of Mathematics, 5734 South University Avenue, Chicago, IL 60636, U.S.A. [email protected] Guixiang Xu: Institute of Applied Physics and Computational Mathematics, Beijing, China [email protected] 2010Mathematics Subject Classification. 35L05, 35B40 The first author was partially supported by NSF grant DMS-1600779, and grant DMS-1128155 through IAS.The second author was supported by the NSF of China (No. 11601017) and a startup grant from Peking University. The third author was partially supported by the NSF, DMS-1500696. The fourth author was partially supported by the NSF of China (No. 11671046 ). [ Hao Jia, Baoping Liu, Wilhelm Schlag, Guixiang Xu December 30, 2023 =====================================================§ INTRODUCTION Fix β>2. DefineY:={V∈ C(^3): sup_x∈^3(1+|x|)^β|V(x)|<∞}.We study solutions to∂_ttu-Δ u-Vu+u^5=0,with initial data u(0)=(u_0,u_1)∈(^3). Since for a short time the term Vu can be considered as a small perturbation, by adaptations of results in <cit.> we know for any initial data (u_0,u_1)∈Ḣ^1× L^2(^3), there exists a unique solutionu(t)∈ C([0,∞),Ḣ^1)∩ L^5_tL^10_x([0,T)×^3)for any T<∞ to equation (<ref>). Moreover, the energyℰ(u(t))=ℰ(u(t),∂_tu(t)):=∫_^3[|∇ u|^2/2+(∂_tu)^2/2-Vu^2/2+u^6/6](x,t) dxis constant for all time. If V^+(x) =max(V(x),0) is large enough,then the operator -Δ -V may have negative eigenvalues. In this case,the equation admits a unique nontrivial ground state Q>0 which is the global minimizer ofJ(ϕ):=∫_^3[|∇ϕ|^2/2-Vϕ^2/2+ϕ^6/6]dx.In addition to the ground states Q and -Q, there can be a number of “excited states"with higher energies (see Appendix A of <cit.>), which are changing sign steady states to equation (<ref>) and decay like c/(1+|x|). Small excited states are always unstable, but large excited states may be stable. These steady states play a fundamental role in understanding the long time dynamics for finite energy solutions to equation (<ref>) withinitial data of arbitrary energy. In the radial case we proved in <cit.> that if we consider generic radial potential V∈ Y such that the equationadmits only finitely many steady states, which are all hyperbolic [This means thatthe linearized operator _ϕ:=Δ -V +5ϕ^4 around any steady state (ϕ, 0) has no zero eigenvalue nor zero resonance], then generic data will lead to solutions that scatter to one of the stable steady states [We say a steady state (ϕ,0) is stable if the linearized operator _ϕ has no negative eigenvalue.], while each unstable steady state will attract a finite codimensional C^1 manifold in the energy space. The result in <cit.> satisfactorily characterized the global dynamical behavior of all finite energy solutions to equation (<ref>) in the radial case. The proof in <cit.> relies crucially on the channel of energy estimate for the linear wave equation which was first developed by Duyckaerts-Kenig-Merle <cit.>. The channel of energy estimate works best for wave equation in dimension 3 with radial data. In this case for many nonlinear problems, itcharacterizes the steady states as the only solutions that do not radiate energy in either time direction. It is an essential ingredient in the work of Duyckaerts, Kenig and Merle <cit.> wherethey established the “soliton resolution" for all type II solutions (i.e. solutions that stay bounded in energy norm up to time infinity or finite blow up time.) for focusing energy critical wave equation with radial data in ^3.In the nonradial case or other dimensions, there are only weaker versions of the channel of energy estimate available <cit.>, and they have been used to establish similar resolution results for focusing energy critical wave equations either under size restriction for the initial data <cit.>, or along a sequence of times <cit.>. All the results mentioned here belong to a larger effort that aims to understand the long time dynamics for solutions of dispersive equations in the presence of nontrivial coherent structures [As defined in <cit.>, coherent structures are solutions that are localized in space, uniformly in time.Examples are solitons, kinks, vortices, monopoles, breathers, etc.]. Due to the limitation of techniques to deal with problems beyond the perturbative regime,we are still at an early stage of understanding of this type of problem. Hence the current interest is to workon carefully chosen models in order to develop our intuition and technique. We refer the reader to <cit.> and references therein for the related results on equivariant wave maps, and to <cit.> forresults on nonlinear Schrödinger equations with potential. In this paper, we consider nonradial solutions to (<ref>) and construct the global center stable manifold for unstable excited states.This gives us a better understandingof the non-generic behavior of solutions. More precisely, our result shows that solutions that scatter to unstable excited states form a finite co-dimentional manifold in the energy space and hence such solutions are non-generic in a very precise, topological sense.Although such results are expected, it is often not easy to rigorously confirm them, in a non-perturbative setting. More precisely, we say a solution u scatters tosteady state (ϕ,0)as t→ +∞ ifthere exists a finite energy free wave u^L (solution to the linear wave equation) such thatu(t) -(ϕ,0) -u^L(t)_→ 0,as t→ +∞.We establish the following result.Let Ω be an open dense subset of Y such that equation (<ref>) with V∈Ω has only finitely many steady states which are all hyperbolic.[We know that such Ω exists by easy adaptions of arguments in <cit.>.] Let Σ be the set ofsteady states.Denote u(t):=S(t)(u_0,u_1) as the solution to equation (<ref>) withinitial data (u_0,u_1)∈(^3).For each (ϕ,0)∈Σ, define_ϕ:={(u_0,u_1)∈(^3): S(t)(u_0,u_1) scattersto(ϕ,0)ast→ +∞}.Denoteℒ_ϕ:=-Δ -V+5ϕ^4as the linearized operator around ϕ. If ℒ_ϕ has no negative eigenvalues, then _ϕ is an open set ⊆(^3). If ℒ_ϕhas n negative eigenvalues, then _ϕ is a path connected C^1 manifold ⊂(^3) of co-dimension n.We note that there is no smallness assumption in the theorem, and the manifold can extend arbitrarily far away from the unstable steady state relative to the norm in Ḣ^1× L^2(^3). Theorem <ref> characterizes all solutions that scatter to a steady state. We expect that generically all solutions scatter to steady states. In the radial case, it was proved that for generic potential all finite energy solutions scatter to one of the steady states, but the proof depends on a particular form of the channel of energy inequality which is valid only in three dimensions and in the radial case. In the nonradial case, it remains an open problem how to characterize the generic behavior. It is perhaps worth pointing out that all nonradial large data results in the study of dynamics of nonlinear dispersive equations depend crucially on monotonicity formulae which are sensitive to algebraic features of the equation. There are currently no effective monotonicity formulae known for equation (<ref>) in the nonradial case.Compared with the radial case <cit.>, we have two main difficulties in constructing the manifold:(i) Consider any solution U that scatters to unstable excited states (ϕ,0). When we perturb around U, i.e., we write the solution as U+η, the resulting nonlinearity contains quadratic terms like U(t)^3η^2 which have a component that behaves like ϕ^3η^2. Standard Strichartz estimate requires control of the nonlinearity in spaces such as L^1_tL^2_x, which forces us to estimate η in the endpoint Strichartz norm L^2_tL^∞_x. However, the endpoint Strichartz estimate for free waves was shown to be false for general data in <cit.>. To overcome this technical obstacle, we use thereversed Strichartz estimate due toBeceanu and Goldberg <cit.>.By reverse Strichartz estimates, we mean estimates in the space ·_L^p_xL^q_t. That is, we first integrate in time and then in space, which is the reverse order of integration for the usual Strichartz estimates. This order of integration arises naturallyin the context of KdV and derivative nonlinearSchrödinger equations, where the local smoothing effect needs to be exploited. For the wave equation the advantages of space-time reversal are less well-known, see however Proposition 3.1 in <cit.> for an example of an L^∞_x L^1_t estimate which fails for L^1_tL^∞_x. In that reference as well as in our case, the main feature is that the fundamental solution for linear wave equation in three dimensions is nonnegative and is integrable in time:∫_0^∞1/tδ(|x|-t) dt=1/|x|.This property can be used to trade decay in space for decay in time.For theϕ^3η^2 term, which is only quadratic in η,there is not enough decay in time to use the standard Strichartz estimates. On the other hand, there is enough decay in space thanks to the ϕ^3 term. This is exactly the right kind of problem where the reverse Strichartz estimates are more effective.Using the reverse Strichartz estimates, we can follow the same techniques in <cit.> to construct a local, finite co-dimensional center stable manifold ℳ near U(0) with the property that if a solution u starts on the manifold, i.e., u(0)∈ℳ, then u(t) stays close to U(t) for all t≥ 0 and scatters to (ϕ, 0) as t→∞; if on the other hand, u(0) is close to U(0) but not on the manifold, then no matter how small u(0)-U(0)_ is, u(t) will deviate from U(t) by a fixed amount at a future time. (ii) The local manifold construction ensures that anysolution u(t) starting off the local manifold, i.e., u(0)∈ B_ϵ(U(0))\ℳ_ϕ, will leave the time dependent neighborhood B_ϵ(U(t)) eventually. Up to this point, the argument is still essentially based on perturbative techniques. However, perturbative arguments alone are not sufficient to determine the dynamics when u(t) and U(t) separate from each other. In order to obtain information on the dynamics for all times, we use the channel of energy inequality introduced by Duyckaerts-Kenig-Merle <cit.> to show that the solution u necessarily radiates energy into the far field after it leaves B_ϵ(U(t)).This is the crucial global component in our paper.The channel of energy inequality we use here works for nonradial solutions and is not sensitive to the dimension. For another channel of energy inequality which applies in the nonradial case and in all dimensions, see the one for outgoing waves in <cit.> and <cit.>.More precisely,since U scatters to ϕ, at large timeswe know that U(t) can essentially be identified as a free radiation at large distances and(ϕ,0) in the finite region. If we take initial data u(0) and U(0) close enough so that at a given large time t the solutions are still sufficiently close,we can conclude that locally u(t) is essentially (ϕ,0) plus a small but nontrivial perturbation.We will show that the perturbation contains a nontrivial unstable mode, which grows exponentially. Hence at a later time, when the unstable mode dominates all other modes, we use the channel of energy estimate in Lemma <ref> to conclude thatu will send out a fixed amount of energy into large distances and hence the energy leftin the finite region isstrictly less than that of (ϕ, 0). From this we know that u cannot scatter to (ϕ,0). It is interesting to note that our argument shows that a solution, which starts close, but off of the manifold and far away from the unstable steady state, exhibits two types of radiation: a first radiation so that locally in space it is close to the unstable steady state at large times,and a second radiation which eventually pulls it off the steady state forever.In effect, this second step is in the nature of a one-pass theorem, see <cit.>. While a virial identity is the key for the one-pass theorem in those references, here it is an exterior energy estimate.Our paper is organized as follows.In Section <ref>,we construct the local center stable manifold for each solution that scatters to ϕ.In Section <ref> we recall the perturbation lemma,prove the channel of energy estimate and also prove a result on the growth of the unstable modes. Lastly,in Section <ref> we prove our main result Theorem <ref>.§ CONSTRUCTION OF THE LOCAL CENTER-STABLEMANIFOLDWe begin with some notation. We use c,C>0to denote positiveconstants that may be different from line to line.For nonnegative quantities X and Y, we write X≲ Y when X≤ CY for some non-essential C>0.When a given operator L has negative eigenvalues, we denote theseas -k^2 with k>0. Since we work with fixed potentials, we allow all constants to depend on the potential.Let us first recall the definition of Lorentz spaces L_x^p,q(^3) for 0<p<∞ and 0<q≤∞f_L^p,q_x(^3):=p^1/qλμ{|f|≥λ}^1/p_L^q(^+, dλ/λ).Hereμ is the standard Lebesgue measure on ^3. Clearly L^p,p=L^p for any 0<p<∞. We adopt the usual convention that L^∞, ∞=L^∞. Notice that L^p,q⊂ L^p,r whenever q<r. The Hölder inequality still holds for Lorentz spaces <cit.>, viz. fg_L^r,s≤r' f_L^p_1,q_1 g_L^p_2,q_2provided 1/p_1+1/p_2= 1/r<1,1/q_1+1/q_2≥1/sand the endpointfg_L^1≤f_L^p,q_1 g_L^p',q_2, 1/q_1+1/q_2≥1.Young's inequalities read as follows:f∗g_L^r,s≤3r f_L^p_1,q_1 g_L^p_2,q_2 provided 1/p_1+1/p_2= 1/r+1>1, 1/q_1+1/q_2≥1/sand the endpoint f∗g_L^∞≤f_L^p,q_1 g_L^p',q_2, 1/q_1+1/q_2≥1.SinceY⊂ L^3/2, 1_x(^3),Theorem 3, Theorem 1 and Corollary 2 of <cit.> imply the following reversed Strichartz estimate for wave equations with a potential V∈ Yin ^3. Take V∈ Y such that the operator -Δ -V has no zero eigenvalues or zero resonance. Denote by P^⊥ the projection operator onto the continuous spectrum of -Δ-V. Letω:=√(P^⊥(-Δ -V)).Let I be a time interval with t_0∈ I. Then for any (f,g)∈(^3) and F∈(^3× I), the solution γ(t)=(γ(t), ∂_tγ(t)) to the equation∂_ttγ+ω^2γ=P^⊥F,(t,x)∈ I×^3,with γ(t_0)=P^⊥(f,g) satisfies(γ,γ_t)_C_t^0 (Ḣ^1× L^2)+ γ_∩( ^3× I)≤ C ((f,g)_ +F_ ( ^3 × I)).The appearance of Lorentz spaces here is both natural and essential. Indeed, |x|^-1∈ L^3,∞(^3), and by (<ref>) or (<ref>), sup_y∈^3∫_^3 |f(x)| |x-y|^-1dx ≤ C f_L^3/2,1(^3),cf. (<ref>).Our main goal in this section is to prove the following result on the local center stable manifold.Let Ω be a dense open subset of Y such that equation (<ref>) has only finitely many steady states, all of which are hyperbolic. Let V∈Ω⊂ Y. Suppose that U(t) is a finite energy solution to equation (<ref>) which scatters to an unstable steady state (ϕ,0). Let-k_1^2≤-k_2^2≤⋯≤-k_n^2<0be the negative eigenvaluesof _ϕ = -Δ-V+5^4 (counted with multiplicity) with orthonormaleigenfunctions ρ_1, ρ_2,…,ρ_n, respectively. We denote by P_i the projection operator onto the i-th eigenfunction and by P^⊥the projection operatoronto the continuous spectrum, i.e., P_i =ρ_i ⊗ρ_i , P^⊥ =I -∑_i=1^nρ_i ⊗ρ_i.Decompose(^3)=X_cs⊕ X_u,whereX_cs={(u_0,u_1)∈(^3): ⟨ k_ju_0+ u_1,ρ_j⟩_L^2=0,for all 1≤ j≤ n},andX_u= span {(ρ_j,k_jρ_j),1≤ j≤ n}.Then there exist ϵ_0>0, T sufficiently large, a ball B_ϵ_0((0,0))⊂(^3), and a smooth mappingΨ: U(T)+(B_ϵ_0((0,0))∩ X_cs)⟶,satisfying Ψ(U(T))=U(T), with the following property. Let ℳ be the graph of Ψ and set ℳ=S(-T)ℳ, where S(t) denotes the solution map for equation (<ref>). Then any solution to equation (<ref>) with initial data (u_0,u_1)∈ℳ scatters to (,0). Moreover, there is an ϵ_1 with 0<ϵ_1<ϵ_0, such that if a solution u(t) with initial data (u_0,u_1)∈ B_ϵ_1(U(0))⊂(^3) satisfiesu(t)-U(t)_Ḣ^1× L^2<ϵ_1forallt≥ 0,then (u_0,u_1)∈ℳ. Remark:Ω as in the theorem exists, see <cit.>. The proof of Theorem <ref>closely follows the argument for the local manifold in the radial case in <cit.>. However, there is an important additional technical difficulty: in order to control the quadratic nonlinear term ϕ^3η^2 in η, we need to use reversed Strichartz estimates instead of the endpoint version of the standard Strichartz estimates — which do not hold in the nonradial case. We note that if ϵ_1 satisfies the theorem, then any smaller ϵ_1 will also suffice.By the assumption that U scatters to ϕ, there exists a free radiation U^L ∈(^3), such thatlim_t→∞U(t)-(ϕ,0)- U^L(t)_Ḣ^1× L^2=0.We now divide the construction of the center-stablemanifold into the following four steps as those in <cit.>.Step 1: L^6 decay for free waves.We observe that for any finite energy free radiation U^L, we haveU^L(t)_L^6_x→ 0ast→∞.This is a simple consequence of the dispersive estimate for smooth free waves, and an approximation argument. Step 2: reversed space-time estimates for the radiation term U-. Denote h(t,x) =U(t,x) -(x), then the radiation term h satisfiesh_tt-Δ h -V(x)h + 5^4 h +N(, h)=0,whereN(, h) =(+h)^5 -ϕ^5 -5^4h= 10 ^3h^2 + 10 ^2 h^3 +5h^4 +h^5. In what follows, we will show thath_L^6,2_xL^∞_t∩ L^∞_xL^2_t ( ^3×[T,∞)) <∞,for sufficiently large T. From Agmon's estimate in <cit.>, the eigenfunctions {ρ_i}_i decay exponentially. Decomposingh = λ_1(t)ρ_1 +⋯ + λ_n(t)ρ_n +γ,with γ⊥ρ_i for i=1,⋯ ,n,and plugging this into equation (<ref>), we obtain∑_i=1^n(λ̈_i(t) -k_i^2 λ_i(t))ρ_i + γ̈ +ℒ_γ = N(,h),where ℒ_=-Δ-V+5ϕ^4. By orthogonality between γ(t) and ρ_i, i=1,…, n, we derive the following equations for λ_i(t) and γ(t,x): {λ̈_i(t) -k_i^2 λ_i(t) = P_i N(,h):=N_ρ_i, i=1,… n γ̈ +ω^2 γ =P^⊥ N(,h) := N_c, ω:=√(P^⊥ℒ_).. By the decay of the potential V and the steady state , we know that -V+5^4 in the linearized operator ℒ_ decays likeO(1/(1+|x|)^min{β,4}), which is better than the critical rate O(1/|x|^2) as |x|→∞. Hence we canapply the result of Proposition 6 in <cit.> and conclude that the reversed Strichartz estimates as in Lemma <ref> hold for solutions to the equationγ_tt+ω^2 γ =F,where F satisfies the compatibility condition P^⊥F=F.From (<ref>) and (<ref>), we know thatlim_T→∞h(t,x)_L^∞_tL^6_x([T,∞)×^3) =0.Also by the exponential decay of ρ_i, we have|λ_i(t)|=|ρ_i| h |≤ρ_i_L^6/5h(t,x)_L^6_x(^3)→ 0as t→∞ .Let Γ(t) be the solution operator for theequation γ_tt+ω^2 γ=0, i.e.,Γ(t-t_0)(γ(t_0), γ̇(t_0)) = cos (ω (t-t_0))γ(t_0)+1/ωsin (ω (t-t_0))γ̇(t_0).We claim: lim_T→ +∞Γ(t-T)(γ(T), γ̇(T))_∩( ^3×[T,∞)) =0. We postpone the proof of Claim <ref> to the endof this section. Hence given a small positive number ϵ≪ 1, which will be chosen later,we can pick a large time T=T(ϵ,U), such thath_∞6([T,∞)×^3) ≤ϵ λ_i(t)_L^∞_t ([T,∞)) ≤ϵ Γ(t-T)(γ(T), γ̇(T))_∩( ^3×[T,∞)) ≤ϵ.From (<ref>) and (<ref>), by the reverse Strichartz estimates in Lemma <ref>, it follows that the linear solution h^L to∂_tth^L-Δ h^L +5ϕ^4 h^L -Vh^L=0,with initial data h^L(T)=h(T) satisfies thath^L_∩( ^3×[T,T))≤K/2ϵ,if T is sufficiently close to T.We can then use standard perturbation arguments to show that h∈∩( ^3×[T,T)) withh_∩( ^3×[T,T))≤ Kϵ,as long as we choose ϵ to be sufficiently small. Here we take K large enough so that it dominates any constants appearing in the reverse Strichartz estimates. By a continuity argument, we shall prove thath_∩( ^3×[T,T))≤ Kϵ,for all T, not just for T that are close to T. Suppose that (<ref>) holds for T, we shall show that for a small δ>0,(<ref>) holds for T+δ.Let h be a solution to the equation (<ref>) withh_∩( ^3×[T,T̃) )≤ Kϵandh_L^∞_tL^6_x(ℝ^3×[T,T̃))≤ϵ.Suppose that K>10. If ϵ is sufficiently small, then for δ>0 sufficiently small, we have h_∩( ^3×[T,T̃+δ) )≤ C_1Kϵ,where C_1 is a constant that only depends on V. The claim will be proved at the end of the theorem. We note that due to the use of L^6,2_xL^∞_t type spaces, the continuity in time is not obvious. From the equation for λ_i(t) in (<ref>) and the uniform bound (<ref>) on λ_i, we conclude that for t≥ Tλ_i(t)=cosh (k_i(t-T))λ_i(T)+1/k_isinh(k_i(t-T)) λ̇_i(T)+1/k_i∫_T^t sinh (k_i(t-s))N_ρ_i(s)ds=e^k_i(t-T)/2[λ_i(T)+1/k_iλ̇_i(T) +1/k_i∫_T^t e^k_i(T-s)N_ρ_i(s)ds]+ℛ(t),where ℛ(t) denotes a termthat remains bounded for bounded N_ρ_i(s). By (<ref>) and the above formula,we obtain the following stability conditionλ̇_i(T)= - k_iλ_i(T)- ∫_T^∞ e^k_i(T-s)N_ρ_i(s)ds.Under this condition we can rewrite equation (<ref>) as the following integral equation{λ_i(t) =e^-k_i(t-T)[λ_i(T)+1/2k_i∫_T^∞ e^k_i (T-s)N_ρ_i(s)ds]-1/2k_i∫_T^∞ e^-k_i|t-s|N_ρ_i(s)ds =e^-k_i(t-T)[λ_i(T)+1/2k_i∫_T^T+δ e^k_i (T-s)N_ρ_i(s)ds] -1/2k_i∫_T^T+δ e^-k_i|t-s|N_ρ_i(s)ds+e^-k_i(t-T)1/2k_i∫_T+δ^∞ e^k_i (T-s)N_ρ_i(s)ds -1/2k_i∫_T+δ^∞e^-k_i|t-s|N_ρ_i(s)ds, γ(t) =cos (ω (t-T))γ(T)+1/ωsin (ω (t-T))γ̇(T)+∫_T^tsin (ω(t-s))/ωN_c (s)ds. .By (<ref>) and the reversed Strichartz estimates in Lemma <ref>, we get thatλ_i(t)_L^2([T, T+δ))≤ C( |λ_i(T)| + N_ρ_i_L^2_t([T,T+δ)) + N_ρ_i_L^∞_t([T+δ,∞))), and γ_∩(^3× [T,T+δ)) ≤C (Γ(t-T)(γ(T), γ̇(T))_∩( ^3×[T,T+δ) ) + N_c_( ^3×[T,T+δ) )).Here the constant C depends on the L^1 and L^2 integrals of e^-k_it and on the constants in the reversed Strichartz estimates[Instead of estimating the energy norm (^3) of (γ(T), γ̇(T)) in (<ref>), which may not be small, we estimate its free evolution in ∩(^3× [T,T+δ)). Consequently,we can obtain smallness thanks to (<ref>).]. On the one hand, by the fact thatN_ρ_i=ρ_i | N(,h), N_ρ=∑_i N_ρ_i ρ_i,N_c=N-N_ρand the exponential decay of ρ_i, we haveN_ρ_i_L^2_t([T,T+δ))+ N_c_(^3× [T,T+δ))≤C N(,h)_(^3×[T,T+δ)).By the Hölder inequality in Lorentz spaces, noting that ϕ does not depend on time, we have^3 h^2_(^3×[T,T+δ)) ≲_L^6_x^3( h^2_(^3×[T,T+δ)) + h_(^3×[T,T+δ)) h_(^3×[T,T+δ))),^2 h^3_(^3×[T,T+δ)) ≲_L^6_x^2( h^3_(^3×[T,T+δ)) + h^2_(^3×[T,T+δ)) h_(^3×[T,T+δ))), as well ash^4_(^3×[T,T+δ)) ≲_L^6_x( h^4_(^3×[T,T+δ)) + h^3_(^3×[T,T+δ)) h_(^3×[T,T+δ))),h^5_(^3×[T,T+δ))≲h^5_(^3×[T,T+δ)) + h^4_(^3×[T,T+δ)) h_(^3×[T,T+δ)) . ConsequentlyN_ρ_i_L^2_t([T,T+δ))+ N_c_( ^3×[T,T+δ))≤ C ∑^5_j=2h^j_∩ (^3×[T,T+δ) ).On the other hand, by (<ref>) and the exponential decay of ρ_i, we have N_ρ_i_L^∞_t([T+δ,∞)) ≤ C ρ_i_L^6_x(^3)N(,h)_∞6/5([T+δ,∞)×^3)≤ C ∑_i=2^5_L^6_x^5-ih_∞6( [T+δ,∞)×^3)^i ≤ Cϵ^2.The bounds on λ_i and γ imply an estimate on h viah=∑_iλ_iρ_i+γ. In fact, combining estimates (<ref>), (<ref>), (<ref>)-(<ref>),with (<ref>), (<ref>) and Claim <ref>,one concludes thath_L^6,2_xL^∞∩ L^∞_xL^2_t(ℝ^3×[T,T+δ))≤K/2ϵ+C{∑^5_j=2(4C_1Kϵ +ϵ)^j+ϵ^2 },here C only depends on the constants in the reversed Strichartz inequalities and_L^6_x and ρ_i_L^∞_x∩ L^6,2_x.If we choose ϵ≪ 1, which can be achieved by taking T sufficiently large,such thatϵ +∑_j=2^5 (4C_1K+1)^j ϵ^j-1 <1,say, then it follows thath_∩( ^3×[T,T+δ) )≤ Kϵ.Hence, by a standard continuity argument, we conclude that (<ref>) holds for all T>T and h∈ L^6,2_xL^∞_t∩ L^∞_xL^2_t(ℝ^3×[T, ∞)).Step 3: construction of the center-stable manifold near a solution U.Given a finite energy solution U to (<ref>) satisfying (<ref>), we consider another finite energy solution u, with U(T)-u(T)_(^3) small for a fixed large time T, taken from Step 2. We write u= U + η, then η satisfiesη_tt-Δη - V(x)η +(U+η)^5-U^5=0, (t,x)∈(T,∞).With U= +h, we can rewrite the equation asη_tt + ℒ_η +Ñ(, h, η)=0, (t,x)∈(T,∞),withÑ(, h, η) = ( +h +η)^5- ( + h)^5 - 5^4η.Note that Ñ containsterms which are linear in η. However, a further inspection shows that the coefficients of the linear terms in η contains the factor hand hence decay in both space and time, and can be made small if we choose T sufficiently large. First decompose η asη =_1(t)ρ_1 +⋯+ _n(t)ρ_n +, ⊥ρ_ifor i=1,⋯, n. We shall use similar arguments as in step 2 to obtain a solution η which stays small for all large, positive times, with given (_1(T),⋯,_n(T)) and (,)(T). Note that in order to determine the solution η, we still have to determine (T). We can obtain equations for _i, similar to (<ref>). Since we seeka forward solution which grows at most polynomially, we obtain a similarnecessary and sufficient stability condition as (<ref>)_i(T)= - k_i_i(T)- ∫_T^∞ e^k_i(T-s)Ñ_ρ_i(s)ds.Using equations (<ref>) and (<ref>) we arrive at the system of equations for _i and ,{_i(t) =e^-k_i(t-T)[_i(T)+1/2k_i∫_T^∞ e^k_i (T-s)Ñ_ρ_i(s)ds]-1/2k_i∫_T^∞ e^-k_i|t-s|Ñ_ρ_i(s) ds(t) =cos (ω (t-T))(T)+1/ωsin (ω (t-T))(T)+1/ω∫_T^tsin (ω(t-s))Ñ_c (s) ds. .Define(_1,…, _n, )_X:= ∑_i=1^n_i(t)_L^∞_t ∩ L_t^2([T,∞)) +_∩ (^3× [T,∞) ).Estimating system (<ref>), we obtain that_i(t)_L^∞∩ L^2([T, ∞))≲|_i(T)| + Ñ_ρ_i_L^∞_t∩ L^2_t([T,∞)) ≲|_i(T)| +Ñ_( ^3×[T,∞) ),and_∩(^3× [T,∞))≲ ((T), (T))_ + Ñ_( ^3×[T,∞) ).Note that|Ñ| ≲∑_j=1^4 |^4-j h^j η |+ ∑_k≥ 2, i+j+k=5 |^i h^j η^k|.For the linear term in η,by the Hölder inequalities inLorentz spaces (<ref>), we get that^3 hη_( ^3×[T,∞) ) ≤_L^6_x^3 h_(^3×[T,∞) )η_∩(^3×[T,∞) ), ^2 h^2η_( ^3×[T,∞) ) ≤_L^6_x^2 h^2_(^3×[T,∞) )η_∩(^3×[T,∞) ),h^3η_( ^3×[T,∞) ) ≤_L^6_xh^3_(^3×[T,∞) )η_∩(^3×[T,∞) ),h^4η_( ^3×[T,∞) ) ≤h^4_(^3×[T,∞) )η_∩(^3×[T,∞) ).By (<ref>), we have∑_j=1^4 ^4-j h^j η_( ^3×[T,∞) )≤ C ϵη_∩(^3×[T,∞) ).The higher order terms in η are easier to estimate. Similar to the above, we can always estimate h in , hence∑_k≥ 2, i+j+k=5^i h^j η^k_( ^3×[T,∞) )≤ C ∑_k=2^5η^k_∩(^3×[T,∞) ). By definition of X, η_∩(^3×[T,∞) )≤ C (_1, ⋯, _n, )_X. We can combine (<ref>), (<ref>) and (<ref>), (<ref>) to get(_1,⋯, _n, )_X≤L (∑_i=1^n |_i(T)| + ((T), (T))_) + Lϵ(_1,⋯, _n, )_X + L ∑_k=2^5(_1,⋯, _n, )_X^k,where L>1 is a constant only depending on the constants in the reversed Strichartz estimates,ϕ_L^6(^3) and ρ_i_L^∞_x∩ L^6,2_x (for convenience of later use, we will also assume L>n). This inequality implies that if we take ϵ=ϵ_0 sufficiently small (which can be achieved by choosing T suitably large),with∑_i=1^n |_i(T)| + ((T), (T))_≤ϵ_0,such that L^3ϵ_0<1/32, then the map defined by the right-hand side of system (<ref>) takes a ballB_2Lϵ_0(0)⊆ X into itself. Moreover, we can check by the same argument that this map is in facta contraction on B_2Lϵ_0(0)⊆ X. Thus for any given small (_1(T), ⋯_n(T), (T)) satisfying (<ref>), we obtain a unique fixed point of (<ref>). It follows thatu(t,x):=U(t,x) +∑_i=1^k _i(t)ρ_i+(t,x) solves (<ref>) on ^3× [T,∞), satisfyingu-U_L_t^∞( [T,∞);)≲∑_i=1^n |_i(T)| + ((T), (T))_ with Lipschitz dependence on the data _i(T) and ((T),(T)). By the smoothness of the nonlinearity Ñ, the integral terms in (<ref>) depend on _i, smoothly. Hence _i(t),(t,x) and the solution u(t,x) actually have smooth dependence on the data.Step 4: Proof of scattering.In this step, we prove thatthe solution u constructed in step 3scatters to the same steady state (ϕ, 0) as U. For each solution u with the decomposition(<ref>) and any time T'≥ T,we denote (_1,…, _n, )_X[T',∞):= ∑_i=1^n_i(t)_L^∞_t ∩ L_t^2([T',∞)) +_∩ (^3× [T',∞) ). Here X[T,∞) is the space X from step 3, and from the construction we know that(_1,⋯, _n, )_X[T,∞) <2Lϵ_0 <1/16 We will show that(_1,⋯, _n, )_X[T',∞)→ 0 asT'→∞.We shall need the following property of the linear evolution, which will be proved towards the end of this section:For(f_0, f_1)∈ P^⊥(Ḣ^1× L^2),denotef(t,x) = cos (ω t)f_0+1/ωsin (ω t )f_1,then we have lim_T_0→∞f(t,x)_∩( ^3×[T_0,∞) )= 0.And there exists a free wave f^L(t,x) with data f^L(0)∈,such that lim_t→ +∞f(t,x) - f^L(t,x)_ =0.Using (<ref>) in Claim <ref>,for the ϵ_0 chosen in step 3, we can take T_1>T large enough such that e^-k_i(t-T)_i(T)_L^∞_t ∩ L_t^2([T_1,∞))< ϵ_0^2,cos (ω (t-T))(T)+1/ωsin (ω (t-T))(T)__∩( ^3×[T_1,∞) )<ϵ_0^2.We control the system (<ref>) on the interval [T_1, ∞) in the following fashion:we estimate the linear part on the interval [T_1,∞) using (<ref>)(<ref>), and then estimate the nonlinear (integral) term over the larger interval[T,∞). This yields_i(t)_L^∞∩ L^2([T_1, ∞))≲ e^-k_i(t-T)_i(T)_L^∞_t ∩ L_t^2([T_1,∞))+Ñ_( ^3×[T,∞) ),_∩(^3× [T,∞))≲ cos (ω (t-T))(T)+1/ωsin (ω (t-T))(T)_∩( ^3×[T_1,∞) )+ Ñ_( ^3×[T,∞) ).Combing these estimates with (<ref>), (<ref>), (<ref>), we infer that (notice we assumed L>n)(_1,⋯, _n, )_X[T_1,∞)≤(n+1) ϵ_0^2 + Lϵ_0 (_1,⋯, _n, )_X[T,∞) + L ∑_k=2^5(_1,⋯, _n, )_X[T,∞)^k, ≤L ϵ_0^2+ 2L^2ϵ_0^2 + L ∑_k=2^5 (2Lϵ_0)^k < 2L (2Lϵ_0)^2. Next,fix our choice of T_1 and rewrite system (<ref>) by breaking the integral into finer pieces,{_i(t)= e^-k_i(t-T)[_i(T)+1/2k_i∫_T^T_1 e^k_i (T-s)Ñ_ρ_i(s)ds]-1/2k_i∫_T^T_1 e^-k_i|t-s|Ñ_ρ_i(s) ds+ e^-k_i(t-T_1)1/2k_i∫_T_1^∞ e^k_i (T_1-s)Ñ_ρ_i(s)ds -1/2k_i∫_T_1^∞ e^-k_i|t-s|Ñ_ρ_i(s) ds,(t)= cos (ω (t-T))(T)+1/ωsin (ω (t-T))(T)+1/ω∫_T^T_1sin (ω(t-s))Ñ_c (s) ds+ 1/ω∫_T_1^tsin (ω(t-s))Ñ_c (s) ds. .We can pick T_2>T_1 large enough such that the first line in the expression of _i is small in L^∞_t ∩ L_t^2([T_2,∞)), also the first line in the expression ofis small in ∩( ^3×[T_2,∞) ). We can require that they are bounded by ϵ_0^3. Note that we used Claim <ref> for the term 1/ω∫_T^T_1sin (ω(t-s))Ñ_c (s) ds, which can be viewed as a superposition of linear evolutions.Then estimating the second line of _i andover the larger interval [T_1,∞),we obtain (_1,⋯, _n, )_X[T_2,∞)≤(n+1)ϵ_0^3 +L(ϵ_0 (_1,⋯, _n, )_X[T_1,∞) +∑_k=2^5(_1,⋯, _n, )_X[T_1,∞)^k ), ≤2L(2Lϵ_0)^3. It is clear that this process can be repeated indefinitely:once we fix T_j, we can rewrite the system(<ref>) as in (<ref>), and find T_j+1 >T_j such that the first lineis bounded by ϵ_0^j+1,which implies the estimate (_1,⋯, _n, )_X[T_j+1,∞)≤2L(2Lϵ_0)^j+1.In view of (<ref>), (<ref>), (<ref>), (<ref>) we conclude that lim_T'→ +∞(_1,⋯, _n, )_X[T',∞)=0,lim_T'→∞η_∩( ^3×[T',∞) )= 0,lim_T'→∞Ñ_( ^3×[T',∞) ) = 0. These asymptotics allow us to writethe asymptotic profile of γ̃in the form γ̃_∞(t)=cos (ω (t-T))(T)+1/ωsin (ω (t-T))(T)+1/ω∫_T^∞sin (ω(t-s))Ñ_c (s) ds,with the property thatγ̃ -γ̃_∞_L^∞_t Ḣ× L^2 [T',∞)≲Ñ_( ^3×[T',∞) )→ 0as T'→ +∞.γ̃_∞(t) can be further replaced with a free wave by (<ref>) in Claim <ref>. Combing the preceding with the fact _i(t)_L^∞_t ∩ L_t^2([T',∞))→ 0as T'→ +∞, weconclude thatu scatters to the same steady state(ϕ, 0) as U.We can now defineΨ: U(T)+(B_ϵ_0((0,0))∩ X_cs)⟶Ḣ^1× L^2,as follows: for any (_0,_1)∈ P^⊥((^3)) and _i∈ such thatξ:=∑_i=1^n_i(ρ_i,-k_iρ_i)+(_0,_1)+ U(T)∈U(T)+(B_ϵ_0((0,0))∩ X_cs),set_i(T)=_i, fori=1,…,n and((T),(T))=(_0,_1).Then with _i(T) given by(<ref>), we defineΨ(ξ):=(∑_i=1^n_i(T)ρ_i+_0,∑_i=1^n_i(T)ρ_i+_1) +U(T).If ϵ_0 is chosen sufficiently small, then _i is uniquely determined by contraction mapping in the above. We define ℳ as the graph of Ψ and let ℳ be S(-T)(ℳ). We can then check that Ψ, ℳ, ℳ verify the requirements of the theorem. Since S(T) is a diffeomorphism, ℳ is a C^1 manifold.Step 5: unconditional uniqueness. Now suppose that a solution u to equation (<ref>)satisfies u-U_L^∞( [0,∞);)≤ϵ_1≪ϵ_0. We need to show that u(T)∈ℳ. We denote η(t,x) = u(t,x)-U(t,x) = ∑_i=1^n _i(t)ρ_i +(t,x), thenη∈ L^∞_t([0,∞);) with norm smaller than ϵ_1. Using similar arguments as in Step 2, we can conclude that for sufficiently large T and T which is bigger than but close to T, _i(t)_L_t^∞([T,∞)) +(t,x)_ L_t^∞( [T,∞);) ≤ C ϵ_1,_i(t)_L^2([T, T))≤ Cϵ_1, _∩( ^3×[T,T̃) )≤ Cϵ_1. Notice the L^∞ bound on _i implies that the stability condition (<ref>) must hold true,we are again reduced to (<ref>).Now we wish to show that _i(t)∈ L^2([T,∞)),(t,x)∈∩ (^3×[T,∞) ),with a small norm, which together with the fixed point theorem imply u(T)∈ℳ. Pulling back from T to 0, we can obtain the desired result. To show (<ref>), we follow similar arguments as in step 2. Define the norm(_1,⋯, _n, )_X([T,T̃)): = ∑_i=1^n_i(t)_L_t^2([T, T̃)) + _∩ (^3×[T,T̃) ).Similar to (<ref>), (<ref>), (<ref>) and (<ref>), we get∑_i=1^n_i(t)_L^2([T, T̃)) + _∩( ^3×[T,T̃)) ≤C(∑_i=1^n |_i(T)| + ((T), (T))_ +Ñ_( ^3×[T,T̃)) + Ñ_L^∞_tL^6/5_x([T̃,∞))) ≤ C ϵ_1+ C ϵ_0 η_∩( ^3×[T,T̃)) +C ∑_k=2^5η^k_∩(^3×[T,T̃))+ C ∑_i+j+k=5, k≥ 1_L^6_x^ih^j_L^∞_t L^6_x( [T̃,∞))η^k_L^∞_tL^6_x( [T̃,∞)) ≤ C ϵ_1+ C ϵ_0 η_∩(^3× [T,T̃)) +C ∑_k=2^5η^k_∩(^3×[T,T̃)), where the constant C may change from line to line. Hence by (<ref>), we have(_1,⋯, _n, )_X([T,T̃))≤ C ϵ_1 + Cϵ_0 (_1,⋯, _n, )_X([T,T̃))+L ∑_k=2^5(_1,⋯, _n, )_X[T,T̃)^k.By a continuity argument similar to the one used in Step 2, we can conclude that(_1,⋯, _n, )_X([T,∞))≤lim inf_T̃→∞(_1,⋯, _n, )_X([T,T̃))≤ C ϵ_1 <ϵ_0,We omit the routine details.Now we give the proof for Claim <ref>. Claim <ref> will be proved as a consequence of the following lemma.Let U^L be afinite energy free radiation and (ϕ, 0) be a steady state to equation (<ref>). Recall thatω =√(P^⊥(-Δ - V +5ϕ^4)).Let γ be the solution to{∂_ttγ +ω^2 γ =0,in[T,∞)×^3,γ(T) = P^⊥ (U^L(T)). .For any ϵ>0, if we take T=T(ϵ, U^L)>0 sufficiently large, thenγ_∩ (^3×[T,∞) )<ϵ.For a given ϵ>0, fix 0<δ≪ϵ to be determined below. We can take a smooth compactly supported (in space) free radiation U^L such thatU^L(0)-U^L(0)_(^3)≤δ.Let us assume that supp(U^L(0)) ⋐ B_R(0) for some R>0. Hence by the strong Huygens' principle, for large time t we have|U^L(t,x)|≤C/tχ_[t-R≤|x|≤ t+R],for t>R.Now for T≫ R, by direct computation we get thatU^L(t,x)_L^6,2_xL^∞_t(ℝ^3× [T,∞)) ≲1/tχ_[t-R≤ |x|≤ t+R]_L^6,2_xL_t^∞(ℝ^3×[T,∞)) ≲1/|x|-Rχ_[|x|>T-R]_L^6,2_x ≲1/|x|χ_[|x|>T/2]_L^6,2_x≲1/√(T).Similarly,U^L(t,x)_L^∞_xL^2_t(ℝ^3×[T,∞)) ≲1/tχ_[t-R≤ |x|≤ t+R]_L^∞_xL^2_t(ℝ^3×[T,∞)) ≲(1/|x|^2χ_[|x|>T-R]· R)^1/2_L^∞_x≲_R 1/T.Hence lim_T→∞U^L_∩ (^3×[T,∞) )=0 . Since U^L is a free radiation, we see that∂_ttU^L -ΔU^L - VU^L + 5ϕ^4U^L = - VU^L +5ϕ^4U^L,in(0,∞)×^3. By the decay property of V, 5ϕ^4and (<ref>), simple calculations show that lim_T→∞- V U^L +5ϕ^4 U^L_ ( ^3 × [T,∞)) =0. Choose T sufficiently large, such that U^L_∩ (^3×[T,∞) )≤ δ,- VU^L +5ϕ^4U^L_ ( ^3 × [T,∞))≤ δ. Note that v:= γ - P^⊥U^L solves ∂_tt v +ω^2v = - P^⊥(- VU^L +5ϕ^4U^L ),(t,x)∈ [T, ∞)×^3, with initial data v(T)= P^⊥(U^L(T) - P^⊥U^L(T)). [By definition, it is clear that P^⊥ is bounded in L^6/5,2_xL^∞_t∩ L^3/2,1_xL^2_t.] It is clear fromthe bounds (<ref>) andenergy conservation for the free radiation thatv(T)_≤ Cδ.By (<ref>)and reversed Strichartz estimates from Lemma <ref>, we can conclude that v_∩ (^3×[T,∞) )≤ Cδ. Combining bounds (<ref>) and (<ref>), and fixing δ small, the lemma is proved.Now the proof of Claim <ref> is easy. Note that due to the fact thatlim_T→∞U(T) - (ϕ,0) -U^L(T)_(^3) =0,we see that the initial data for γ satisfieslim_T→∞γ(T)-P^⊥U^L(T)_(^3) =0.HenceClaim <ref> follows from the above lemma and reversed Strichartz estimates. Proof of Claim <ref>: From the boundh_∩ (^3×[T,T) )≤ Kϵ,we check as in the proof of Theorem <ref> thatf_L^6/5,2_xL^∞_t∩ L^3/2,1_xL^2_t(^3×[T,T))≲ K^5ϵ^2,where f=N(ϕ,h). h satisfies∂_tth-Δ h-Vh+5ϕ^4h+f=0,and thus h̃:=P^⊥h satisfies∂_tth̃-Δh̃-Vh̃+5ϕ^4h̃+P^⊥f=0.By reverse Strichartz estimates and the estimates (<ref>) on f, we conclude that the solution h̃^L to∂_tth̃^L-Δh̃^L-Vh̃^L+5ϕ^4h̃^L=0with h̃^L(T)=P^⊥(h(T)) satisfies thath̃^L-h̃_∩ (^3×[T,T)) ≤ CK^5ϵ^2,and henceh̃^L_∩ (^3×[T,T)) ≤ C_0Kϵ+CK^5ϵ^2.Using approximation by smooth and compactly supported data, it is easy to show that there exists sufficiently small δ>0 such thath̃^L_∩ (^3×[T,T+δ)) ≤ C_0Kϵ+2CK^5ϵ^2.Hence, by taking δ smaller if necessary so that the growth of the unstable modes can be controlled, we can conclude that the solution h^L to∂_tth^L-Δ h^L-Vh^L+5ϕ^4h^L=0with h^L(T)=h(T) satisfies thath^L_∩ (^3×[T,T+δ)) ≤ C_0Kϵ+ϵ+4CK^5ϵ^2.Then by a standard perturbation argument, we see that if ϵ is sufficiently small, thenh_∩ (^3×[T,T+δ)) ≤ C_0Kϵ+ϵ+8CK^5ϵ^2.Combining the above with estimates of h on the interval [T,T) and choosing C_1≫ C_0, the claim is proved.Proof of Claim <ref>:Fromthe proofof Lemma <ref>, we know that for free wave U^L with smooth compactly supported data, we havelim_T_0→∞U^L_∩ (^3×[T_0,∞) )=0.Then by approximation, (<ref>) holds true for any free wave with finite energy. Now letf(t,x) be a solution to the equation (recall ω^2 =P^⊥(-Δ -V+5ϕ^4)){∂_ttf +ω^2 f=0,in[0,∞)×^3,u(0) = (f_0, f_1)∈ P^⊥(Ḣ^1× L^2) ..For any given ϵ>0, we first take smoothand compactly supporteddata (f̃_0, f̃_1) such that(f_0, f_1) - (f̃_0, f̃_1)_≲ϵ,which further implies (f_0, f_1) - P^⊥ (f̃_0, f̃_1)_≤(f_0, f_1) - (f̃_0, f̃_1)_≲ϵ.We take g(t,x) to be the solution to the equation{∂_ttg +ω^2 g=0,in[0,∞)×^3,g(0) =(g_0, g_1) : = P^⊥ (f̃_0, f̃_1)..From Strichartz estimates, we haveg∈∩ (^3×[0,∞) ). Let us recallanestimate from the proof of <cit.>(page 27 in the journal version)which is slightly stronger than the estimate stated in the main result <cit.>.Notice that it in fact follows from interpolation between the bounds in <cit.>. For0≤θ_1, θ_2 ≤ 1 and θ_1+θ_2≤ 1,t^1-θ_1-θ_2(cos (tω)g_0+sin tω/ω g_1)_(𝒦^θ_2)^*_xL^∞_t≲Δ g_0_𝒦^θ_1 +∇ g_1_𝒦^θ_1 It is not necessary for us to give the detailed definition of 𝒦^θ and (𝒦^θ)^*,as we only need the embedding propertyL^3/3-θ,1⊂𝒦^θ,(𝒦^θ)^* ⊂ L^3/θ,∞.Hence we can take θ_2=1/2 and θ_1=0,andobtain the estimate we need, viz. cos (tω)g_0+sin tω/ω g_1_L^6,∞_x L^∞_t[T_0,∞)≲ T_0^-1/2(Δ g_0_L^1 +∇ g_1_L^1) Notice that eigenfunctions ρ_i to _ϕ=-Δ -V +5ϕ^4 decay exponentially and ρ_i∈ W^2,p, 1≤ p ≤∞.Together with the fact (f̃_0, f̃_1)issmooth and compactly supported and (g_0, g_1)= P^⊥ (f̃_0, f̃_1),we have Δ g_0, ∇ g_1∈L^1. Define the matrix operator J(t) =[ cos (t|∇|) |∇|^-1sin (t|∇|); -|∇| sin(t|∇|)cos(t|∇|) ],and consider the free wave g^L(t,x)with the initial data [ g^L(0); g^L_t(0) ] =[ g_0; g_1 ] + ∫_0^∞ J(-s)[0; (V-5ϕ^4)g(s) ] ds.We wish to compare g and g^L. By the decay property of V, 5ϕ^4 andthe Strichartz estimate (<ref>), we know the integral term in (<ref>) converges in .Then we have [ g(t); g_t(t) ] - [ g^L(t); g^L_t(t) ] = - ∫_t^∞ J(t-s)[0; (V-5ϕ^4)g(s) ] ds.In particularg(t) = g^L(t) - ∫_t^∞sin ((t-s)|∇|)/|∇|( (V-5ϕ^4)g(s) ) ds.Since g∈ (^3×[0,∞) ), by continuity of the norm in the time variable, we haveg_ (^3×[T_0,∞) )→ 0 as T_0→∞. Together with the fact V-5ϕ^4 ∈ L^3/2,1 and from Hölder, we obtain (V-5ϕ^4)g_L^3/2,1_xL^2_t( ^3 × [T_0,∞))→0asT_0→∞.From(<ref>), we also have (V-5ϕ^4)g_L^6/5,2_xL^∞_t( ^3 × [T_0,∞))≲V-5ϕ^4_L^3/2, 2_xg_L^6,∞_xL^∞_t[T_0,∞)→ 0, T_0→ +∞. Now we can apply the Strichartz estimate (<ref>)to (<ref>) which impliesg_∩ (^3×[T_0,∞) )≲g^L_∩ (^3×[T_0,∞) ) +(V-5ϕ^4)g_ ( ^3 × [T_0,∞))⟶0asT_0→ +∞. Hence we can pick T_* large enough such thatg_∩ (^3×[T_0,∞) ) <ϵ for T_0>T_*.Combining this with the difference estimate f(t,x) -g(t,x)_∩ (^3×[T_0,∞) )≲(f_0,f_1) -(g_0, g_1)_≲ϵ,we getf_∩ (^3×[T_0,∞) ) <ϵ,forT_0>T_*.We have proved (<ref>). In a similar fashion, weconsiderthefree wave f^L(t,x)with the initial data [ f^L(0); f^L_t(0) ] =[ f_0; f_1 ] + ∫_0^∞ J(-s)[0; (V-5ϕ^4)f(s) ]ds.We knowthe integral term here converges inand f(t) = f^L(t) - ∫_t^∞sin ((t-s)|∇|)/|∇|( (V-5ϕ^4)f(s) ) ds.Now that we have already proved f_∩ (^3×[T_0,∞) )→ 0 as T_0→ +∞,we can apply Strichartz to obtain f(t,x) -f^L(t,x)_ ≲(V-5ϕ^4)f(s,x)_L^6/5,2_xL^∞_s∩ L^3/2,1_xL^2_s ( ^3 × [t,∞))≲V-5ϕ^4_L^3/2,1_xf(s,x)_L^6,2_xL^∞_s∩ L^∞_xL^2_s ( ^3 × [t,∞))→0 ast→ +∞. This establishes (<ref>). Due to the near optimal decay assumption on our potential V, we can not apply the structure formula from <cit.> to obtain scattering for solutions to the wave equation with potential. The proof above seems to provide a new perspective:scattering to a free wave occurs because the potential term becomes negligible for large times. This insight requires the use of reverse Strichartz estimates. § CHANNEL OF ENERGY INEQUALITYIn this section, we first prove the channel of energy estimate for solutions to the linear wave equation with potential if the initial data has a dominatingdiscrete mode. Then we show this estimate also holds for equation (<ref>) as long as the initial data is small enough.Finally, for data which has a nontrivial but not dominantdiscrete mode, weprove a growth lemma which ensures thatonce we require the initial data to be sufficiently small, we can find a large time at which the solution is still small and thediscrete mode becomes dominant.For the following basic perturbation result, we refer the reader to <cit.> for proof.Let 0∈ I⊂ be an interval of time. Suppose ũ(t,x)∈ C_t(I,Ḣ^1(^3)) with ũ_L^5_tL^10_x(I×^3)≤ M<∞, a_L^5/4_tL^5/2_x(I×^3)≤β<∞ and e(t,x), f(t,x)∈ L^1_tL^2_x(I×^3), satisfy∂_ttũ-Δũ+a(t,x)ũ+ũ^5=e,with initial data ũ(0)=(ũ_0,ũ_1)∈Ḣ^1× L^2. Suppose for some sufficiently small positive ϵ<ϵ_0=ϵ_0(M,β),|e|+|f|_L^1_tL^2_x(I×^3)+(u_0,u_1)-(ũ_0,ũ_1)_Ḣ^1× L^2<ϵ.Then there is a unique solution u∈ C(I,Ḣ^1) with u_L^5_tL^10_x(I×^3)<∞, satisfying the equation∂_ttu-Δ u+a(t,x)u+u^5=f,with initial data u(0) =(u_0,u_1). Moreover, we have the following estimatesup_t∈ Iu(t)-ũ(t)_Ḣ^1× L^2+u-ũ_L^5_tL^10_x(I×^3)<C(M,β)ϵ. We also need the following result on the precise asymptotics of eigenfunctions corresponding to negative eigenvalues of the Schrödinger operator -Δ-V, which is a consequence of Theorem 4.2 inMeshkov <cit.>.Let V satisfy sup_x∈^3 (1 + |x|)^β |V (x)| <∞ for some β >2, and suppose that ρ≢0is an eigenfunction corresponding to the eigenvalue -k^2 of -Δ - V. Then there exists f ∈ L^2(𝕊^2) which does not vanish identically, such thatρ(x) = e^-k|x||x|^-1(f(x/|x|)+ω(x)),where ω(x) satisfies∫_𝕊^2|ω(Rθ)|^2dσ(θ) = O(R^-1/2 ),as R → +∞. An important observation in <cit.> is that the above precise asymptotics implies the following channel of energy inequality for the associated linear wave equation. Let V satisfy sup_x∈^3 (1 + |x|)^β |V (x)| <∞ for some β >2, and suppose that ρ≢0 is an eigenfunction corresponding to the eigenvalue -k^2 of the operator -Δ - V. Suppose that u solves the equationu_tt-Δ u -Vu=0with u(0) = μ^+(ρ, kρ), then for any R>0 the following channel of energy estimate holds for some constant c(ρ, V,R)>0∫_|x|≥ t+R|∂_t u|^2(x,t)dx ≥ c(ρ, V,R) |μ^+|^2,fort≥ 0.Similarly,if u(0)=μ^- (ρ, -kρ),then∫_|x|≥ |t|+R|∂_t u|^2(x,t)dx ≥ c(ρ, V,R) |μ^-|^2, fort≤ 0.We first prove the lemma for initial data u(0) = μ^+(ρ, kρ). In this casethe solution u has the explicit formu(t, x) = μ^+ e^ktρ.From(<ref>), we can take r_0 large enough such that when r>r_0, we have ∫_𝕊^2|ω(rθ)|^2dσ(θ)< 1/10∫_𝕊^2|f(θ)|^2 dσ(θ).By the asymptotics of ρ in (<ref>), we get that∫_|x|≥ t+R |∂_t u|^2(x,t)dx≥ ∫_r≥ t+R+r_0∫_𝕊^2 |μ^+ k|^2 e^-2k(r-t)(f(θ) + ω(rθ))^2 dσ(θ) dr≳ ∫_R+r_0^∞∫_𝕊^2|μ^+k|^2e^-2kr|f(θ)|^2 dσ(θ) dr.Then (<ref>) follows.The case whenu(0) = μ^-(ρ, -kρ) is similar, and we omit the detail.Lemma <ref>can be generalized to the case when the initial data has finitely many discrete modes. Let V satisfy sup_x∈^3 (1 + |x|)^β |V (x)| <∞ for some β >2,and suppose that-Δ -Vhas negativeeigenvalues -k_1^2 ≤ -k_2^2 ≤…≤ -k_n^2 <0 with corresponding orthonormal eigenmodes ρ_1, ρ_2, …, ρ_n. Suppose that u solves the equationu_tt-Δu -Vu=0with initial datau(0) = ∑_i=1^n μ^+_i(ρ_i,k_iρ_i), then for any R>0, there exists a constant c(R)>0 such that we have the following channel of energy estimate forward in time ∫_|x|≥ t+R|∂_t u|^2(x,t)dx ≥c(R) ∑_i=1^n |μ_i^+|^2,fort >0.Similarly, if we consider data of the formu(0) = ∑_i=1^n μ^-_i (ρ_i, -k_iρ_i), the channel of energy estimate holds backward in time.It suffices to prove the lemma for sufficiently large R>0. By normalizing the coefficients, we will prove (<ref>) when ∑_i=1^n |μ^+_i|^2=1. We divide the proof into several steps.Step 0: Computing the asymptotics.First notice thatthe solution has an explicit formulau=∑_i=1^n μ_i^+e^k_itρ_i . From Lemma <ref>, we know that each ρ_i has the following asymptoticρ_i = e^-k_i|x|1/|x|(f_i(x/|x|)+ω_i(x)) with f_i∈ L^2(𝕊^2) which does not vanishing identically, andω_i satsifies (<ref>). Now given any R>0, using Lemma <ref> we havelim_t→ +∞∫_|x|≥ t+R|∂_t u|^2(x,t)dx = lim_t→ +∞∫_r>t+R∫_θ∈𝕊^2[∑_i=1^n μ_i^+k_i e^-k_i(r-t)(f_i(θ)+ω_i(rθ)) ]^2 dσ(θ)dr = lim_t→ +∞∫_r>R∫_θ∈𝕊^2[∑_i=1^n μ_i^+k_i e^-k_ir(f_i(θ)+ω_i((r+t)θ)) ]^2 dθ dr =∫_r>R∫_θ∈𝕊^2[∑_i=1^n μ_i^+k_i e^-k_irf_i(θ) ]^2 dσ(θ)dr.Here we used the decay condition (<ref>) for ω_i. Step 1: lower bound for the asymptotics. We claim thatfor any R≥0 fixed, there exists constant c(R)>0 such thatfor any μ^+_i satisfying ∑_i=1^n |μ^+_i|^2=1, we have∫_r>R∫_θ∈𝕊^2[∑_i=1^n μ_i^+k_i e^-k_irf_i(θ) ]^2 dσ(θ)dr ≥ c(R).Suppose (<ref>) is not true, then forany N>0, we findμ_i^+(N)satisfying ∑_i=1^n |μ^+_i(N)|^2=1such that∫_r>R∫_θ∈𝕊^2[∑_i=1^n μ_i^+(N)k_i e^-k_irf_i(θ) ]^2 dσ(θ)dr < 1/N. Using thatμ^+_i(N) are bounded,we can extract a convergent subsequence. Hence we can assume that μ^+_i(N)→ a_i as N→∞, and ∑_i=1^n a_i^2=1.By the dominated convergence theorem,we pass to the limit in (<ref>) andget∫_r>R∫_θ∈𝕊^2[∑_i=1^n a_ik_i e^-k_irf_i(θ) ]^2 dσ(θ)dr=0 whichimplies that∑_i=1^n a_ik_i e^-k_irf_i(θ) =0forr>R,θ∈𝕊^2.Now we consider the problem in several cases:Case 1: if k_i are different, then in (<ref>)we first multiply with e^-k_nr and let r→∞, we conclude a_nf_n=0, and similarly we concludea_i f_i(θ)=0, for1≤ i≤ n andθ∈𝕊^2.Since f_i_L^2(𝕊^2)≠0, we conclude that a_i=0, which is a contradiction to ∑ a_i^2=1.Case 2: If one of the eigenvalues has multiplicity more than 1, say, k_i_0 with multiplicity m, i.e.,k_i_0= k_i_0+1=k_i_0+m-1≠k_j for any j∈{1, …, n}\{i_0, i_0+1, … i_0+m-1}. All other eigenvalue still have multiplicity 1.Then (<ref>) now reads as a_1k_1 e^-k_1rf_1(θ) + …+ e^-k_i_0rk_i_0[∑_i=i_0^i_0+m-1 a_if_i(θ)] + … +a_nk_n e^-k_nrf_n(θ) =0for r >R, θ∈𝕊^2. Applying the same method as in Case 1, we conclude that a_1 f_1(θ)=0, …,∑_i=i_0^i_0+m-1 a_i f_i(θ)=0, … ,a_n f_n(θ) =0,forθ∈𝕊^2,which implies a_i=0,for any i∈{1, …, n}\{i_0, i_0+1, … i_0+m-1}. Now we consider the part∑_i=i_0^i_0+m-1 a_i f_i(θ)=0 and prove that all a_i=0. Denote L=-Δ-V. By L_ϕρ_i =-k_i_0^2ρ_i,i_0≤ i≤ i_0+m-1,we see that L(∑_i=i_0^i_0+m-1 a_iρ_i) = -k_i_0^2(∑_i=i_0^i_0+m-1 a_i ρ_i). Assuming towards a contradiction that not all a_i=0, we conclude that ∑_i=i_0^i_0+m-1 a_iρ_i is aneigenfunction for L with eigenvalue -k_i_0^2.On the other hand, ∑_i=i_0^i_0+m-1 a_iρ_i= e^-k_i_0|x|1/|x| [∑_i=i_0^i_0+m-1 a_i w_i(x)].Thiscontradicts Lemma <ref>, in particular (<ref>).Hence we conclude that a_i=0, 1≤ i≤ n, which is a contradiction to ∑ a_i^2=1.Case 3: In general, we could have several eigenvalues that have multiplicitymore than 1. In that case we repeatthe argument in Case 2 as needed.Hence we conclude that our claim (<ref>) is true. Step 2: Refining the lower bound for asymptotics. Next we refine (<ref>) by obtaininga better lower bound.Let α_i = k_ie^-k_ir f_i(θ), 1≤ i≤ n for r>0, θ∈𝕊^2,and< α_i ,α_j>:=∫_r>0∫_θ∈𝕊^2α_i α_j dσ(θ) dr, 𝒜_n× n:=[< α_i ,α_j>]_1≤ i, j≤ n.Then (<ref>) with R=0 implies that 𝒜 is a positive definite matrix. And for any v⃗∈^n, v⃗=1, one hasv⃗^t A v⃗≥ c(0) >0.Now for any R>0, wechange variables r=s+Rin (<ref>) to wit ∫_r>R∫_θ∈𝕊^2[∑_i=1^n μ_i^+k_i e^-k_irf_i(θ) ]^2 dσ(θ)dr =∫_s>0∫_θ∈𝕊^2[∑_i=1^n μ_i^+k_i e^-k_i s e^-k_iRf_i(θ) ]^2 dσ(θ)ds =∑_i,jμ^+_i e^-k_iRμ^+_j e^-k_jR<α_i,α_j> ≥c(0) ∑_i=1^n|μ^+_i e^-k_iR|^2. Step 3: Channel of energy estimateNow we prove(<ref>). The computation from Step 0 implies that ∫_|x|≥t+R|∂_t u|^2(x,t)dx = ∫_r>R∫_θ∈𝕊^2 [∑_i=1^n μ_i^+k_i e^-k_ir(f_i(θ)+ω_i((r+t)θ)) ]^2 dσ(θ)dr Expanding the square, this further equals =∑_i,j=1^n∫_r>R∫_θ∈𝕊^2 μ_i^+μ_j^+k_ik_j e^-(k_i+k_j)r f_i(θ) f_j(θ) dσ(θ)dr + ∑_i,j=1^n ∫_r>R∫_θ∈𝕊^2μ_i^+μ_j^+k_ik_j e^-(k_i+k_j)r[ f_i(θ)ω_j((r+t)θ) + f_j(θ)ω_i((r+t)θ) ] dθ dr+ ∑_i,j=1^n ∫_r>R∫_θ∈𝕊^2μ_i^+μ_j^+k_ik_j e^-(k_i+k_j)r ω_i((r+t)θ) ω_j((r+t)θ)dσ(θ)dr.Using the decay estimate of ω_j in (<ref>) and Cauchy-Schwarz inequality, we infer that∑_i,j=1^n|∫_r>R∫_θ∈𝕊^2μ_i^+μ_j^+k_ik_j e^-(k_i+k_j)r f_i(θ)ω_j((r+t)θ)dσ(θ)dr| ≤∑_i,j=1^n ( ∫_r>R∫_θ∈𝕊^2|μ_i^+k_i|^2e^-2k_ir|f_i(θ)|^2dσ(θ)dr)^1/2×( ∫_r>R∫_θ∈𝕊^2|μ_j^+k_j|^2e^-2k_jr|ω_j((r+t)θ)|^2dσ(θ)dr)^1/2≲ R^-1/4∑_i=1^n|μ_i^+|^2e^-2k_iR.Similarly,we have∑_i,j=1^n∫_r>R∫_θ∈𝕊^2μ_i^+μ_j^+k_ik_j e^-(k_i+k_j)r ω_i((r+t)θ) ω_j((r+t)θ)dσ(θ)dr≲R^-1/2∑_i=1^n|μ_i^+|^2e^-2k_iR.Together with (<ref>) we obtain ∫_|x|≥ t+R|∂_t u|^2(x,t)dx≥ c(0) ∑_i=1^n|μ^+_i|^2 e^-2k_iR-C(R^-1/4 +R^-1/2)∑_i=1^n|μ^+_i|^2 e^-2k_iR≥ c(0)/2∑_i=1^n|μ^+_i e^-k_iR|^2≥c(0)/2e^-2k_1R,where R is sufficiently large. The lemma is proved.Next we consider the case when there are several negative eigenvalues and prove that if one of the discrete modes is dominant, then we still have the channel of energy estimate.Let V satisfy sup_x∈^3 (1 + |x|)^β |V (x)| <∞ for some β >2,and suppose that-Δ -V has no zero eigenvalue or zero resonance, and that it has negativeeigenvalues -k_1^2 ≤ -k_2^2 ≤…≤ -k_n^2 <0 with corresponding orthonormal eigenmodes ρ_1, ρ_2, …, ρ_n.Let u(t) bea solution to (<ref>) with initial datau(0) =(γ_0, γ_1) + ∑_i=1^n [μ^+_i (ρ_i, k_iρ_i) +μ^-_i(ρ_i, -k_iρ_i)]satisfying theorthogonal conditions ∫ρ_iγ_0dx =∫ρ_iγ_1dx = 0, 1≤ i≤ n. (1) For any R≥ 0, if wehave|μ^+_i_0|> K_0[(γ_0, γ_1)_+∑_i=1^n|μ_i^-|]for sufficiently large constant K_0:=K_0(R)>0,then there exists a constant c(R)>0 such that∫_|x|≥ t+R |∂_t u|^2(x,t)dx ≥c(R)|μ^+_i_0|^2,forall t≥ 0. (2) For any R≥ 0, if we have|μ^-_i_0|> K_0[(γ_0, γ_1)_+∑_i=1^n|μ_i^+|]for sufficiently large fixed constant K_0:=K_0(R)>0, thenthere exists a constant c(R)>0 such that∫_|x|≥ |t|+R |∂_t u|^2(x,t)dx ≥c(R) |μ^-_i_0|^2,forall t≤ 0. To prove (1), first note that the solution is of the formu= ∑_i=1^n μ_i^+ e^k_itρ_i +μ_i^-e^-k_itρ_i +γ(t,x)with the continuous part γ solving the equationγ_tt +P^⊥(-Δ -V)γ=0Hence from Lemma <ref> and the Strichartz estimate for γ (<ref>), we get for t≥ 0∫_|x|>t+R |∂_t u(x,t)|^2dx ≥ 1/2∫_|x|≥ t+R|∑_i=1^nμ_i^+k_i e^k_itρ_i|^2dx -2 ∑_i=1^n∫_|x|≥ t+R|μ_i^- k_ie^-k_itρ_i|^2 dx-2 ∫_|x|≥ t+R |∂_t γ|^2 dx ≥c(R) ∑_i=1^n |μ_i^+|^2 - C ∑_i=1^n |μ_i^-|^2- C (γ_0,γ_1)^2_≥ c(R) /2 |μ_i_0^+|^2,if K_0 in (<ref>) is sufficiently large.Case (2) follows from (1) by time reversal. Next we shall see that the channel of energy estimate is stable with respect to nonlinear perturbations.In particular, the following lemma shows that if the initial data is very close to a steady state, and one discrete eigenmode of the initial data is dominant,then the solution will radiate energy outside the light cone either forward or backward in time.Fix any R>0. Consider a finite energy solution u to the nonlinearequation (<ref>) with initial data (u_0, u_1)∈. Given a stationary solution ϕ and_ϕ = -Δ -V +5ϕ^4 with orthonormal eigenmodes ρ_1, ρ_2, …, ρ_n corresponding to negative eigenvalues -k_1^2 ≤ -k_2^2 ≤…≤ -k_n^2 <0.(1) Let(u_0, u_1) be of the formu(0) = (ϕ, 0) + (h_0,h_1) with (h_0,h_1) =(γ_0, γ_1) + ∑_i=1^n [μ^+_i (ρ_i, k_iρ_i) +μ^-_i(ρ_i, -k_iρ_i)]and ∫ρ_i γ_0dx=∫ρ_i γ_1dx =0 for all 1≤ i≤ n. Assume that|μ_i_0^+|:=max{|μ_i^+|, i=1,…,n}and that|μ^+_i_0|>K[(γ_0, γ_1)_ +∑_i=1 ^n |μ_i^-|],as well as (h_0,h_1)_<ϵ_∗,for some sufficiently large constants K≫ 1 and sufficiently small ϵ_∗>0 that only depend on the potential V and R. Thenthe solution satisfies the channel of energy estimate∫_|x|≥ t+R |∂_t u|^2(x,t)dx ≥ c(R) |μ^+_i_0|^2,fort≥ 0for some constant c(R)>0. (2) Assume that (u_0, u_1) has the decomposition u(0) = (ϕ, 0) +(h_0,h_1) with (h_0,h_1) = ∑_i=1^nμ^+_i (ρ_i, k_iρ_i) +(ℛ_0, ℛ_1).Furthermore, suppose that for|μ_i_0^+|:=max{|μ_i^+|, i=1,…,n}, we have |μ^+_i_0| > K(ℛ_0, ℛ_1)_ and (h_0,h_1)_<ϵ_∗, for sufficiently large K≫ 1 and sufficiently small ϵ_∗>0 that depend only on V, R.Thenthe solution u satisfies the channel of energy estimate ∫_|x|≥ t+R |∂_t u|^2(x,t)dx ≥ c(R) |μ^+_i_0|^2,fort≥ 0for someconstant c(R)>0. (3) Similar results hold when we switchμ_i^- with μ_i^+ in (1), (2) andconsider t≤ 0. (1)Write u=ϕ +h.Then hsolves the equation h_tt +(-Δ -V +5ϕ^4) h =𝒩(h,ϕ) with 𝒩(h,ϕ)=-(ϕ+h)^5 + ϕ^5 +5ϕ^4h.Leth^L be the solution to the linear equationh^L_tt +(-Δ -V +5ϕ^4) h^L =0.DefineV(x,t):= {V(x)if |x|≥ |t|, 0if |x| < |t|, .and ϕ(x,t):= {ϕ(x)if |x|≥ |t|, 0if |x| < |t|, . respectively.Let h̃^L and h̃ be the solution to the linear and nonlinear wave equation with truncated potential, viz.h̃^L_tt +(-Δ -V +5ϕ^4) h̃^L =0, h̃_tt +(-Δ -V +5ϕ^4) h̃ =𝒩(h̃,ϕ).It is easy to check thatV, ϕ∈ L^5/4_tL^5/2_x(ℝ^3×ℝ).We take the initial datah(0)=h^L(0)=h̃^L(0) =h̃(0)= (γ_0, γ_1) + ∑_i=1^n [μ^+_i (ρ_i, k_iρ_i) +μ^-_i(ρ_i, -k_iρ_i)]which satisfy the condition(<ref>) with a large constant K to be chosen later.By finite speed of propagation,t∈,h=h̃, h^L =h̃^L for |x|>|t|. In view ofLemma <ref>,sup_t∈ [0,∞)h̃(t)_ + h̃_L_t^5L_x^10([0,∞)×^3)≲h̃(0)_≲ |μ^+_i_0|andsup_t∈ [0,∞)h̃(t)- h̃^L(t)_ + h̃ -h̃^L_L_t^5L_x^10([0,∞)×^3)≲ |μ^+_i_0|^2,if ϵ_∗ is chosen sufficiently small depending on V. Take K>K_0(R) where K_0(R) is the constant from part (1) of Corollary <ref>, then we get that the linear solution h^L satisfies the channel estimate,∫_|x|≥ t+R|∂_t h^L(x,t)|^2dx ≥ c(R) |μ_i_0^+|^2, fort≥ 0. Hence,for all t≥ 0∫_|x|≥ t+R |∂_t h|^2(x,t) dx = ∫_|x|≥ t+R |∂_t h̃|^2(x,t) dx≥ ∫_|x|≥ t+R |∂_t h̃^L|^2(x,t) dx- C |μ^+_i_0|^4= ∫_|x|≥ t+R|∂_th^L|^2(x,t) dx- C|μ^+_i_0|^4 ≥ c(R)|μ^+_i_0|^2 -C|μ^+_i_0|^4 ≥c(R)/2|μ^+_i_0|^2.The last line holds providedϵ_∗=ϵ_∗(R)≳|μ_i_0^+| is small enough.(2) Consider two solutions to equation (<ref>) u and v, with datau(0)=(ϕ, 0) + ∑_i=1^n μ^+_i (ρ_i, k_iρ_i) + (ℛ_0, ℛ_1), v(0)=(ϕ, 0) + ∑_i=1^n μ^+_i (ρ_i, k_iρ_i),respectively.If we set u=ϕ +h and v =ϕ +ℓ, then h, ℓ satisfyh_tt +(-Δ -V +5ϕ^4) h = 𝒩(h,ϕ) ℓ_tt +(-Δ -V +5ϕ^4) ℓ =𝒩(ℓ,ϕ)with initial datah(0)=∑_i=1^n μ^+_i (ρ_i, k_iρ_i)+(ℛ_0, ℛ_1), ℓ(0) = ∑_i=1^n μ^+_i (ρ_i, k_iρ_i). Asin the proof for(1), we define V, ϕ and consider truncated versions h̃, ℓ̃that satisfy the equation (<ref>), with datah̃(0) =h(0), ℓ̃(0) =ℓ(0). Then from finite speed of propagation we inferh=h̃, ℓ=ℓ̃ for |x|≥ |t|. The perturbation lemma <ref> and (<ref>) yield the boundsup_t∈ [0,∞)h̃(t)-ℓ̃(t)_≤ Ch̃(0)-ℓ̃(0)_≤C/K|μ^+_i_0|.Note that ℓ(0)_≲ |μ_i_0^+|.Frompart (1) we know thatthere exists ϵ_∗(R)>0 small enough,such that if ℓ(0)<ϵ_∗, thenℓ(t,x) satisfy the channel of energy inequality∫_|x|≥ t+R |∂_t ℓ |^2(x,t) dx≥ c(R) |μ^+_i_0|^2 fort≥ 0.Hence we getfor t≥ 0, ∫_|x|≥ t+R |∂_t h|^2(x,t) dx = ∫_|x|≥ t+R |∂_th̃|^2(x,t)dx ≥ ∫_|x|>t+R |∂_tℓ̃|^2(x,t)dx - C^2/K^2|μ_i_0^+|^2 =∫_|x|≥ t+R|∂_t ℓ |^2(x,t) dx - C^2/K^2|μ_i_0^+|^2 ≥c(R) |μ_i_0^+|^2 - C^2/K^2|μ_i_0^+|^2 ≥c(R)/2 |μ_i_0^+|^2.The last line holds if we pick K:=K(R) large enough. (3) The proofis similar to(1) and (2) and we omit the details here. Initially,the discrete spectral component may not be large enough as required by (<ref>). But since anyeigenmode grows exponentially either forward or backward in time, we might expect that it will take over the dispersiveterm for large times as long as it is not too small initially.The following lemma makes this logic precise.Given a steady state solution ϕto the nonlinear equation (<ref>),suppose that _ϕ = -Δ -V +5ϕ^4 has orthonormal eigenmodes ρ_1, ρ_2, …, ρ_n corresponding to eigenvalues -k_1^2 ≤ -k_2^2 ≤…≤ -k_n^2 <0.Suppose that u is a solution to equation (<ref>) with initial datau(0) =(ϕ, 0) + ∑_i=1^n [μ^+_i (ρ_i, k_iρ_i) + μ^-_i (ρ_i, - k_iρ_i)]+ (γ_0, γ_1)obeying the orthogonalityconditions ∫ρ_i γ_0dx =∫ρ_iγ_1dx =0, for all 1≤ i ≤ n. Write the solution asu(t) =(ϕ, 0) + h(t). (1) Suppose |μ_i_0^+|:=max{|μ_i^+|, i=1,…,n}≥κh(0)_for some constant κ>0. Then for any ϵ_∗>0, K>1, there exist ε(κ, ϵ_∗, K)>0 sufficiently small and T(κ, K)>0 sufficiently large,such thatif h(0)_<ε thenh(T) = ∑_i=1^n e^k_iTμ_i^+(ρ_i, k_iρ_i) + (ℛ_0, ℛ_1),withh(T) _<ϵ_∗and(ℛ_0,ℛ_1)_≤1/Ke^k_i_0T |μ_i_0^+|(ρ_i_0, k_i_0ρ_i_0)_.(2) Suppose |μ_i_0^-|:=max{μ_i^-: i=1,…,n}≥κh(0)_for some constant κ>0. Then for any ϵ_∗>0, K>1, there exist ε(κ, ϵ_∗, K)>0 sufficiently small and T(κ, K)>0 sufficiently large,such thatif h(0)_<ε thenh(-T) = ∑_i=1^n e^k_iTμ_i^-(ρ_i, -k_iρ_i) + (ℛ_0, ℛ_1),withh(-T) _<ϵ_∗ and(ℛ_0, ℛ_1)_≤1/Ke^k_i_0T |μ_i_0^-|(ρ_i_0, -k_i_0ρ_i_0)_.The proof of (2) is again the time reversal of (1), so it suffices to consider the latter.Step 1: bound on h. Writing u=ϕ+h, we see thath solves the equation (with 𝒩 as above)h_tt +(-Δ -V +5ϕ^4) h =𝒩(h,ϕ).Leth^Lbe the solution to the linear equationh^L_tt +(-Δ -V +5ϕ^4) h^L =0with datah^L(0)=h(0). We denote by S(t)gthe solution to the linear equation (<ref>) with data (0,g) for any g∈ L^2. By decomposing the data into continuous and discrete modes,the Strichartz estimates (<ref>) for the continuous modes, and the explicit formula for the evolution of discrete modes, we can find absolute constants C, A≥ 1 such thatsup_τ∈ [0,t)S(τ)g_ +S(τ)g_L^5_tL^10_x([0,t)×^3)≤C e^k_1tg_L^2 sup_τ∈ [0,t)h^L(τ)_ +h^L(τ)_L^5_tL^10_x([0,t)×^3)≤ A/8 e^k_1th^L(0)_Denote ϵ:=h(0)_<ε. Now on an interval [0,T) with e^3k_1Tε sufficiently small, we will use a continuity argument to show that for t∈ [0,T)sup_τ∈ [0,t)h(τ)_ +h(τ)_L^5_tL^10_x([0,t)×^3)≤A e^k_1th(0)_.In fact,assuming that the bound (<ref>) holds for 0≤ t≤ t_0 with some 0< t_0<T, we will show that we actually havesup_τ∈ [0,t)h(τ)_ +h(τ)_L^5_tL^10_x([0,t)×^3)≤A/2 e^k_1th(0)_,for all0≤ t≤ t_0.Then a simple continuity argument finishes the proof of proof of (<ref>). From Duhamel's formulah(τ) =h^L(τ) +∫_0^τ S(τ-s)𝒩(h,ϕ)(s)dsDenoteF(τ,x)= ∫_0^τ S(τ-s)𝒩(h,ϕ)(s)ds, then from (<ref>) we get sup_τ∈[0,t)(F(τ)_Ḣ^1 + ∂_τF(τ)_L^2) ≤ sup_τ∈[0,t) C ∫_0^τe^k_1(τ-s)𝒩(h,ϕ)(s)_L^2 ds≤sup_τ∈[0,t)Ce^k_1τ𝒩(h,ϕ)(s)_L^1_sL^2_x([0,τ)×^3)≤Ce^k_1t𝒩(h,ϕ)(s)_L^1_sL^2_x([0,t)×^3)andF_L^5_τL^10_x([0,t)×^3) ≤∫_0^t χ_{τ-s ≥0}S(τ-s)𝒩(h,ϕ)(s)_L^10_xds_L^5_τ[0,t) ≤∫_0^tS(τ-s)𝒩(h,ϕ)(s)_L^5_τL^10_x([s,t)×^3)ds ≤C ∫_0^t e^k_1(t-s)𝒩(h,ϕ)(s)_L^2ds≤Ce^k_1t𝒩(h,ϕ)(s)_L^1_sL^2_x([0,t)×^3).Note that |𝒩(h,ϕ)(s)| ≲∑_j=2^5 |ϕ|^5-j |h|^j.Assuming the bound (<ref>) on [0,t_0),for any t∈ [0,t_0) we pick an integer J_0≥ 0 such that J_0 <t≤ J_0+1.This leads to ϕ^3h^2_L^1_sL^2_x([0,t)×^3)=∑_q=0^J_0ϕ^3h^2_L^1_sL^2_x([q,q+1)×^3) +ϕ^3h^2_L^1_sL^2_x([J_0,t)×^3) ≲ ∑_q=0^J_0h^2_L^5_sL^10_x([q,q+1)×^3) + h^2_L^5_sL^10_x([J_0,t)×^3) ≲ ∑_q=0^J_0(Ae^k_1(q+1)ϵ)^2 +(Ae^k_1t ϵ)^2 ≲A^2ϵ^2 e^2k_1t We can control the other terms in 𝒩(h,ϕ) in an analogous fashion, whence𝒩(h,ϕ)(s)_L^1_sL^2_x([0,t)×^3)≲∑_j=2^5(A ϵe^k_1t)^jfor0≤ t<t_0.Using (<ref>),we therefore obtainsup_τ∈ [0,t)h⃗(τ)_ +h_L^5_tL^10_x([0,t)×^3)≤sup_τ∈ [0,t)h^L(τ)_ +h^L_L^5_tL^10_x([0,t)×^3) + Ce^k_1t𝒩(h,ϕ)(s)_L^1_s L^2_x([0,t)×^3)≤A/8e^k_1tϵ + Ce^k_1t[∑_j=2^5(Ae^k_1t ϵ)^j]≤A/2 e^k_1tϵ,provided e^3k_1Tε≪ 1 is sufficiently small. Hence (<ref>) holds on [0,T) as long asT, ε satisfy the relation e^3k_1Tε< ϵ_1 with a small fixed constant ϵ_1.Step 2: Decide the constants.Now we consider the linear solution h^L with datah^L(0)=∑_i=1^n [ μ^+_i (ρ_i, k_iρ_i) + μ^-_i (ρ_i, - k_iρ_i) ]+ (γ_0, γ_1), then we have the explicit formulafor the linear solutionh^L(t) = ∑_i=1^n μ^+_i e^k_it (ρ_i, k_iρ_i) + ∑_i=1^n μ^-_i e^-k_it(ρ_i, -k_iρ_i) + γ(t).For any given κ, K, we can choose a large constant T(κ, K) such that∑_i=1^n μ^-_i e^-k_iT(ρ_i, -k_iρ_i) + γ(T)_≲T/κ|μ^+_i_0|≤1/2K |μ^+_i_0| e^k_i_0T(ρ_i_0, k_i_0ρ_i_0)_Next from Duhamel's formula and the estimate of 𝒩 in step 1, we have h(T)-h^L(T)_ =∫_0^T S(T-s)𝒩(h,ϕ)(s)ds _ ≤Ce^k_1T[∑_j=2^5(Aϵ e^k_1T)^j] <1/2K|μ^+_i_0| ,if e^3k_1Tε is sufficiently small. Hence we have h(T) =∑_i=1^n μ^+_i e^k_iT (ρ_i, k_iρ_i)+(ℛ_0, ℛ_1) with(ℛ_0, ℛ_1)= ∑_i=1^n μ^-_i e^-k_it(ρ_i, -k_iρ_i) + γ(t) + h(T)-h^L(T)and(ℛ_0, ℛ_1)_≤1/K|μ^+_i_0| e^k_i_0T(ρ_i_0, k_i_0ρ_i_0)_.We also haveh(T)_≤∑_i=1^n e^k_iT|μ_i^+| +(ℛ_0, ℛ_1)_≲ e^k_1Tε<ϵ_∗,by choosing ε sufficiently small.While part (1) of Lemma <ref> guarantees that at time T the unstable modee^k_i_0Tμ_i_0^+(ρ_i_0,k_i_0ρ_i_0)dominates the continuous part and the stable mode, we cannot be sure of its size compared to the other unstable modes, which might grow faster. However, we can easily conclude that the largest mode at time T,say e^k_jTμ_j^+(ρ_j,k_jρ_j),satisfies e^k_jTμ_j^+(ρ_j,k_jρ_j)_≥1/n+1h(T)_.§ GLOBAL CENTER STABLE MANIFOLD OF UNSTABLE EXCITED STATES In this section we prove our main result. Before giving the detailed proof, let us briefly summarize the main ideas in physical terms. The crucial fact that we establish can be explained roughly as follows. Take any solution U(t) which scatters to an unstable steady state ϕ. We have shown in Section 2 that in a small neighborhood of U(0) in the energy space , there exists a local, finite co-dimensional manifold ℳ such that if u(t) starts on the manifold, i.e., ifu(0)∈ℳ, then u(t) stays close to U(t) for all positive times and scatters to (ϕ,0). On the other hand, if u(t) starts in a small neighborhood of U(0) but off the manifold, thensup_t≥ 0u(t)-U(t)_≥ϵ_1>0,no matter how small u(0)-U(0)_ is.Suppose that u(0)-U(0)_ is sufficiently small, then dynamically u(t) will stay close to U(t) for a long time, say for 0≤ t≤ T_0. Since U(t) scatters to (ϕ,0), we can write (in the energy space)U(t)≈ (ϕ, 0)+U^L(t)for large times. Hence for large t≤ T_0,u(t)≈ (ϕ, 0)+U^L(t)in the energy space. After time T_0, u(t) starts to deviate from U(t) as u(0)∉ℳ. By an expansion of the energy functional near the steady state, we shall show that the deviation is due to growth in the unstable mode. Then it is not hard to conclude that at a large time T_1>T_0, u(t)-(ϕ,0)-U^L(t) concentrates most of its energy in the discrete mode and has energy ≳ϵ_1. These argumentsfinally set the stage for us to apply the channel of energy inequalities proved in the previous section. We will show that besides the radiated energy that U^L carries to spatial infinity, u(t) emits a second radiation. The total radiated energy for u(t) will therefore exceed the radiated energy for U(t) by a fixed amount. Now note that u(t) has almost the same amount of energy as U(t), a comparison argument of the energy in the local region then implies that u(t), having strictly less energy than (ϕ,0) in the local region, can no longer scatter to (ϕ,0). Hence, locally the set ℳ_ϕ of all initial data for which the solution scatters to (ϕ,0) coincide with ℳ. Thus the set ℳ_ϕ has a manifold structure. This is the key property showing that scattering to unstable steady states is non-generic.Now we turn to the main argument. Let us first compute the expansion of energy around any steady state (ϕ,0).Let(u_0,u_1)=(ϕ,0)+(Λ_0,Λ_1), where (Λ_0,Λ_1)∈.Assume thatΛ_0_L^6(^3) < β≪ 1,then we haveℰ((u_0,u_1)) = ℰ(ϕ, 0) +1/2(_ϕΛ_0, Λ_0) +1/2 (Λ_1, Λ_1) + O(β^3),where _ϕ = -Δ -V +5ϕ^4.Suppose _ϕ has orthonormal eigenmodes ρ_1, ρ_2, …, ρ_n corresponding to eigenvalues -k_1^2 ≤ -k_2^2 ≤…≤ -k_n^2 <0. If we further decompose (Λ_0,Λ_1)=(X_0,X_1)+ (w_0,w_1),(w_0,w_1)=∑_i=1^n [μ^+_i (ρ_i, k_iρ_i) +μ_i^- (ρ_i, -k_iρ_i)] + (γ_0, γ_1), with (X_0, X_1)∈ and the orthogonality condition ∫ρ_jγ_0dx= ∫ρ_jγ_1dx =0, for all 1≤ j≤ n. Then we have ℰ((u_0,u_1)) = ℰ(ϕ, 0)+ 1/2[ (_ϕX_0, X_0) +(X_1,X_1) ]+1/2[ (_ϕγ_0,γ_0)+ (γ_1,γ_1)]- ∑_i=1^n 2μ^+_iμ^-_ik_i^2 +(_ϕ X_0 , w_0) + (X_1, w_1)+ O(β^3).The proof is by direct computationℰ((u_0,u_1)) = ℰ((ϕ, 0)+(Λ_0,Λ_1)) = ∫|∇ϕ +∇Λ_0 |^2/2 +| Λ_1|^2/2 -V(ϕ+Λ_0)^2/2 +(ϕ+Λ_0)^6/6 dx= ∫1/2(|∇ϕ|^2 - 1/2 Vϕ^2+1/6ϕ^6 )dx +∫(-Δϕ- Vϕ + ϕ^5)Λ_0dx+ ∫1/2[ |∇Λ_0|^2- 1/2 VΛ_0^2+5/2ϕ^4Λ_0^2 + 1/2|Λ_1 |^2 +1/6∑_j≥ 3C_6^j ϕ^6-jΛ_0^j ] dx= ℰ(ϕ, 0) +1/2(_ϕΛ_0, Λ_0) +1/2 (Λ_1, Λ_1) + O(β^3).This finishes the proof of (<ref>).Next we furtherexpand the energy functional using (<ref>) ℰ((u_0,u_1)) =ℰ(ϕ, 0) +1/2(_ϕ (X_0+w_0), X_0+w_0) +1/2(X_1+w_1, X_1+w_1) + O(β^3)=ℰ(ϕ, 0) + 1/2[(_ϕ w_0,w_0) + (w_1, w_1)] + 1/2[ (_ϕX_0, X_0) +(X_1,X_1) ] + (_ϕ X_0 , w_0)+ (X_1, w_1) + O(β^3) .Since _ϕρ_i = -k^2_i ρ_i,we get(_ϕ w_0,w_0)= -∑_i=1^n (μ^+_i+μ^-_i)^2k_i^2+ (_ϕγ_0,γ_0),(w_1, w_1) =∑_i=1^n (μ^+_i-μ^-_i)^2k_i^2+(γ_1,γ_1).Combining the calculations above, we get (<ref>). Now we are ready to present the main idea of our paper, which is crucial to conclude that the set of initial data for which the solution scatters to an unstable steady state (ϕ,0) has a manifold structure, and hence is a “thin set".Let V∈ Y be a potential such that equation (<ref>) has only finitely many steady states, all of which are hyperbolic. [By a simple adaptation of the result in <cit.>, we know that such V are dense in Y.] Suppose thatthe finite energy solution U(t) to equation (<ref>) scatters to an unstable excited state (ϕ,0). Let ℳ be the local center-stablemanifold around U(0) and let ϵ_0, ϵ_1 be as defined in Theorem <ref>. Then there exist ϵ with 0<ϵ<ϵ_1<ϵ_0 and δ(ϵ_1)≫ϵ, such that for any solution u with finite energy initial data (u_0,u_1)∉ℳ with(u_0,u_1)-U(0)_Ḣ^1× L^2<ϵ,we can find A>0 such that for all t≥ A∫_|x|≥ t-A[|∇ u|^2/2+(∂_tu)^2/2](x,t) dx≥ℰ(U(t))-ℰ((ϕ,0))+δ.As a consequence, u(t) willnot scatter to (ϕ,0). We divide our proof into several steps.Step 1: Set up the parameters.By the local center-stablemanifold theorem of Section <ref>, the locally defined finite co-dimensional manifold ℳ satisfies the property that any solution to equation (<ref>) with initial data on ℳ scatters to (ϕ,0). Moreover, if a solution u(t) with initial data (u_0,u_1)∈ B_ϵ_1(U_0) satisfiesu(t)-U(t)_Ḣ^1× L^2<ϵ_1forallt≥ 0,then (u_0,u_1)∈ℳ. Take ϵ<ϵ_1 sufficiently small to be chosen below. Since the solution U(t) scatters to (ϕ,0) as t→∞, denotingby U^Lthe scattered linear wave, we have the property thatlim_t→∞U(t)-U^L(t)-(ϕ,0)_Ḣ^1× L^2=0. This implies that ℰ(U)= ℰ(ϕ,0) + 1/2U^L_^2By (<ref>), the fact that ϕ∈Ḣ^1(^3) and U^L∈ L^5_tL^10_x( [0,∞)×^3), for any small δ_1>0, we can first fix some large L and then choose T_1>L sufficiently large, such that for all t≥ T_1, * (Free wave small in L^6 norm)U^L(t)_L^6(^3)≤δ_1 * (Closeness of U to U^L+(ϕ,0) and choice of the bounded region)U(t)-U^L(t)-(ϕ,0)_Ḣ^1× L^2 +U^L(0)_(|x|≥ L) +ϕ_Ḣ^1(|x|≥ L)≤δ_1; * (Most energy of the free radiation is exterior)∫_|x|≥ t-T_1+L|∇_x,tU^L|^2(t,x) dx≥∫_^3|∇_t,xU^L|^2(t,x) dx-δ_1^2; * (Control on the Strichartz norm of the radiation) LetD:={(x,t): |x|≤ T_1+L-t, 0≤ t≤ T_1}.Then we haveU^L_L^5_tL^10_x((0,∞)×^3\ D)<δ_1.We remark that (<ref>) isa consequence of the strong Huygens principle and approximation by free waves with compactly supported initial data. (<ref>) ensures that U^L can essentially be taken as zero for our purposes inside the region |x|≤ t-T_1+L for t≥ T_1, which will be important to keep in mind later, in order todistinguish the second piece of radiation. By the continuous dependence of the solution to equation (<ref>) on the initial data in (^3) and by finite speed of propagation,if we take ϵ sufficiently small and initial data (u_0,u_1)∈\ℳ with(u_0,u_1)-U(0)_<ϵ,thenu(T_1)-U(T_1)_can be made sufficiently small. Hence, noting that V_L^5/4_tL^5/2_x(|x|≥ |t|) is finite, we can apply Lemma <ref> to conclude thatu(t)-U(t)_(|x|≥ t-T_1)≤δ_1,forallt≥ T_1.(<ref>) means that we can effectively identify u with U in the exterior region|x|≥ t-T_1, t≥ T_1.Hence by (<ref>), we see thatu(t)-U^L(t)_(|x|≥ t-T_1+L)≤ 3δ_1,that is, we can also identify u with U^L in the exterior region |x|≥ t-T_1+L, t≥ T_1.In order to avoid any possibility of confusion due to the many parameters, we remark that δ_1 and ϵ can be made as small as we wish, and will be chosen later. T_1, L depend on δ_1 and U only. ϵ is a small free parameter below some threshold determined by δ_1. The key point for us is that ϵ_1>0 is fixed no matter how small ϵ is chosen, see (<ref>).Since (u_0,u_1)∉ℳ, there exists an exit time T_2>0 from the ϵ_1 ball, i.e.,such thatu(T_2)-U(T_2)_(^3)= ϵ_1.Note that the choices of T_1 and L donot depend on ϵ. Therefore, by the continuous dependence of the solution on its initial data in Ḣ^1× L^2, if we choose ϵ sufficiently small, we can assume T_2>2(L+T_1+1). Step 2: Analyze the size of discrete mode at time T_2.Let us analyze u(T_2) in more detail. By the estimates (<ref>) and (<ref>) we can writeu(T_2)=(ϕ,0)+U^L(T_2)+(w_0,w_1),where w=(w_0,w_1)∈ satisfies2ϵ_1≥ϵ_1+δ_1≥w_(^3)≥ϵ_1-δ_1≥ϵ_1/2,if δ_1 is chosen smaller than ϵ_1/2. We nowlist several facts:(i) From (<ref>), we infer that|ℰ(u) - ℰ(U)| ≲ϵ(ii)Rewrite the decompostion (<ref>) in the form u(T_2)= (ϕ,0)+(Λ_0,Λ_1), (Λ_0,Λ_1)=U^L(T_2)+ (w_0,w_1),(w_0,w_1)=∑_i=1^n [μ^+_i (ρ_i, k_iρ_i) +μ_i^- (ρ_i, -k_iρ_i)] + (γ_0, γ_1), with orthogonality conditions ∫ρ_0γ_jdx= ∫ρ_1γ_j dx=0, for all 1≤ j≤ n. (<ref>) implies thatϵ_1^2 ≲(γ_0, γ_1)_^2 + ∑_i=1^n [k_i^2(μ^+_i -μ^-_i)^2+ (μ^+_i +μ^-_i)^2]. (iii) Expand the energy functional at T_2. Since Λ_0= U^L(T_2) +w_0, from (<ref>),(<ref>) and our a priori choice δ_1<1/2ϵ_1, we haveΛ_0_L^6≲ϵ_1. We now applyLemma <ref> and obtainℰ(u(T_2)) =ℰ(ϕ, 0)+ 1/2[ (_ϕU^L(T_2), U^L(T_2)) +(U^L_t(T_2),U^L_t(T_2)) ]- ∑_i=1^n 2μ^+_iμ^-_ik_i^2+1/2[ (_ϕγ_0,γ_0)+ (γ_1,γ_1)] +(_ϕU^L(T_2) ,w_0)+ (U^L_t(T_2), w_1)+ O(ϵ_1^3).Note that using the L^6 estimate of U^L in (<ref>), we further have1/2[ (_ϕU^L(T_2), U^L(T_2)) +(U^L_t(T_2),U^L_t(T_2)) ] = 1/2U^L(T_2)_^2 + ((-V+5ϕ^4)U^L(T_2), U^L(T_2))=1/2U^L(T_2)_^2 + O(δ_1^2)In view of (<ref>),(<ref>)together with (<ref>) implies thatw_0_Ḣ^1(|x|≥ T_2-T_1+L) +w_1_L^2(|x|≥ T_2-T_1+L)≤ 4δ_1.Thus (w_0,w_1) is small inside the region {|x|≥ T_2-T_1+L}, while (<ref>) implies that U^L is small inside the region {|x|<T_2-T_1+L}:U^L(T_2)_(|x| < T_2-T_1+L)≤δ_1.Hence we get that|(U^L_t(T_2), w_1) |=|∫_{|x|≥ T_2-T_1+L}∪{|x| < T_2-T_1+L} U^L_t(x,T_2)w_1(x) dx |≲δ_1 ;and that |(_ϕU^L(T_2) ,w_0)|= | ∫ (-Δ -V +5ϕ^4)U^L(x,T_2)w_0(x)dx | =| ∫_{|x|≥ T_2-T_1+L}∪{|x| < T_2-T_1+L}∇ U^L(T_2) ·∇ w_0dx+ ∫ ( -V +5ϕ^4) U^L(T_2)w_0dx|≲δ_1 Now let us combineestimates (<ref>), (<ref>), (<ref>) with(<ref>),noting (<ref>), we deduceℰ(u) = ℰ(U)+1/2[ (_ϕγ_0,γ_0)+ (γ_1,γ_1)] - ∑_i=1^k 2μ^+_iμ^-_ik_i^2 +O(δ_1 +ϵ_1^3)(iv) Since (γ_0, γ_1) is in the continuous spectrum and ℒ_ϕ has no zero eigenvalues or zero resonance, we have (_ϕγ_0,γ_0)+ (γ_1,γ_1) ≳(γ_0,γ_1)^2_In combination with (<ref>), (<ref>), and(<ref>), this coercivityyields ℰ(u) - ℰ(U) + ∑_i=1^n 2μ^+_iμ^-_ik_i^2 =1/2[ (_ϕγ_0,γ_0)+ (γ_1,γ_1)] + O(δ_1 +ϵ_1^3)≳(γ_0,γ_1)^2_ +O(δ_1 +ϵ_1^3)≳ cϵ_1^2 -∑_i=1^n[k_i^2(μ^+_i -μ^-_i)^2+ (μ^+_i +μ^-_i)^2] +O(δ_1 +ϵ_1^3).This implies that∑_i=1^n 2μ^+_iμ^-_ik_i^2 + C/2∑_i=1^nk_i^2(μ^+_i -μ^-_i)^2+ C/2∑_i=1^n (μ^+_i +μ^-_i)^2≳ϵ_1^2 -|ℰ(U) - ℰ(u)| - O(δ_1+ϵ_1^3) ≳ϵ_1^2 - Cϵ - O(δ_1+ϵ_1^3).Since all the constants depend only on U, we can choose δ_1, ϵ≪ϵ_1^2 and conclude that ∑_i=1^n |μ^+_i|^2 +|μ^-_i|^2≳ϵ_1^2Now we denote |μ_max|=max{|μ_i^+|, |μ_i^-|,1≤ i≤ n}. We can find 1≤ i_0≤ nsuch that either |μ_i_0^+| =|μ_max| or |μ^-_i_0|=|μ_max|. From (<ref>) and (<ref>), we get2n|μ_max|^2 ≥ cϵ_1^2 ≥c/4w_^2,hence |μ_max|≥√(c/8n)w_. The constant c only depends on V and ϕ.Step 3: Show the second emission of energy and finish the proof. Case 1: |μ_max|=|μ_i_0^+|. Consider the solution ũ to equation (<ref>) with ũ(T_2) = (ϕ,0)+(w_0, w_1),Take κ=√(C/8n),K and ϵ_∗corresponding to R=0 in Lemma <ref>. Note that both parameters depend only on V.With these choices of parameters, we get ε(κ,K,ϵ_∗)>0 from Lemma <ref>. Shrinking ϵ_1 if necessary, we can assume that ϵ_1<ε(κ,K,ϵ_∗). We emphasize that none of these parameters depend on δ_1 or ϵ, which are free parameters at this point. This is very important of course, in order not to run into a circular argument. We also note that T=T(κ,K,V) from Lemma <ref>does not depend on δ_1 or ϵ. We can now apply part (1) of Lemma <ref> and part (2) of Lemma <ref> to conclude that∫_|x|>t-(T_2+T) |∂_t ũ|^2(x,t)dx ≥ c(ϵ_1)>0,for t≥ T_2+T.Denote Ξ:= R^3×[T_2, T_2+T]⋃{(x,t): |x|>t-T_2-T, t≥ T_2+T}. Note that(|V|+ϕ^4)χ_Ξ∈ L^5/4_tL^5/2_x(ℝ^3×ℝ),and that U^L+ũ is an approximate solution to (<ref>) with a right hand side f with f_L^1_tL^2_x≲δ_1. By bound (<ref>) and Lemma <ref> (by treating u as perturbation of U^L+ũ), if we choose δ_1 sufficiently small, thenfor (x,t)∈Ξ,u(t,x)=U^L(x,t)+ũ(x,t)+r(x,t),where the remainder term r satisfiessup_t∈r(t)_(^3)≤ Cδ_1.The estimate (<ref>), decomposition (<ref>) and the estimate on the remainder term (<ref>) imply for t≥ T_2 (in particular, t≥ T_2+T)∫_|x|≥ t-T_1+L|∇_t,xũ|^2(x,t)dx≤ Cδ_1,this combined with (<ref>) implies that for t≥ T_2+T∫_t-T_1+L≥ |x|≥ t-(T_2+T)|∇_t,xũ|^2(x,t)dx≥ c(ϵ_1)-Cδ_1.Hence by estimating u(t)_ in different regions {|x|≥ t-T_1+L} and {t-T_1+L≥ |x|≥ t-(T_2+T)}, we get thatu^2_(|x|≥ t-(T_2+T))≥U^L+ũ^2_(|x|≥ t-(T_2+T) -C_1δ_1≥U^L^2_(|x|≥ t-T_1+L) + c(ϵ_1)-C_2δ_1≥U^L^2_(^3)+c(ϵ_1)-C_3δ_1≥U^L^2_(^3)+1/2 c(ϵ_1).The last line holds when we choose δ_1 sufficiently small. (<ref>) is then proved with A=T_2+T and δ =1/2c(ϵ_1)>0.Now we prove that u cannot scatter to (ϕ, 0)t→ +∞. Suppose it does so with free radiation u^L, i.e., u(t)-(ϕ,0) -u^L(t) _→ 0, ast→ +∞.Then (<ref>) implies thatu^L(t)^2_(^3)≥lim_t→∞u(t)^2_(|x|≥ t-A)≥U^L(t)^2_(^3)+δ.Note thatU^L^2_=ℰ(U)-ℰ(ϕ,0),u^L^2_=ℰ(u)-ℰ(ϕ,0).We have reached contradiction with (<ref>) if u(0)-U(0)_(^3) is chosen small, and thus have proved the theorem in case 1.Case 2:|μ_max|=|μ_i_0^-|. We will show this is impossible if we take ϵ small enough.In fact, again applying part (1) of Lemma <ref> and part (2) of Lemma <ref>,consider the solution ũ to equation (<ref>) with dataũ(T_2) = (ϕ,0)+(w_0,w_1).We can find a time T>0 such that∫_|x|>|t-(T_2-T)| |∂_t ũ|^2(x,t)dx ≥ c(ϵ_1)>0,for t≤ T_2-T.By taking ϵ sufficiently small, we can assume T_2>2T. Now setting time t=0 in (<ref>), we get ũ(0)_(|x|>1/2 T_2) >c(ϵ_1).Introduce the setΞ':=R^3×[T_2-T, T_2]⋃{(x,t): |x|>|t-(T_2-T)|, 0≤ t≤ T_2-T}.In analogyto case 1, if we choose δ_1>0 sufficiently small, then for (x,t)∈Ξ'we haveu(t,x)=U^L(t,x)+ũ(t,x)+r(t,x),with the remainder term r satisfyingsup_t∈r(t)_(^3)≤ Cδ_1.From (<ref>) and by our choice of T_2, i.e.,T_2>2(L+T_1+1) and T_2>2T, we have U^L(0)_(|x|>1/2 T_2)<δ_1 and(u_0,u_1)_(|x|>1/2 T_2)≥∇_x,tũ(0)_(|x|>1/2 T_2) - Cδ_1 > c(ϵ_1)-Cδ_1 >1/2c(ϵ_1)The last inequality holds provided we take δ_1 small enough.This yieldsa contradiction to the finite energy of U(0) by choosing ϵ sufficiently small and T_2 sufficiently large. Hence case 2 does not arise and we are done. Next we prove the property of path connectedness.For any unstable excited state (ϕ, 0), the corresponding center-stable manifold ℳ_ϕ is path connected.Given data (u_0, u_1), (ũ_0,ũ_1)∈ℳ_, we denote the correspondingsolutions by u, ũ. Write h=u-, ℓ=ũ-.Repeat step 1 and step 2inthe proof of Theorem <ref>. Then given any ϵ≪ 1, we can find T=T(ϵ, u, ũ), such that h_L^6,2_xL^∞_t∩ L^∞_xL^2_t ( ^3×[T,∞)) <ϵ,ℓ_L^6,2_xL^∞_t∩ L^∞_xL^2_t ( ^3×[T,∞)) <ϵ.Now we seek a function w(θ, t, x) of the form w(θ, t, x)=(1-θ) u +θũ+η=+ (1-θ) h +θℓ +∑_i=1^nλ_i(θ, t)ρ_i +γ(θ, t,x)such thatfor all θ∈ [0,1], γ(θ, t,x)⊥ρ_i, i=1,…, n andw(θ,t,x) is a solution to equation (<ref>) that scatters to . For θ∈[0,1] fixed, the equation satisfied by η = ∑_i=1^nλ_i(θ, t)ρ_i +γ(θ, t,x) is: η_tt-Δη -V(x)η + 5^4η + N(θ,h,ℓ,,η)=0,where N(θ,h,ℓ,,η) = ( + (1-θ) h +θℓ +η )^5 -(1-θ) (+h)^5 -θ ( +ℓ)^5 - 5^4 η.Nowwecan repeat the stability condition (<ref>) and obtainthe reduced system of the form (<ref>). In N(θ,h,ℓ,,η), the terms independent of η are of the form ( + (1-θ) h +θℓ )^5 -(1-θ) (+h)^5 -θ ( +ℓ)^5 =∑_i+j+k=5, i≤ 3 C(θ, i, j, k) ^i h^j ℓ^k.Notice that there are noterms^5 or ^4 h,ϕ^4ℓ.Also, the linear term of η inN(θ,h,ℓ,,η) is 5( + (1-θ) h +θℓ )^4 η - 5^4 ηhenceall linear terms involve a factor of h or ℓ.Now we can repeat estimates (<ref>)(<ref>),then (<ref>) for the linear term in η, (<ref>) for higher order terms in η. We also havethe following estimate on terms independent of η∑_i+j+k=5, i≤ 3 C(θ, i, j, k)^i h^j ℓ^k_12( [T,∞)×^3)≲ϵ^2To sum up, using the X norm defined in (<ref>), we conclude that(λ_1,⋯, λ_n, γ)_X([T,∞))≤ Lϵ^2 +L( ∑_i=1^n |λ_i(θ, T)| + (γ(θ, T), γ̇(θ,T))_) + L ϵ(λ_1,⋯, λ_n, γ)_X([T,∞)) + L ∑_k=2^5(λ_1,⋯, λ_n, γ)_X([T,∞))^k,where L>1 is a constant only depending on the constants in the reversed Strichartz estimates,ϕ_L^6(^3) and ρ_i_L^∞_x∩ L^6,2_x. Moreover, in a similar fashion one sees that the difference of two solutions satisfies a similar estimate in which the first two terms disappear.Following step 3 of the proof for Theorem <ref>, wecan use the contraction mapping principle and conclude that forsufficiently smalldata∑_i=1^n |λ_i(θ,T)| + (γ(θ,T), γ̇(θ, T))_≤δthere isa solutionw as in (<ref>) which solves (<ref>). We can alsocheck that w scatters to ϕ as in step 4 of the proof for Theorem <ref>. In particular, let us takeλ_i(θ,T)=1/nδθ(1-θ) and γ⃗(θ,T,x)=0⃗. We claim that the corresponding solution w(θ, t, x)satisfies the following relationw(0,t,x)=u(t,x), w(1,t,x)=ũ(t,x), for allt∈.In fact, notice that λ_i(0,T)=0, γ⃗(0,T,x)=0⃗ implies λ_i(0,t)=0, γ⃗(0,t,x)=0⃗ for t≥ T, which further implies w(0,t,x)=u(t,x), t≥ T. Similarly we have w(1,t,x)=ũ(t,x), t≥ T. Then (<ref>) follows from theuniqueness of solutions to equation (<ref>).Hence {w⃗(θ, 0, x), θ∈ [0,1]} is a path in ℳ_ϕ connecting the two data (u_0, u_1), (ũ_0, ũ_1).Now we can finish the proof for our main theorem.We only consider the case in which (ϕ,0) is unstable; stable (ϕ,0) can be handled using standard perturbation arguments and the reversed Strichartz estimates. We only note that due to the lack of local wellposedness of equation (<ref>) in the reverse Strichartz space L^6,2_xL^∞_t∩ L^∞_xL^2_t, we need to use the fact thatlim_T→∞U-ϕ_L^6,2_xL^∞_t∩ L^∞_xL^2_t(ℝ^3×[T,∞))=0,if U(t) scatters to ϕ as t→∞. This fact can be easily deduced by using the same argument as in Claim <ref>. In some small neighborhood of any point U(0) on ℳ_ϕ, ℳ_ϕ coincides with the local center-stablemanifold ℳ of codimension n which we constructed in Section <ref>. By Theorem <ref>, ℳ_ϕ is thus a global manifold of co-dimension n. The path-connectedness follows from Theorem <ref>. amsplain | http://arxiv.org/abs/1706.09284v2 | {
"authors": [
"Hao Jia",
"Baoping Liu",
"Wilhelm Schlag",
"Guixiang Xu"
],
"categories": [
"math.AP"
],
"primary_category": "math.AP",
"published": "20170627005858",
"title": "Global center stable manifold for the defocusing energy critical wave equation with potential"
} |
^1Fakultät für Physik, Munich Quantum Center, and Center for NanoScience (CeNS), Ludwig-Maximilians-Universität München, Geschwister-Scholl-Platz 1, D-80539 München, Germany^2Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, U.S.A^3Center for Integrated Nanotechnologies, Materials Physics and Applications Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, U.S.A We study both experimentally and theoretically the fundamental interplay of exciton localization and polarization in semiconducting single-walled carbon nanotubes. From Stark spectroscopy of individual carbon nanotubes at cryogenic temperatures we identify localized excitons as permanent electric dipoles with dipole moments of up to 1 eÅ. Moreover, we demonstrate field-effect doping of localized excitons with an additional charge which results in defect-localized trions. Our findings provide not only fundamental insight into the microscopic nature of localized excitons in carbon nanotubes, they also signify their potential for sensing applications and may serve as guidelines for molecular engineering of exciton-localizing quantum dots in other atomically thin semiconductors including transition metal dichalcogenides. Dipolar and charged localized excitons in carbon nanotubes Jan T. Glückert^1, Lyudmyla Adamska^2, Wolfgang Schinner^1, Matthias S. Hofmann^1, Stephen K. Doorn^3, Sergei Tretiak^2, and Alexander Högele^1 December 30, 2023 ===================================================================================================================================================Optical transitions of semiconducting carbon nanotubes (CNTs) are dominated by excitons <cit.> which exhibit strong antibunching in the photoluminescence (PL) in their localized limit <cit.>. Exciton localization can arise at unintentional defects with shallow potentials <cit.> or incidental proximal charges <cit.> and ensure non-classical emission statistics up to room-temperature<cit.> for excitons bound to deep traps of oxygen side-wall dopants <cit.>. Along with oxygen functionalization <cit.>, covalent side-wall chemistry with aryl and alkyl functionality <cit.> provides a versatile molecular means to engineer the photophysics of semiconducting CNTs. Introduced in a moderate concentration, the decoration of CNT side-walls with covalent defects results in substantial modifications such as brightening of nanotube emission and increased quantum yields <cit.>, axially pinned PL <cit.> and inhibited diffusion <cit.>.Defect-localized excitons in CNTs represent a viable resource for applications in quantum sensing and quantum cryptography. For the latter technology, CNTs may facilitate the development of robust single-photon sources with room-temperature operation in the telecom band by utilizing discrete optical transitions of defect-localized excitons <cit.>. Covalent chemistry is readily available to fine-tune the exciton PL energy <cit.>, and recent successful integration of CNTs into optical cavities <cit.> has demonstrated Purcell enhancement and directional coupling of single-photon emission as means to increase the single-photon emission efficiency. Moreover, the interplay of chemical modification and charge doping facilitates photoemission from trions <cit.>, which can be utilized to interface photons with the CNT spin degree of freedom <cit.> via schemes of spin-tagged optical transitions analogous to charged semiconductor quantum dots and nitrogen vacancy (NV) centers in diamond <cit.>. This spin-photon interface in turn should enable all-optical sensing of magnetic fields in analogy to magnetometry based on charged NV color centers <cit.>. A corresponding nanoscale sensor for the measurement of the electric field <cit.> with sensitivity down to the elementary electron charge <cit.> could utilize the electric dipole moment associated with localized excitons.Our work identifies both integral elements - dipolar localized excitons and voltage-controlled trions - for the development of sensing devices based on carbon nanotubes. By embedding CNTs in a field-effect (FET) device, we performed Stark spectroscopy of localized nanotube excitons in a transverse electric field at cryogenic temperatures. Our experiments demonstrate that exciton localization is accompanied by static exciton polarization irrespective of the details of the localizing potential. An average localization-induced electric dipole moment of ∼ 0.3 eÅ found experimentally is in good quantitative agreement with ab-initio model calculations for excitons bound by oxygen defects on the side-wall of a (6,5) nanotube. Moreover, we found that defect potential traps can bind an additional charge to promote PL from defect-localized trions <cit.>, with control over the charging state provided by the gate voltage.To subject nanotubes to a transverse electric field we fabricated FET devices based on a metal-oxide-semiconductor sequence as illustrated in Fig. <ref>a. The FET devices were fabricated starting with a p^+-doped silicon back gate terminated by an insulating layer of d_1=100 nm thermal SiO_2 that was cleaned with standard solvents and subsequently exposed to an oxygen plasma before spin-coating micelle-encapsulated CoMoCat CNTs with a spatial density below μm^-2.The CNT layer was subsequently covered by sputter deposition with an insulating layer of Al_2O_3 of variable thickness d_2 (with d_2=7, 17, 39 and 42 nm in four different sample layouts), and a semitransparent NiCr layer of 5 nm thickness. A gate voltage V_g applied between the top and the ground electrode resulted in a homogeneous transverse electric field F through F=V_g/d, with d being the total thickness of the oxide layers. The functionality of our FET devices with break-down voltages of ± 80 V at low temperatures, corresponding to transverse electric field strengths of up to ± 1 V/nm, was confirmed with capacitance-voltage spectroscopy <cit.>.Individual CNTs embedded in a FET device were studied with photoluminescence (PL) spectroscopy in a home-built confocal microscope at the temperature of liquid helium of 4.2 K. A Ti:sapphire laser tuned in the range of 730 - 900 nm was used to excite the PL via phonon sidebands in continuous wave mode. The PL of individual CNTs was dispersed with a monochromator and recorded with a low-noise nitrogen-cooled silicon CCD. Our experiments focused on (6,4) and (9,1) chiral nanotubes with emission in the spectral range of 1.35 - 1.43 eV <cit.>. Characteristic PL signatures of individual nanotubes in our device are shown in Fig. <ref>b and c. Most of the CNTs were found to exhibit either a single-peak PL emission with an asymmetric lineshape (labelled as X in Fig. <ref>b) characteristic of disorder-localized excitons <cit.> or a two-peak emission spectrum (denoted as X and X^* in Fig. <ref>c). In our experiments, we assign one-peak spectra to excitons localized by environmental disorder, and two-peak spectra to exciton PL from oxygen-dopant sites introduced on CNT side-walls by sputter deposition of Al_2O_3 <cit.>. The evolution of the CNT spectra with a single-peak and a double-peak spectrum as a function of the transverse electric field are shown in Fig. <ref>b and c, respectively. The electric field strength and orientation was varied proportional to the zig-zag voltage ramp shown in Fig. <ref>a. The gate voltage was changed in discrete steps between maximum positive and negative values, with V_max ranging between 15 V and 30 V depending on the device. After each voltage step a PL spectrum was acquired for an incremental build-up of PL intensity false-color plots as in Fig. <ref>b for a single-peak emission, and in Fig. <ref>c for the X and X^* peaks. We repeated this procedure on more than 50 individual CNTs. Roughly one third of the tubes we have investigated showed irregular responses such as non-monotonic energy jumps or irreversible intensity fluctuations and were discarded from further analysis. The more regular responses as in Fig. <ref>b and c are representative for CNT excitons localized by incidental and oxygen-specific defect traps, respectively.The vast majority of the nanotubes in our devices exhibited linear energy dispersions in response to the transverse electric field ramp, and both blue- and red-shifts were observed <cit.> for different peaks (Fig. <ref>b and c). The linear slope, associated with the first-order Stark response of a permanent dipole, is in striking contrast to the second-order Stark effect expected for pristine CNTs. From a fitting procedure of CNT PL with single- and double-peak emission spectra as a function of the electric field strength according to E(F)= E_0 - p F (red solid lines in Fig. <ref>b and c) we extracted the transverse dipole moment p of localized excitons with emission energy E_0 at V_g=0 V. For the CNT in Fig. <ref>b, we obtained p_X=-0.38 eÅ, and for the two states X and X^* of Fig. <ref>c we determined p_X=0.36 eÅ and p_X^*=-0.26 eÅ from linear fits to the data.The results of the fitting procedure for all other CNTs with single- and double-peak emission are summarized in the histogram of Fig. <ref>d. It shows the distribution of the absolute value of the transverse permanent dipole moments determined for different CNTs and devices. The maximum value of the distribution at |p| ≃ 0.7 eÅ corresponds to an electron-hole separation of ∼ 10% of the CNT diameter, a remarkably large value for a permanent dipole moment that is absent in pristine CNTs according to symmetry considerations. Another remarkable trend in our data are the anti-correlated signs of the dipole moments associated with X and X^* peaks (data points within the grey-shaded quadrants in Fig. <ref>e). Among the tubes with two-peak spectra, the majority exhibited positive p_X and negative p_X^* permanent dipole values (corresponding to data points in the lower right quadrant).Our experimental observations suggest an intimate interplay of exciton localization and polarization which we confirmed by atomistic calculations of a (6,5) model nanotube in transverse electric field <cit.>. In our calculations, the nanotube was embedded in a homogeneous medium with permittivity ε_r=6.3 to account for the effective dielectric environment composed of Al_2O_3 (ε_r=9.3) and Si_2O_2 (ε_r=3.9) layers at the top and bottom of the tube and micellar encapsulation. First, we modelled the response of a pristine tube and found a quadratic energy dispersion of the bright luminescent state with transverse polarizability α_⊥≃ 7.7 Å^2 in accord with previous estimates both from tight-binding <cit.> and first-principles calculations <cit.>.In stark contrast, for both bright peaks associated with an oxygen side-wall defect in ether-d configuration <cit.>, our calculations yield predominantly linear dispersions (Fig. <ref>) when subjected to a transverse electric field of up to 0.2V/nm. The slopes and signs depend on the position of the defect on the nanotube side-wall as indicated by inset schematics in Fig. <ref>. Our calculations predict red- and blue-shifts for the X and X^* emission, respectively, with corresponding maximum values for the permanent dipole moments of 0.35 eÅ and -0.58 eÅ for a defect placed at the apex of the tube (left panel of Fig. <ref>). This defect geometry is expected to dominate our experiments with side-wall dopants introduced preferentially from the top by oxide sputtering, whereas localizing sites at the nanotube base caused by proximal charges <cit.> at the SiO_2 surface should be less frequent. Consistently, our experimental data of Fig. <ref>e reflects both the anti-correlated signs of the two-peak dispersions predicted by theory, and the different likelihood for defects to occur at the top and the bottom of the tubes (in the latter case the respective slopes would remain anti-correlated but interchange their signs). Experimental observation of dispersions as in the right panel of Fig. <ref> should be rare because of the peripheral configuration of the related defects in the top-down sputter deposition process.Both experiment and theory suggest that the radial symmetry of the electron-hole charge distribution is imbalanced at the exciton-localizing defect sites by field gradients associated with defect traps, and both the strength and the orientation of the respective dipole moment depend on the specifics of the localizing defect. The defect potentials should also act as traps for individual charges <cit.> and, in the presence of photoexcited electron-hole pairs, give rise to emission from energetically lower-lying trions <cit.>. Indeed, we observed signatures of such red-shifted PL satellites for some nanotubes within limited gate voltage ranges of our devices. Fig. <ref>a shows the PL response of a CNT to the gate voltage ramp as in Fig. <ref>a. The PL intensity is represented on a logarithmic false-color scale to enhance the visibility of the weak lowest-energy satellite which we assign to defect-localized trion PL emission (denoted in Fig. <ref>a as T; the sharp horizontal features unaffected by the gate voltage correspond to Raman scattered laser photons). For this specific tube, the T peak was observed around 1.24 eV in addition to X and X^* emission only at negative gate voltages. Other CNTs exhibited similar features only for positive voltages <cit.> indicating that the polarity of the defect excess charge trapped out of the optically excited charge reservoir <cit.> depends on the defect potential details. Akin to previous experiments <cit.>, the trion emission emerges at the expense of the main peak PL intensity (compare the relative intensities of T and X peaks at 0 V and -10 V in Fig. <ref>b).Further confirmation for the assignment of the voltage-induced satellite to trion emission comes from the inspection of the trion binding energy. We extract the energy scale associated with the binding of an excess charge to the lowest defect-localized state by taking the energy splitting Δ_TX^* between the T and X^* emission peaks. This splitting, shown for all CNTs with charging signatures in the inset histogram of Fig. <ref>b, varies between 20 and 60 meV for the (6,4) and (9,1) narrow-diameter tubes in the spectral region of our experiment. This trion binding energy is not to be confused with earlier experiments measuring the splitting between the trion peak and the E_11 emission energy with excess contribution from exchange interactions <cit.>. It should be rather compared with the theoretical estimate of the bare trion binding energy <cit.>, or with the energy splitting observed between the neutral and charged defect-localized emission peaks in diazonium-functionalized CNTs <cit.>. Theory predicts a trion binding energy of about 30 meV for a (6,5) nanotube in a dielectric medium with ε_r =6 <cit.>. In aqueous suspension with ε_r ≃ 2, the corresponding experimental value of ∼ 100 meV <cit.> was found in accord with the scaling of the trion binding energy with the dielectric constant as ε_r^-1.56 <cit.>. Given the relatively high effective dielectric constant of the CNT environment in our FET devices and same diameters of (6,5) and (9,1) CNTs, we find very good agreement between our lower values of Δ_TX^* and theory. Consistently, the larger values in the distribution of Fig. <ref>b are associated with (6,4) oxygen-doped nanotubes because of the inverse dependence of the trion binding energy on the tube diameter <cit.>.Our observation of defect-localized emission in combination with voltage-controlled charging places CNTs alongside semiconductor quantum dots <cit.> and NV centers <cit.> with charge-tunable emission characteristics and spin-projective optical transitions <cit.>. An intriguing advantage of CNTs for spin-based applications is expected to arise from prolonged electron spin coherence time in an isotopically engineered nuclear-spin free lattice <cit.>. Moreover, the absence of dangling bonds in sp^2-hybridized CNTs could enable long spin coherence times of electrons localized at engineered nanotube side-wall defects with immediate environmental proximity - a key factor for nanoscale-magnetometry <cit.> where near-surface color centers in diamond currently encounter major limitations due to unsaturated sp^3-bonds of the diamond crystal surface <cit.>. Finally, our results could inspire efforts to create chemically engineered quantum dots for in-plane confinement of excitons in emergent two-dimensional transition metal dichalcogenide semiconductors <cit.>.We thank J. P. Kotthaus, S. Rotkin, I. Bondarev and V. Perebeinos for useful discussions, P. Altpeter and R. Rath for assistance in the clean room, and P. Stallhofer from Wacker AG for providing the wafer material. This work was performed in part at the Center for Integrated Nanotechnologies, a U.S. Department of Energy, Office of Science user facility. 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Nanotechnol. volume7, pages699 (year2012).§ SUPPLEMENTARY ONLINE MATERIAL Field-effect device characteristicsIn order to apply transverse electric fields to individual nanotubes we fabricated metal-oxide-semiconductor (MOS) devices as illustrated in Fig. <ref>a. Highly p^+-doped silicon substrate terminated by an insulating layer of d_1=100 nm thermal silicon oxide (SiO_2) was used as the ground electrode. The sample surface was cleaned with standard solvents and subsequently exposed to an oxygen plasma. Commercial CoMoCat-nanotubes (SouthWest NanoTechnologies) encapsulated in sodium dodecylbenzenesulfonate (SDS) were dispersed out of an aqueous suspension on SiO_2 substrates. The average CNT length was ∼ 500 nm, the spin coating parameters were adjusted to yield a CNT density below 1 μm^-2 which was confirmed by AFM imaging like in Fig. <ref>a. The CNT layer was subsequently covered with a second insulating layer of aluminum oxide (Al_2O_3) of variable thickness d_2, yielding a total oxide thickness d=d_1+d_2. A semitransparent top electrode of 3-5 nm nickel chromium completed the MOS structure.To determine the strength of the homogeneous electric field in our MOS devices with d_2=7, 17, 39 and 42 nm we performed capacitance-voltage (CV) measurements at 4.2 K. We used a differential CV measurement technique by admixing a sinusoidal modulation voltage with amplitude δ V=10 mV and frequency f in the rage of 4-450 Hz to the dc gate voltage. The resulting ac capacitive current was demodulated with a lock-in amplifier and scaled to the current of a reference capacitance. A representative CV curve recorded for a device with d_2=7 nm is shown in Fig. <ref>b. At high negative voltages the MOS device response was dominated by hole accumulation where we obtained the maximum capacitance C_i ≃ 1 nF in accord with the geometry of our device. For more positive values of V_g the CV curve showed a reduction of the capacitance to C_min/C_i ≃ 0.93 without recovery despite further biasing and slow modulation (black CV traces in Fig. <ref>b), indicating that the limit of strong inversion was suppressed in our device because of slow generation-recombination rates of minority charge carriers at 4.2 K. This feature as well as the hysteresis at positive V_g are characteristic for non-ideal p-type MOS-capacitors with inhibited inversion <cit.> and distinct charging and discharging dynamics of charge traps at the Si-SiO_2 interface <cit.>. We note, however, that this non-ideal capacitance change as a function of the gate voltage is sufficiently small (<10 %) to establish a linear relation between the macroscopic electric field strength, F, and the gate voltage, V_g, through F=V_g/d. With this relation we estimate that field strengths of up to ± 1 V/nm were routinely accessible with voltages of ± 80 V applied to our devices at low temperatures without break-down. The details of charge traps were investigated with CV spectroscopy with in situ illumination. In the presence of photo-generated electrons by diffuse illumination, inversion was recovered for low-frequency modulation (blue circles in Fig. <ref>b) as opposed to high-frequency modulation (orange circles in Fig. <ref>b). From modeling of the CV characteristics in the limiting case of low (high) modulation frequency <cit.> shown as blue (orange) solid line in Fig. <ref>c we determined the characteristic charge impurity density of Q_tot=3.6 · 10^12 cm^-2 in our devices. It includes both the surface states at the Si-SiO_2 interface and the charge states in the insulating oxide volume <cit.>. Devices of different oxide thicknesses were used to determine the volume density of the oxide states, Q_oxide/d = 3.0 · 10^17 cm^-3. On average this number implies the presence of a charge trap state within the volume of a cylinder with ∼ 2 nm radius around a 200 nm long CNT. These charge traps likely constitute the charge reservoir for photoactivated charge doping of nanotubes that exhibited trion emission.Stark spectroscopy and trion photoluminescencePhotoluminescence (PL) Stark spectroscopy was performed in response to transverse electric field as detailed in the main text and the Methods section. Individual nanotubes exhibited distinct PL energy shifts according to dipole moments p of different defect configurations and geometries. Fig. <ref> highlights the case of different responses to a gate voltage ramp as in Fig. 2a of the main text. The data in Fig. <ref>a is reproduced from Fig. 2c of the main text for direct comparison with another nanotube shown in Fig. <ref>b with a double-peak spectrum yet different linear Stark shifts of the PL energy. In Fig. <ref> we exemplify the response of different individual CNTs to charge doping. Two nanotubes exhibited trion emission satellites at positive gate voltages (Fig. <ref>a and b) in contrast to a nanotube with similar emission characteristics at zero volts and stable trion emission at negative gate voltages (Fig. <ref>c).Quantum chemistry calculationsThe computations were performed using Gaussian09 software suite <cit.> with B3LYP functional <cit.> and STO-3G basis set. The dielectric environment due to silicon oxide and aluminum oxide surroundings of the nanotubes was taken into account as solvent with a dielectric constant of ε_r=6.3, which is the average dielectric constant of the two oxides. The solvent effects were simulated in the framework of continuum polarizable conductor-like medium <cit.>; 8-nm long segments of (6,5) carbon nanotube with hydrogen-terminated ends were used in these calculations. Pristine and oxygen-doped CNTs were geometry optimized in solvent. The optical transition energies were calculated using Time-Dependent Density Functional Theory (TD-DFT).Electric field was applied in transverse direction and optical transition energies were computed without additional geometry optimization. The transition energies, shifted by the energy of E_11 transition in pristine CNT, are shown in Fig. <ref>. In order to establish a quantitative measure of exciton wave functions, exciton plots were built for relevant optical transitions and shown in the left panels of Fig.s <ref>, <ref> and <ref> (see Ref.s <cit.> for more details on exciton plot characterization of excited states in one-dimensional structures). In these contour plots, the bright spot elongated along the diagonal signifies where the exciton wave function is located along the CNT axis, and the "width" of the elongated plot is the electron-hole correlation length (also referred to as the exciton size). Our analysis shows that excitons are localized on sub-10 nm length scale (to about 3 nm for the E_11^* state, denoted as X^* in the main text, and about 6 nm for the E_11^- state denoted as X in the main text) with an electron-hole correlation length of about 2 nm. We also projected the exciton wave function onto the basis of atomic orbitals in order to provide a qualitative real-space visual measure of exciton wave function. The corresponding plots are shown in the right panels of Fig.s <ref>, <ref> and <ref>.9 natexlab#1#1bibnamefont#1#1bibfnamefont#1#1citenamefont#1#1url<#>1urlprefixURL[Sze(1981)]Sze authorS. M. Sze, titlePhysics of Semiconductor Devices (publisherJohn Wiley & Sons, year1981), edition2nd ed.[Grove et al.(1965)Grove, Deal, Snow, and Sah]Grove authorA. Grove, authorB. Deal, authorE. Snow, and authorC. Sah, journalSolid-State Electronics volume8, pages145 (year1965).[Goetzberger. and Irvin(1968)]Goetzberger authorA. Goetzberger. and authorJ. C. Irvin, journalIEEE Trans. Electron Dev. volume15, pages1009 (year1968).[Frisch et al.(2015)Frisch, Trucks, Schlegel, Scuseria, Robb, Cheeseman, Scalmani, Barone, Mennucci, Petersson et al.]Frisch2009 authorM. Frisch, authorG. Trucks, authorH. Schlegel, authorG. Scuseria, authorM. Robb, authorJ. Cheeseman, authorG. Scalmani, authorV. Barone, authorB. Mennucci, authorG. 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"authors": [
"Jan T. Glückert",
"Lyudmyla Adamska",
"Wolfgang Schinner",
"Matthias S. Hofmann",
"Stephen K. Doorn",
"Sergei Tretiak",
"Alexander Högele"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170626124742",
"title": "Dipolar and charged localized excitons in carbon nanotubes"
} |
[email protected] Atominstitut, Technische Universität Wien, Stadionallee 2, A-1020 Wien, Austria [email protected] Atominstitut, Technische Universität Wien, Stadionallee 2, A-1020 Wien, AustriaDepartment of Physics, New Mexico State University, Las Cruces, New Mexico 88003, USA [email protected] Atominstitut, Technische Universität Wien, Stadionallee 2, A-1020 Wien, Austria [email protected] Atominstitut, Technische Universität Wien, Stadionallee 2, A-1020 Wien, Austria FH Campus Wien, University of Applied Sciences, Favoritenstraße 226, 1100 Wien, Austria [email protected] Peter the Great St. Petersburg Polytechnic University, Polytechnicheskaya 29, 195251, Russian Federation In the Standard Model (SM) we calculate the decay rate of the neutron radiative β^-–decay to order O(α^2/π^2 ∼ 10^-5), where α is the fine–structure constant, and radiative corrections to order O(α/π∼ 10^-3). The obtained results together with the recent analysis of the neutron radiative β^-–decay to next–to–leading order in the large proton–mass expansion, performed by Ivanov et al.Phys. Rev. D 95, 033007 (2017), describe recent experimental data by the RDK II Collaboration (Bales et al., Phys. Rev. Lett. 116, 242501 (2016)) within 1.5 standard deviations.We argue a substantial influence of strong low–energy interactions of hadrons coupled to photons on the properties of the amplitude of the neutron radiative β^-–decay under gauge transformations of real and virtual photons.12.15.Ff, 13.15.+g, 23.40.Bw, 26.65.+t Precision Theoretical Analysisof Neutron Radiative Beta Decay to Order O(α^2/π^2) Ya. A. Berdnikov December 30, 2023 =====================================================================================§ INTRODUCTION During a long period the radiative β^-–decay of a free neutron n → p + e^- + ν̅_e + γ was used as an auxiliary process in the analysis of the radiative corrections to the neutron β^-–decay for the cancellation of infrared divergences, coming from the virtual photon exchanges <cit.>–<cit.>. Only starting from 1996 it has been accepted as a physical process because of the work by Gaponov and Khafizov <cit.>, who made first calculation of the energy spectrum and the decay rate. Then, the neutron radiative β^-–decay was reinvestigated in <cit.> and <cit.>. The first experimental data BR_βγ = 3.13(35)× 10^-3 and BR_βγ = 3.09(32)× 10^-3, measured by Nico et al. <cit.> and Cooper et al.<cit.>, for the photon–energy region 15keV≤ω≤ 340keV, were in agreement within one standard deviation with the theoretical values BR_βγ = 2.87 × 10^-3 <cit.> and BR_βγ = 2.85× 10^-3, calculated by Gardner <cit.> using the theoretical decay rate, published in <cit.>.Recently new precise experimental values of the branching ratios of the radiative β^-–decay of a free neutron have been reported by the RDK II Collaboration Bales et al.<cit.>: BR^(exp)_βγ = 3.35(16)× 10^-3 and BR^(exp)_βγ = 5.82(66)× 10^-3, measured for the photon–energy regions 14keV≤ω≤ 782keV and 0.4keV≤ω≤ 14keV, respectively.Recently <cit.> the rate of the neutron radiative β^-–decay has been recalculated in the Standard Model (SM) and in the tree–approximation to next–to–leading order in the large proton mass expansion by taking into account the contributions of the weak magnetism and proton recoil. As has been found the new theoretical values of the branching ratios BR_βγ = 3.04 × 10^-3 and BR_βγ = 5.08 × 10^-3, calculated for experimental photon–energy regions 14keV≤ω≤ 782keV and 0.4keV≤ω≤ 14keV, respectively, agree with new experimental values BR^(exp)_βγ = 3.35(16)× 10^-3 and BR^(exp)_βγ = 5.82(66)× 10^-3 only within 2 and 1.2 standard deviations.As has been shown in <cit.> the relative contributions of the weak magnetism and proton recoil to the branching ratios of the neutron radiative β^-–decay are of about 0.7 %. Of course, these contributions are small compared to the error bars of the experimental values but they are by a factor 4 larger than the contribution of the weak magnetism and proton recoil 0.16 % to the rate of the neutron β^-–decay <cit.>. As has been pointed out in <cit.> the contributions to the rate of the neutron radiative β^-–decay, calculated in the SM and in the tree–approximation to next–to–leading order in the large baryon mass expansion including the contributions of baryon resonances (see, for example, Bernard et al.<cit.>), cannot in principle exceed 1.5 %. So one may expect some tangible contributions only beyond the tree–approximation, taking into account, for example, one–virtual–photon exchanges to leading order in the large proton mass expansion, i.e. the radiative corrections of order O(α/π).We would like to remind that radiative corrections of order O(α/π) change the rate of the neutron β^-–decay by about 3.75 % <cit.>. Because of an enhancement of the contributions of order 1/M, where 2M = m_n + m_p is an averaged nucleon mass <cit.>, to the rate of the neutron radiative β^-–decay, one may also expect an enhancement of the relative contributions of the radiative corrections of order O(α/π).For the first time the radiative corrections of order O(α/ π) for the analysis of T–odd momentum correlations in the neutron radiative β^-–decay to order O(α^2/π^2) have been calculated by Gardner and He <cit.>. In this paper we give a complete analysis of the radiative corrections to order O(α/π) to the rate of the neutron radiative β^-–decay, caused by pure Quantum Electrodynamics (QED), where photons couple to point–like proton and electron with a contribution of strong low–energy interactions defined by the axial couping constant λ only. A complete set of Feynman diagrams, describing the amplitude of the neutron radiative β^-–decay in the tree and one–loop approximation, are shown in Fig. <ref>, Fig. <ref>, Fig. <ref>, Fig. <ref> and Fig. <ref>, respectively.In Fig. <ref>, Fig. <ref>, Fig. <ref> and Fig. <ref> the states of real and virtual photons with 4–momenta k and q, respectively, are described by the polarization vector ε^*_λ'(k) with λ' = 1,2 and a Green function D_αβ(q) = (η_αβ - (1 - ξ) q_αq_β/q^2)/(q^2 + i0) <cit.>, where the polarization vector obeys the constraint ε^*_λ'(k) · k = 0 with k^2 = 0 and ξ is a gauge parameter. The Feynman diagrams in Fig. <ref> describe a neutron radiative β^--decay with two real photons in the final state. Integrating over degrees of freedom of one of the photons one obtains the contribution of order O(α^2/π^2) to the rate of the neutron radiative β^-–decay with one real photon in the final state.The contributions of strong low–energy hadronic interactions in the Feynman diagrams Fig. <ref>, Fig. <ref>, Fig. <ref>, Fig. <ref> and Fig. <ref> (see also Fig. <ref>) are denoted by shaded regions.The contributions of pure QED are given by the Feynman diagrams in Fig. <ref>a, Fig. <ref>b, Fig. <ref> and Fig. <ref>, where real and virtual photons couple to the point–like proton and electron and strong low–energy hadronic and electromagnetic interactions are factorized. The contribution of strong low–energy interactions is described by the axial coupling constant λ only. In the diagram Fig <ref>c a real photon is emitted by a hadronic block. In spite of a possible dependence of the contribution of this diagram on electron and photon energies it has been neglected in the first calculations of the neutron radiative β^-–decay by Gaponov and Khafizov <cit.> and in the subsequent calculations by Bernard et al.<cit.> and Ivanov et al. <cit.>. In this paper we also accept such an approximation. We neglect the contributions of all Feynman diagrams, where even if one photon (real or virtual) is emitted or absorbed by a hadronic block. In section <ref> we propose a justification of the neglect of the contribution of the diagram in Fig. <ref>c. However, an analysis of contributions of strong low–energy hadronic interactions in the diagrams in Fig. <ref> and Fig. <ref> demands a special consideration and goes beyond the scope of this paper.It is well known that the amplitude of the neutron radiative β^-–decay should be gauge invariant. This means that when making a gauge transformation of a real photon wave function, i.e. replacing the photon polarization vector ε^*_λ'(k) by ε^*_λ'(k) →ε^*_λ'(k) + c k, where c is an arbitrary constant, the contribution proportional to c k should vanish <cit.> (see also <cit.>). In Appendices A and B of the Supplemental Material we investigate the properties of Feynman diagrams in Fig. <ref> and Fig. <ref> with respect to a gauge transformation ε^*_λ'(k) →ε^*_λ'(k) + c k. By means of a direct calculation we show that in Fig. <ref> the sum of the diagrams Fig. <ref>a and Fig. <ref>b is gauge invariant. This implies that the diagram Fig. <ref>c should be gauge invariant by itself. In turn, in Fig. <ref> the diagrams with photons coupled to the proton (Fig. <ref>a, Fig. <ref>b and Fig. <ref>c) and electron (Fig. <ref>d, Fig. <ref>e and Fig. <ref>f) are invariant under a gauge transformation ε^*_λ'(k) →ε^*_λ'(k) + c k separately. We show that invariance of the diagrams in Fig. <ref> with respect to a gauge transformation ε^*_λ'(k) →ε^*_λ'(k) + c k leads to Ward identities, which impose well–known constraints on the renormalization parameters <cit.> and certain constraints on the structure functions (see Appendix B of the Supplemental material). It is important to emphasize that to leading order in the large proton mass expansion the contribution of the diagram in Fig. <ref>a is proportional to the time–component of the photon polarization vector ε^0*_λ'(k), which vanishes in the physical gauge ε^*_λ'(k) = (0, ε⃗^ *_λ'(k⃗ )), where the polarization vector ε⃗^ *_λ'(k⃗ ) obeys the constraint k⃗·ε⃗^ *_λ'(k⃗ ) = 0 <cit.> (see also <cit.>). As has been shown in <cit.> the contribution of the diagrams in Fig. <ref>, taken to leading order in the large proton mass expansion with a real photon in the physical gauge ε^*_λ'(k) = (0, ε⃗^ *_λ'(k⃗ )), describes well the main part of the branching ratio of the neutron radiative β^-–decay (see Table I).As regards the diagrams in Fig. <ref>, to leading order in the large proton mass expansion the contribution of the diagrams Fig. <ref>a, Fig. <ref>b and Fig. <ref>c becomes proportional to ε^0*_λ'(k) and vanishes in the physical gauge ε^*_λ'(k) = (0, ε⃗^ *_λ'(k⃗ )). As a result, only the diagrams Fig. <ref>d, Fig. <ref>e and Fig. <ref>f give a contribution to the amplitude of the neutron radiative β^-–decay, calculated to leading order in the large proton mass expansion with a real photon in the physical gauge ε^*_λ'(k) = (0, ε⃗^ *_λ'(k⃗ )) (see Appendix B of the Supplemental Material).According to Sirlin <cit.>, the contributions of the Feynman diagrams with one–loop corrections, which are shown in Fig. <ref>, Fig. <ref> and Fig. <ref>, should be also invariant under a gauge transformation of a virtual photon, which reduces to a redefinition of a longitudinal part of a photon Green function D_αβ(q) → D_αβ(q) + c(q^2) q_α q_β, where c(q^2) is an arbitrary function of q^2 <cit.>. In Appendix B of the Supplemental Material we show that the contributions of the diagrams in Fig. <ref> are invariant also under a gauge transformation D_αβ(q) → D_αβ(q) + c(q^2) q_α q_β.Unlike the Feynman diagrams in Fig. <ref> the properties and calculation of the set of Feynman diagrams in Fig. <ref> and Fig. <ref> are not so simple and transparent. In Appendix C of the Supplemental Material we show that the contributions of the diagrams Fig. <ref>a and Fig. <ref>b, where strong low–energy and electromagnetic interactions are factorized, vanish after renormalization of masses and wave functions of the proton and electron. In turn, the diagrams Fig. <ref>c and Fig. <ref>d cannot be treated separately from the diagrams in Fig. <ref>, since by themselves they are not invariant under gauge transformations ε^*_λ'(k) →ε^*_λ'(k) + c k and D_αβ(q) → D_αβ(q) + c(q^2) q_α q_β. Following Sirlin <cit.> we assume that required gauge invariance can be fulfilled only for a sum of the Feynman diagrams Fig. <ref>c, Fig. <ref>d and Fig. <ref>, where strong low–energy hadronic and electromagnetic interactions are overlapped and photons (real and virtual) are emitted or absorbed by a hadronic block.Such an assertion is not proved but based on the following observation. After a removal of the lines of a real photon emission the diagrams Fig. <ref>c and Fig. <ref>d reduce themselves to the diagram Fig. <ref>a, which, as has been shown by Sirlin <cit.>, gives the main contribution of the radiative corrections of order O(α/π) to the rate of the neutron β^-–decay. However, the diagram Fig. <ref>a by itself is not invariant under a gauge transformation D_αβ(q) → D_αβ(q) + c(q^2) q_α q_β. As has been pointed out by Sirlin <cit.>, only a sum of the diagrams in Fig. <ref> should be gauge invariant. However, an exact calculation of the diagrams Fig. <ref>b and Fig. <ref>c demands a certain model of strong low–energy interactions of hadrons coupled to photons at low energies.Nevertheless, Sirlin, using the current algebra approach <cit.>, has succeeded in showing that the contributions of the diagrams Fig. <ref>b and Fig. <ref>c do not depend on the electron energy E_e. Such a remarkable property of these diagrams has allowed Sirlin to decompose the contribution of the diagram Fig. <ref>a into invariant and non–invariant parts with respect to a gauge transformation D_αβ(q) → D_αβ(q) + c(q^2) q_α q_β in such a way that a gauge–non–ivariant part does not depend on the electron energy. Then, a constant gauge–non–invariant part has been merely absorbed by formal renormalization of the Fermi weak coupling constant G_F and the axial coupling constant λ. We would like to emphasize that, unfortunately, the diagrams Fig. <ref>c and Fig. <ref>d do not possess such a remarkable property. Nevertheless, it is obvious that different insertions of real photon lines transform the diagrams in Fig. <ref> into a set of Feynman diagrams Fig. <ref>c, Fig. <ref>d and Fig. <ref> and should not destroy gauge properties of these diagrams with respect to a gauge transformation D_αβ(q) → D_αβ(q) + c(q^2) q_αq_β. As a result, the analytical analysis of the diagrams Fig. <ref>c, Fig. <ref>d and Fig. <ref>, which is performed in Appendices C and D of the Supplemental Material, runs as follows. Firstly, we show that to leading order in the large proton mass expansion the diagram Fig. <ref>d, calculated with the contribution of strong low–energy hadronic interactions given by the axial coupling constant λ only, vanishes in the physical gauge of a real photon ε^*_λ'(k) = (0, ε⃗^ *_λ'(k⃗ )). Secondly, we calculate the diagram Fig. <ref>c to leading order in the large proton mass expansion and in the physical gauge of a real photon.After that we decompose the contribution of the diagram Fig. <ref>c into invariant and non–invariant part with respect to a gauge transformation D_αβ(q) → D_αβ(q) + c(q^2) q_α q_β. Keeping only the part, that is invariant under a gauge transformation D_αβ(q) → D_αβ(q) + c(q^2) q_α q_β, and removing from it a part independent of the electron E_e and photon ω energy by renormalization of the Fermi weak coupling and axial coupling constant, we obtain a contribution, which can be accepted as a physical contribution of the diagram Fig. <ref>c to the amplitude and rate of the neutron radiative β^-–decay to order O(α^2/π^2). What then is the role of the Feynman diagrams in Fig. <ref> ?As regards the diagrams in Fig. <ref>, since the contribution of them cannot be calculated in a model–independent way, we follow Sirlin <cit.> and assume that the diagrams in Fig. <ref> i) cancel a gauge–non–invariant part of the diagram Fig. <ref>c, determined relative to a gauge transformation D_αβ(q) → D_αβ(q) + c(q^2) q_α q_β, and the rest ii) either vanishes to leading order in the proton mass expansion in the physical gauge of a real photon ε^*_λ'(k) = (0, ε⃗^ *_λ'(k⃗ )) (see Appendix D of the Supplemental Material) or iii) is a constant, which can be absorbed by renormalization of the Fermi coupling constant G_F and the axial coupling constant λ. This agrees also well with an assumption that different insertions of real photons' lines into the diagrams in Fig. <ref> do not corrupt the properties of the Feynman diagrams in Fig. <ref>c, Fig. <ref>d and Fig. <ref> under a gauge transformation of a photon Green function D_αβ(q) → D_αβ(q) + c(q^2) q_α q_β even if to leading order in the large proton mass expansion. In Appendix D of the Supplemental Material we analyse the contributions of the diagrams Fig. <ref>f and Fig. <ref>g, where strong low–energy interactions are given by the axial coupling constant λ only. We show that to leading order in the large proton mass expansion the contributions of these diagrams vanish.Hence, an important contribution, which may cancel a gauge–non–invariant part of the diagram Fig. <ref>c, is able to come only from the diagrams, where a real or virtual photon couple to a hadronic block.The diagram in Fig. <ref> defines one of a set of Feynman diagrams of the neutron radiative β^-–decay with emission of two real photons.Such a process with one undetected photon can imitate a contribution of order O(α^2/π^2) to the rate of the neutron radiative β^-–decay. All diagrams of the neutron radiative β^-–decay with emission of one or two photons by the proton, calculated to leading order in the large proton mass expansion, do not contribute to the rate of the neutron radiative β^-–decay in the physical gauge of real photons. Then, the contributions of the diagrams with emission of photons from the hadronic blocks are neglected (see a discussion in section <ref>). Thus, in the accepted approximation the main contribution to the rate of the neutron radiative β^-–decay is defined by the Feynman diagrams in Fig. <ref> with the account for the contributions, caused by symmetry of the final state with respect to symmetry properties of the two photons in the final state of the decay. For the analytical calculation of the diagram in Fig. <ref> the contribution of strong low–energy interactions is defined by the axial coupling only. The analytical calculation of the diagrams in Fig. <ref> is given in Appendix E of the Supplemental Material. The paper is organized as follows. In section <ref> we give a short description of the renormalization procedure of effective low–energy electroweak interactions for the neutron radiative β^-–decay. In section <ref> we adduce the contributions of the Feynman diagrams in Fig. <ref>, Fig. <ref>, Fig. <ref>, Fig. <ref> and Fig. <ref> to the rate of the neutron radiative β^-–decay . The numerical values of the branching ratio of the neutron radiative β^-–decay for the three regions of photon energies i)15keV≤ω≤ 350keV, ii) 14keV≤ω≤ 782keV and iii) 0.4keV≤ω≤ 14keV, are given in Table I. In section <ref> we discuss the obtained results. In the Supplemental Material we give i) detailed analytical calculations and analysis of the contributions of Feynman diagrams in Fig. <ref>, Fig. <ref>, Fig. <ref>, Fig. <ref> and Fig. <ref> to the amplitude and rate of the neutron radiative β^-–decay.Of course, we have to confess that the main problem of our analysis of the radiative corrections to order O(α/π), defining corrections to order O(α^2/π^2) to the rate of the neutron radiative β^-–decay, concerns the contributions of diagrams with real or virtual photons coupled to a hadronic block. A justification of our assumption concerning the properties of these diagrams within a certain model of strong low–energy interactions of hadrons coupled to photons should be important for a confirmation of the approximation accepted in this paper and the results obtained therein. We would like to accentuate that unlike a passive role of strong low–energy hadronic interactions in the radiative corrections of order O(α/π) to the rate of the neutron β^-–decay, strong low–energy interactions of hadrons coupled to real and virtual photons in the diagrams in Fig. <ref>, should play a more important role, going beyond a formal renormalization of the Fermi weak coupling and axial coupling constant, but give some contributions, which depend on the electron and photon energies and momenta, and should cancel a gauge–non–invariant part of the diagram Fig. <ref>c. The observed peculiarities of the Feynman diagrams Fig. <ref> and Fig. <ref> agree well with an important role of strong low–energy hadronic interactions in decay processes that have been already pointed out by Weinberg <cit.>. Thus, the problem of strong low–energy hadronic interactions in the neutron radiative β^-–decay to order O(α^2/π^2) demands a special analysis and we are planning to perform such a model–dependent analysis of the neutron radiative β^-–decay to order O(α^2/π^2) in our forthcoming publication. § RENORMALIZATION PROCEDURE OF EFFECTIVE LOW–ENERGY ELECTROWEAKINTERACTIONS FOR THE NEUTRON RADIATIVE Β^-–DECAYIn the Standard Model of electroweak interactions the neutron radiative β^-–decay, defined in the one–loop approximation with one–virtual–photon exchanges, is described by the following interactionsL_ int(x) =L_ W(x) +L_ em(x),where L_ W(x) is the effective Lagrangian of low–energy V-A interactions with a real axial coupling constant λ = - 1.2750(9) <cit.> (see also <cit.>)L_ W(x) = - G_F/√(2) V_ud [ψ̅_p(x)γ_μ(1 + λγ^5)ψ_n(x)][ψ̅_e(x)γ^μ(1 - γ^5)ψ_ν(x)],where G_F = 1.1664 × 10^-11MeV^-2 is the Fermi coupling constant, and |V_ud| = 0.97417(21) is the Cabibbo–Kobayashi–Maskawa matrix element <cit.>. Then, ψ_p(x), ψ_n(x), ψ_e(x) and ψ_ν(x) are the field operators of the proton, neutron, electron and antineutrino, respectively, and γ^μ and γ^5 are the Dirac matrices <cit.>. Since we calculate the radiative corrections of order O(α/π) to the neutron radiative β^-–decay to leading order in the large proton mass expansion, in the effective Lagrangian L_ W(x) we do not take into account the contribution of the weak magnetism proportional to 1/M, where 2M = m_n + m_p is an averaged nucleon mass <cit.>.For the calculation of the radiative corrections to order O(α/π) the Lagrangian of the electromagnetic interaction L_ em(x) we take in the following formL_ em(x)=- 1/4 F^(0)_μν(x)F^(0)μν(x) - 1/2ξ_0 (∂_μA^(0)μ(x))^2 + ψ̅_0e(x)(iγ^μ∂_μ - m_0e)ψ_0e(x) - (- e_0) ψ̅_0e(x)γ^μψ_0e(x)A^(0)_μ(x)+ ψ̅_0p(x)(iγ^μ∂_μ - m_0p)ψ_0p(x) - (+ e_0) ψ̅_0p(x)γ^μψ_0p(x) A^(0)_μ(x),where F^(0)_μν(x) = ∂_μA^(0)_ν(x) - ∂_νA^(0)_μ(x) is the electromagnetic field strength tensor of the bare (unrenormalized) electromagnetic field operator A^(0)_μ(x); ψ_0e(x) and ψ_0p(x) are bare operators of the electron and proton fields with bare masses m_0e and m_0p, respectively; - e_0 and + e_0 are bare electric charges of the electron and proton, respectively. Then, ξ_0 is a bare gauge parameter. After the calculation of the one–loop corrections of order O(α/π) a transition to the renormalized field operators, masses and electric charges is defined by the LagrangianL_ em(x)=- 1/4 F_μν(x)F^μν(x) - 1/2ξ (∂_μA^μ(x))^2+ ψ̅_e(x)(iγ^μ∂_μ - m_e)ψ_e(x) - (- e) ψ̅_e(x) γ^μψ_e(x) A_μ(x)+ ψ̅_p(x)(iγ^μ∂_μ - m_p )ψ_p(x) - (+ e) ψ̅_p(x) γ^μψ_p(x) A_μ(x) + δ L_ em(x),where A_μ(x), ψ_e(x) and ψ_p(x) are the renormalized operators of the electromagnetic, electron and proton fields, respectively; m_e and m_p are the renormalized masses of the electron and proton; e is the renormalized electric charge; and ξ is the renormalized gauge parameter. The Lagrangian δ L_ em(x) contains a complete set of the counterterms <cit.>,δ L_ em(x)=- 1/4 (Z_3 - 1) F_μν(x)F^μν(x) - Z_3 - 1/Z_ξ 1/2ξ (∂_μA^μ(x))^2+ (Z^(e)_2 - 1) ψ̅_e(x)(iγ^μ∂_μ - m_e)ψ_e(x) - (Z^(e)_1 - 1) (- e) ψ̅_e (x)γ^μψ_e(x) A_μ(x) - Z^(e)_2 δ m_e ψ̅_e(x)ψ_e(x) + (Z^(p)_2 - 1) ψ̅_p(x)(iγ^μ∂_μ - m_p )ψ_p(x) - (Z^(p)_1 - 1) ( + e)ψ̅_p(x) γ^μψ_p(x) A_μ(x) - Z^(p)_2 δ m_p ψ̅_p(x) ψ_p(x),where Z_3, Z^(e)_2, Z^(e)_1, Z^(p)_2, Z^(p)_1, δ m_e and δ m_p are the counterterms. Here Z_3 is the renormalization constant of the electromagnetic field operator A_μ, Z^(e)_2 and Z^(e)_1 are the renormalization constants of the electron field operator ψ_e and the electron–electron–photon (e^-e^-γ) vertex, respectively; Z^(p)_2 and Z^(p)_1 are the renormalization constants of the proton field operator ψ_p and the proton–proton–photon (p p γ) vertex, respectively. Then, (- e) and (+ e), m_e and m_p and δ m_e and δ m_p are the renormalized electric charges and masses and the mass–counterterms of the electron and proton, respectively. Rescaling the field operators <cit.>√(Z_3)A_μ(x) = A^(0)_μ(x),√(Z^(e)_2) ψ_e(x) = ψ_0e(x),√(Z^(p)_2) ψ_p(x) = ψ_0p(x)and denoting m_e + δ m_e = m_0e, m_p + δ m_p = m_0p and Z_ξξ = ξ_0 we arrive at the LagrangianL_ em(x)=- 1/4 F^(0)_μν(x)F^(0)μν(x) - 1/2ξ_0 (∂_μA^(0)μ(x))^2 + ψ̅_0e(x)(iγ^μ∂_μ - m_0e)ψ_0e(x) - ( - e) Z^(e)_1 (Z^(e)_2)^-1 Z^-1/2_3 ψ̅_0e(x)γ^μψ_0e(x)A^(0)_μ(x)+ ψ̅_0p(x)(iγ^μ∂_μ - m_0p)ψ_0p(x) - (+ e)Z^(p)_1(Z^(p)_2)^-1 Z^-1/2_3 ψ̅_0p(x)γ^μψ_0p(x)A^(0)_μ(x).Because of the Ward identities Z^(e)_1 = Z^(e)_2 and Z^(p)_1 = Z^(p)_2 <cit.>, we may replace (-e) Z^-1/2_3 = - e_0 and (+ e) Z^-1/2_3 = + e_0. This brings Eq.(<ref>) to the form of Eq.(<ref>). We would like to emphasize that to order O(α/π) the renormalization constant Z_3 is equal to unity, i.e., Z_3 = 1. This is because of the absence of closed fermion loops, giving contributions of order O(α^2/π^2) to the amplitude of the neutron radiative β^-–decay that goes beyond the accepted approximation O(α/π) for the amplitude and O(α^2/π^2) for the rate of the neutron radiative β^-–decay. Hence, to order O(α/π) the bare e_0 and renormalized e electric charges are equal, i.e. e_0 = e. Now we may proceed to the discussion of the contributions of the radiative corrections of order O(α/π), where α = e^2/4π = 1/137.036 is the fine–structure constant <cit.>, to the amplitude and rate of the neutron radiative β^-–decay. The detailed calculations and analysis of the Feynman diagrams in Fig. <ref>, Fig. <ref>, Fig. <ref> and Fig. <ref>, defining a complete set of radiative corrections of order O(α/π), we give in the Supplemental Material. In section <ref> we adduce the analytical expressions for the contributions of the diagrams in Fig. <ref>, Fig. <ref>, Fig. <ref> and Fig. <ref> to the rate of the neutron radiative β^-–decay. The numerical values are collected in Table I. For completeness we take into account the tree–level contribution, given by the Feynman diagrams in Fig. <ref> and calculated in <cit.> to order 1/M, including corrections of the weak magnetism and proton recoil. § RATE OF NEUTRON RADIATIVE Β^-–DECAY WITH ONE DETECTEDPHOTONThe rate of the neutron radiative β^-–decay with a photon, detected in the photon energy region ω_ min≤ω≤ω_ max, is given byλ_βγ(ω_ max, ω_ min) = ∑^5_j = 1λ^( Fig j)_βγ(ω_ max, ω_ min),where λ^( Fig j)_βγ(ω_ max, ω_ min) are the rates, caused by the contributions of the diagrams in Fig. j for j = 1,2,…,5.They are calculated in the Supplemental Material. To leading order in the large proton mass expansion the contribution of the diagrams in Fig. <ref> is equal to <cit.>λ^( Fig. <ref>)_βγ(ω_ max,ω_ min)=(1 + 3 λ^2) α/π G^2_F|V_ud|^2/2π^3∫^ω_ max_ω_ mindω/ω∫^E_0 - ω_m_edE_e√(E^2_e - m^2_e) E_e F(E_e, Z = 1) (E_0 - E_e - ω)^2 ×{(1 + ω/E_e + 1/2ω^2/E^2_e) [1/β ℓ n(1 + β/1 - β) - 2] + ω^2/E^2_e},where E_0 = (m^2_n - m^2_p + m^2_e)/2 m_n is the end–point energy of the electron–energy spectrum of the neutron β^-–decay <cit.>; ω is a photon energy; β = k_e/E_e = √(E^2_e - m^2_e)/E_e is a velocity of the electron with a momentum k_e; and F(E_e, Z = 1) is the relativistic Fermi function, describing the Coulomb proton–electron interaction in the final state of the decay. It is equal toF(E_e, Z = 1 ) =(1 + 1/2γ) 4(2 r_pm_eβ)^2γ/Γ^2(3 + 2γ)e^ πα/β/(1 - β^2)^γ |Γ(1 + γ +i α/β)|^2,where γ = √(1 - α^2) - 1, r_p is the electric radius of the proton and α = 1/137.036 is the fine–structure constant.In numerical calculations we shall use r_p = 0.841fm <cit.>. The rate of the neutron radiative β^-–decay, calculated to next–to–leading order in the large proton mass expansion, taking into account the contributions of the weak magnetism and proton recoil to order 1/M, where 2M = m_n + m_p is the averaged nucleon mass, has been calculated in <cit.>. The result isλ^( Fig. <ref>)_βγ(ω_ max,ω_ min)=(1 + 3 λ^2)α/πG^2_F |V_ud|^2/2π^3∫^ω_ max_ω_ mindω/ω∫^E_0 - ω_m_e dE_e E_e√(E^2_e - m^2_e)(E_0 - E_e - ω)^2 × F(E_e, Z = 1) ρ^( Fig. <ref>)_βγ(E_e,ω).The function ρ^( Fig. <ref>)_βγ(E_e,ω) is given by the integral <cit.>ρ^( Fig. <ref>)_βγ(E_e,ω) = ∫dΩ_eγ/4π {[1 + 2 ω/M E_e - k⃗_e·n⃗_k⃗/E_0 - E_e - ω + 3/M (E_e + ω - 1/3 E_0) + λ^2 - 2(κ + 1)λ + 1/1 + 3λ^2 E_0 - E_e - ω/M] × [(1 + ω/E_e) k^2_e - (k⃗_e·n⃗_k⃗)^2/(E_e - k⃗_e·n⃗_k⃗)^2 + ω^2/E_e 1/E_e - k⃗_e·n⃗_k⃗] + 3λ^2 - 1/1 + 3 λ^2 1/M (k^2_e + ωk⃗_e·n⃗_k⃗/E_e [k^2_e - (k⃗_e·n⃗_k⃗)^2/(E_e - k⃗_e·n⃗_k⃗)^2 + ω/E_e - k⃗_e·n⃗_k⃗]+ (ω + k⃗_e·n⃗_k⃗)[(1 + ω/E_e)ω/E_e - k⃗_e·n⃗_k⃗ - m^2_e/E_e ω/(E_e - k⃗_e·n⃗_k⃗)^2]) - λ^2 + 2 (κ + 1)λ - 1/1 + 3λ^2 1/M [k^2_e + ω^2 + 2ωk⃗_e·n⃗_k⃗/E_e × k^2_e - (k⃗_e·n⃗_k⃗)^2/(E_e - k⃗_e·n⃗_k⃗)^2 + ω/E_e k^2_e - (k⃗_e·n⃗_k⃗)^2/E_e - k⃗_e·n⃗_k⃗ + ω^2/E_e ω + k⃗_e·n⃗_k⃗/E_e - k⃗_e·n⃗_k⃗] - λ(λ - 1)/1 + 3λ^2 1/M [ω/E_e k^2_e - (k⃗_e·n⃗_k⃗)^2/E_e - k⃗_e·n⃗_k⃗ + 3 ω^2/E_e]},where κ = κ_p - κ_n = 3.70589 is the isovector anomalous magnetic moment of the nucleon <cit.>, dΩ_eγ is an infinitesimal solid angle of the electron–photon momentum correlations k⃗_e·n⃗_k⃗ = k_e cosθ_eγ and n⃗_k⃗ = k⃗/ω is a unit vector along the photon 3–momentum <cit.>. The contribution of the diagrams in Fig. <ref> is equal to (see Appendix B of the Supplemental Material)λ^( Fig. <ref>)_βγ(ω_ max,ω_ min) = (1 + 3λ^2) α^2/π^2 G^2_FV_ud|^2/4π^3∫^ω_ max_ω_ min dω∫^E_0 - ω_m_e dE_e (E_0 - E_e - ω)^2 √(E^2_e - m^2_e) × F(E_e, Z = 1)∫dΩ_eγ/4π {k^2_e - (k⃗_e ·n⃗_k⃗)^2/(E_e - k⃗_e·n⃗_k⃗)^2Re F_4 + ω/E_e - k⃗_e·n⃗_k⃗Re (2 F_2 - F_3 - 2F_4)},where F_2, F_3 and F_4 are given in Eq.(B-71) of the Supplemental Material as functions of k_e· k = ω (E_e - k⃗_e·n⃗_k⃗). The contribution of the diagrams in Fig. <ref> and Fig. <ref> we define as (see Appendix C of the Supplemental Material)λ^( Fig. <ref>)_βγ(ω_ max,ω) = (1 + 3λ^2) α^2/π^2 G^2_F|V_ud|^2/4π^3∫^ω_ max_ω_ mindω/ω∫^E_0 - ω_m_e dE_e F(E_e, Z = 1) (E_0 - E_e - ω)^2 √(E^2_e - m^2_e) × ∫dΩ_eγ/4π {f_1(E_e, k⃗_e, ω, k⃗ ) [(E_e + ω) k^2_e - (k⃗_e ·n⃗_k⃗)^2/(E_e - k⃗_e ·n⃗_k⃗)^2 + ω^2/E_e - k⃗_e ·n⃗_k⃗] + f_2(E_e, k⃗_e, ω, k⃗ ) [(2(E_e + ω)^2 - m^2_e- ω (E_e - k⃗_e ·n⃗_k⃗)) k^2_e - (k⃗_e ·n⃗_k⃗)^2/(E_e - k⃗_e ·n⃗_k⃗)^2 + 2(E_e + ω) ω^2/E_e - k⃗_e ·n⃗_k⃗ - ω^2]},where the functions f_1(E_e, k⃗_e, ω, k⃗ ) and f_2(E_e, k⃗_e, ω, k⃗ ) are given in Eq.(<ref>) of the Supplemental material.They are defined by the contribution of the diagram Fig. <ref>c, since to leading order in the large proton mass expansion and in the physical gauge of a real photon the contribution of the diagram in Fig. <ref>d vanishes. Then, the rate λ^( Fig. <ref>)_βγ(ω_ max,ω_ min) is defined by a part of the diagram Fig. <ref>c, which is invariant under a gauge transformation D_αβ(q) → D_αβ(q) + c(q^2) q_αq_β. A non–invariant part of the diagram Fig. <ref>c is absorbed by the diagrams in Fig. <ref>. We assume that the contribution of the diagrams in Fig. <ref>, calculated to leading order in the large proton mass expansion and in the physical gauge of a real photon, contains only i) an electron–photon–energy dependent part, cancelling a part of the diagram Fig. <ref>c that is non–invariant under the gauge transformation D_αβ(q) → D_αβ(q) + c(q^2) q_αq_β, and ii) a constant, which can be absorbed by renormalization of the Fermi weak coupling constant G_F and the axial coupling constant λ similar to Sirlin's analysis of the radiative corrections to the rate of the neutron β^-–decay <cit.>. Of course, our assumption is much stronger than Sirlin's one. Nevertheless, we believe that it is correct and it might be confirmed by a model–dependent way within a model of strong interactions of hadrons coupled to photons at low energies (see a discussion in section <ref>).The contribution of the diagrams in Fig. <ref> of the neutron radiative β^-–decay with two real photons and only one detected photon is equal to (see Appendix E of the Supplemental Material)λ^( Fig. <ref>)_βγ(ω_ max, ω_ min)=(1 + 3λ^2) α^2/π^2 G^2_F|V_ud|^2/16π^3∫^ω_ max_ω_ mindω∫^E_0 - ω_m_edE_e√(E^2_e - m^2_e)∫^E_0 - E_e - ω_0 dq_0 (E_0 - E_e - ω - q_0)^2× F(E_e, Z = 1) ∫dΩ_eγ/4π∫dΩ_eγ'/4π (ρ^(1)_eγγ'(E_e,k⃗_e, ω, n⃗_k⃗, q_0, n⃗_q⃗) + ρ^(2)_eγγ'(E_e,k⃗_e, ω, n⃗_k⃗, q_0, n⃗_q⃗) + ρ^(2)_eγγ'(E_e,k⃗_e, q_0, n⃗_q⃗, ω, n⃗_k⃗)),where q_0 is the energy of an undetected photon and n⃗_q⃗ = q⃗/q_0 is a unit vector along its 3–momentum q⃗. The functions ρ^(1)_eγγ'(E_e,k⃗_e, ω, n⃗_k⃗, q_0, n⃗_q⃗), ρ^(2)_eγγ'(E_e,k⃗_e, ω, n⃗_k⃗, q_0, n⃗_q⃗) and ρ^(2)_eγγ'(E_e,k⃗_e, q_0, n⃗_q⃗, ω, n⃗_k⃗) are given by Eq.(<ref>), Eq.(<ref>) and Eq.(<ref>) in the Supplemental Material.The numerical values of the branching ratios BR^( Fig.j)_βγ = τ_n λ_βγ(ω_ max, ω_ min)_ Fig j for j = 1,2,…,5 and their total contribution are given in Table I for the three photon–energy regions i) 15keV≤ω≤ 340keV, ii) 14keV≤ω≤ 782keV and iii) 0.4keV≤ω≤ 14keV. The branching ratios BR^( Fig.j)_βγ are obtained relative to the neutron lifetime τ_n = 879.6(1.1)s, calculated in <cit.> and agreeing well with the world–averaged value τ_n = 880.2(1.0)s <cit.>.§ CONCLUSIONWe have proposed a precision analysis of the rate of the neutron radiative β^-–decay n → p + e^- + ν̅_e + γ to order O(α^2/π^2), defined by the 1/M corrections, caused by the weak magnetism and proton recoil <cit.>, and radiative corrections of order O(α/π) in the one–virtual–photon approximation, and the contribution of the neutron radiative β^-–decay with two real photons n → p + e^- + ν̅_e + γ + γ. Integrating over degrees of freedom of one of two photons one arrives at the contribution of order O(α^2/π^2) to the rate of the neutron radiative β^-–decay n → p + e^- + ν̅_e + γ. The contributions of the one–virtual–photon exchanges we have classified by the Feynman diagrams in Fig. <ref>, Fig. <ref> and Fig. <ref>. In the diagrams in Fig. <ref> the contributions of strong low–energy and electromagnetic interactions are factorized, and both the real and virtual photons couple to the point–like proton and electron. The contribution of strong low–energy interactions of hadrons is given by the axial coupling constant λ only. All divergences, caused by virtual photon exchanges, are absorbed by renormalization of masses and wave functions of the proton and electron, the proton–proton–photon (ppγ) and electron–electron–photon (e^-e^-γ) vertices. Therewith, the counterterms of renormalization of the wave functions and vertices obey standard Ward identities <cit.>. The diagrams in Fig. <ref> are invariant under gauge transformations ε^*_λ'(k) →ε^*_λ'(k) + c k of a real photon wave function and D_αβ(q) → D_αβ(q) + c(q^2) q_αq_β of a photon Green function, respectively. The structure functions, defining the renormalized contribution of the Feynman diagrams in Fig. <ref> to the amplitude of the neutron radiative β^-–decay of order O(α/π), obey Ward identities. The contribution of the Feynman diagrams in Fig. <ref> to the branching ratio is of order 10^-7 (see Table I).The dominant but most problematic contribution comes from the Feynman diagrams in Fig. <ref> and Fig. <ref>. For the calculation of the contribution of these diagrams we follow Sirlin's assumption for the calculation of the radiative corrections of order O(α/π) to the rate of the neutron β^-–decay <cit.>. This means that we assume that the contribution of the diagrams in Fig. <ref>, which survives to leading order in the large proton mass expansion in the physical gauge of a real photon, contains i) a part of the diagram Fig. <ref>c, which is not invariant under gauge transformations of a real photon wave function ε^*_λ'(k) →ε^*_λ'(k) + c k and of a photon Green function D_αβ(q) → D_αβ(q) + c(q^2) q_αq_β, respectively, and ii) a part independent of the electron and photon energies, which can be absorbed by renormalization of the Fermi weak coupling constant G_F and the axial coupling constant λ. This is, of course, an extended interpretation of Sirlin's assumption, since in the neutron β^-–decay the Feynman diagrams similar to the diagrams in Fig. <ref> (see Fig. <ref>b and Fig. <ref>c) have been found independent of the electron energy, the contribution of which has been absorbed by renormalization of the Fermi weak and axial coupling constants. A confirmation of our assumption, concerning the properties of the Feynman diagrams in Fig. <ref> and Fig. <ref> might be supported by the fact that all possible insertions of real photon external lines transform the Feynman diagrams in Fig. <ref> to the Feynman diagrams Fig. <ref>c, Fig. <ref>d and Fig. <ref>. It is obvious that all possible insertions of real photon external lines should not change the properties of the diagrams with respect to a gauge transformation D_αβ(q) → D_αβ + c(q^2) q_αq_β. Hence, all diagrams in Fig. <ref> should play an auxiliary role for the diagram Fig. <ref>c to leading order in the large proton mass expansion. Thus, such an extended Sirlin's assumption, applied to the calculation of the Feynman diagrams in Fig. <ref>c, Fig. <ref>d and Fig. <ref>, we have realized as follows. Firstly, we have shown that in Fig. <ref> only the diagram Fig. <ref>c survives to leading order in the large proton mass expansion in the physical gauge of a real photon. Secondly, we have decomposed the contribution of the diagram Fig. <ref>c into invariant and non–invariant parts with respect to a gauge transformation of a photon Green function D_αβ(q) → D_αβ(q) + c(q^2) q_αq_β. Finally, we have omitted a gauge–non–invariant part and the contributions, independent of the electron and photon energies, we have removed by renormalization of the Fermi weak coupling constant G_F and the axial coupling constant λ, respectively. The contribution of the diagrams in Fig. <ref> and Fig. <ref> to the branching ratio of the neutron radiative β^-–decay, obtained in such a way, is of order 10^-4 (see Table I).It is important to emphasize that the renormalized contribution of the diagram Fig. <ref>c, which we have defined in terms of the functions f_1(E_e, k⃗_e, ω, k⃗ ) and f_2(E_e, k⃗_e, ω, k⃗ ) (see Eq.(<ref>) of the Supplemental Material), does not depend on the infrared cut–of μ, which is introduced as a photon mass <cit.>. This is unlike the contribution of the diagram Fig. <ref>a to the rate of the neutron β^-–decay, which has been found as a function of the infrared cut–off μ <cit.>. A μ–dependence of the radiative corrections, caused by the diagram Fig. <ref>a, has been cancelled only by the diagram Fig. <ref>b (see <cit.>).We would like to accentuate that the contribution of the diagrams in Fig. <ref>, describing a neutron radiative β^-–decay with two real photons in the final state, is also infrared stable. Having integrated over the momentum and energy of one of two photons we have obtained the contribution of order O(α^2/π^2) to the rate of the neutron radiative β^-–decay with a photon, detected in the energy region ω_ min≤ω≤ω_ max. The contribution of the neutron radiative β^-–decay with two real photons in the final state, described by the diagrams in Fig. <ref>, is of order 10^-5 (see Table I).Total contributions of the radiative corrections of order O(α/π) to the rate of the neutron radiative β^-–decay are about 2.42 %, 2.44 % and 3.66 % for three photon–energy regions 15keV≤ω≤ 340keV, 14keV≤ω≤ 782keV and 0.4keV≤ω≤ 14keV, respectively. They are commensurable with the radiative correction 3.75 % to the rate of the neutron β^-–decay <cit.>. However, they are not enhanced with respect to the contribution of the radiative corrections to the rate of the neutron β^-–decay as we have expected because of an enhancement of the corrections of order 1/M, caused by the weak magnetism and proton recoil <cit.>. The theoretical values of the branching ratios (see Table I) do not contradict the experimental data within the experimental error bars. Nevertheless, deviations of about 4.21 %, 7.05 % and 9.52 % of the mean values of the experimental data from the theoretical values for three photon–energy regions 15keV≤ω≤ 340keV, 14keV≤ω≤ 782keV and 0.4keV≤ω≤ 14keV, respectively, might only imply that such a distinction cannot be covered by the contributions of interactions beyond the Standard Model. Therefore, apart from the experimental error bars one may expect a better agreement between theory and experiment only from the contributions of strong low–energy interactions of hadrons beyond the axial coupling constant λ. One may expect that they might be caused by the contributions of diagrams in Fig. <ref>, where real and virtual photon couple to a hadronic block.In this connection we may confess that there are two problems of our precision analysis of the rate of the neutron radiative β^-–decay to order O(α^2/π^2). They are i) a justification of a neglect of the diagrams with photons coupled to hadronic blocks such as the diagram Fig. <ref>c and so on and ii) a justification of Sirlin's assumption for an extraction of a physical contribution from the Feynman diagrams in Fig. <ref> and Fig <ref>. As we have mentioned above both of these problems can be investigated only by a model–dependent way within certain models of strong low–energy interactions of hadrons coupled to photons.However, very likely that the contribution of the diagram Fig. <ref>c is really not important. One may show this at the tree–level using the following effective low–energy electromagnetic interactions of the neutron and proton <cit.>δ L_ em(x) = κ_n e/4 M ψ̅_n(x)σ_μνψ_n(x) F^μν(x) + κ_p e/4 M ψ̅_p(x)σ_μνψ_p(x) F^μν(x),where κ_n = - 1.91304 and κ_p = 1.79285 are anomalous magnetic moments of the neutron and proton <cit.>, respectively, and σ_μν = (i/2)(γ_μγ_ν - γ_νγ_μ) are Dirac matrices <cit.>. The contribution of the diagram Fig. <ref>c to the amplitude of the neutron radiative β^-–decay is equal toM_ Fig. <ref>c(n → p e^- ν̅_e γ)_λ' = = - κ_n/2M [u̅_p(k⃗_p, σ_p) γ^μ(1 + λγ^5) 1/m_n - k̂_n + k̂ - i0 i σ_αβ k^αε^β *_λ'u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)] + κ_p/2M [u̅_p(k⃗_p, σ_p) i σ_αβ k^αε^β *_λ' γ^μ(1 + λγ^5) 1/m_n - k̂_p - k̂ - i0u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)].One may see that the amplitude Eq.(<ref>) is invariant under a gauge transformation ε^*_λ'(k) →ε^*_λ'(k) + c k.The contribution of the diagram Fig. <ref>c to the branching ratio is given byB^( Fig. <ref>c)_βγ(ω_ max, ω_ min)= α/π G^2_F|V_ud|^2/2π^3 M (2λ^2 (κ_p + κ_n) - λ (κ_p - κ_n)) ×∫^ω_ max_ω_ mindω ω∫^E_0 - ω_m_edE_e √(E^2_e - m^2_e) (E_0 - E_e - ω)^2 F(E_e, Z = 1). For the three photon energy regions (see Table I) the branching ratio is equal to B^( Fig. <ref>c)_βγ = 0.97× 10^-10, B^( Fig. <ref>c)_βγ = 1.25× 10^-10 and B^( Fig. <ref>c)_βγ = 4.90× 10^-13, respectively. This may testify that the diagram Fig. <ref>c can actually be neglected. Such a neglect does not violate invariance of the diagrams Fig. <ref>a and Fig. <ref>b with respect to a gauge transformation ε^*_λ'(k) →ε^*_λ'(k) + c k. Our justification of a possible neglect of the contribution of the diagram Fig. <ref>c confirms also a neglect of all diagrams with emission of a real photon by a hadronic block in the radiative neutron β^-–decay with two real photons in the final state, given by the diagram in Fig. <ref>.Hence, the main contribution of strong low–energy interactions we may expect only from the diagrams in Fig. <ref> and Fig. <ref>. As a first step on the way of the analysis of these diagrams we are planning to use the Standard Model of electroweak interactions supplemented by the linear σ–model of strong low–energy nucleon–pion interactions by Gell–Mann and Levy <cit.> (see also <cit.>). It is well–known that a linear σ–model is a renormalizable one <cit.>. Renormalization of an extended version of a linear σ–model has been investigated in <cit.>. The observed peculiar properties of strong low–energy hadronic interactions in the neutron radiative β^-–decay to order of O(α^2/π^2) agree well with assertion, pointed out by Weinberg <cit.>, about the important role of strong low–energy hadronic interactions in decay processes.We would like to emphasize that analysis of the rate of the neutron radiative β^-–decay to order O(α^2/π^2) is a first step toward the analysis of the neutron β^-–decay to order O(α^2/π^2). One of the most intriguing theoretical features of this analysis, which we anticipate, is a cancellation of the infrared dependences in the sum of the contributions of the diagrams with only virtual photon exchanges and the diagrams of the neutron radiative β^-–decay with one and two photons in the final state. For the analytical investigation of this problem the results, obtained in this paper, are of great deal of importance. The calculation of the neutron β^-–decay to order α^2/π^2 ∼ 10^-5 together with the contributions of order (α/π) (E_e/M) ∼ 3× 10^-6 and E^2_e/M^2 ∼ 10^-6 should give a new level of theoretical precision for the experimental search of interactions beyond the Standard Model <cit.>.It is well known that in the limit m_σ→∞, where m_σ is a scalar σ–meson mass, a linear σ–model is equivalent to current algebra <cit.>.This means that the results, obtained in a linear σ–model and taken in the limit m_σ→∞, should reproduce the results, obtained in current algebra <cit.>, i.e. in a model–independent approach. This bridges between the results, which we are planning to obtain for the contributions of strong low–energy interactions to the radiative corrections of order O(α^2/π^2) for the neutron radiative and neutron β^-–decays, and the results, obtained by Sirlin <cit.> for the contributions of strong low–energy interactions to the radiative corrections of order O(α/π) for the neutron β^-–decay.§ ACKNOWLEDGEMENTS The work of A. N. Ivanov was supported by the Austrian “Fonds zur Förderung der Wissenschaftlichen Forschung” (FWF) under contracts I689-N16, I862-N20, P26781-N20 and P26636-N20,“Deutsche Förderungsgemeinschaft” (DFG) AB 128/5-2 and by the ÖAW within the New Frontiers Groups Programme, NFP 2013/09. The work of R. Höllwieser was supported by the Erwin Schrödinger Fellowship program of the Austrian Science Fund FWF (“Fonds zur Förderung der wissenschaftlichen Forschung”) under Contract No. J3425-N27. The work of M. Wellenzohn was supported by the MA 23 (FH-Call 16) under the project “Photonik - Stiftungsprofessur für Lehre”. § SUPPLEMENTAL MATERIAL§ APPENDIX A: THE AMPLITUDE AND RATE OF THE NEUTRONRADIATIVE Β^-–DECAY IN THE TREE–APPROXIMATION, DESCRIBED BY THE FEYNMAN DIAGRAM IN FIG. <REF> The amplitude of the neutron radiative β^-–decay, represented by the diagrams in Fig. <ref>, Fig. <ref>, Fig. <ref>, Fig. <ref> and Fig. <ref>, we define as followsM(n → p e^- ν̅_e γ)_λ' = e G_F/√(2) V_ud∑^5_j = 1M_ Fig. j(n → p e^- ν̅_e γ)_λ',where M_ Fig. j(n → p e^- ν̅_e γ)_λ' is the contribution of the diagram in Fig. j for j = 1,2,…,5. The amplitude M_ Fig.<ref>(n → p e^- ν̅_e γ)_λ' of the neutron radiative β^-–decay, defined by the diagrams in Fig.<ref>, is <cit.>M_ Fig.<ref>(n → p e^- ν̅_e γ)_λ' = [u̅_p(k⃗_p, σ_p) γ^μ(1 + λγ^5)u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e,σ_e) 1/2k_e· k Q_e γ_μ (1 - γ^5) v_ν(k⃗_ν, + 1/2)]- [u̅_p(k⃗_p, σ_p) Q_p1/2k_p · k γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)][u̅_e(k⃗_e,σ_e) γ^μ (1 - γ^5) v_ν(k⃗_ν, + 1/2)],where Q_e and Q_p are given byQ_e = 2 ε^*_λ'· k_e + ε̂^*_λ'k̂ ,Q_p = 2 ε^*_λ'· k_p + ε̂^*_λ'k̂.For the derivation of Eq.(<ref>) we have used the Dirac equations for the free proton and electron. Replacing ε^*_λ'→ k and using k^2 = 0 we get M(n → p e^- ν̅_e γ)|_ε^*_λ'→ k = 0. This confirms invariance of the amplitude Eq.(<ref>) with respect to a gauge transformation ε^*_λ'(k) →ε^*_λ'(k) + c k, where c is an arbitrary constant.To leading order in the large proton mass expansion the amplitude of the neutron radiative β^-–decay, calculated in the tree–approximation, is equal to <cit.>M_ Fig.<ref>(n → p e^- ν̅_e γ)_λ' = 2m_n × {[φ^†_pφ_n] [u̅_e(k⃗_e,σ_e) 1/2k_e· k Q_e γ^0 (1 - γ^5) v_ν(k⃗, + 1/2)] - λ [φ^†_p σ⃗ φ_n] ·[u̅_e(k⃗_e,σ_e) 1/2k_e· k Q_e γ⃗ (1 - γ^5) v_ν(k⃗, + 1/2)]- ε^0*_λ'/ω[φ^†_p φ_n] [u̅_e(k⃗_e,σ_e) γ^0 (1 - γ^5) v_ν(k⃗, + 1/2)] + ε^0*_λ'/ω λ [φ^†_p σ⃗ φ_n] ·[u̅_e(k⃗_e,σ_e) γ⃗ (1 - γ^5) v_ν(k⃗, + 1/2)]}.The hermitian conjugate amplitude is M^†_ Fig.<ref>(n → p e^- ν̅_e γ)_λ' = 2m_n × {[φ^†_n φ_p] [v̅_ν(k⃗, + 1/2) 1/2k_e· k γ^0 Q̅_e (1 - γ^5) u_e(k⃗_e,σ_e)] - λ [φ^†_n σ⃗ φ_p] ·[v̅_ν(k⃗, + 1/2) 1/2k_e· k γ⃗ Q̅_e (1 - γ^5) u_e(k⃗_e,σ_e)]- ε^0_λ'/ω[φ^†_n φ_p] [v̅_ν(k⃗, + 1/2) γ^0 (1 - γ^5) u_e(k⃗_e,σ_e)] + ε^0_λ'/ω λ [φ^†_n σ⃗φ_p] ·[v̅_ν(k⃗, + 1/2) γ⃗ (1 - γ^5) u_e(k⃗_e,σ_e)]},where Q̅_e = 2 k_e·ε_λ' + k̂ ε̂_λ' <cit.>. The amplitudes Eq.(<ref>) and Eq.(<ref>) are invariant under a gauge transformation ε^*_λ'(k) →ε^*_λ'(k) + c k or ε^0*_λ'(k) →ε^0*_λ'(k) + c ω and ε⃗^ *_λ'(k) →ε⃗^ *_λ'(k) + c k⃗.As has been shown in <cit.>, for the calculation of the rate of the neutron radiative β^-–decay one may set ε^0*_λ' = 0 and deal with only physical degrees of freedom of a photon <cit.> such as ε^*_λ' = (0, ε⃗^ *_λ'), which obey the relations <cit.> (see also <cit.>)k⃗·ε⃗^ *_λ' = k⃗·ε⃗_λ' = 0 , ε⃗^ *_λ'·ε⃗_λ” = δ_λ'λ”, ∑_λ' = 1,2ε⃗^ i *_λ'ε⃗^ j_λ' = δ^ij - k⃗^ ik⃗^ j/ω^2 = δ^ij - n⃗^ i_k⃗n⃗^ j_k⃗,where n⃗_k⃗ = k⃗/ω is a unit vector along the photon 3–momentum <cit.>. The rate of the neutron radiative β^-–decay with a photon, emitted in the energy region ω_ min≤ω≤ω_ max and calculated to leading order in the large proton mass expansion, is <cit.>λ^( Fig. <ref>)_βγ(ω_ max,ω_ min)= α/π (1 + 3 λ^2) G^2_F|V_ud|^2/2π^3∫^ω_ max_ω_ mindω/ω∫^E_0 - ω_m_edE_e√(E^2_e - m^2_e) E_e F(E_e, Z = 1) (E_0 - E_e - ω)^2 ×{(1 + ω/E_e + 1/2ω^2/E^2_e) [1/β ℓ n(1 + β/1 - β) - 2] + ω^2/E^2_e},where β = k_e/E_e = √(E^2_e - m^2_e)/E_e is a velocity of the electron with a 3–momentum k_e, the Fermi function F(E_e, Z = 1) describes the proton–electron Coulomb interaction in the final state of the decay <cit.> and E_0 = (m^2_n - m^2_p + m^2_e)/2m_n = 1.2927MeV is the end–point energy of the electron–energy spectrum of the neutron β^-–decay <cit.>. The rate of the neutron radiative β^-–decay, taken to next–to–leading order in the large proton mass expansion and taking into account the contributions of the weak magnetism and proton recoil, has been calculated in <cit.>. It is given by <cit.>λ^( Fig <ref>)_βγ(ω_ max,ω_ min)=(1 + 3 λ^2)α/πG^2_F |V_ud|^2/2π^3∫^ω_ max_ω_ mindω/ω∫^E_0 - ω_m_e dE_e E_e√(E^2_e - m^2_e)(E_0 - E_e - ω)^2 × F(E_e, Z = 1) ρ^( Fig <ref>)_βγ(E_e,ω).The function ρ^( Fig <ref>)_βγ(E_e,ω) is given by the integral <cit.>ρ^( Fig <ref>)_βγ(E_e,ω) = ∫dΩ_eγ/4π {[1 + 2 ω/M E_e - k⃗_e·n⃗_k⃗/E_0 - E_e - ω + 3/M (E_e + ω - 1/3 E_0) + λ^2 - 2(κ + 1)λ + 1/1 + 3λ^2 E_0 - E_e - ω/M] × [(1 + ω/E_e) k^2_e - (k⃗_e·n⃗_k⃗)^2/(E_e - k⃗_e·n⃗_k⃗)^2 + ω^2/E_e 1/E_e - k⃗_e·n⃗_k⃗] + 3λ^2 - 1/1 + 3 λ^2 1/M (k^2_e + ωk⃗_e·n⃗_k⃗/E_e [k^2_e - (k⃗_e·n⃗_k⃗)^2/(E_e - k⃗_e·n⃗_k⃗)^2 + ω/E_e - k⃗_e·n⃗_k⃗]+ (ω + k⃗_e·n⃗_k⃗)[(1 + ω/E_e)ω/E_e - k⃗_e·n⃗_k⃗ - m^2_e/E_e ω/(E_e - k⃗_e·n⃗_k⃗)^2]) - λ^2 + 2 (κ + 1)λ - 1/1 + 3λ^2 1/M [k^2_e + ω^2 + 2ωk⃗_e·n⃗_k⃗/E_e × k^2_e - (k⃗_e·n⃗_k⃗)^2/(E_e - k⃗_e·n⃗_k⃗)^2 + ω/E_e k^2_e - (k⃗_e·n⃗_k⃗)^2/E_e - k⃗_e·n⃗_k⃗ + ω^2/E_e ω + k⃗_e·n⃗_k⃗/E_e - k⃗_e·n⃗_k⃗] - λ(λ - 1)/1 + 3λ^2 1/M [ω/E_e k^2_e - (k⃗_e·n⃗_k⃗)^2/E_e - k⃗_e·n⃗_k⃗ + 3 ω^2/E_e]},where dΩ_eγ is an infinitesimal solid angle of the electron–photon momentum correlations k⃗_e·n⃗_k⃗ = k_e cosθ_eγ. The numerical value of the rate Eq.(<ref>) is given in Table I. It has been calculated in <cit.>.§ APPENDIX B: CONTRIBUTIONS OF FEYNMAN DIAGRAMS IN FIG. <REF> TO THE AMPLITUDE AND RATE OF THE NEUTRON RADIATIVE Β^-–DECAY The contribution of the Feynman diagrams in Fig. <ref> to the amplitude of the neutron radiative β^-–decay we define as followsM_ Fig. <ref>(n → p e^- ν̅_e γ)_λ' = ∑_j = a,b,c M^(p)_ Fig. <ref>j(n → p e^- ν̅_e γ)_λ' + ∑_j = d,e,f M^(e)_ Fig. <ref>j(n → p e^- ν̅_e γ)_λ',where the amplitudes M^(p)_ Fig. <ref>j(n → p e^- ν̅_e γ)_λ' and M^(e)_ Fig. <ref>j(n → p e^- ν̅_e γ)_λ' are given by the following analytical expressionsM^(p)_ Fig. <ref>a(n → p e^- ν̅_e γ)_λ' = [u̅_p(k⃗_p, σ_p) ε^*_λ'·Λ_p(k_p, k) 1/m_p - k̂_p - k̂ - i0 γ^μ(1 + λγ^5)u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e,σ_e) γ_μ (1 - γ^5)v_ν(k⃗_ν, + 1/2)],where Λ^α_p(k_p, k) is the proton–proton–photon (ppγ) vertex functionΛ^α_p(k_p, k) = (Z^(p)_1 - 1) γ^α + e^2∫d^4q/(2π)^4i γ^σ 1/m_p - k̂_p - q̂ - i0 γ^α 1/m_p - k̂_p - q̂ - k̂ - i0 γ^β D_σβ(q) andM^(p)_ Fig. <ref>b(n → p e^- ν̅_e γ)_λ' = [u̅_p(k⃗_p, σ_p)ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0 Σ^(p)(k_p,k) 1/m_p - k̂_p - k̂ - i0 γ^μ(1 + λγ^5)u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e,σ_e) γ_μ (1 - γ^5)v_ν(k⃗_ν, + 1/2)],where Σ^(p)(k_p, k) is the proton self–energy correctionΣ^(p)(k_p, k) = - δ m_p - (Z^(p)_2 - 1) (m_p - k̂_p - k̂) + e^2∫d^4q/(2π)^4i γ^σ1/m_p - k̂_p - q̂ - k̂ - i 0γ^β D_σβ(q),and M^(p)_ Fig. <ref>c(n → p e^- ν̅_e γ)_λ' = [u̅_p(k⃗_p, σ_p)Σ^(p)(k_p) 1/m_p - k̂_p - i0 ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0 γ^μ(1 + λγ^5)u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e,σ_e)γ_μ (1 - γ^5)v_ν(k⃗_ν, + 1/2)],where Σ^(p)(k_p) is the proton self–energy correctionΣ^(p)(k_p) = - δ m_p - (Z^(p)_2 - 1) (m_p - k̂_p ) + e^2∫d^4q/(2π)^4i γ^σ1/m_p - k̂_p - q̂ - i 0γ^β D_σβ(q),and M^(e)_ Fig. <ref>d(n → p e^- ν̅_e γ)_λ' =- [u̅_p(k⃗_p, σ_p)γ^μ(1 + λγ^5)u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e,σ_e) ε^*_λ'·Λ_e(k_e, k) 1/m_e - k̂_e - k̂ - i0 γ_μ (1 - γ^5)v_ν(k⃗_ν, + 1/2)],where Λ^α_e(k_e, k) is the electron–electron–photon (e^-e^-γ) vertex functionΛ^α_e(k_e, k) = (Z^(e)_1 - 1) γ^α + e^2∫d^4q/(2π)^4i γ^σ 1/m_e - k̂_e - q̂ - i0 γ^α 1/m_e - k̂_e - q̂ - k̂ - i0 γ^β D_σβ(q),and M^(e)_ Fig. <ref>e(n → p e^- ν̅_e γ)_λ' = - [u̅_p(k⃗_p, σ_p)γ^μ(1 + λγ^5)u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i0 Σ^(e)(k_e,k) 1/m_e - k̂_e - k̂ - i0 γ_μ (1 - γ^5)v_ν(k⃗_ν, + 1/2)],where Σ^(e)(k_e, k) is the electron self–energy correctionΣ^(e)(k_e, k) = - δ m_e - (Z^(e)_2 - 1) (m_e - k̂_e - k̂) + e^2∫d^4q/(2π)^4i γ^σ1/m_e - k̂_e - q̂ - k̂ - i 0γ^β D_σβ(q),and M^(e)_ Fig. <ref>f(n → p e^- ν̅_e γ)_λ' = -[u̅_p(k⃗_p, σ_p)γ^μ(1 + λγ^5)u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e,σ_e) Σ^(e)(k_e) 1/m_e - k̂_e - i0 ε̂^*_λ' 1/m_e - k̂_e - k̂ - i0 γ_μ (1 - γ^5) v_ν(k⃗_ν, + 1/2)],where Σ^(e)(k_e) is the electron self–energy correctionΣ^(e)(k_e) = - δ m_e - (Z^(e)_2 - 1) (m_p - k̂_e) + e^2∫d^4q/(2π)^4i γ^β1/m_p - k̂_e - q̂ - i 0γ^β D_σβ(q),where D_σβ(q) = (η_σβ -(1 - ξ)q_σq_β/q^2)/(q^2 + i 0) is a photon Green function and ξ is a gauge parameter.Now we may analyse the properties of the Feynman diagrams in Fig. <ref> with respect to gauge transformations ε^*_λ'(k) →ε^*_λ'(k) + c k and D_σβ(q) → D_σβ(q) + c(q^2) q_σq_β. Making a gauge transformation ε^*_λ'(k) →ε^*_λ'(k) + c k for the scalar product k·Λ_p(k_p,k) we obtain the following expressionk·Λ_p(k_p,k) = (Z^(p)_1 - 1)((m_p - k̂_p) - (m_p - k̂_p - k̂)) + e^2∫d^4q/(2π)^4i γ^σ1/m_p - k̂_p - q̂ - k̂ - i 0γ^β D_σβ(q) - e^2∫d^4q/(2π)^4i γ^σ1/m_p - k̂_p - q̂ - i 0γ^β D_σβ(q) = [ - δ m_p - (Z^(p)_1 - 1) (m_p - k̂_p - k̂)+ e^2∫d^4q/(2π)^4i γ^σ1/m_p - k̂_p - q̂ - k̂ - i 0γ^β D_σβ(q)] - [ - δ m_p - (Z^(p)_1 - 1) (m_p - k̂_p)+ e^2∫d^4q/(2π)^4i γ^σ1/m_p - k̂_p - q̂- i 0γ^β D_σβ(q)].Since Z^(p)_1 = Z^(p)_2 <cit.>, Eq.(<ref>) can be transcribed into the standard form of the Ward identity <cit.>k·Λ_p(k_p,k) = Σ^(p)(k_p, k) - Σ^(p)(k_p),where Σ^(p)(k_p, k) and Σ^(p)(k_p) are given by Eq.(<ref>) and Eq.(<ref>), respectively. Then, because of the relation Z^(e)_1 = Z^(e)_2 <cit.>, we get the Ward identity <cit.>k·Λ_e(k_e,k) = Σ^(e)(k_e, k) - Σ^(e)(k_e),where Σ^(e)(k_e, k) and Σ^(e)(k_e) are given by Eq.(<ref>) and Eq.(<ref>), respectively.Making a gauge transformation of a photon Green function D_σβ(q) → D_σβ(q) + c(q^2) q_σq_β we obtain the following correction to the (ppγ) vertex diagramε^*_λ'·δΛ_p(k_p,k) = e^2 ∫d^4q/(2π)^4i c(q^2) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0 - e^2 ∫d^4q/(2π)^4i c(q^2) ε̂^*_λ' 1/m_p - k̂_p - q̂ - k̂ - i0,where we have used the Dirac equation for a free proton. The self–energy diagrams acquire the correctionsε̂^*_λ' 1/m_p - k̂_p - k̂ - i0 δΣ^(p)(k_p,k) 1/m_p - k̂_p - k̂ - i0 = - e^2 ∫d^4q/(2π)^4i c(q^2) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0 q̂ 1/m_p - k̂_p - k̂ - i0- e^2 ∫d^4q/(2π)^4i c(q^2) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0 + e^2 ∫d^4q/(2π)^4i c(q^2) ε̂^*_λ' 1/m_p - k̂_p - q̂ - k̂ - i0.Since the first term in Eq.(<ref>) is equal to zero, we getε̂^*_λ' 1/m_p - k̂_p - k̂ - i0 δΣ^(p)(k_p,k) 1/m_p - k̂_p - k̂ - i0 = - e^2 ∫d^4q/(2π)^4i c(q^2) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0+ e^2 ∫d^4q/(2π)^4i c(q^2) ε̂^*_λ' 1/m_p - k̂_p - q̂ - k̂ - i0.Then, the self–energy correction Σ^(p)(k_p) is invariant under the gauge transformation D_σβ(q) → D_σβ(q) + c(q^2) q_σq_β. We getδΣ^(p)(k_p) = - e^2 ∫d^4q/(2π)^4i c(q^2) q̂ = 0,where we have used the Dirac equation for a free proton. Plugging Eq.(<ref>) and Eq.(<ref>) into Eq.(<ref>) and Eq.(<ref>) one may see that the acquired corrections cancel each other in the first term of Eq.(<ref>). This confirms invariance of the Feynman diagrams in Fig. <ref>a, Fig. <ref>b and Fig. <ref>c under a gauge transformation D_σβ(q) → D_σβ(q) + c(q^2) q_σq_β. It is obvious that the Feynman diagrams in Fig. <ref>d, Fig. <ref>e and Fig. <ref>f are also invariant under a transformation D_σβ(q) → D_σβ(q) + c(q^2) q_σq_β of a photon Green function. The observed gauge invariance allows to make calculations of the Feynman diagrams in Fig. <ref> in the Feynman gauge ξ = 1 <cit.>.For the calculation of Λ^α_p(k_p, q) we rewrite the right-hand-side (r.h.s.) of Eq.(<ref>) as followsΛ^α_p(k_p, k) = (Z^(p)_1 - 1) γ^α + e^2∫d^4q/(2π)^4iγ^β(m_p + k̂_p + q̂)γ^α(m_p + k̂_p + q̂ + k̂)γ_β/[m^2_p - (k_p + q)^2 - i0][m^2_p - (k_p + q + k)^2 - i0] 1/q^2 + i0.Since the integral over q diverges, we have to regularize it.For this aim we use the Pauli–Villars regularization and make in Eq.(<ref>) a replacement <cit.> (see also <cit.>)1/q^2 + i 0→1/Λ^2 - q^2 - i0 - 1/μ^2 - q^2 - i0,where Λ and μ are the ultraviolet and infrared cut–off, respectively, which should be finally taken in the limit Λ→∞ and μ→ 0.The next step is to merge the denominators. Merging the denominators of the proton propagators we get1/[m^2_p - (k_p + q)^2 - i0]1/[m^2_p - (k_p + q + k)^2 - i0] = ∫^1_0dx/[m^2_p- k^2 x(1 - x) - (q + k_p + k x)^2 - i0]^2.Then, we have to take into account the contribution of the regularized photon propagator. Using the formula <cit.> 1/A^2 B= ∫^1_02y dy/[A y + B (1 -y)]^3we obtain1/[m^2_p - (k_p + q)^2 - i0]1/[m^2_p - (k_p + q + k)^2 - i0]( 1/Λ^2 - q^2 - i0 - 1/μ^2 - q^2 - i0) = = ∫^1_0∫^1_0 2y dx dy/[Λ^2(1 - y) + m^2_p y^2 - 2 k_p· k x y (1 - y) - k^2 x y (1 - x y) - (q - (k_p + k x)y)^2 - i0]^3 - ∫^1_0∫^1_0 2y dx dy/[μ^2(1 - y) + m^2_py^2 - 2 k_p· k x y (1 - y) - k^2 x y (1 - x y) - (q - (k_p + k x)y)^2 - i0]^3.Now we take into account that k^2 = 01/[m^2_p - (k_p + q)^2 - i0]1/[m^2_p - (k_p + q + k)^2 - i0]( 1/Λ^2 - q^2 - i0 - 1/μ^2 - q^2 - i0) = = ∫^1_0∫^1_0 2y dx dy/[Λ^2(1 - y) + m^2_p y^2 - 2 k_p· k x y (1 - y) - (q + (k_p + k x)y)^2 - i0]^3 - ∫^1_0∫^1_0 2y dx dy/[μ^2(1 - y) + m^2_py^2 - 2 k_p· k x y (1 - y) - (q + (k_p + k x)y)^2 - i0]^3.For the numerator of the integrand of Eq.(<ref>) we get the expressionγ^β(m_p + k̂_p + q̂)γ^α(m_p + k̂_p + q̂ + k̂)γ_β =- 2m ^2_p γ^α + 4 m_p (2 k_p + 2 q + k)^α + 2 (k_p + q)^2 γ^α + 2 k̂ (k̂_p + q̂) γ^α- 4 (k̂_p + q̂ + k̂) (k_p + q)^α.Then, making a change of variables q + (k_p + k x)y → q and integrating over the solid angle in the 4–dimensional q–space we arrive at the expressionγ^β(m_p + k̂_p + q̂)γ^α(m_p + k̂_p + q̂ + k̂)γ_β→ - 2m ^2_p γ^α + 4 m_p (2 k_p + 2 q - 2(k_p + k x)y + k)^α+ 2 (k_p + q - (k_p + k x)y)^2 γ^α + 2 k̂ (k̂_p + q̂ - (k̂_p + k̂ x)y) γ^α - 4 (k̂_p + q̂ - (k̂_p + k̂ x) y + k̂) × (k_p + q - (k_p + k x)y)^α.Having integrated over the directions of the 4–vector q we getγ^β(m_p + k̂_p + q̂)γ^α(m_p + k̂_p + q̂ + k̂)γ_β→[q^2 - 2m^2_py(2 - y) + 4k_p· k (1 - xy) (1 - y) - 2 m_p k̂ (1 - y)- 2 k^2 x y (1 - xy)] γ^α + [4 m_p k^α_p (1 - y^2) + 4 m_p k^α (1 - x y(1 + y)) - 4 k^α_p k̂ (1 - xy)(1 - y) + 4 k̂ k^α x y(1 - x y)].Now we may remove the terms, which vanish because of the relations ε^*_λ'· k = 0 and k^2 = 0. This gives the following representation for the (ppγ) vertexΛ^α_p(k_p, k) = (Z^(p)_1 - 1) γ^α + e^2∫^1_0dx∫^1_0dy 2y ∫d^4q/(2π)^4i( 1/[Λ^2(1 - y) + m^2_p y^2 - 2 k_p· k x y (1 - y) - q^2 - i0]^3 - 1/[μ^2(1 - y) + m^2_py^2 - 2 k_p· k x y (1 - y) - q^2 - i0]^3){[q^2 - 2m^2_py(2 - y) + 4k_p· k (1 - xy) (1 - y) - 2 m_p k̂ (1 - y)] γ^α+ [4 m_p k^α_p (1 - y^2) - 4 k^α_p k̂ (1 - xy)(1 - y)]}.Making a Wick rotation <cit.> we obtain the expressionΛ^α_p(k_p, k) = (Z^(p)_1 - 1) γ^α + e^2∫^1_0dx∫^1_0dy 2y ∫d^4q/(2π)^4( 1/[Λ^2(1 - y) + m^2_p y^2 - 2 k_p· k x y (1 - y) + q^2]^3 - 1/[μ^2(1 - y) + m^2_py^2 - 2 k_p· k x y (1 - y) + q^2]^3){[- q^2 - 2m^2_py(2 - y) + 4k_p· k (1 - xy) (1 - y) - 2 m_p k̂ (1 - y)] γ^α+ [4 m_p k^α_p (1 - y^2) - 4 k^α_p k̂ (1 - xy)(1 - y)]},which we transcribe into the formΛ^α_p(k_p, k) = (Z^(p)_1 - 1) γ^α - e^2 γ^α∫^1_0dx∫^1_0dy 2y ×∫d^4q/(2π)^4( 1/[Λ^2(1 - y) + m^2_p y^2 - 2 k_p· k x y (1 - y) + q^2]^2 - 1/[μ^2(1 - y) + m^2_py^2 - 2 k_p· k x y (1 - y) + q^2]^2)+ e^2∫^1_0dx∫^1_0dy 2y ∫d^4q/(2π)^4 1/[Λ^2(1 - y) + m^2_p y^2 - 2 k_p· k x y (1 - y) + q^2]^3 {[Λ^2(1 - y) - m^2_py(4 - 3y)+ 2k_p· k (2 - 3xy) (1 - y) - 2 m_p k̂ (1 - y)] γ^α + [4 m_p k^α_p (1 - y^2) - 4 k^α_p k̂ (1 - xy)(1 - y)]}- e^2∫^1_0dx∫^1_0dy 2y ∫d^4q/(2π)^4 1/[μ^2(1 - y) + m^2_p y^2 - 2 k_p· k x y (1 - y) + q^2]^3 {[μ^2(1 - y) - m^2_py(4 - 3y)+ 2k_p· k (2 - 3xy) (1 - y) - 2 m_p k̂ (1 - y)] γ^α + [4 m_p k^α_p (1 - y^2) - 4 k^α_p k̂ (1 - xy)(1 - y)]}Having integrated over q we getΛ^α_p(k_p, k) = (Z^(p)_1 - 1) γ^α + γ^α e^2/16π^2∫^1_0dx∫^1_0dy 2y ℓ n(Λ^2(1 - y) + m^2_p y^2 - 2 k_p· k x y (1 - y)/μ^2(1 - y) + m^2_py^2 - 2 k_p· k x y (1 - y))+ e^2/32π^2∫^1_0dx∫^1_0dy 2y 1/Λ^2(1 - y) + m^2_p y^2 - 2 k_p· k x y (1 - y) {[Λ^2(1 - y) - m^2_py(4 - 3y)+ 2k_p· k (2 - 3xy) (1 - y) - 2 m_p k̂ (1 - y)] γ^α + [4 m_p k^α_p (1 - y^2) - 4 k^α_p k̂ (1 - xy)(1 - y)]}- e^2/32π^2∫^1_0dx∫^1_0dy 2y 1/μ^2(1 - y) + m^2_p y^2 - 2 k_p· k x y (1 - y) {[μ^2(1 - y) - m^2_py(4 - 3y)+ 2k_p· k (2 - 3xy) (1 - y) - 2 m_p k̂ (1 - y)] γ^α + [4 m_p k^α_p (1 - y^2) - 4 k^α_p k̂ (1 - xy)(1 - y)]}.Then, we make a change of variables xy → x. This givesΛ^α_p(k_p, k) = (Z^(p)_1 - 1) γ^α + γ^α e^2/8π^2∫^1_0dy∫^y_0dxℓ n(Λ^2(1 - y) + m^2_p y^2 - 2 k_p· k x(1 - y)/μ^2(1 - y) + m^2_py^2 - 2 k_p· k x (1 - y))+ e^2/16π^2∫^1_0dy∫^y_0dx 1/Λ^2(1 - y) + m^2_p y^2 - 2 k_p· k x (1 - y) {[Λ^2(1 - y) - m^2_py(4 - 3y)+ 2k_p· k (2 - 3 x) (1 - y) - 2 m_p k̂ (1 - y)] γ^α + [4 m_p k^α_p (1 - y^2) - 4 k^α_p k̂ (1 - x)(1 - y)]}- e^2/16π^2∫^1_0dy∫^y_0dx 1/μ^2(1 - y) + m^2_p y^2 - 2 k_p· k x (1 - y) {[μ^2(1 - y) - m^2_py(4 - 3y)+ 2k_p· k (2 - 3 x) (1 - y) - 2 m_p k̂ (1 - y)] γ^α + [4 m_p k^α_p (1 - y^2) - 4 k^α_p k̂ (1 - x)(1 - y)]}.Taking into account the limit Λ≫ m_p we getΛ^α_p(k_p, k) = (Z^(p)_1 - 1) γ^α + γ^α e^2/8π^2∫^1_0dy∫^y_0dx ℓ n(Λ^2(1 - y) /μ^2(1 - y) + m^2_py^2 - 2 k_p· k x (1 - y)) + e^2/8π^2∫^1_0dy∫^y_0dx 1/μ^2(1 - y) + m^2_p y^2 - 2 k_p· k x (1 - y) {[ m^2_py(2 - y) - 2 k_p· k (1 - x) (1 - y)+ m_p k̂ (1 - y)] γ^α + [- 2 m_p k^α_p (1 - y^2) + 2 k^α_p k̂ (1 - x)(1 - y)]}.To leading order in the large proton mass expansion we obtain the following resultΛ^α_p(k_p, q) = (Z^(p)_1 - 1) γ^α + e^2/8π^2 (ℓ n(Λ/m_p) + 5/4) γ^α + e^2/8π^2 (2 ℓ n(- 2 k_p· k/m^2_p) - 1) k^α_p/m_p(see Eq.(<ref>) with the replacement m_e → m_p and k_e → k_p, where we have dropped the terms of order O(1/m_p)). The calculation of the proton self–energy corrections in Eq.(<ref>) and Eq.(<ref>) runs as follows. For the calculation of the integrals over q we regularize the photon propagator1/q^2 + i 0→1/Λ^2 - q^2 - i0 - 1/μ^2 - q^2 - i0.Then, we rewrite Eq.(<ref>) and Eq.(<ref>) as followsΣ^(p)(k_p, k)=- δ m_p - (Z^(p)_2 - 1) (m_p - k̂_p - k̂) + e^2∫d^4q/(2π)^4i γ^β(m_p + k̂_p + k̂ + q̂)γ_β/[m^2_p - (k_p + k + q)^2 - i 0] × ( 1/Λ^2 - q^2 - i0 - 1/μ^2 - q^2 - i0)and Σ^(p)(k_p)=- δ m_p - (Z^(p)_2 - 1) (m_p - k̂_p) + e^2∫d^4q/(2π)^4i γ^β(m_p + k̂_p + q̂)γ_β/[m^2_p - (k_p + q)^2 - i 0] × ( 1/Λ^2 - q^2 - i0 - 1/μ^2 - q^2 - i0).Using the algebra of the Dirac matrices we getΣ^(p)(k_p, k)=- δ m_p - (Z^(p)_2 - 1) (m_p - k̂_p - k̂) + e^2∫d^4q/(2π)^4i 4m_p - 2(k̂_p + k̂ + q̂)/[m^2_p - (k_p + k + q)^2 - i 0] × ( 1/Λ^2 - q^2 - i0 - 1/μ^2 - q^2 - i0)and Σ^(p)(k_p)=- δ m_p - (Z^(p)_2 - 1) (m_p - k̂_p) + e^2∫d^4q/(2π)^4i 4 m_p - 2(k̂_p + q̂)/[m^2_p - (k_p + q)^2 - i 0] × ( 1/Λ^2 - q^2 - i0 - 1/μ^2 - q^2 - i0).Merging the proton and photon propagators we obtain1/[m^2_p - (k_p+ k + q)^2 - i 0]( 1/Λ^2 - q^2 - i0 - 1/μ^2 - q^2 - i0) = = ∫^1_0 dx (1/[Λ^2 (1 - x ) + m^2_p x - (k_p + k)^2 x (1 - x) - (q + (k_p + k)x)^2 - i0]^2- 1/[μ^2 (1 - x ) + m^2_p x - (k_p + k)^2 x (1 - x) - (q + (k_p + k)x)^2 - i0]^2and 1/[m^2_p - (k_p + q)^2 - i 0]( 1/Λ^2 - q^2 - i0 - 1/μ^2 - q^2 - i0) = = ∫^1_0 dx (1/[Λ^2 (1 - x ) + m^2_p x - k^2_p x (1 - x) - (q + k_p x)^2 - i0]^2- 1/[μ^2 (1 - x ) + m^2_p x - k^2_p x (1 - x) - (q + k_p x)^2 - i0]^2Plugging Eq.(<ref>) and Eq.(<ref>) into Eq.(<ref>) and Eq.(<ref>), respectively, making a shift of variables q + (k_p + k)x → q and q + k_p x → q and integrating over the 4–dimensional solid angle in the q–space we arrive at the expressionsΣ^(p)(k_p, k)=- δ m_p - (Z^(p)_2 - 1) (m_p - k̂_p - k̂)+ e^2/8π^2∫^1_0dx∫d^4q/π^2i {m_p(1 + x) + (m_p - k̂_p - k̂)(1 - x) /[Λ^2 (1 - x ) + m^2_p x - (k_p + k)^2 x (1 - x) - q^2 - i0]^2- m_p(1 + x) + (m_p - k̂_p - k̂)(1 - x) /[μ^2 (1 - x ) + m^2_p x - (k_p + k)^2 x (1 - x) - q^2 - i0]^2}and Σ^(p)(k_p) = - δ m_p - (Z^(p)_2 - 1)(m_p - k̂_p)+ e^2/8π^2∫^1_0dx∫d^4k/π^2i {m_p(1 + x) + (m_p - k̂_p)(1 - x) /[Λ^2 (1 - x ) + m^2_p x - k^2_p x (1 - x) - q^2 - i0]^2- m_p(1 + x) + (m_p - k̂_p )(1 - x) /[μ^2 (1 - x ) + m^2_p x - k^2_p x (1 - x) - q^2 - i0]^2}.Making the Wick rotation and integrating over q^2 we getΣ^(p)(k_p, k)=- δ m_p - (Z^(p)_2 - 1)(m_p - k̂_p - k̂) - e^2/8π^2∫^1_0dx (m_p(1 + x) + (m_p - k̂_p - k̂)(1 - x)) × ℓ n(Λ^2 (1 - x ) + m^2_p x - (k_p + k)^2 x (1 - x)/μ^2 (1 - x ) + m^2_p x - (k_p + k)^2 x (1 - x))and Σ^(p)(k_p)=- δ m_p - (Z^(p)_2 - 1) (m_p - k̂_p) - e^2/8π^2∫^1_0dx(m_p(1 + x) + (m_p - k̂_p)(1 - x)) × ℓ n(Λ^2 (1 - x ) + m^2_p x - k^2_p x (1 - x)/μ^2 (1 - x ) + m^2_p x - k^2_p x (1 - x) ).Since Λ≫ m_p, we may reduces the integrands to the formΣ^(p)(k_p, k)=- δ m_p - (Z^(p)_2 - 1)(m_p - k̂_p - k̂) - e^2/8π^2∫^1_0dx (m_p(1 + x) + (m_p - k̂_p - k̂)(1 - x)) × {ℓ n(Λ^2 /μ^2 (1 - x ) + m^2_p x - (k_p + k)^2 x (1 - x)) + ℓ n(1 - x)}and Σ^(p)(k_p)=- δ m_p - (Z^(p)_2 - 1) (m_p - k̂_p) - e^2/8π^2∫^1_0dx(m_p(1 + x) + (m_p - k̂_p)(1 - x)) × {ℓ n(Λ^2/μ^2 (1 - x ) + m^2_p x - k^2_p x (1 - x) ) + ℓ n(1 - x)}.The next step is to rewrite Eq.(<ref>) and Eq.(<ref>) in the following formΣ^(p)(k_p, k)=- δ m_p - (Z^(p)_2 - 1)(m_p - k̂_p - k̂) - e^2/8π^2∫^1_0dx (m_p(1 + x) + (m_p - k̂_p - k̂)(1 - x)) × {ℓ n(Λ^2 /m^2_p x^2 + μ^2 (1 - x )) + ℓ n(1 - x) - ℓ n(1 + (m^2_p - (k_p + k)^2)x(1 - x)/m^2_p x^2 + μ^2 (1 - x )) }and Σ^(p)(k_p)=- δ m_p - (Z^(p)_2 - 1) (m_p - k̂_p) - e^2/8π^2∫^1_0dx(m_p(1 + x) + (m_p - k̂_p)(1 - x)) × {ℓ n(Λ^2/m^2_p x^2 + μ^2 (1 - x )) + ℓ n(1 - x) - ℓ n(1 + (m^2_p - k^2_p)x(1 - x)/m^2_p x^2 + μ^2 (1 - x ))}.For the proton on–mass shell k^2_p = m^2_p we transcribe Eq.(<ref>) and Eq.(<ref>) into the formΣ^(p)(k_p, k) = - δ m_p - (Z^(p)_2 - 1)(m_p - k̂_p - k̂) - m_pe^2/8π^2∫^1_0dx(1 + x){ℓ n(Λ^2 /m^2_p x^2 + μ^2 (1 - x )) + ℓ n(1 - x)}- (m_p - k̂_p - k̂) e^2/8π^2∫^1_0dx (1 - x) {ℓ n(Λ^2 /m^2_p x^2 + μ^2 (1 - x )) + ℓ n(1 - x) - ℓ n(1 - 2 k_p· k x(1 - x)/m^2_p x^2 + μ^2 (1 - x )) }andΣ^(p)(k_p)=- δ m_p - (Z^(p)_2 - 1) (m_p - k̂_p) - m_pe^2/8π^2∫^1_0dx(1 + x){ℓ n(Λ^2 /m^2_p x^2 + μ^2 (1 - x )) + ℓ n(1 - x)}- (m_p - k̂_p ) e^2/8π^2∫^1_0dx (1 - x) {ℓ n(Λ^2 /m^2_p x^2 + μ^2 (1 - x )) +ℓ n(1 - x)}.Keeping only the leading order contributions in the large proton mass expansion we getΣ^(p)(k_p, k)=- δ m_p - (Z^(p)_2 - 1)(m_p - k̂_p - k̂) - 3 m_p e^2/8π^2 (ℓ n(Λ/m_p) + 1/4)- (m_p - k̂_p - k̂) e^2/8π^2 (ℓ n(Λ/m_p) + 5/4) + e^2/8π^2 k_p· k/m_p (2 ℓ n(- 2 k_p· k/m_p) - 1)(see Eq.(<ref>) with the replacement m_e → m_p and k_e → k_p, where we have dropped the terms of order O(1/m_p)) andΣ^(p)(k_p)=- δ m_p - (Z^(p)_2 - 1) (m_p - k̂_p) - 3 m_pe^2/8π^2 (ℓ n(Λ/m_p) + 1/4)- (m_p - k̂_p ) e^2/8π^2 (ℓ n(Λ/m_p) + 5/4).As a result, the renormalization parameters Z^(p)_1, δ m_p and Z^(p)_2 are given byδ m_p=- 3 m_pe^2/8π^2 (ℓ n(Λ/m_p) + 1/4) Z^(p)_1=Z^(p)_2 = 1 - e^2/8π^2 (ℓ n(Λ/m_p) + 5/4).They are calculated in agreement with the Ward identity Z^(p)_1 = Z^(p)_2, required by gauge invariance.Thus, the renormalized (ppγ) vertex Λ̅^α_p(k_p, k) and proton self–energy corrections Σ̅^(p)(k_p,k) and Σ̅^(p)(k_p) are equal toΛ̅^α_p(k_p,k)= e^2/8π^2 (2ℓ n(- 2 k_p · k/m^2_p) - 1) k^α_p/m_p, Σ̅^(p)(k_p,k)= e^2/8π^2 k_p · k/m_p (2 ℓ n(- 2 k_p · k/m^2_p) - 1), Σ̅^(p)(k_p)=0,The sum of the renormalized amplitudes in Fig. <ref>a, <ref>b and <ref>c is given by∑_j =a,b,c M_ Fig. <ref>j(n → p e^- ν̅_e γ)_λ' = [u̅_p(k⃗_p, σ_p) ε^*_λ'·Λ̅_p(k_p, k) 1/m_p - k̂_p - k̂ - i0 γ^μ(1 + λγ^5)u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e,σ_e)γ_μ (1 - γ^5)v_ν(k⃗_ν, + 1/2)]+ [u̅_p(k⃗_p, σ_p)ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0 Σ̅^(p)(k_p,k) 1/m_p - k̂_p - k̂ - i0 γ^μ(1 + λγ^5)u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e,σ_e)γ_μ (1 - γ^5)v_ν(k⃗_ν, + 1/2)],where we have taken into account that Σ̅^(p)(k_p) = 0. Since the renormalized (ppγ) vertex Λ̅^α_p(k_p,k) and proton self–energy correction Σ̅^(p)(k_p,k) obey the Ward identity <cit.>k ·Λ̅_p(k_p,k) = Σ̅^(p)(k_p,k),the amplitude Eq.(<ref>) is invariant under the gauge transformation ε^*_λ'(k) →ε^*_λ'(k) + c k. This confirms the correctness of the calculation of the diagrams in Fig. <ref>a, <ref>b and <ref>c.Now we may proceed to the calculation of the Feynman diagrams in Fig. <ref>d, <ref>e and <ref>f.Skipping intermediate calculations, which are similar to those we have performed for the diagrams in Fig. <ref>a, <ref>b and <ref>c, we arrive at the following expressions for the (e^-e^-γ) vertex function and electron self–energy correctionsΛ^α_e(k_e, k) = (Z^(e)_1 - 1) γ^α + γ^α e^2/8π^2∫^1_0dy∫^y_0dx ℓ n(Λ^2(1 - y) /μ^2(1 - y) + m^2_ey^2 - 2 k_e· k x (1 - y)) + e^2/8π^2∫^1_0dy∫^y_0dx 1/μ^2(1 - y) + m^2_e y^2 - 2 k_e· k x (1 - y) {[m^2_ey(2 - y) - 2 k_e· k (1 - x) (1 - y)+ m_e k̂ (1 - y)] γ^α + [- 2 m_e k^α_e (1 - y^2) + 2 k^α_e k̂ (1 - x)(1 - y)]}and Σ^(e)(k_e, k)=- δ m_e - (Z^(e)_2 - 1)(m_e - k̂_e - k̂) - e^2/8π^2∫^1_0dx (m_e(1 + x) + (m_e - k̂_e - k̂)(1 - x)) × {ℓ n(Λ^2 /m^2_e x^2 + μ^2 (1 - x )) + ℓ n(1 - x) - ℓ n(1 - 2k_e· k x(1 - x)/m^2_e x^2 + μ^2 (1 - x )) }and Σ^(e)(k_e) = - δ m_e - (Z^(e)_2 - 1) (m_e - k̂_e) - 3 m_ee^2/8π^2 (ℓ n(Λ/m_e) + 1/4) - (m_e - k̂_e ) e^2/8π^2 (ℓ n(Λ/m_e) + 5/4),where we have kept the electron on–mass shell k^2_e = m^2_e. For the calculation of Λ^α_e(k_e,k) we propose to transcribe it into the formΛ^α_e(k_e, k) = (Z^(e)_1 - 1) γ^α + e^2/8π^2 γ^α∫^1_0dy∫^y_0 dx {ℓ n(Λ^2/m^2_e y^2 + μ^2(1 - y)) + ℓ n(1 - y)} + e^2/8π^2 γ^α∫^1_0dy∫^y_0 dx ℓ n(m^2_e y^2 + μ^2(1 - y)/m^2_e y^2 + μ^2(1 - y) - 2 k_e· kx (1 - y)) + e^2/8π^2∫^1_0dy∫^y_0dx 1/m^2_e y^2 + μ^2(1 - y) - 2 k_e· kx (1 - y) {[m^2_e y (2 - y) - 2 k_e· k(1 - x) (1 - y)+m_e k̂(1 - y)] γ^α+ [ - 2 m_e k^α_e (1 - y^2) + 2 k^α_e k̂(1 - x)(1 - y)]}.The result of the calculation of the first integral in Eq.(<ref>) is equal toγ^α∫^1_0dy∫^y_0 dx {ℓ n(Λ^2/m^2_e y^2 + μ^2(1 - y)) + ℓ n(1 - y)} = γ^α (ℓ n(Λ/m_e) + 5/4).For the last two integrals in Eq.(<ref>) we obtain the following resultγ^α∫^1_0dy∫^y_0 dx ℓ n(m^2_e y^2 + μ^2(1 - y)/m^2_e y^2 + μ^2(1 - y) - 2 k_e· kx (1 - y)) + ∫^1_0dy∫^y_0dx 1/m^2_e y^2 + μ^2(1 - y) - 2 k_e· kx (1 - y) × {[m^2_ey(2 - y) - 2 k_e· k(1 - x) (1 - y) + m_e k̂(1 - y)] γ^α + [ - 2 m_e k^α_e (1 - y^2) + 2 k^α_e k̂ (1 - x)(1 - y)]}== γ^α{1/2 + ℓ n(- 2 k_e· k/m^2_e) [ m^2_e + k_e· k/m^2_e + 2 k_e· k - m^2_e/2 k_e· k ℓ n(1 + 2k_e· k/m^2_e)] - m^2_e/2 k_e· kLi_2(- 2k_e· k/m^2_e) + m_e k̂/m^2_e + 2k_e· k × ℓ n(- 2 k_e· k/m^2_e)} + k^α_e/m_e{-m^2_e/m^2_e + 2 k_e· k + 2 m^2_e(m^2_e + 3 k_e· k)/(m^2_e + 2 k_e· k)^2 ℓ n(- 2 k_e· k/m^2_e)} + k^α_ek̂/k_e· k {m^2_e + k_e· k/m^2_e + 2 k_e· k+ ℓ n(- 2 k_e· k/m^2_e) [- (m^2_e + k_e· k)(m^2_e + 4 k_e· k)/(m^2_e + 2 k_e· k)^2 + m^2_e/2 k_e· k ℓ n(1 + 2k_e· k/m^2_e)] + m^2_e/2 k_e· kLi_2(- 2k_e· k/m^2_e)},where we have used a relation k̂γ^α = - γ^αk̂ + 2k^α and omitted the term 2k^α.Then, Li_2(- 2k_e · k/m^2_e) is the Spence function. Summing up the contributions, for the vertex Λ^α_e(k_e,k) we obtain the following analytical expressionΛ^α_e(k_e, k) = (Z^(e)_1 - 1) γ^α + e^2/8π^2 γ^α (ℓ n(Λ/m_e) + 5/4)+ e^2/8π^2 γ^α{ -1 +ℓ n(- 2 k_e· k/m^2_e) [m^2_e + k_e· k/m^2_e + 2 k_e· k -m^2_e/2 k_e· k ℓ n(1 + 2k_e·k/m^2_e)] - m^2_e/2 k_e· k Li_2(- 2k_e· k/m^2_e)+ m_e k̂/m^2_e + 2k_e· k ℓ n(- 2 k_e· k/m^2_e)} + e^2/8π^2 k^α_e/m_e{- m^2_e/m^2_e + 2 k_e· k + 2 m^2_e(m^2_e + 3 k_e· k)/(m^2_e + 2 k_e· k)^2 ℓ n(- 2 k_e· k/m^2_e)}+ e^2/8π^2 k^α_ek̂/k_e· k {m^2_e + k_e· k/m^2_e + 2 k_e· k - ℓ n(- 2 k_e· k/m^2_e) [(m^2_e + k_e· k)(m^2_e + 4 k_e· k)/(m^2_e + 2 k_e· k)^2 - m^2_e/2 k_e· k ℓ n(1 + 2k_e· k/m^2_e)]+ m^2_e/2 k_e· kLi_2(- 2k_e· k/m^2_e)}.Now we may proceed to the calculation of the electron self–energy correction Σ^(e)(k_e,k), defined by Eq.(<ref>). The result isΣ^(e)(k_e, k) = - δ m_e - (Z^(e)_2 - 1)(m_e - k̂_e - k̂) - 3 m_e e^2/8π^2 (ℓ n(Λ/m_e) + 1/4) - (m_e - k̂_e - k̂) e^2/8π^2 (ℓ n(Λ/m_e) + 5/4) + (m_e - k̂_e - k̂) e^2/8π^2 [k_e· k/m^2_e + 2 k_e· k + 2 (k_e· k)(m^2_e + k_e· k)/(m^2_e + 2 k_e· k)^2 ℓ n(- 2 k_e· k/m^2_e)] + m_e e^2/8π^2[-k_e· k/m^2_e + 2 k_e · k+ 2 (k_e· k)(m^2_e + 3 k_e· k)/(m^2_e + 2 k_e · k)^2 ℓ n(- 2 k_e· k/m^2_e)].The renormalization parameters Z^(e)_1, δ m_e and Z^(e)_2 are given byδ m_e=- 3 m_ee^2/8π^2 (ℓ n(Λ/m_e) + 1/4) Z^(e)_1=Z^(e)_2 = 1 - e^2/8π^2 (ℓ n(Λ/m_e) + 5/4),where the counterterms Z^(e)_1 and Z^(e)_2 obey the Ward identity Z^(e)_1 = Z^(e)_2 <cit.>. Thus, for the renormalized (e^-e^-γ) vertex Λ̅^α_e(k_e, k) and electron self–energy corrections Σ̅^(e)(k_e,k) and Σ̅^(e)(k_e) we obtain the following expressionsε^*_λ'·Λ̅_e(k_e, k) = e^2/8π^2 [ε̂^*_λ' F_1(k_e· k) + ε̂^*_λ'k̂/m_e F_2(k_e· k) + ε^*_λ'· k_e/m_e F_3(k_e· k) + ε^*_λ'· k_e/k_e· k k̂ F_4(k_e· k)], Σ̅^(e)(k_e,k) = m_e e^2/8π^2 F_5(k_e· k) + (m_e - k̂_e - k̂) e^2/8π^2 F_6(k_e· k), Σ̅^(e)(k_e) = 0,where the functions F_j(k_e· k) for j = 1,2,…,6 are equal toF_1(k_e· k) = -1 + ℓ n(- 2 k_e· k/m^2_e) [ m^2_e + k_e· k/m^2_e + 2 k_e· k - m^2_e/2 k_e· k ℓ n(1 + 2k_e· k/m^2_e)] - m^2_e/k_e· kLi_2(- 2k_e· k/m^2_e), F_2(k_e· k) =m^2_e/m^2_e + 2k_e· k ℓ n(- 2k_e· k/m^2_e), F_3(k_e· k) = - m^2_e/m^2_e + 2 k_e· k + 2 m^2_e(m^2_e + 3 k_e· k)/(m^2_e + 2 k_e· k)^2 ℓ n(- 2k_e· k/m^2_e), F_4(k_e· k) = m^2_e + k_e· k/m^2_e + 2 k_e· k - ℓ n(- 2 k_e· k/m^2_e) [(m^2_e + k_e· k)(m^2_e + 4 k_e· k)/(m^2_e + 2 k_e· k)^2 - m^2_e/2 k_e· k ℓ n(1 + 2k_e· k/m^2_e)]+ m^2_e/2 k_e· kLi_2(- 2k_e· k/m^2_e), F_5(k_e· k) = - k_e· k/m^2_e + 2 k_e · k + 2 (k_e· k)(m^2_e + 3 k_e· k)/(m^2_e + 2 k_e · k)^2 ℓ n(- 2 k_e· k/m^2_e), F_6(k_e· k) = k_e· k/m^2_e + 2 k_e· k + 2 (k_e· k)(m^2_e + k_e· k)/(m^2_e + 2 k_e· k)^2 ℓ n(- 2 k_e· k/m^2_e).After renormalization the sum of the diagrams in Fig. <ref>d, Fig. <ref>e and Fig. <ref>f is equal to∑_j =d,e,f M_ Fig. <ref>j(n → p e^- ν̅_e γ)_λ' = - [u̅_p(k⃗_p, σ_p)γ^μ(1 + λγ^5)u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e,σ_e) ε^*_λ'·Λ̅_e(k_e, k) 1/m_e - k̂_e - k̂ - i0 × γ_μ (1 - γ^5)v_ν(k⃗_ν, + 1/2)] - [u̅_p(k⃗_p, σ_p)γ^μ(1 + λγ^5)u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i0 Σ̅^(e)(k_e,k) × 1/m_e - k̂_e - k̂ - i0 γ_μ (1 - γ^5)v_ν(k⃗_ν, + 1/2)],where we have taken into account that Σ̅^(e)(k_e) = 0. The correctness of the calculation of the renormalized (e^-e^-γ) vertex function and electron self–energy corrections we verify by using the Ward identity <cit.> (see also <cit.>). Indeed, one may show that the amplitude Eq.(<ref>) is invariant under the gauge transformation ε^*_λ'(k) →ε^*_λ'(k) + c k if Λ̅^α_e(k_e,k) and Σ̅^(e)(k_e, k) obey the Wald identityk·Λ̅_e(k_e,k) = Σ̅^(e)(k_e,k),multiplied by u̅_e(k⃗_e, σ_e) or at k̂_e = m_e <cit.>, The Ward identity Eq.(<ref>) imposes the following relations between the functions F_j(k_e· k):F_1(k_e,k) + F_4(k_e· k)=- F_6(k_e· k), k_e· k/m_e F_3(k_e· k)=m_e F_5(k_e· k).The functions F_j(k_e· k) with j = 1,3,4,5,6 in Eq.(<ref>) fulfil the constraints Eq.(<ref>).Since to leading order in the large proton mass expansion and in the physical gauge of a photon the diagrams Fig. <ref>a, Fig. <ref>b and Fig. <ref>c vanish, the contribution of the diagrams in Fig. <ref> to the amplitude of the neutron radiative β^-–decay is defined fully by the diagrams Fig. <ref>d, Fig. <ref>e and Fig. <ref>f, respectively.In the non–relativistic limit the contribution of the diagrams Fig. <ref> is given byM_ Fig. <ref>(n → p e^- ν̅_e γ)_λ' = 2 m_n e^2/8π^2 1/2k_e· k {[φ^†_pφ_n] {u̅_e(k⃗_e,σ_e) [2 ε_λ'· k_e(F_1 + F_3 + F_4 - m^2_e/k_e· k F_5 + F_6)+ ε̂^*_λ' k̂ (F_1 + 2F_2 - m^2_e/k_e· k F_5 + F_6) + 2k_e· k/m_e ε̂^*_λ' (F_2 - m^2_e/2 k_e· k F_5) + ε^*_λ'· k_e/m_e k̂ (- 2 F_2 + F_3)] γ^0×(1 - γ^5)v_ν(k⃗_ν, + 1/2)} - λ[φ^†_pσ⃗ φ_n] ·{u̅_e(k⃗_e,σ_e) [2 ε_λ'· k_e(F_1 + F_3 + F_4 -m^2_e/k_e· k F_5 + F_6)+ ε̂^*_λ' k̂ (F_1 + 2F_2 - m^2_e/k_e· k F_5 + F_6) + 2k_e· k/m_e ε̂^*_λ' (F_2 - m^2_e/2 k_e· k F_5) + ε^*_λ'· k_e/m_e k̂ (- 2 F_2 + F_3)] γ⃗ ×(1 - γ^5)v_ν(k⃗_ν, + 1/2)]}}.Using the relations Eq.(<ref>), which are imposed by gauge invariance, we transcribe Eq.(<ref>) into the formM_ Fig. <ref>(n → p e^- ν̅_e γ)_λ' = 2 m_n e^2/8π^2 1/2k_e· k {[φ^†_pφ_n] {u̅_e(k⃗_e,σ_e) [ ε̂^*_λ' k̂ (2 F_2 - F_3 - F_4) + k_e· k/m_e ε̂^*_λ' (2 F_2 - F_3)+ ε^*_λ'· k_e/m_e k̂ (- 2 F_2 + F_3)] γ^0(1 - γ^5)v_ν(k⃗_ν, + 1/2)} - λ [φ^†_pσ⃗ φ_n] ·{u̅_e(k⃗_e,σ_e) [ ε̂^*_λ' k̂ (2 F_2 - F_3 - F_4)+ k_e· k/m_e ε̂^*_λ' (2 F_2 - F_3) + ε^*_λ'· k_e/m_e k̂ (- 2 F_2 + F_3)]γ⃗(1 - γ^5)v_ν(k⃗_ν, + 1/2)}}.The amplitude Eq.(<ref>) is gauge invariant. It vanishes after the replacement ε^*_λ'→ k for a photon on–mass shell k^2 = 0. The contribution of the diagrams in Fig. <ref> to the rate of the neutron radiative β^-–decay with photon energies from the interval ω_ min≤ω≤ω_ max is given byλ^( Fig <ref>)_βγ(ω_ max,ω_ min) = (1 + 3λ^2) α^2/π^2 G^2_FV_ud|^2/32π^3∫^ω_ max_ω_ min dω∫^E_0 - ω_m_e dE_e F(E_e, Z = 1) (E_0 - E_e - ω) √(E^2_e - m^2_e) ω × ∫dΩ_eγ/4π∫dΩ_ν/4π [1/2∑_ pol, λ'1/1 + 3λ^2( M^†_ Fig. <ref>(n → p e^- ν̅_e γ)_λ'M̃_ Fig. <ref>(n → p e^- ν̅_e γ)_λ' +h.c.)]|_E_ν = E_0 - E_e - ω,where the abbreviation h.c. means “hermitian conjugate”. Then, M̃_ Fig. <ref> = (8π^2/2m_n e^2)M_ Fig. <ref> and dΩ_eγ and dΩ_ν are infinitesimal solid angle elements of the electron–photon momentum correlations and antineutrino, respectively. The sum over polarizations of interacting particles is defined by the following traces over Dirac matrices∑_ pol, λ'1/1 + 3λ^2( M^†_ Fig. <ref>(n → p e^- ν̅_e γ)_λ'M̃_ Fig. <ref>(n → p e^- ν̅_e γ)_λ' +h. c.)= 1/1 + 3λ^2 2/(2k_e· k)^2 × ∑_λ'{ tr{(m_e + k̂_e) [ ε̂^*_λ' k̂ (2 F_2 - F_3 - F_4) + k_e· k/m_e ε̂^*_λ' (2 F_2 - F_3) + ε^*_λ'· k_e/m_e k̂ (- 2 F_2 + F_3)] γ^0(1 - γ^5) × k̂_νγ^0 (2 k_e·ε_λ' + k̂ε̂_λ') (1 - γ^5)} + λ δ^ijtr{(m_e + k̂_e) [ ε̂^*_λ' k̂ (2 F_2 - F_3 - F_4) + k_e· k/m_e ε̂^*_λ' (2 F_2 - F_3)+ ε^*_λ'· k_e/m_e k̂ (- 2 F_2 + F_3)] γ⃗^ i (1 - γ^5) k̂_νγ⃗^ j(2 k_e·ε_λ' + k̂ε̂_λ') (1 - γ^5)} +h.c.}.Having integrated over directions of the antineutrino momentum k⃗_ν we get∫dΩ_ν/4π ∑_ pol, λ'1/1 + 3λ^2( M^†_ Fig. <ref>(n → p e^- ν̅_e γ)_λ'M̃_ Fig. <ref>(n → p e^- ν̅_e γ)_λ' +h. c.)= E_ν/(k_e· k)^2 × ∑_λ'{ tr{(m_e + k̂_e) [ ε̂^*_λ' k̂ (2 F_2 - F_3 - F_4) + k_e· k/m_e ε̂^*_λ' (2 F_2 - F_3) + ε^*_λ'· k_e/m_e k̂ (- 2 F_2 + F_3)] × γ^0 (2 k_e·ε_λ' + k̂ε̂_λ') (1 - γ^5)} +h.c.}.The traces over the Dirac matrices are equal to ∑_λ' tr{(m_e + k̂_e) ε̂^*_λ' k̂ γ^0 (2 k_e·ε_λ' + k̂ε̂_λ') (1 - γ^5)} =8 ω (k^2_e - (k⃗_e·n⃗_k⃗)^2) + 16 ω^2 (E_e - k⃗_e·n⃗_k⃗), ∑_λ' tr{(m_e + k̂_e) ε̂^*_λ' γ^0 (2 k_e·ε_λ' + k̂ε̂_λ') (1 - γ^5)} =- 8 m_eω, ∑_λ'(k_e·ε^*_λ')tr{(m_e + k̂_e) k̂ γ^0 (2 k_e·ε_λ' + k̂ε̂_λ') (1 - γ^5)} = 8 m_eω (k^2_e - (k⃗_e·n⃗_k⃗)^2).Thus, we get∫dΩ_ν/4π 1/2 ∑_ pol, λ'1/1 + 3λ^2( M^†_ Fig. <ref>(n → p e^- ν̅_e γ)_λ'M̃_ Fig. <ref>(n → p e^- ν̅_e γ)_λ' +h. c.) = E_ν/(k_e· k)^2 × {8 ωRe F_4 (k^2_e - (k⃗_e·n⃗_k⃗)^2) + 8 ω^2Re (2 F_2 - F_3 - 2F_4) (E_e - k⃗_e ·n⃗_k⃗)},where ReF_4 and Re(2 F_2 - F_3 - 2F_4) are the real parts of the functions F_2, F_3 and F_4, i.e. Re F_j = (F_j + F^*_j)/2 for j = 2,3,4. Plugging Eq.(<ref>) into Eq.(<ref>) we obtainλ^( Fig <ref>)_βγ(ω_ max,ω_ min)=(1 + 3λ^2) α^2/π^2 G^2_FV_ud|^2/4π^3∫^ω_ max_ω_ min dω∫^E_0 - ω_m_e dE_e F(E_e, Z = 1) (E_0 - E_e - ω)^2 √(E^2_e - m^2_e) × ∫dΩ_eγ/4π {k^2_e - (k⃗_e ·n⃗_k⃗)^2/(E_e - k⃗_e·n⃗_k⃗)^2Re F_4 + ω/E_e - k⃗_e·n⃗_k⃗Re (2 F_2 - F_3 - 2F_4)},where F_2, F_3 and F_4 are given in Eq.(<ref>) as functions of k_e· k = ω (E_e - k⃗_e·n⃗_k⃗). It is important to emphasize that the contribution of the diagrams in Fig. <ref> to the rate of the neutron radiative β^-–decay is not infrared divergent. § APPENDIX C: THE AMPLITUDE OF THE NEUTRONRADIATIVE Β^-–DECAY, DESCRIBED BY FEYNMAN DIAGRAMS IN FIG. <REF> Since, according to our calculations in Appendix B, the renormalized self–energy corrections Σ^(p)(k_p) and Σ^(e)(k_e) of the proton and electron vanish, the contributions of the Feynman diagrams Fig. <ref>a and Fig. <ref>b to the amplitude of the neutron radiative β^-–decay vanish. At first glimpse, non–trivial contributions of the diagrams in Fig. <ref> are given by the diagrams in Fig. <ref>c and Fig. <ref>d. The analytical expression for the diagrams Fig. <ref>c and Fig. <ref>d areM_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' = =e^2∫d^4q/(2π)^4i [u̅_p(k⃗_p,σ_p) γ^α1/m_p - k̂_p + q̂ -i0 J^μ(k_p, k_p + k)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ^β1/m_e - k̂_e - k̂- q̂ - i0γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] D_αβ(q)andM_ Fig. <ref>d(n → p e^- ν̅_eγ)_λ' = = - e^2∫d^4q/(2π)^4i [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0γ^α1/m_p - k̂_p + q̂ - k̂ -i0 J^μ(k_p, k_p + k)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ^β1/m_e - k̂_e - q̂ - i0γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] D_αβ(q),where J^μ(k_p, k_p + k) is a hadronic current, described by the shaded region of the diagrams. In case of strong interactions, defined by only the axial coupling constant λ, the hadronic current J^μ(k_p, k_p + k) is equal to J^μ(k_p, k_p + k) = γ^μ(1 + λ γ^5). Let us check invariance of the Feynman diagrams Fig. <ref>c and Fig. <ref>d under gauge transformations ε^*_λ'(k) →ε^*_λ'(k) + ck and D_αβ(q) → D_αβ(q) + c(q^2) q_αq_β. Making, first, a gauge transformation ε^*_λ'(k) →ε^*_λ'(k) + ck for the contributions of the term ck we get the following expressionsM_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ'|_ε^*_λ'(k)→ k = = - e^2∫d^4q/(2π)^4i [u̅_p(k⃗_p,σ_p) γ^α1/m_p - k̂_p + q̂ -i0 J^μ(k_p, k_p + k)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ^β1/m_e - k̂_e - k̂- q̂ - i0γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] D_αβ(q)and M_ Fig. <ref>d(n → p e^- ν̅_eγ)_λ'|_ε^*_λ'→ k = = + e^2∫d^4q/(2π)^4i [u̅_p(k⃗_p,σ_p) γ^α1/m_p - k̂_p + q̂ - k̂ -i0 J^μ(k_p, k_p + k)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ^β1/m_e - k̂_e - q̂ - i0γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] D_αβ(q),where we have used the Dirac equations for the free proton and electron. One may see that the sum of the diagrams Fig. <ref>c and Fig. <ref>d is not invariant under a gauge transformation ε^*_λ'(k) →ε^*_λ'(k) + ck. It is obvious that the sum of the diagrams Fig. <ref>c and Fig. <ref>d is not also invariant under a gauge transformation D_αβ(q) → D_αβ(q) + c(q^2) q_α q_β.Now we may proceed to the calculation of the Feynman diagrams Fig. <ref>c and Fig. <ref>d. For this aim we replace J^μ(k_p, k_p + k) by J^μ(k_p, k_p + k) →γ^μ(1 + λ γ^5). Then, merging denominators and skipping standard intermediate calculations we transcribe the r.h.s. of Eq.(<ref>) and Eq.(<ref>) into the formM_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' = e^2 ∫^1_0dx∫^1_0 dy 2 y ∫d^4q/(2π)^4i [u̅_p(k⃗_p,σ_p) γ^α(m_p + k̂_p - k̂_p(x) y)γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_α(m_e + k̂_e + k̂ + k̂_p(x)y) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × (1/[q^2 - k^2_p(x)y^2 + 2(k_e· k)(1 - x)y - μ^2 (1 - y) + i 0]^3 - 1/[q^2 - k^2_p(x)y^2 + 2(k_e· k)(1 - x)y - Λ^2 (1 - y) + i 0]^3)- e^2 1/4 [u̅_p(k⃗_p,σ_p) γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)][u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] ×∫^1_0dx∫^1_0 dy 2 y ∫d^4q/(2π)^4i (q^2/[q^2 - k^2_p(x)y^2 + 2(k_e· k)(1 - x)y - μ^2 (1 - y) + i 0]^3 - q^2/[q^2 - k^2_p(x)y^2 + 2(k_e· k)(1 - x)y - Λ^2 (1 - y) + i 0]^3),andM_ Fig. <ref>d(n → p e^- ν̅_eγ)_λ' = = - e^2∫^1_0dx∫^1_0 dy 2 y ∫d^4q/(2π)^4i [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0γ^α(m_p + k̂_p + k̂+ k̂_e(x) y) γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_α(m_e + k̂_e - k̂_e(x)y) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × (1/[q^2 - k^2_e(x)y^2 + 2(k_p· k)(1 - x) y - μ^2 (1 - y) + i 0]^3 - 1/[q^2 - k^2_e(x)y^2 + 2(k_p· k)(1 - x) y - Λ^2 (1 - y) + i 0]^3)+ e^2 1/4 [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_e - k̂_p - k̂ - i 0 γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)][u̅_e(k⃗_e,σ_e) γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] ×∫^1_0dx∫^1_0 dy 2 y ∫d^4q/(2π)^4i (q^2/[q^2 - k^2_e(x)y^2 + 2(k_p· k)(1 - x)y - μ^2 (1 - y) + i 0]^3- q^2/[q^2 - k^2_e(x)y^2 + 2(k_p· k)(1 - x)y - Λ^2 (1 - y) + i 0]^3),where we have denoted k_p(x) = k_p x - (k_e + k) (1-x) and k_e(x) = k_e x - (k_p + k) (1 - x), respectively. Making the Wick rotation and integrating over q^2 we arrive at the expressionsM_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' = - e^2/32π^2∫^1_0dx∫^1_0 dy 2 y[u̅_p(k⃗_p,σ_p) γ^α(m_p + k̂_p - k̂_p(x) y)γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_α(m_e + k̂_e + k̂ + k̂_p(x)y) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × (1/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y) - 1/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + Λ^2 (1 - y))- e^2/64π^2 [u̅_p(k⃗_p,σ_p) γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)][u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] ×∫^1_0dx∫^1_0 dy 2 y ℓ n(k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + Λ^2 (1 - y)/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y))andM_ Fig. <ref>d(n → p e^- ν̅_eγ)_λ' = = + e^2∫^1_0dx∫^1_0 dy 2 y [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0γ^α(m_p + k̂_p + k̂+ k̂_e(x) y) γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_α(m_e + k̂_e - k̂_e(x)y) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × (1/k^2_e(x)y^2 - 2(k_p· k)(1 - x) y + μ^2 (1 - y) - 1/k^2_e(x)y^2 - 2(k_p· k)(1 - x) y + Λ^2 (1 - y))+ e^2/64π^2 [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_e - k̂_p - k̂ - i 0 γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)][u̅_e(k⃗_e,σ_e) γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] ×∫^1_0dx∫^1_0 dy 2 y ℓ n(k^2_e(x)y^2 - 2(k_p· k)(1 - x)y + Λ^2 (1 - y)/k^2_e(x)y^2 - 2(k_p· k)(1 - x)y + μ^2 (1 - y)).For Λ≫ m_p we getM_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' = - e^2/32π^2∫^1_0dx∫^1_0 dy 2 y[u̅_p(k⃗_p,σ_p) γ^α(m_p + k̂_p - k̂_p(x) y)γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_α(m_e + k̂_e + k̂ + k̂_p(x)y) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × 1/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y)- e^2/64π^2 [u̅_p(k⃗_p,σ_p) γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)][u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] ×∫^1_0dx∫^1_0 dy 2 y ℓ n(Λ^2 (1 - y)/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y))andM_ Fig. <ref>d(n → p e^- ν̅_eγ)_λ' = = + e^2/32π^2∫^1_0dx∫^1_0 dy 2 y [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0γ^α(m_p + k̂_p + k̂+ k̂_e(x) y) γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_α(m_e + k̂_e - k̂_e(x)y) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × 1/k^2_e(x)y^2 - 2(k_p· k)(1 - x) y + μ^2 (1 - y)+ e^2/64π^2 [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_e - k̂_p - k̂ - i 0 γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)][u̅_e(k⃗_e,σ_e) γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] ×∫^1_0dx∫^1_0 dy 2 y ℓ n(Λ^2 (1 - y)/k^2_e(x)y^2 - 2(k_p· k)(1 - x)y + μ^2 (1 - y)).First, in the last terms of Eq.(<ref>) and Eq.(<ref>) we integrate over x and y. Keeping only the leading terms in the large proton mass expansion we getM_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' = - e^2/32π^2∫^1_0dx∫^1_0 dy 2 y[u̅_p(k⃗_p,σ_p) γ^α(m_p + k̂_p - k̂_p(x) y)γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_α(m_e + k̂_e + k̂ + k̂_p(x)y) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × 1/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y) - e^2/32π^2 (ℓ n(Λ/m_p) + 3/4) [u̅_p(k⃗_p,σ_p) γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )]andM_ Fig. <ref>d(n → p e^- ν̅_eγ)_λ' = = + e^2/32π^2∫^1_0dx∫^1_0 dy 2 y [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0γ^α(m_p + k̂_p + k̂+ k̂_e(x) y) γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_α(m_e + k̂_e - k̂_e(x)y) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] 1/k^2_e(x)y^2 - 2(k_p· k)(1 - x) y + μ^2 (1 - y)+ e^2/32π^2 (ℓ n(Λ/m_p) + 3/4) [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_e - k̂_p - k̂ - i 0 γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )].One may see that the terms dependent on the ultra–violet cut–off Λ are invariant under a gauge transformation ε^*_λ'(k) →ε^*_λ'(k) + c k. For the integration over x and y in the first terms of Eq.(<ref>) and Eq.(<ref>) we transcribe them as followsM_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' = - e^2/32π^2 [u̅_p(k⃗_p,σ_p) γ^α(m_p + k̂_p)γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_α(m_e + k̂_e + k̂) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] ×∫^1_0dx∫^1_0 dy 2 y1/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y) + e^2/32π^2∫^1_0dx∫^1_0 dy 2 y^2[u̅_p(k⃗_p,σ_p) γ^αk̂_p(x) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_α(m_e + k̂_e + k̂) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × 1/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y) - e^2/32π^2∫^1_0dx∫^1_0 dy 2 y^2[u̅_p(k⃗_p,σ_p) γ^α(m_p + k̂_p)γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_αk̂_p(x) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × 1/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y) + e^2/32π^2∫^1_0dx∫^1_0 dy 2 y^3[u̅_p(k⃗_p,σ_p) γ^αk̂_p(x) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_αk̂_p(x) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × 1/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y)- e^2/32π^2 (ℓ n(Λ/m_p) + 3/4) [u̅_p(k⃗_p,σ_p) γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )]and M_ Fig. <ref>d(n → p e^- ν̅_eγ)_λ' = + e^2/32π^2[u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0γ^α(m_p + k̂_p + k̂) γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_α(m_e + k̂_e) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] ∫^1_0dx∫^1_0 dy 2 y 1/k^2_e(x)y^2 - 2(k_p· k)(1 - x) y + μ^2 (1 - y) +e^2/32π^2∫^1_0dx∫^1_0 dy 2 y^2 [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0γ^αk̂_e(x) γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_α(m_e + k̂_e) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] 1/k^2_e(x)y^2 - 2(k_p· k)(1 - x) y + μ^2 (1 - y) -e^2/32π^2∫^1_0dx∫^1_0 dy 2 y^2 [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0γ^α(m_p + k̂_p + k̂) γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_αk̂_e(x) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] 1/k^2_e(x)y^2 - 2(k_p· k)(1 - x) y + μ^2 (1 - y) -e^2/32π^2∫^1_0dx∫^1_0 dy 2 y^3 [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0γ^αk̂_e(x) γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_αk̂_e(x) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] 1/k^2_e(x)y^2 - 2(k_p· k)(1 - x) y + μ^2 (1 - y)+ e^2/32π^2 (ℓ n(Λ/m_p) + 3/4) [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_e - k̂_p - k̂ - i 0 γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )].Using the Dirac equations for the free proton and electron we getM_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' = - e^2/16π^2 [u̅_p(k⃗_p,σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 k̂_p(m_e + k̂_e + k̂) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] ×∫^1_0dx∫^1_0 dy 2 y1/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y) + e^2/32π^2∫^1_0dx∫^1_0 dy 2 y^2[u̅_p(k⃗_p,σ_p) γ^αk̂_p(x) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_α(m_e + k̂_e + k̂) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × 1/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y) - e^2/16π^2∫^1_0dx∫^1_0 dy 2 y^2[u̅_p(k⃗_p,σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 k̂_pk̂_p(x) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × 1/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y) + e^2/32π^2∫^1_0dx∫^1_0 dy 2 y^3[u̅_p(k⃗_p,σ_p) γ^αk̂_p(x) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_αk̂_p(x) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × 1/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y)- e^2/32π^2 (ℓ n(Λ/m_p) + 3/4) [u̅_p(k⃗_p,σ_p) γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )]and M_ Fig. <ref>d(n → p e^- ν̅_eγ)_λ' = + e^2/16π^2[u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0 k̂_e (m_p + k̂_p + k̂) γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] ∫^1_0dx∫^1_0 dy 2 y 1/k^2_e(x)y^2 - 2(k_p· k)(1 - x) y + μ^2 (1 - y) +e^2/16π^2∫^1_0dx∫^1_0 dy 2 y^2 [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0 k̂_e k̂_e(x) γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] 1/k^2_e(x)y^2 - 2(k_p· k)(1 - x) y + μ^2 (1 - y) -e^2/32π^2∫^1_0dx∫^1_0 dy 2 y^2 [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0γ^α(m_p + k̂_p + k̂) γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_αk̂_e(x) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] 1/k^2_e(x)y^2 - 2(k_p· k)(1 - x) y + μ^2 (1 - y) -e^2/32π^2∫^1_0dx∫^1_0 dy 2 y^3 [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0γ^αk̂_e(x) γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_αk̂_e(x) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] 1/k^2_e(x)y^2 - 2(k_p· k)(1 - x) y + μ^2 (1 - y)+ e^2/32π^2 (ℓ n(Λ/m_p) + 3/4) [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_e - k̂_p - k̂ - i 0 γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )].It is convenient to rewrite Eq.(<ref>) and Eq.(<ref>) as followsM_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' = - e^2/16π^2 [u̅_p(k⃗_p,σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)]{[u̅_e(k⃗_e,σ_e) ε̂^*_λ' k̂_p γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )]+ 2 k_p· (k_e + k) [u̅_e(k⃗_e,σ_e) ε̂^*_λ'1/m_e - k̂_e - k̂ - i 0 γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )]} ×∫^1_0dx∫^1_0 dy 2 y1/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y) + e^2/32π^2∫^1_0dx∫^1_0 dy 2 y^2[u̅_p(k⃗_p,σ_p) γ^αk̂_p(x) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] {[u̅_e(k⃗_e,σ_e) ε̂^*_λ' γ_α γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )]+ 2(k_e + k)_α[u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )]} × 1/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y) - e^2/16π^2∫^1_0dx∫^1_0 dy 2 y^2[u̅_p(k⃗_p,σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 k̂_pk̂_p(x) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × 1/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y) + e^2/32π^2∫^1_0dx∫^1_0 dy 2 y^3[u̅_p(k⃗_p,σ_p) γ^αk̂_p(x) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_αk̂_p(x) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × 1/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y)- e^2/32π^2 (ℓ n(Λ/m_p) + 3/4) [u̅_p(k⃗_p,σ_p) γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )]and M_ Fig. <ref>d(n → p e^- ν̅_eγ)_λ' = + e^2/16π^2{[u̅_p(k⃗_p,σ_p) ε̂^*_λ' k̂_e γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)]+ 2k_e· (k_p + k) [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0 γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)]} × [u̅_e(k⃗_e,σ_e) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] ∫^1_0dx∫^1_0 dy 2 y 1/k^2_e(x)y^2 - 2(k_p· k)(1 - x) y + μ^2 (1 - y) +e^2/16π^2∫^1_0dx∫^1_0 dy 2 y^2 [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0 k̂_e k̂_e(x) γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] 1/k^2_e(x)y^2 - 2(k_p· k)(1 - x) y + μ^2 (1 - y) -e^2/32π^2∫^1_0dx∫^1_0 dy 2 y^2 {[u̅_p(k⃗_p,σ_p) ε̂^*_λ' γ^α γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)]+ 2(k_p + k)_α[u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0 γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)]} × [u̅_e(k⃗_e,σ_e) γ_αk̂_e(x) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] 1/k^2_e(x)y^2 - 2(k_p· k)(1 - x) y + μ^2 (1 - y) -e^2/32π^2∫^1_0dx∫^1_0 dy 2 y^3 [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_p - k̂_p - k̂ - i0γ^αk̂_e(x) γ^μ(1 + λ γ^5)u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_αk̂_e(x) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] 1/k^2_e(x)y^2 - 2(k_p· k)(1 - x) y + μ^2 (1 - y)+ e^2/32π^2 (ℓ n(Λ/m_p) + 3/4) [u̅_p(k⃗_p,σ_p) ε̂^*_λ' 1/m_e - k̂_p - k̂ - i 0 γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )].The calculation of the integrals over x and y to leading order in the large proton mass expansion runs as follows∫^1_0dx∫^1_0 dy 2 yk^α_p/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y) = 2k^α_p∫^1_0dx/k^2_p(x) ℓ n(- k^2_p(x)/2k_e· k)== k^α_p/m_p 1/|k⃗_e + k⃗ |{π^2/3 - ℓ n(2|k⃗_e + k⃗ |^2/- k_e· k) ℓ n(E_e + ω + |k⃗_e + k⃗ |/E_e + ω - |k⃗_e + k⃗ |) - 1/2 ℓ n^2(E_e + ω + |k⃗_e + k⃗ |/E_e + ω - |k⃗_e + k⃗ |)-2Li_2(E_e + ω + |k⃗_e + k⃗ |/E_e + ω - |k⃗_e + k⃗ |)}, ∫^1_0dx∫^1_0 dy 2 y^2k̂_p/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y) = 2∫^1_0dx/k^2_p(x)k̂_p = 1/m_p {k̂_p/m_p [ℓ n(m^2_p/(k_e + k)^2)- E_e + ω/|k⃗_e + k⃗ | ℓ n(E_e + ω + |k⃗_e + k⃗ |/E_e + ω - |k⃗_e + k⃗ |)] + k̂_e + k̂/|k⃗_e + k⃗ | ℓ n(E_e + ω + |k⃗_e + k⃗ |/E_e + ω - |k⃗_e + k⃗ |)},∫^1_0dx∫^1_0 dy 2 y^2k̂_pk̂_p(x)/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y) = 2∫^1_0 dx/k^2_p(x) k̂_pk̂_p(x) = [ℓ n(m^2_p/(k_e + k)^2)- E_e + ω/|k⃗_e + k⃗ | ℓ n(E_e + ω + |k⃗_e + k⃗ |/E_e + ω - |k⃗_e + k⃗ |)] + k̂_p(k̂_e + k̂)/m_p |k⃗_e + k⃗ | ℓ n(E_e + ω + |k⃗_e + k⃗ |/E_e + ω - |k⃗_e + k⃗ |), ∫^1_0dx∫^1_0 dy 2 y^3k^α_p(x) k^β_p(x)/k^2_p(x)y^2 - 2(k_e· k)(1 - x)y + μ^2 (1 - y) = ∫^1_0dx/k^2_p(x) k^α_p(x) k^β_p(x) = η^0αη^0β.One may see that in Eq.(<ref>) the second integral from above, calculated to leading order in the large proton mass expansion, does not contribute to the amplitude Eq.(<ref>). Then, to leading order in the large proton mass expansion the contribution of the diagram Fig. <ref>d is proportional to ε^0*_λ' and vanishes in the physical gauge ε^*_λ' = (0, ε⃗^ *_λ') (see Appendix A and <cit.>). Thus, below we may discuss the diagram Fig. <ref>c only.For the extraction of a physical contribution of the diagram Fig. <ref>c we have to investigate the property of this diagram with respect to the gauge transformation D_αβ(q) → D_αβ(q) + c(q^2) q_αq_β. For this aim we rewrite Eq.(<ref>) as follows <cit.>M_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' = -e^2∫d^4q/(2π)^4i D_αβ(q) [u̅_p(k⃗_p,σ_p) (2 k^α_p - q^α + i σ^αρq_ρ) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)]/q^2 - 2 k_p· q + i0 × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ^β1/m_e - k̂_e - k̂- q̂ - i0 γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )],where σ^αρ = i/2(γ^αγ^ρ - γ^ργ^α) are the Dirac matrices <cit.> and the amplitudeM^(1)_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' = -e^2∫d^4q/(2π)^4i D_αβ(q) [u̅_p(k⃗_p,σ_p)i σ^αρq_ρ γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)]/q^2 - 2 k_p· q + i0 × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ^β1/m_e - k̂_e - k̂- q̂ - i0γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )]is invariant under a gauge transformation D_αβ(q) → D_αβ(q) + c(q^2) q_αq_β.Now we consider the expressionM_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' -M^(1)_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ'= = -e^2∫d^4q/(2π)^4i D_αβ(q) [u̅_p(k⃗_p,σ_p) (2 k^α_p - q^α) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)]/q^2 - 2 k_p· q + i0 × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ^β1/m_e - k̂_e - k̂- q̂ - i0 γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )],which we transcribe into the formM_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' -M^(1)_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ'= =e^2∫d^4q/(2π)^4i D_αβ(q) [u̅_p(k⃗_p,σ_p) (2 k^α_p - q^α) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)]/q^2 - 2 k_p· q + i0 × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 (m_e - k̂_e - k̂) γ^β + 2 (k_e + k)^β + q^β - i σ^βφq_φ/q^2 + 2 (k_e + k)· q + 2 k_e · k + i0 γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )].One may see that the amplitudeM^(2)_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' = -e^2∫d^4q/(2π)^4i D_αβ(q) [u̅_p(k⃗_p,σ_p) (2 k^α_p - q^α) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)]/q^2 - 2 k_p· q + i0 × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0i σ^βφq_φ/q^2 + 2 (k_e + k)· q + 2 k_e · k+ i0γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )]is also invariant under a gauge transformation D_αβ(q) → D_αβ(q) + c(q^2) q_αq_β. Now we discuss the following expressionM_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' -M^(1)_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' -M^(2)_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' = = -e^2∫d^4q/(2π)^4i D_αβ(q) [u̅_p(k⃗_p,σ_p) (2 k^α_p - q^α) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)]/q^2 - 2 k_p· q + i0 × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 (m_e - k̂_e - k̂) γ^β + 2 (k_e + k)^β + q^β/q^2 + 2(k_e + k)· q + 2 k_e · k + i0 γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )],which we transcribe into the formM_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' -M^(1)_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' -M^(2)_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' =[u̅_p(k⃗_p,σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' γ^β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )]( - e^2)∫d^4q/(2π)^4i D_αβ(q) (2 k^α_p - q^α)/q^2 - 2 k_p· q + i0 1/q^2 + 2(k_e + k)· q + 2 k_e · k + i0 + [u̅_p(k⃗_p,σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × (-e^2) ∫d^4q/(2π)^4i D_αβ(q) (2 k^α_p - q^α)/q^2 - 2 k_p· q + i0 2 (k_e + k)^β + q^β/q^2 + 2(k_e + k)· q + 2 k_e · k + i0.Then, we propose to rewrite Eq.(<ref>) as followsM_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' - ∑^3_j =1 M^(j)_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' = [u̅_p(k⃗_p,σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] e^2 ∫d^4q/(2π)^4i D_αβ(q) (2 k^α_p - q^α)(2 k^β_p - q^β)/(q^2 - 2 k_p· q + i0)^2- [u̅_p(k⃗_p,σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] [u̅_e(k⃗_e,σ_e) ε̂^*_λ' γ^β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × e^2 ∫d^4q/(2π)^4i D_αβ(q) (2 k^α_p - q^α)/q^2 - 2 k_p· q + i0 1/q^2 + 2(k_e + k)· q + 2 k_e · k + i0+ [u̅_p(k⃗_p,σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × e^2 ∫d^4q/(2π)^4i D_αβ(q) q^β/q^2 (2 k^α_p - q^α)/q^2 - 2 k_p· q + i0 2 k_e· k/q^2 + 2(k_e + k)· q + 2 k_e · k + i0,where M^(3)_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' is the amplitude, invariant under a gauge transformation D_αβ(q) → D_αβ(q) + c(q^2) q_αq_β, defined byM^(3)_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' =- [u̅_p(k⃗_p,σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)] × [u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] × e^2 [∫d^4q/(2π)^4i D_αβ(q) (2 k^α_p - q^α)/q^2 - 2 k_p· q + i02 (k_e + k)^β + q^β + q^β/q^2 2k_e· k/q^2 + 2(k_e + k)· q + 2 k_e · k + i0 + ∫d^4q/(2π)^4i D_αβ(q) (2 k^α_p - q^α)(2 k^β_p - q^β)/(q^2 - 2 k_p· q + i0)^2].The r.h.s. of Eq.(<ref>) is not invariant under a gauge transformation D_αβ(q) → D_αβ(q) + c(q^2) q_αq_β. It is important to emphasize that unlike a gauge non–invariant part of the Feynman diagram Fig. <ref>a, which is independent of the electron energy, a gauge non–invariant part of the diagram Fig. <ref>c, given by the r.h.s. of Eq.(<ref>), has a constant part and a part dependent on the electron and photon energies and momenta. The results of the calculation are given by e^2 ∫d^4q/(2π)^4i D_αβ(q) (2 k^α_p - q^α)(2 k^β_p - q^β)/(q^2 - 2 k_p· q + i0)^2 = e^2/8π^2 (ξ ℓ n(Λ/m_p) + (3 - ξ) ℓ n(μ/m_p) + 3/2).For the second integral we obtain the following expressione^2 ∫d^4q/(2π)^4i D_αβ(q) (2 k^α_p - q^α)/q^2 - 2 k_p· q + i0 1/q^2 + 2(k_e + k)· q + 2k_e· k + i0 = - e^2/8π^2 k_pβ∫^1_0dx/k^2_p(x) ℓ n(- k^2_p(x)/2 k_e· k)+ e^2/16π^2 ∫^1_0dx k_pβ(x)/k^2_p(x) + e^2/16π^2 (1 - ξ) (k_e + k)_β/(k_e + k)^2[1 + m^2_e/(k_e + k)^2 ℓ n(- 2 k_e · k/m^2_e)],where k_p(x) = k_p x - (k_e + k)(1 - x). The third integral is equal toe^2 ∫d^4q/(2π)^4i D_αβ(q) q^β/q^2 (2 k^α_p - q^α)/q^2 - 2 k_p· q + i0 2 k_e· k/q^2 + 2(k_e + k)· q + 2 k_e · k + i0 = = - e^2/8π^2 ξ [ℓ n(m_e/μ) + (1 - k_e· k/(k_e + k)^2) ℓ n(- 2k_e· k/m^2_e)].Thus, a part of the amplitude M_ Fig. <ref>c(n → p e^- ν̅_e γ)_λ', invariant under a gauge transformation D_αβ(q) → D_αβ(q) + c(q^2) q_αq_β, is given byM_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' == - e^2/16π^2 [u̅_p(k⃗_p,σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)][u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] ×f_1(E_e,k⃗_e,ω, k⃗ ) - e^2/16π^2 [u̅_p(k⃗_p,σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n,σ_n)][u̅_e(k⃗_e,σ_e) ε̂^*_λ' 1/m_e - k̂_e - k̂ - i 0 γ^0 (k̂_e + k̂) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] ×f_2(E_e,k⃗_e,ω, k⃗ ),where the functions f_1(E_e,k⃗_e,ω, k⃗ ) and f_2(E_e,k⃗_e,ω, k⃗ ) are defined byf_1(E_e,k⃗_e,ω, k⃗ )= ℓ n(m^2_p/m^2_e + 2 k_e· k) - 2 (1 - k_e· k/(k_e + k)^2) ℓ n(- 2k_e· k/m^2_e) - E_e + ω/|k⃗_e + k⃗ | ℓ n(E_e + ω + |k⃗_e + k⃗ |/E_e + ω - |k⃗_e + k⃗ |), f_2(E_e,k⃗_e,ω, k⃗ )= 1/|k⃗_e + k⃗ | ℓ n(E_e + ω + |k⃗_e + k⃗ |/E_e + ω - |k⃗_e + k⃗ |).Independent of electron and photon energies and momenta contributions of M_ Fig. <ref>c(n → p e^- ν̅_e γ)_λ' are removed by renormalization of the Fermi weak G_F and axial λ coupling constant <cit.>.In the non–relativistic limit for the proton the amplitude Eq.(<ref>) is equal to1/2m_nM_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' = = e^2/16π^2 [φ^†_pφ_n]{[u̅_e(k⃗_e,σ_e) (2 k_e ·ε^*_λ' + ε̂^*_λ'k̂)γ^0 (1 - γ^5) v_ν(k⃗_ν, + 1/2 )] 1/2k_e · kf_1(E_e,k⃗_e,ω, k⃗ ) + [u̅_e(k⃗_e,σ_e) (2 k_e ·ε^*_λ' + ε̂^*_λ'k̂) γ^0 (k̂_e + k̂) γ^0 (1 - γ^5) v_ν(k⃗_ν, + 1/2 )] 1/2k_e· kf_2(E_e,k⃗_e,ω, k⃗ )} -e^2/16π^2 λ [φ^†_pφ_n]·{[u̅_e(k⃗_e,σ_e) (2 k_e ·ε^*_λ' + ε̂^*_λ'k̂)γ⃗ (1 - γ^5) v_ν(k⃗_ν, + 1/2 )] 1/2 k_e · kf_1(E_e,k⃗_e,ω, k⃗ ) + [u̅_e(k⃗_e,σ_e) (2 k_e ·ε^*_λ' + ε̂^*_λ'k̂) γ^0 (k̂_e + k̂) γ⃗(1 - γ^5) v_ν(k⃗_ν, + 1/2 )] 1/2k_e· kf_2(E_e,k⃗_e,ω, k⃗ )}.The hermitian conjugate amplitude is 1/2m_nM^†_ Fig. <ref>c(n → p e^- ν̅_eγ)_λ' == e^2/16π^2 [φ^†_nφ_p]{[v̅_ν(k⃗_ν,+ 1/2) γ^0 (2 k_e ·ε_λ' + k̂ε̂_λ')(1 - γ^5)u_e(k⃗_e, σ_e)] 1/2k_e · kf^*_1(E_e,k⃗_e,ω, k⃗ ) + [v̅_ν(k⃗_ν,+ 1/2) γ^0 (k̂_e + k̂) γ^0 (2 k_e ·ε_λ' + k̂ ε̂_λ') (1 - γ^5) u_e(k⃗_e, σ_e)] 1/2k_e· kf^*_2(E_e,k⃗_e,ω, k⃗ )} -e^2/16π^2 λ [φ^†_pφ_n]·{[v̅_ν(k⃗_ν,+ 1/2) γ⃗ (2 k_e ·ε_λ' + k̂ ε̂_λ') (1 - γ^5)u_e(k⃗_e, σ_e)] 1/2 k_e · kf^*_1(E_e,k⃗_e,ω, k⃗ )+ [v̅_ν(k⃗_ν,+ 1/2) γ⃗ (k̂_e + k̂) γ^0 (2 k_e·ε_λ' + k̂ ε̂_λ')(1 - γ^5) u_e(k⃗_e, σ_e) ] 1/2k_e· kf^*_2(E_e,k⃗_e,ω, k⃗ )}.The contribution of the diagrams in Fig. <ref> to the rate of the neutron radiative β^-–decay is defined byλ^( Fig. <ref>)_βγ(ω_ max,ω_ min) = (1 + 3λ^2) α^2/π^2 G^2_FV_ud|^2/64π^3∫^ω_ max_ω_ min dω∫^E_0 - ω_m_e dE_e F(E_e, Z = 1) (E_0 - E_e - ω) √(E^2_e - m^2_e) ω × ∫dΩ_eγ/4π∫dΩ_ν/4π [1/2∑_ pol, λ'1/1 + 3λ^2( M^†_ Fig. <ref>(n → p e^- ν̅_e γ)_λ'M̃_ Fig. <ref>(n → p e^- ν̅_e γ)_λ' +h.c.)]|_E_ν = E_0 - E_e - ω,where M̃_ Fig. <ref> = (16π^2/2m_n e^2)M_ Fig. <ref>c.The sum over polarizations of the proton, electron and photon, averaged over polarizations of the neutron, is defined by the following traces over Dirac matrices1/2∑_ pol, λ'1/1 + 3λ^2( M^†_ Fig. <ref>(n → p e^- ν̅_e γ)_λ'M̃_ Fig. <ref>(n → p e^- ν̅_e γ)_λ' +h. c.) = 1/1 + 3λ^2 1/2 (k_e· k)^2 × {f_1tr{(m_e + k̂_e) (2 k_e ·ε^*_λ' + ε̂^*_λ' k̂) γ^0k̂_νγ^0(2 k_e ·ε_λ' + k̂ ε̂_λ')(1 - γ^5)}+ λ^2 δ^ijf_1 tr{(m_e + k̂_e)(2 k_e ·ε^*_λ' + ε̂^*_λ' k̂) γ⃗^ i k̂_ν γ⃗^ j(2 k_e ·ε_λ' + k̂ ε̂_λ')(1 - γ^5)}} + 1/1 + 3λ^2 1/2 (k_e· k)^2 × {f_2tr{(m_e + k̂_e) (2 k_e ·ε^*_λ' + ε̂^*_λ' k̂) γ^0 (k̂_e + k̂ ) γ^0k̂_ν γ^0 (2 k_e ·ε_λ' + k̂ ε̂_λ')(1 - γ^5)}+ λ^2 δ^ijf_2 tr{(m_e + k̂_e)(2 k_e ·ε^*_λ' + ε̂^*_λ' k̂) γ^0 (k̂_e + k̂ ) γ⃗^ i k̂_ν γ⃗^ j(2 k_e ·ε_λ' + k̂ ε̂_λ')(1 - γ^5)}} +h. c. .Having integrated over directions of the antineutrino momentum k⃗_ν we arrive at the expression∫dΩ_ν/4π 1/2∑_ pol, λ'1/1 + 3λ^2( M^†_ Fig. <ref>(n → p e^- ν̅_e γ)_λ'M̃_ Fig. <ref>(n → p e^- ν̅_e γ)_λ' +h. c.) = E_ν/1 + 3λ^2 1/2 (k_e· k)^2 × {f_1tr{(m_e + k̂_e) (2 k_e ·ε^*_λ' + ε̂^*_λ' k̂) γ^0 (2 k_e ·ε_λ' + k̂ ε̂_λ')(1 - γ^5)}+ λ^2 δ^ijf_1 tr{(m_e + k̂_e)(2 k_e ·ε^*_λ' + ε̂^*_λ' k̂) γ⃗^ i γ^0 γ⃗^ j(2 k_e ·ε_λ' + k̂ ε̂_λ')(1 - γ^5)}} + E_ν/1 + 3λ^2 1/2 (k_e· k)^2 × {f_2tr{(m_e + k̂_e) (2 k_e ·ε^*_λ' + ε̂^*_λ' k̂) γ^0 (k̂_e + k̂ ) γ^0 (2 k_e ·ε_λ' + k̂ ε̂_λ')(1 - γ^5)}+ λ^2 δ^ijf_2 tr{(m_e + k̂_e)(2 k_e ·ε^*_λ' + ε̂^*_λ' k̂) γ^0 (k̂_e + k̂ ) γ⃗^ i γ^0γ⃗^ j(2 k_e ·ε_λ' + k̂ ε̂_λ')(1 - γ^5)}} +h. c. .Since δ^ijγ⃗^ i γ^0 γ⃗^ j = γ⃗· γ⃗ = 3 γ^0, we get∫dΩ_ν/4π 1/2∑_ pol, λ'1/1 + 3λ^2( M^†_ Fig. <ref>(n → p e^- ν̅_e γ)_λ'M̃_ Fig. <ref>(n → p e^- ν̅_e γ)_λ' +h. c.) = = E_ν/2 (k_e· k)^2 {f_1 ∑_λ' tr{(m_e + k̂_e) (2 k_e ·ε^*_λ' + ε̂^*_λ' k̂) γ^0 (2 k_e ·ε_λ' + k̂ ε̂_λ')(1 - γ^5)}+ f_2 ∑_λ' tr{(m_e + k̂_e) (2 k_e ·ε^*_λ' + ε̂^*_λ' k̂) γ^0 (k̂_e + k̂ ) γ^0 (2 k_e ·ε_λ' + k̂ ε̂_λ')(1 - γ^5)} +h. c.}.The traces are equal to∑_λ' tr{(m_e + k̂_e) (2 k_e ·ε^*_λ' + ε̂^*_λ' k̂) γ^0 (2 k_e ·ε_λ' + k̂ ε̂_λ')(1 - γ^5)} = 16(E_e + ω)(k^2_e - (k⃗_e ·n⃗_k⃗)^2)+ 16 ω^2(E_e - k⃗_e ·n⃗_k⃗), ∑_λ' tr{(m_e + k̂_e) (2 k_e ·ε^*_λ' + ε̂^*_λ' k̂) γ^0 (k̂_e + k̂ ) γ^0 (2 k_e ·ε_λ' + k̂ ε̂_λ')(1 - γ^5)} = = 16 (2 E_e (E_e + ω) - ω(E_e - k⃗_e ·n⃗_k⃗) - m^2_e) (k^2_e - (k⃗_e ·n⃗_k⃗)^2) + 32 ω (E_e + ω) (k^2_e - (k⃗_e ·n⃗_k⃗)^2)+ 32 ω^2 (E_e + ω) (E_e - k⃗_e ·n⃗_k⃗) - 16 ω^2(E_e - k⃗_e ·n⃗_k⃗)^2.Plugging Eq.(<ref>) into Eq.(<ref>) for the contribution of the diagram in Fig. <ref> to the rate of the neutron radiative β^-–decay with a photon from the energy region ω_ min≤ω≤ω_ max we obtain the following expressionλ^( Fig. <ref>)_βγ(ω_ max,ω_ min) = (1 + 3λ^2) α^2/π^2 G^2_F|V_ud|^2/4π^3∫^ω_ max_ω_ mindω/ω∫^E_0 - ω_m_e dE_e F(E_e, Z = 1) (E_0 - E_e - ω)^2 √(E^2_e - m^2_e) × ∫dΩ_eγ/4π { Ref_1(E_e, k⃗_e, ω, k⃗ ) [(E_e + ω) k^2_e - (k⃗_e ·n⃗_k⃗)^2/(E_e - k⃗_e ·n⃗_k⃗)^2 + ω^2/E_e - k⃗_e ·n⃗_k⃗] +Ref_2(E_e, k⃗_e, ω, k⃗ ) [(2(E_e + ω)^2 - m^2_e- ω (E_e - k⃗_e ·n⃗_k⃗)) k^2_e - (k⃗_e ·n⃗_k⃗)^2/(E_e - k⃗_e ·n⃗_k⃗)^2 + 2(E_e + ω) ω^2/E_e - k⃗_e ·n⃗_k⃗ - ω^2]},where the functions f_1(E_e, k⃗_e, ω, k⃗ ) and f_2(E_e, k⃗_e, ω, k⃗ ) are given in Eq.(<ref>). § APPENDIX D: THE AMPLITUDE OF THE NEUTRONRADIATIVE Β^-–DECAY, DESCRIBED BY FEYNMAN DIAGRAMS IN FIG. <REF> The analytical expressions for the diagrams Fig. <ref>f and Fig. <ref>g are given byM_ Fig. <ref>a(n → p e^- ν̅_eγ)_λ' = = - e^2∫d^4q/(2π)^4 i [u̅_p(k⃗_p, σ_p) γ^α 1/m_p - k̂_p + q̂ - i 0 ε̂^*_λ' 1/m_p - k̂_p - k̂ + q̂ - i 0 γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e, σ_e) γ^α 1/m_e - k̂_e - q̂ - i 0 γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)] 1/q^2 + i 0and M_ Fig. <ref>b(n → p e^- ν̅_eγ)_λ' = = e^2∫d^4q/(2π)^4 i [u̅_p(k⃗_p, σ_p) γ^α 1/m_p - k̂_p + q̂ - i 0 γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e, σ_e) γ_α 1/m_e - k̂_e - q̂ - i 0 ε̂^*_λ' 1/m_e - k̂_e - k̂ - q̂ - i 0 γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)] 1/q^2 + i 0For the calculation of the integrals over q we rewrite Eq.(<ref>) and Eq.(<ref>) as follows M_ Fig. <ref>a(n → p e^- ν̅_eγ)_λ' = = e^2∫d^4q/(2π)^4 i [u̅_p(k⃗_p, σ_p) (2k^α_p - γ^αq̂)/q^2 - 2k_p· q + i0 ε̂^*_λ' (m_p + k̂_p + k̂ - q̂)/q^2 - 2(k_p + k)· q + 2 k_p· k + i 0 γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e, σ_e) (2k_eα + γ_αq̂)/q^2 + 2k_e· q + i 0 γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)] 1/q^2 + i 0and M_ Fig. <ref>b(n → p e^- ν̅_eγ)_λ' = = -e^2∫d^4q/(2π)^4 i [u̅_p(k⃗_p, σ_p) (2 k^α_p - γ^αq̂)/q^2 - 2 k_p· q + i 0 γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e, σ_e) (2 k_eα + γ_αq̂)/q^2 + 2 k_e· q + i 0 ε̂^*_λ' (m_e+ k̂_e + k̂ + q̂)/q^2 + 2 (k_e + k)· q + 2 k_e · k + i 0 γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)] 1/q^2 + i 0,where we have used the properties of the Dirac γ–matrices and Dirac equations for the free proton and electron <cit.>, keeping the proton and electron on–mass shell k^2_p = m^2_p and k^2_e = m^2_e. Then, we merge the denominators <cit.>1/q^2 - 2k_p· q + i0 1/q^2 - 2(k_p + k)· q + 2 k_p · k + i 0 1/q^2 + 2k_e· q + i 0 1/q^2 - μ^2 + i 0 = = ∫^1_0dx∫^1_0dy 2 y∫^1_0dz 3 z^2 1/[(q - k_p(x,y)z)^2 - k^2_p(x,y)z^2 + 2 k_p· k x y z - μ^2 (1 - z) + i 0]^4, and1/q^2 - 2 k_p· q + i 0 1/q^2 + 2 k_e· q + i 0 1/q^2 + 2 (k_e + k)· q + 2 k_e · k + i 0 1/q^2 - μ^2 + i 0 == ∫^1_0dx∫^1_0dy 2 y∫^1_0dz 3 z^2 1/[(q + k_e(x,y)z)^2 - k^2_e(x,y)z^2 + 2 k_e · kx y z - μ^2 (1 - z) + i 0]^4,where k_p(x,y) = k_p y - k_e(1 - y) + k x y, k_e(x,y) = k_e y - k_p (1 - y) + k x y and μ is a photon mass, regularizing infrared divergences <cit.>. Making the shifts of variables q - k_p(x,y)z → q and q + k_e(x,y)z → q in Eq.(<ref>) and Eq.(<ref>), respectively, and integrating over the directions of the virtual 4–momentum q we arrive at the expressionsM_ Fig. <ref>a(n → p e^- ν̅_eγ)_λ' = = e^2 ∫^1_0dx∫^1_0dy 2 y∫^1_0dz 3 z^2 ∫d^4q/(2π)^4 i 1/[q^2 - k^2_p(x,y)z^2 + 2 k_p· k x y z - μ^2 (1 - z) + i 0]^4 × {[u̅_p(k⃗_p, σ_p) (2 k^α_p - γ^αk̂_p(x)z) ε̂^*_λ' (m_p + k̂_p + k̂ - k̂_p(x,y)z) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e, σ_e) (2 k_eα + γ_αk̂_p(x,y)z) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]- 1/2 q^2 [u̅_p(k⃗_p, σ_p) γ^α ε̂^*_λ' γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) (2 k_eα + γ_αk̂_p(x,y)z) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]- 1/4 q^2 [u̅_p(k⃗_p, σ_p) γ^α γ^β ε̂^*_λ' (m_p + k̂_p + k̂ - k̂_p(x,y)z) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e, σ_e) γ_α γ_β γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]- 1/4 q^2 [u̅_p(k⃗_p, σ_p) (2 k^α_p - γ^αk̂_p(x,y)z) ε̂^*_λ' γ^β γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)]}and M_ Fig. <ref>b(n → p e^- ν̅_eγ)_λ' = = -e^2 ∫^1_0dx∫^1_0dy 2 y∫^1_0dz 3 z^2 ∫d^4q/(2π)^4 i 1/[q^2 - k^2_e(x,y)z^2 + 2 k_e· k x y z - μ^2 (1 - z) + i 0]^4 × {[u̅_p(k⃗_p, σ_p) (2 k^α_p + γ^αk̂_e(x,y)z) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e, σ_e) (2 k_eα - γ_αk̂_e(x,y)z) ε̂^*_λ' (m_e + k̂_e + k̂ - k̂_e(x,y)z) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)]- 1/2 q^2 [u̅_p(k⃗_p, σ_p) (2 k^α_p + γ^αk̂_e(x,y)z) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ_α ε̂^*_λ' γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]- 1/4 q^2 [u̅_p(k⃗_p, σ_p) γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e, σ_e) γ_αγ_β ε̂^*_λ' (m_e + k̂_e + k̂ - k̂_e(x,y)z) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]- 1/4 q^2 [u̅_p(k⃗_p, σ_p) γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) (2 k_eα - γ_αk̂_e(x,y)z)ε̂^*_λ' γ_β γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)].Making a Wick rotation and integrating over q^2 we arrive at the following expressions for the diagrams Fig. <ref>f and Fig. <ref>gM_ Fig. <ref>a(n → p e^- ν̅_eγ)_λ' = e^2/96π^2∫^1_0dx∫^1_0dy 2 y∫^1_0dz 3 z^2 1/[k^2_p(x,y)z^2 - 2 k_p· k x y z+ μ^2 (1 - z)]^2 × [u̅_p(k⃗_p, σ_p) (2 k^α_p - γ^αk̂_p(x, y) z) ε̂^*_λ' (m_p + k̂_p + k̂ - k̂_p(x,y)z) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e, σ_e) (2 k_eα + γ_αk̂_p(x,y)z) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]+ e^2/48π^2∫^1_0dx∫^1_0dy 2 y∫^1_0dz 3 z^2 1/k^2_p(x,y)z^2 - 2 k_p · kxyz+ μ^2 (1 - z) × {1/2 [u̅_p(k⃗_p, σ_p) γ^α ε̂^*_λ' γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) (2 k_eα + γ_αk̂_p(x,y)z) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]+ 1/4 [u̅_p(k⃗_p, σ_p) γ^α γ^β ε̂^*_λ' (m_p + k̂_p + k̂ - k̂_p(x,y)z) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e, σ_e) γ_α γ_β γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)] + 1/4 [u̅_p(k⃗_p, σ_p) (2 k^α_p - γ^αk̂_p(x,y)z) ε̂^*_λ' γ^β γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)]}and M_ Fig. <ref>b(n → p e^- ν̅_eγ)_λ' = - e^2/96π^2∫^1_0dx∫^1_ody 2 y∫^1_0dz 3 z^2 1/[k^2_e(x,y)z^2 - 2 k_e· kxyz+ μ^2 (1 - z)]^2 × {[u̅_p(k⃗_p, σ_p) (2 k^α_p + γ^αk̂_e(x,y)z) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e, σ_e) (2 k_eα - γ_αk̂_e(x,y)z) ε̂^*_λ' (m_e + k̂_e + k̂ - k̂_e(x,y)z) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)]- e^2/48π^2∫^1_0dx∫^1_0 dy 2 y∫^1_0dz 3 z^2 1/k^2_e(x,y)z^2 - 2 k_e· kxyz + μ^2 (1 - z) × {1/2 [u̅_p(k⃗_p, σ_p) (2 k^α_p + γ^αk̂_e(x,y)z) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ_α ε̂^*_λ' γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]+ 1/4 [u̅_p(k⃗_p, σ_p) γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] × [u̅_e(k⃗_e, σ_e) γ_αγ_β ε̂^*_λ' (m_e + k̂_e + k̂ - k̂_e(x,y)z) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]+ 1/4 [u̅_p(k⃗_p, σ_p) γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) (2 k_eα - γ_αk̂_e(x,y)z)ε̂^*_λ' γ_β γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]}.For the analysis of the integrals over the Feynman parameters x,y and z it is convenient to rewrite Eq.(<ref>) and Eq.(<ref>) as follows M_ Fig. <ref>a(n → p e^- ν̅_eγ)_λ' = e^2/96π^2∫^1_0dx∫^1_0dy 2 y∫^1_0dz 3 z^2 1/[k^2_p(x,y)z^2 - 2 k_p· k x y z+ μ^2 (1 - z)]^2 × { 4 (k_p· k_e)[u̅_p(k⃗_p, σ_p) ε̂^*_λ' (m_p + k̂_p + k̂) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]+ 2 z [u̅_p(k⃗_p, σ_p) ε̂^*_λ' (m_p + k̂_p + k̂) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) k̂_pk̂_p(x,y) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)] - 4 z (k_p· k_e)[u̅_p(k⃗_p, σ_p) ε̂^*_λ' k̂_p(x,y)z γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]- 2 z [u̅_p(k⃗_p, σ_p) k̂_ek̂_p(x, y) ε̂^*_λ' (m_p + k̂_p + k̂) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]+ 2 z^2 [u̅_p(k⃗_p, σ_p) k̂_ek̂_p(x, y) ε̂^*_λ' k̂_p(x,y) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]- z^2[u̅_p(k⃗_p, σ_p) γ^αk̂_p(x, y) ε̂^*_λ' (m_p + k̂_p + k̂) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ_αk̂_p(x,y) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]- 2 z^2 [u̅_p(k⃗_p, σ_p) ε̂^*_λ' k̂_p(x,y) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) k̂_p k̂_p(x,y) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]+ z^3 [u̅_p(k⃗_p, σ_p) γ^αk̂_p(x, y) ε̂^*_λ' k̂_p(x,y) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ_αk̂_p(x,y) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]}+ e^2/48π^2∫^1_0dx∫^1_0dy 2 y∫^1_0dz 3 z^2 1/k^2_p(x,y)z^2 - 2 k_p · kxyz+ μ^2 (1 - z) × {[u̅_p(k⃗_p, σ_p) k̂_e ε̂^*_λ' γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]+ 1/4 [u̅_p(k⃗_p, σ_p) γ^α γ^β ε̂^*_λ' (m_p + k̂_p + k̂) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ_α γ_β γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)] + 1/2 [u̅_p(k⃗_p, σ_p) ε̂^*_λ' γ^β γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) k̂_p γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)] + 1/2 z [u̅_p(k⃗_p, σ_p) γ^α ε̂^*_λ' γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ_αk̂_p(x,y) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]- 1/4 z [u̅_p(k⃗_p, σ_p) γ^α γ^β ε̂^*_λ' k̂_p(x,y) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ_α γ_β γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)] - 1/4 z [u̅_p(k⃗_p, σ_p) γ^αk̂_p(x,y)ε̂^*_λ' γ^β γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ_α γ_β γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)]}andM_ Fig. <ref>b(n → p e^- ν̅_eγ)_λ' = - e^2/96π^2∫^1_0dx∫^1_ody 2 y∫^1_0dz 3 z^2 1/[k^2_e(x,y)z^2 - 2 k_e· kxyz+ μ^2 (1 - z)]^2 × {4 (k_p· k_e) [u̅_p(k⃗_p, σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) ε̂^*_λ' (m_e + k̂_e + k̂) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)] - 2 z [u̅_p(k⃗_p, σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) k̂_p k̂_e(x,y)ε̂^*_λ' (m_e + k̂_e + k̂) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)] + 2 z [u̅_p(k⃗_p, σ_p) k̂_e k̂_e(x,y)z) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) ε̂^*_λ' (m_e + k̂_e + k̂) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)] - 4 z (k_p· k_e) [u̅_p(k⃗_p, σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) ε̂^*_λ' k̂_e(x,y) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)] - 2 z^2 [u̅_p(k⃗_p, σ_p) k̂_e k̂_e(x,y)z) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) ε̂^*_λ' k̂_e(x,y) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)] + 2 z^2 [u̅_p(k⃗_p, σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)][u̅_e(k⃗_e, σ_e) k̂_p k̂_e(x,y) ε̂^*_λ' k̂_e(x,y)γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)] - z^2 [u̅_p(k⃗_p, σ_p) γ^αk̂_e(x,y) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)][u̅_e(k⃗_e, σ_e) γ_αk̂_e(x,y)ε̂^*_λ' (m_e + k̂_e + k̂) γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)] + z^3 [u̅_p(k⃗_p, σ_p) γ^αk̂_e(x,y) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)][u̅_e(k⃗_e, σ_e) γ_αk̂_e(x,y)ε̂^*_λ' k̂_e(x,y)γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)]- e^2/48π^2∫^1_0dx∫^1_0 dy 2 y∫^1_0dz 3 z^2 1/k^2_e(x,y)z^2 - 2 k_e· kxyz + μ^2 (1 - z) × {[u̅_p(k⃗_p, σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) k̂_pε̂^*_λ' γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)] + 1/4 [u̅_p(k⃗_p, σ_p) γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)][u̅_e(k⃗_e, σ_e) γ_αγ_β ε̂^*_λ' (m_e + k̂_e + k̂) γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]+ 1/2 [u̅_p(k⃗_p, σ_p) k̂_eγ^β γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) ε̂^*_λ' γ_β γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]+ 1/2 z [u̅_p(k⃗_p, σ_p) γ^αk̂_e(x,y)γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ_α ε̂^*_λ' γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]- 1/4 z [u̅_p(k⃗_p, σ_p) γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)][u̅_e(k⃗_e, σ_e) γ_αγ_β ε̂^*_λ' k̂_e(x,y)γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]- 1/4 z [u̅_p(k⃗_p, σ_p) γ^α γ^β γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] [u̅_e(k⃗_e, σ_e) γ_αk̂_e(x,y)ε̂^*_λ' γ_β γ^μ(1 + λγ^5) v_ν(k⃗_ν, + 1/2)]}.One may show that in the large proton mass expansion the amplitudes Eq.(<ref>) and Eq.(<ref>) behave as O(1/m_p) or even faster and vanish at m_p →∞. We assume that the contributions of other diagrams in Fig. <ref> either cancel a gauge–non-invariant part of the diagram Fig. <ref>c or is a constant, which can be removed by renormalization of the Fermi weak coupling constant G_F and the axial coupling constant λ <cit.>.§ APPENDIX E: THE AMPLITUDE OF THE NEUTRONRADIATIVE Β^-–DECAY, DESCRIBED BY FEYNMAN DIAGRAMS IN FIG. <REF> In this Appendix we calculate the diagrams in Fig. <ref>. These diagrams describe the process of the neutron radiative β^-–decay with emission of two real photons n → p + e^- + ν̅_e + γ + γ. The contribution of these diagrams to the rate of the neutron β^-–decay is of order O(α^2/π^2) and after the integration over degrees of freedom of one of the photons one may hardly distinguish such a contribution from that of the neutron radiative β^-–decay with an emission of one real photon. Since the contribution of the process, when photons are emitted by the proton, is suppressed to leading order in the large proton mass expansion, we take into account only the emission of photons by the electron. The analytical expression of the diagrams in Fig. <ref> is given byM_ Fig. <ref>(n → p e^- ν̅_eγγ)_λ'λ” = -e [u̅_p(k⃗_p, σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] × {[u̅_e(k⃗_e,σ_e) ε̂^*_λ'(k) 1/m_e - k̂_e - k̂ - i 0 ε̂^*_λ”(q) 1/m_e - k̂_e - k̂ - q̂ - i 0 γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)]+ [u̅_e(k⃗_e,σ_e) ε̂^*_λ”(q) 1/m_e - k̂_e - q̂ - i 0 ε̂^*_λ'(k) 1/m_e - k̂_e - k̂ - q̂ - i 0 γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)]},where the polarization vectors ε^*_λ'(k) and ε^*_λ”(q) are taken in the physical gauge and obey the constraints k⃗·ε⃗^ *_λ'(k) = 0 and q⃗·ε⃗^ *_λ”(q) = 0, respectively, with k^2 = q^2 = 0.Then, we rewrite Eq.(<ref>) as followsM_ Fig. <ref>(n → p e^- ν̅_eγγ)_λ'λ” = - e [u̅_p(k⃗_p, σ_p) γ^μ(1 + λγ^5) u_n(k⃗_n, σ_n)] × {1/2k_e· k + i0 1/2k_e· (k + q) + 2 k· q + i0 × [u̅_e(k⃗_e,σ_e) (2k_e·ε^*_λ' + ε̂^*_λ' k̂) ε̂^*_λ” (m_e + k̂_e + k̂ + q̂)γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)]+ 1/2k_e· q + i0 1/2k_e· (k + q) + 2 k· q + i0 × [u̅_e(k⃗_e,σ_e) (2k_e·ε^*_λ” + ε̂^*_λ” q̂)ε̂^*_λ' (m_e + k̂_e + k̂ + q̂)γ_μ(1 - γ^5) v_ν(k⃗_ν, + 1/2)]}.In the non–relativistic proton approximation Eq.(<ref>) takes the form1/2m_n eM_ Fig. <ref>(n → p e^- ν̅_eγγ)_λ'λ” = - 1/2k_e· (k + q) + 2 k· q + i0 × {{[φ^†_pφ_n] ( 1/2k_e· k + i0 [u̅_e(k⃗_e,σ_e) (2k_e·ε^*_λ' + ε̂^*_λ' k̂) ε̂^*_λ” (m_e + k̂_e + k̂ + q̂)γ^0 (1 - γ^5) v_ν(k⃗_ν, + 1/2)].+ 1/2k_e· q + q^2 + i0 [u̅_e(k⃗_e,σ_e) (2k_e·ε^*_λ” + ε̂^*_λ” q̂)ε̂^*_λ' (m_e + k̂_e + k̂ + q̂)γ^0 (1 - γ^5) v_ν(k⃗_ν, + 1/2)])} - λ {[φ^†_p σ⃗ φ_n]·( 1/2k_e· k + i0 [u̅_e(k⃗_e,σ_e) (2k_e·ε^*_λ' + ε̂^*_λ' k̂) ε̂^*_λ” (m_e + k̂_e + k̂ + q̂)γ⃗ (1 - γ^5) v_ν(k⃗_ν, + 1/2)] . + 1/2k_e· q + i0 [u̅_e(k⃗_e,σ_e) (2k_e·ε^*_λ” + ε̂^*_λ” q̂)ε̂^*_λ' (m_e + k̂_e + k̂ + q̂)γ⃗ (1 - γ^5) v_ν(k⃗_ν, + 1/2)])}}.The hermitian conjugate amplitude is equal to1/2 m_n eM^†_ Fig. <ref>(n → p e^- ν̅_eγγ)_λ'λ” = - 1/2 k_e· (k + q) + 2 k· q - i 0 × {{[φ^†_nφ_p] (1/2k_e· k - i0 [ v̅_ν(k⃗_ν, + 1/2)γ^0 (1 - γ^5) (m_e + k̂_e + k̂ + q̂) ε̂_λ” (2 k_e ·ε_λ' + k̂ ε̂_λ')u_e(k⃗_e,σ_e)].+ 1/2k_e· q - i0 [ v̅_ν(k⃗_ν, + 1/2)γ^0 (1 - γ^5) (m_e + k̂_e + k̂ + q̂) ε̂_λ' (2 k_e ·ε_λ” + q̂ ε̂_λ”)u_e(k⃗_e,σ_e)])} - λ {[φ^†_p σ⃗ φ_n]·( 1/2k_e· k - i0 [v̅_ν(k⃗_ν, + 1/2) γ⃗(1 - γ^5) (m_e + k̂_e + k̂ + q̂) ε̂_λ” (2 k_e ·ε_λ' + k̂ ε̂_λ')u_e(k⃗_e,σ_e) ] . + 1/2k_e· q - i0 [v̅_ν(k⃗_ν, + 1/2) γ⃗(1 - γ^5) (m_e + k̂_e + k̂ + q̂) ε̂_λ' (2 k_e ·ε_λ” + q̂ ε̂_λ”)u_e(k⃗_e,σ_e) ])}}.The squared absolute value of the amplitude Eq.(<ref>) averaged over the neutron spin and summed over the polarizations of the proton and electron 1/2∑_ pol| M_ Fig. <ref>(n → p e^- ν̅_eγγ)_λ'λ”|^2/4 m^2_n e^2 = 1/(2 k_e· (k + q) + 2 k· q)^2{1/(2k_e· k)^2 tr{(m_e + k̂_e)(2k_e·ε^*_λ' + ε̂^*_λ'k̂)ε̂^*_λ” ×(m_e + k̂_e + k̂ + q̂)γ^0(1 - γ^5)k̂_νγ^0(1 - γ^5)(m_e + k̂_e + k̂ + q̂)ε̂_λ”(2k_e·ε_λ' + k̂ε̂_λ')} + 1/2k_e· k 1/2k_e· q × tr{(m_e + k̂_e)(2k_e·ε^*_λ” + ε̂^*_λ”q̂)ε̂^*_λ' (m_e + k̂_e + k̂ + q̂) γ^0 (1 - γ^5) k̂_νγ^0(1 - γ^5) (m_e + k̂_e + k̂ + q̂)ε̂_λ”(2k_e·ε_λ' + k̂ε̂_λ')}+ 1/2k_e· k 1/2k_e· q tr{(m_e + k̂_e)(2k_e·ε^*_λ' + ε̂^*_λ'k̂)ε̂^*_λ” (m_e + k̂_e + k̂ + q̂) γ^0 (1 - γ^5) k̂_νγ^0(1 - γ^5) (m_e + k̂_e + k̂ + q̂) ×ε̂_λ'(2k_e·ε_λ” + q̂ε̂_λ”)}+ 1/(2k_e· q)^2 tr{(m_e + k̂_e)(2k_e·ε^*_λ” + ε̂^*_λ”q̂)ε̂^*_λ' (m_e + k̂_e + k̂ + q̂)γ^0(1 - γ^5)k̂_νγ^0(1 - γ^5) × (m_e + k̂_e + k̂ + q̂)ε̂_λ'(2k_e·ε_λ” + q̂ε̂_λ”)}} + λ^2 δ^ij/(2 k_e · (k + q) + 2 k· q)^2{1/(2k_e· k)^2 tr{(m_e + k̂_e)(2k_e·ε^*_λ' + ε̂^*_λ'k̂)ε̂^*_λ” ×(m_e + k̂_e + k̂ + q̂)γ^i(1 - γ^5)k̂_νγ^j(1 - γ^5)(m_e + k̂_e + k̂ + q̂)ε̂_λ”(2k_e·ε_λ' + k̂ε̂_λ')} + 1/2k_e· k 1/2k_e· q × tr{(m_e + k̂_e)(2k_e·ε^*_λ” + ε̂^*_λ”q̂)ε̂^*_λ' (m_e + k̂_e + k̂ + q̂) γ^i (1 - γ^5) k̂_νγ^j(1 - γ^5) (m_e + k̂_e + k̂ + q̂)ε̂_λ”(2k_e·ε_λ' + k̂ε̂_λ')}+ 1/2k_e· k 1/2k_e· q tr{(m_e + k̂_e)(2k_e·ε^*_λ' + ε̂^*_λ'k̂)ε̂^*_λ” (m_e + k̂_e + k̂ + q̂) γ^j (1 - γ^5) k̂_νγ^i (1 - γ^5) (m_e + k̂_e + k̂ + q̂) ×ε̂_λ'(2k_e·ε_λ” + q̂ε̂_λ”)} + 1/(2k_e· q)^2 tr{(m_e + k̂_e)(2k_e·ε^*_λ” + ε̂^*_λ”q̂)ε̂^*_λ' (m_e + k̂_e + k̂ + q̂)γ^i (1 - γ^5)k̂_νγ^j (1 - γ^5) × (m_e + k̂_e + k̂ + q̂)ε̂_λ'(2k_e·ε_λ” + q̂ε̂_λ”)}}.For the subsequent calculation we omit the traces with the γ^5–matrix, which should not contribute to the rate of the neutron radiative β^--decay with two photons in the final state. This gives1/2∑_ pol| M_ Fig. <ref>(n → p e^- ν̅_eγγ)_λ'λ”|^2/8 m^2_n e^2 = 1/(2 k_e· (k + q) + 2 k· q)^2{1/(2k_e· k)^2 tr{(m_e + k̂_e)(2k_e·ε^*_λ' + ε̂^*_λ'k̂)ε̂^*_λ” ×(m_e + k̂_e + k̂ + q̂)γ^0 k̂_νγ^0 (m_e + k̂_e + k̂ + q̂)ε̂_λ”(2k_e·ε_λ' + k̂ε̂_λ')} + 1/2k_e· k 1/2k_e· q× tr{(m_e + k̂_e)(2k_e·ε^*_λ” + ε̂^*_λ”q̂)ε̂^*_λ' (m_e + k̂_e + k̂ + q̂) γ^0 k̂_νγ^0 (m_e + k̂_e + k̂ + q̂)ε̂_λ”(2k_e·ε_λ' + k̂ε̂_λ')} + 1/2k_e· k 1/2k_e· q tr{(m_e + k̂_e)(2k_e·ε^*_λ' + ε̂^*_λ'k̂)ε̂^*_λ” (m_e + k̂_e + k̂ + q̂) γ^0 k̂_νγ^0 (m_e + k̂_e + k̂ + q̂) ×ε̂_λ'(2k_e·ε_λ” + q̂ε̂_λ”)}+ 1/(2k_e· q)^2 tr{(m_e + k̂_e)(2k_e·ε^*_λ” + ε̂^*_λ”q̂)ε̂^*_λ' (m_e + k̂_e + k̂ + q̂)γ^0 k̂_νγ^0× (m_e + k̂_e + k̂ + q̂) ε̂_λ' (2k_e·ε_λ” + q̂ε̂_λ”)}}+ λ^2 δ^ij/(2 k_e · (k + q) + 2 k· q)^2 {1/(2k_e· k )^2 tr{(m_e + k̂_e)(2k_e·ε^*_λ' + ε̂^*_λ'k̂)ε̂^*_λ” ×(m_e + k̂_e + k̂ + q̂)γ^i k̂_νγ^j (m_e + k̂_e + k̂ + q̂)ε̂_λ”(2k_e·ε_λ' + k̂ε̂_λ')} + 1/2k_e· k 1/2k_e· q × tr{(m_e + k̂_e)(2k_e·ε^*_λ” + ε̂^*_λ”q̂)ε̂^*_λ' (m_e + k̂_e + k̂ + q̂) γ⃗^ ik̂_νγ⃗^ j (m_e + k̂_e + k̂ + q̂) ε̂_λ” (2k_e·ε_λ' + k̂ε̂_λ')}+ 1/2k_e· k 1/2k_e· q tr{(m_e + k̂_e)(2k_e·ε^*_λ' + ε̂^*_λ'k̂)ε̂^*_λ” (m_e + k̂_e + k̂ + q̂) γ⃗^ jk̂_νγ⃗^ i (m_e + k̂_e + k̂ + q̂) ×ε̂_λ'(2k_e·ε_λ” + q̂ε̂_λ”)} + 1/(2k_e· q)^2 tr{(m_e + k̂_e)(2k_e·ε^*_λ” + ε̂^*_λ”q̂)ε̂^*_λ' (m_e + k̂_e + k̂ + q̂) γ⃗^ ik̂_νγ⃗^ j × (m_e + k̂_e + k̂ + q̂)ε̂_λ'(2k_e·ε_λ” + q̂ε̂_λ”)}}.Then, we average over directions of the antineutrino momentum k⃗_ν and sum over polarizations of photons. This gives∫dΩ_ν/4π 1/2∑_ pol, λ', λ”| M_ Fig. <ref>(n → p e^- ν̅_eγγ)_λ'λ”|^2/8 m^2_n e^2 (1 + 3 λ^2) E_ν = 1/(2 k_e· (k + q) + 2 k · q)^2 ×{1/(2k_e· k)^2∑_λ', λ” tr{(m_e + k̂_e)(2k_e·ε^*_λ' + ε̂^*_λ'k̂) ε̂^*_λ”(m_e + k̂_e + k̂ + q̂) γ^0 (m_e + k̂_e + k̂ + q̂) ε̂_λ”(2k_e·ε_λ' + k̂ ε̂_λ')}+ 1/2k_e· k 1/2k_e· q ∑_λ', λ” tr{(m_e + k̂_e)(2k_e·ε^*_λ” + ε̂^*_λ”q̂) ε̂^*_λ'(m_e + k̂_e + k̂ + q̂) γ^0(m_e + k̂_e + k̂ + q̂) ε̂_λ” (2k_e·ε_λ' + k̂ ε̂_λ')}+ 1/2k_e· k 1/2k_e· q∑_λ', λ” tr{(m_e + k̂_e)(2k_e·ε^*_λ' + ε̂^*_λ'k̂) ε̂^*_λ”(m_e + k̂_e + k̂ + q̂) γ^0 (m_e + k̂_e + k̂ + q̂) ε̂_λ'(2k_e·ε_λ” + q̂ ε̂_λ”)} + 1/(2k_e· q)^2∑_λ', λ” tr{(m_e + k̂_e)(2k_e·ε^*_λ” + ε̂^*_λ”q̂) ε̂^*_λ'(m_e + k̂_e + k̂ + q̂) γ^0 (m_e + k̂_e + k̂ + q̂) ε̂_λ'(2k_e·ε_λ” + q̂ ε̂_λ”)}}.For the calculation of the contribution of the diagram Fig. <ref> to the rate of the neutron radiative β^-–decay with two photons in the final state we have to sum over the photon physical degrees of freedom only.For this aim we use the following relations <cit.>k⃗·ε⃗^ *_λ' = k⃗·ε⃗_λ' = 0 , ε⃗^ *_λ'·ε⃗_λ̅' = δ_λ'λ̅' , q⃗·ε⃗^ *_λ” = q⃗·ε⃗_λ” = 0 , ε⃗^ *_λ”·ε⃗_λ̅” = δ_λ”λ̅”, ∑_λ' = 1,2ε⃗^ i *_λ'ε⃗^ j_λ' = δ^ij - k⃗^ ik⃗^ j/ω^2 = δ^ij - n⃗^ i_k⃗ n⃗^ j_k⃗ , ∑_λ” = 1,2ε⃗^ i *_λ”ε⃗^ j_λ” = δ^ij - q⃗^ iq⃗^ j/q^2_0 = δ^ij - n⃗^ i_q⃗ n⃗^ j_q⃗,where ω and q_0 are photon energies and n⃗_k⃗ = k⃗/ω and n⃗_q⃗ = q⃗/q_0 are unit vectors directed along photon momenta. For the traces in Eq.(<ref>) we obtain the following expressions∑_λ', λ” tr{(m_e + k̂_e)(2k_e·ε^*_λ' + ε̂^*_λ'k̂) ε̂^*_λ”(m_e + k̂_e + k̂ + q̂) γ^0 (m_e + k̂_e + k̂ + q̂) ε̂_λ”(2k_e·ε_λ' + k̂ ε̂_λ')} = = 64 q_0 (ω(E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗) - (E_e + ω) ω (1 - n⃗_k⃗·n⃗_q⃗)) (k^2_e - (k⃗_e ·n⃗_k⃗)^2) + 64(E_e + ω + q_0) × (k^2_e - (k⃗_e ·n⃗_k⃗)^2)(k^2_e - (k⃗_e ·n⃗_q⃗)^2) + 128(E_e + ω + q_0) ω (k^2_e - (k⃗_e ·n⃗_k⃗)^2)((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) + 64(E_e + ω + q_0) ω^2(k^2_e - (k⃗_e ·n⃗_k⃗)^2)(1 - (n⃗_k⃗·n⃗_q⃗)^2) - 64(E_e + ω + q_0) ω (E_e - k⃗_e ·n⃗_k⃗) (k^2_e - (k⃗_e ·n⃗_k⃗)^2- (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗) (k⃗_e ·n⃗_q⃗) (n⃗_k⃗·n⃗_q⃗)) + 64(E_e + ω + q_0) ω^2(E_e - k⃗_e ·n⃗_k⃗)(n⃗_k⃗·n⃗_q⃗) ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) ×(n⃗_k⃗·n⃗_q⃗)) - 64(E_e + ω + q_0) ωq_0 (E_e - k⃗_e ·n⃗_k⃗)((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) (n⃗_q⃗·n⃗_k⃗)) + 64(E_e + ω + q_0) ω^2× (E_e - k⃗_e ·n⃗_k⃗)(E_e - (k⃗_e ·n⃗_q⃗) (n⃗_k⃗·n⃗_q⃗) + ω(1 - (n⃗_k⃗·n⃗_q⃗)^2) + q_0(1 - n⃗_k⃗·n⃗_q⃗)) - 64 ω^2(ω (E_e - k⃗_e ·n⃗_k⃗)+ q_0 (E_e - k⃗_e ·n⃗_q⃗) + ωq_0 (1 - n⃗_k⃗·n⃗_q⃗))and ∑_λ', λ” tr{(m_e + k̂_e)(2k_e·ε^*_λ” + ε̂^*_λ”q̂) ε̂^*_λ'(m_e + k̂_e + k̂ + q̂) γ^0(m_e + k̂_e + k̂ + q̂) ε̂_λ”(2k_e·ε_λ' + k̂ ε̂_λ')} == - 32m^2_e (E_e + ω + q_0) (k^2_e - (k⃗_e ·n⃗_k⃗)^2 - (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗)(k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) - 32m^2_eω(E_e + ω + q_0) × ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) - 32m^2_e q_0(E_e + ω + q_0) ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗)(n⃗_q⃗·n⃗_k⃗)) - 32m^2_eωq_0 ×(E_e + ω + q_0) (1 - n⃗_k⃗·n⃗_q⃗) - 32E_e(ω (E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗) + ωq_0(1 - n⃗_k⃗·n⃗_q⃗)) (k^2_e - (k⃗_e ·n⃗_k⃗)^2- (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗)(k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) + 32 ω(E_e + ω + q_0) ((E_e - k⃗_e ·n⃗_k⃗) + q_0(1 - n⃗_k⃗·n⃗_q⃗)) (k^2_e - (k⃗_e ·n⃗_k⃗)^2- (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗)(k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) + 64(E_e + ω + q_0) (k^2_e - (k⃗_e ·n⃗_k⃗)^2) [(k^2_e - (k⃗_e ·n⃗_q⃗)^2) + ω ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗)(n⃗_q⃗·n⃗_k⃗))] + 32 ω (ω(E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗) + ωq_0 (1 - n⃗_k⃗·n⃗_q⃗))(k^2_e - (k⃗_e ·n⃗_q⃗)^2)- 32 ωE_e(ω (E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗) + ωq_0 (1 - n⃗_k⃗·n⃗_q⃗))((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗) (n⃗_k⃗·n⃗_q⃗))- 32 ω (ω (E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗) + ωq_0 (1 - n⃗_k⃗·n⃗_q⃗))(k^2_e - (k⃗_e ·n⃗_k⃗)^2 - (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗)(k⃗_e ·n⃗_q⃗) × (n⃗_k⃗·n⃗_q⃗)) + 32 ω(E_e + ω + q_0) ((E_e - k⃗_e ·n⃗_k⃗) + q_0 (1 - n⃗_k⃗·n⃗_q⃗)) (k^2_e - (k⃗_e ·n⃗_k⃗)^2 - (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗) × (k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) + 32 ω(E_e + ω + q_0) ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗) (n⃗_k⃗·n⃗_q⃗)) [(k^2_e - (k⃗_e ·n⃗_k⃗)^2) + q_0 ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) ×(n⃗_q⃗·n⃗_k⃗))] + 32 ω(E_e + ω + q_0) (m^2_e + ω (E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗))((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗))- 32 ω(E_e + ω + q_0) ((E_e - k⃗_e ·n⃗_k⃗) + q_0(1 - n⃗_k⃗·n⃗_q⃗)) (k^2_e - (k⃗_e ·n⃗_q⃗)^2) + 32 ω(E_e + ω + q_0) (E_e - k⃗_e ·n⃗_k⃗) × [ (k^2_e - (k⃗_e ·n⃗_q⃗)^2) + ω ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗))]- 32 q_0E_e(ω(E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗)+ ωq_0 (1 - n⃗_k⃗·n⃗_q⃗)) ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗)(n⃗_q⃗·n⃗_k⃗)) + 32 q_0(ω(E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗)+ ωq_0 (1 - n⃗_k⃗·n⃗_q⃗)) (k^2_e - (k⃗_e ·n⃗_k⃗)^2) - 32 q_0(ω(E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗) + ωq_0 (1 - n⃗_k⃗·n⃗_q⃗))× (k^2_e - (k⃗_e ·n⃗_k⃗)^2- (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗)(k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) + 32q_0 (E_e + ω + q_0) (m^2_e + ω (E_e - k⃗_e ·n⃗_k⃗)+ q_0(E_e - k⃗_e ·n⃗_q⃗)) ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗)(n⃗_q⃗·n⃗_k⃗)) + 32q_0 (E_e + ω + q_0)(E_e - k⃗_e ·n⃗_k⃗) × [(k^2_e - (k⃗_e ·n⃗_k⃗)^2) + q_0 ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗)(n⃗_q⃗·n⃗_k⃗))] - 32q_0 (E_e + ω + q_0) ((E_e - k⃗_e ·n⃗_q⃗)+ ω(1 - n⃗_k⃗·n⃗_q⃗)) (k^2_e - (k⃗_e ·n⃗_k⃗)^2 ) + 32q_0 (E_e + ω + q_0) ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗)(n⃗_q⃗·n⃗_k⃗))[(k^2_e - (k⃗_e ·n⃗_q⃗)^2)+ ω ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗) (n⃗_k⃗·n⃗_q⃗))] + 32q_0 (E_e + ω + q_0) ((E_e - k⃗_e ·n⃗_q⃗) + ω(1 - n⃗_k⃗·n⃗_q⃗)) (k^2_e - (k⃗_e ·n⃗_k⃗)^2- (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗)(k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) - 8m^2_eωq_0(1 - (n⃗_k⃗·n⃗_q⃗)^2) (E_e (1 + n⃗_k⃗·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗))- 8m^2_eωq_0 (n⃗_k⃗·n⃗_q⃗) ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) (n⃗_q⃗·n⃗_k⃗)) + 8m^2_eωq_0 (E_e - ω)(n⃗_q⃗·n⃗_k⃗)(1 - (n⃗_q⃗·n⃗_k⃗)^2)+ 8 ωq_0 (1 - (n⃗_k⃗·n⃗_q⃗)^2) (m^2_e + 2 ω(E_e - k⃗_e ·n⃗_k⃗) + 2q_0(E_e - k⃗_e ·n⃗_q⃗) + 2 ωq_0(1 - n⃗_k⃗·n⃗_q⃗)) + [8q_0(m^2_e + 2 ω(E_e - k⃗_e ·n⃗_k⃗) + 2q_0(E_e - k⃗_e ·n⃗_q⃗) + 2 ωq_0(1 - n⃗_k⃗·n⃗_q⃗)) - 16 ωq_0(E_e + ω + q_0) × ( (E_e - k⃗_e ·n⃗_q⃗) + ω (1 - n⃗_k⃗·n⃗_q⃗))](n⃗_k⃗·n⃗_q⃗) ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) ( n⃗_q⃗·n⃗_k⃗)) - 8 ωq_0[ (E_e - ω) × (m^2_e + 2 ω(E_e - k⃗_e ·n⃗_k⃗) + 2q_0(E_e - k⃗_e ·n⃗_q⃗) + 2 ωq_0(1 - n⃗_k⃗·n⃗_q⃗)) - 2(E_e + ω + q_0) (m^2_e + ω(E_e - k⃗_e ·n⃗_k⃗)+ q_0(E_e - k⃗_e ·n⃗_q⃗))]( n⃗_k⃗·n⃗_q⃗)(1 - ( n⃗_k⃗·n⃗_q⃗)^2) - 16 ωq_0 (E_e + ω + q_0) (1 - ( n⃗_k⃗·n⃗_q⃗)^2) [(E_e - k⃗_e ·n⃗_q⃗)× ((E_e - k⃗_e ·n⃗_k⃗) + q_0(1 - n⃗_k⃗·n⃗_q⃗)) + (E_e - k⃗_e ·n⃗_k⃗) ((E_e - k⃗_e ·n⃗_q⃗) + ω (1 - n⃗_k⃗·n⃗_q⃗)) - (m^2_e + ω(E_e - k⃗_e ·n⃗_k⃗)+ q_0(E_e - k⃗_e ·n⃗_q⃗))] + 32 ωq_0 (E_e + ω + q_0) (1 - n⃗_k⃗·n⃗_q⃗) [(k^2_e - (k⃗_e ·n⃗_k⃗)^2) + q_0((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) ×(n⃗_q⃗·n⃗_k⃗))] - 32 ωq_0 (E_e + ω + q_0)(E_e - k⃗_e ·n⃗_q⃗) [ω (1 - ( n⃗_k⃗·n⃗_q⃗)^2) + ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗) (n⃗_k⃗·n⃗_q⃗))]+ 32 ωq_0 (E_e + ω + q_0) (1 - n⃗_k⃗·n⃗_q⃗) [(k^2_e - (k⃗_e ·n⃗_q⃗)^2) + ω ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗) (n⃗_k⃗·n⃗_q⃗))] - 16 ωq_0 (E_e + ω + q_0) (1 - n⃗_k⃗·n⃗_q⃗) [(k^2_e - (k⃗_e ·n⃗_k⃗)^2 - (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗)(k⃗_e ·n⃗_q⃗) (n⃗_k⃗·n⃗_q⃗)) - ω(n⃗_k⃗·n⃗_q⃗) × ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗))]- 16 ωq_0 (E_e + ω + q_0)(E_e - k⃗_e ·n⃗_k⃗) (n⃗_k⃗·n⃗_q⃗) [ω (1 - (n⃗_k⃗·n⃗_q⃗)^2) + ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) (n⃗_q⃗·n⃗_k⃗))]- 16 ωq_0 (E_e + ω + q_0) (n⃗_k⃗·n⃗_q⃗)((E_e - k⃗_e ·n⃗_k⃗) + q_0(1 - n⃗_k⃗·n⃗_q⃗)) ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗) (n⃗_k⃗·n⃗_q⃗))- 16 ωq_0 (E_e + ω + q_0) (n⃗_k⃗·n⃗_q⃗)(E_e - k⃗_e ·n⃗_k⃗) [q_0(1 - (n⃗_k⃗·n⃗_q⃗)^2) + ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) (n⃗_q⃗·n⃗_k⃗))]- 16 ωq_0 (E_e + ω + q_0) (1 - n⃗_k⃗·n⃗_q⃗) [(k^2_e - (k⃗_e ·n⃗_k⃗)^2 - (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗)(k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗))- q_0(n⃗_k⃗·n⃗_q⃗) ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗) (n⃗_k⃗·n⃗_q⃗))] - 32 ωq_0 (E_e + ω + q_0)(E_e - k⃗_e ·n⃗_k⃗) [((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) (n⃗_q⃗·n⃗_k⃗)) +q_0 (1- (n⃗_k⃗·n⃗_q⃗)^2)].The last two traces in Eq.(<ref>) can be obtained from Eq.(<ref>) and Eq.(<ref>) by a replacement ω⟷ q_0 and n⃗_k⃗⟷n⃗_q⃗, respectively.The rate of the neutron radiative β^-–decay with two photons in the final state is defined byλ^( Fig. <ref>)_βγγ = 1/2m_n 1/2∫1/2 ∑_ pol.,λ', λ”|M(n → p e^- ν̅_e γγ)_λ'λ”|^2 (2π)^4 δ^(4)(k_n - k_p - k_e - k_ν - k - q) × d^3k_p/(2π)^3 2 E_p d^3k_e/(2π)^3 2 E_e d^3k_ν/(2π)^3 2 E_ν d^3k/(2π)^3 2 ω d^3q/(2π)^3 2 q_0,where the factor 1/2 in front of the integral takes into account the identity of photons in the final state. A relation of the amplitude M(n → p e^- ν̅_e γγ)_λ'λ” to the amplitude M(n → p e^- ν̅_e γγ)_λ'λ” is given by Eq.(<ref>).Having integrated over the degrees of freedom of the photon with 4–momentum q and keeping the energy of the photon with 4–momentum k within the interval ω_ min≤ω≤ω_ max the rate of the two photon radiative β^-–decay of the neutron is given byλ^( Fig. <ref>)_βγγ(ω_ max, ω_ min)=(1 + 3λ^2) α^2/π^2 G^2_F|V_ud|^2/16π^3∫^ω_ max_ω_ min dω∫^E_0 - ω_m_e dE_ek_eF(E_e, Z = 1)∫^E_0 - E_e - ω_0 dq_0 (E_0 - E_e - ω - q_0)^2 ×∫dΩ_eγ/4π∫dΩ_eγ'/4π∫dΩ_ν/4π 1/2∑_ pol, λ', λ”| M_ Fig. <ref>(n → p e^- ν̅_eγγ)_λ'λ”|^2/8 m^2_n e^2(1 + 3λ^2) E_ν,where dΩ_eγ and dΩ_eγ' are the elements of the solid angles of the electron–photon correlations of photons with 3–momenta k⃗ and q⃗, respectively. Then, we introduce the notation∫dΩ_ν/4π 1/2∑_ pol, λ', λ”| M_ Fig. <ref>(n → p e^- ν̅_eγγ)_λ'λ”|^2/8 m^2_n e^2(1 + 3λ^2) E_ν = ρ^(1)_eγγ'(E_e,k⃗_e, ω, n⃗_k⃗, q_0, n⃗_q⃗) + ρ^(2)_eγγ'(E_e,k⃗_e, ω, n⃗_k⃗, q_0, n⃗_q⃗)+ ρ^(2)_eγγ'(E_e,k⃗_e, q_0, n⃗_q⃗, ω, n⃗_k⃗),where we have denotedρ^(1)_eγγ'(E_e,k⃗_e, ω, n⃗_k⃗, q_0, n⃗_q⃗) = 1/ω q_0/((ω(E_e - k⃗_e·n⃗_k⃗) + q_0(E_e - k⃗_e·n⃗_q⃗) + ωq_0 (1 - n⃗_k⃗·n⃗_q⃗)) ^2 1/(E_e - k⃗_e·n⃗_k⃗)^2 × 1/8 ∑_λ', λ” tr{(m_e + k̂_e)(2k_e·ε^*_λ' + ε̂^*_λ'k̂) ε̂^*_λ”(m_e + k̂_e+ k̂ + q̂) γ^0 (m_e + k̂_e + k̂ +q̂) ε̂_λ”(2k_e·ε_λ' + k̂ ε̂_λ')}andρ^(2)_eγγ'(E_e,k⃗_e, ω, n⃗_k⃗, q_0, n⃗_q⃗) = 1/((ω(E_e - k⃗_e·n⃗_k⃗) + q_0(E_e - k⃗_e·n⃗_q⃗) + ωq_0 (1 - n⃗_k⃗·n⃗_q⃗)) ^2 1/E_e - k⃗_e·n⃗_k⃗1/E_e - k⃗_e·n⃗_q⃗ × 1/16 ∑_λ', λ” tr{(m_e + k̂_e)(2k_e·ε^*_λ” + ε̂^*_λ”q̂) ε̂^*_λ'(m_e + k̂_e +k̂ + q̂) γ^0(m_e + k̂_e + k̂ +q̂) ε̂_λ”(2k_e·ε_λ' + k̂ ε̂_λ')}andρ^(2)_eγγ'(E_e,k⃗_e, q_0, n⃗_q⃗, ω, n⃗_k⃗) = 1/((ω(E_e - k⃗_e·n⃗_k⃗) + q_0(E_e - k⃗_e·n⃗_q⃗) + ωq_0 (1 - n⃗_k⃗·n⃗_q⃗)) ^2 1/E_e - k⃗_e·n⃗_k⃗1/E_e - k⃗_e·n⃗_q⃗ × 1/16 ∑_λ', λ” tr{(m_e + k̂_e)(2k_e·ε^*_λ' + ε̂^*_λ'k̂) ε̂^*_λ”(m_e + k̂_e +k̂ + q̂) γ^0 (m_e + k̂_e + k̂ +q̂) ε̂_λ'(2k_e·ε_λ” + q̂ ε̂_λ”)}.For the trace in Eq.(<ref>) we obtain the following expression ∑_λ', λ” tr{(m_e + k̂_e)(2k_e·ε^*_λ' + ε̂^*_λ'k̂) ε̂^*_λ”(m_e + k̂_e + k̂ + q̂) γ^0 (m_e + k̂_e + k̂ + q̂) ε̂_λ'(2k_e·ε_λ” + q̂ ε̂_λ”)} == - 32m^2_e (E_e + ω + q_0) (k^2_e - (k⃗_e ·n⃗_k⃗)^2 - (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗)(k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) - 32m^2_eω(E_e + ω + q_0) × ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) - 32m^2_e q_0(E_e + ω + q_0) ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗)(n⃗_q⃗·n⃗_k⃗)) - 32m^2_eωq_0 ×(E_e + ω + q_0) (1 - n⃗_k⃗·n⃗_q⃗) - 32E_e(ω (E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗) + ωq_0(1 - n⃗_k⃗·n⃗_q⃗)) (k^2_e - (k⃗_e ·n⃗_k⃗)^2- (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗)(k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) + 32q_0(E_e + ω + q_0) ((E_e - k⃗_e ·n⃗_q⃗) + ω (1 - n⃗_k⃗·n⃗_q⃗)) (k^2_e - (k⃗_e ·n⃗_k⃗)^2- (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗)(k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) + 64(E_e + ω + q_0) (k^2_e - (k⃗_e ·n⃗_q⃗)^2) [(k^2_e - (k⃗_e ·n⃗_k⃗)^2) + q_0((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗)(n⃗_q⃗·n⃗_k⃗))] + 32q_0(ω(E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗) + ωq_0 (1 - n⃗_k⃗·n⃗_q⃗))(k^2_e - (k⃗_e ·n⃗_k⃗)^2) - 32q_0 E_e(ω (E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗) + ωq_0 (1 - n⃗_k⃗·n⃗_q⃗))((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) (n⃗_q⃗·n⃗_k⃗)) - 32q_0 (ω (E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗) + ωq_0 (1 - n⃗_k⃗·n⃗_q⃗))(k^2_e - (k⃗_e ·n⃗_k⃗)^2 - (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗)(k⃗_e ·n⃗_q⃗) × (n⃗_k⃗·n⃗_q⃗))+ 32q_0 (E_e + ω + q_0) ((E_e - k⃗_e ·n⃗_q⃗) + ω(1 - n⃗_k⃗·n⃗_q⃗)) (k^2_e - (k⃗_e ·n⃗_k⃗)^2 - (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗) × (k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) + 32q_0 (E_e + ω + q_0) ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) (n⃗_q⃗·n⃗_k⃗)) [(k^2_e - (k⃗_e ·n⃗_q⃗)^2) + ω ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗) ×(n⃗_k⃗·n⃗_q⃗))] + 32q_0 (E_e + ω + q_0) (m^2_e + ω (E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗))((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗)(n⃗_q⃗·n⃗_k⃗)) - 32q_0 (E_e + ω + q_0) ((E_e - k⃗_e ·n⃗_q⃗) + ω(1 - n⃗_k⃗·n⃗_q⃗)) (k^2_e - (k⃗_e ·n⃗_k⃗)^2) + 32q_0 (E_e + ω + q_0) (E_e - k⃗_e ·n⃗_q⃗) × [ (k^2_e - (k⃗_e ·n⃗_k⃗)^2) + q_0 ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗)(n⃗_q⃗·n⃗_k⃗))] - 32ω E_e(ω(E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗)+ ωq_0 (1 - n⃗_k⃗·n⃗_q⃗)) ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) + 32 ω (ω(E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗)+ ωq_0 (1 - n⃗_k⃗·n⃗_q⃗)) (k^2_e - (k⃗_e ·n⃗_q⃗)^2) - 32 ω (ω(E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗) + ωq_0 (1 - n⃗_k⃗·n⃗_q⃗))× (k^2_e - (k⃗_e ·n⃗_k⃗)^2- (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗)(k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) + 32 ω(E_e + ω + q_0) (m^2_e + ω (E_e - k⃗_e ·n⃗_k⃗) + q_0(E_e - k⃗_e ·n⃗_q⃗)) ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) + 32 ω(E_e + ω + q_0)(E_e - k⃗_e ·n⃗_q⃗) × [(k^2_e - (k⃗_e ·n⃗_q⃗)^2) + ω ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗))] - 32 ω(E_e + ω + q_0) ((E_e - k⃗_e ·n⃗_k⃗)+ q_0 (1 - n⃗_k⃗·n⃗_q⃗)) (k^2_e - (k⃗_e ·n⃗_q⃗)^2 ) + 32 ω(E_e + ω + q_0) ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗))[(k^2_e - (k⃗_e ·n⃗_k⃗)^2)+ q_0 ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) (n⃗_q⃗·n⃗_k⃗))] + 32 ω(E_e + ω + q_0) ((E_e - k⃗_e ·n⃗_k⃗) + q_0 (1 - n⃗_k⃗·n⃗_q⃗)) (k^2_e - (k⃗_e ·n⃗_k⃗)^2- (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗)(k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗)) - 8m^2_eωq_0(1 - (n⃗_k⃗·n⃗_q⃗)^2) (E_e (1 + n⃗_k⃗·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗))- 8m^2_eωq_0 (n⃗_k⃗·n⃗_q⃗) ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗)) (n⃗_k⃗·n⃗_q⃗)) + 8m^2_eωq_0 (E_e - q_0)(n⃗_q⃗·n⃗_k⃗)(1 - (n⃗_q⃗·n⃗_k⃗)^2)+ 8 ωq_0 (1 - (n⃗_k⃗·n⃗_q⃗)^2) (m^2_e + 2 ω(E_e - k⃗_e ·n⃗_k⃗) + 2q_0(E_e - k⃗_e ·n⃗_q⃗) + 2 ωq_0(1 - n⃗_k⃗·n⃗_q⃗)) + [8 ω (m^2_e + 2 ω(E_e - k⃗_e ·n⃗_k⃗) + 2q_0(E_e - k⃗_e ·n⃗_q⃗) + 2 ωq_0(1 - n⃗_k⃗·n⃗_q⃗)) - 16 ωq_0(E_e + ω + q_0) × ( (E_e - k⃗_e ·n⃗_k⃗) + q_0(1 - n⃗_k⃗·n⃗_q⃗))](n⃗_k⃗·n⃗_q⃗) ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗) ( n⃗_k⃗·n⃗_q⃗)) - 8 ωq_0[ (E_e - q_0) × (m^2_e + 2 ω(E_e - k⃗_e ·n⃗_k⃗) + 2q_0(E_e - k⃗_e ·n⃗_q⃗) + 2 ωq_0(1 - n⃗_k⃗·n⃗_q⃗)) - 2(E_e + ω + q_0) (m^2_e + ω(E_e - k⃗_e ·n⃗_k⃗)+ q_0(E_e - k⃗_e ·n⃗_q⃗))]( n⃗_k⃗·n⃗_q⃗)(1 - ( n⃗_k⃗·n⃗_q⃗)^2) - 16 ωq_0 (E_e + ω + q_0) (1 - ( n⃗_k⃗·n⃗_q⃗)^2) [(E_e - k⃗_e ·n⃗_q⃗)× ((E_e - k⃗_e ·n⃗_k⃗) + q_0(1 - n⃗_k⃗·n⃗_q⃗)) + (E_e - k⃗_e ·n⃗_k⃗) ((E_e - k⃗_e ·n⃗_q⃗) + ω (1 - n⃗_k⃗·n⃗_q⃗)) - (m^2_e + ω(E_e - k⃗_e ·n⃗_k⃗)+ q_0(E_e - k⃗_e ·n⃗_q⃗))] + 32 ωq_0 (E_e + ω + q_0) (1 - n⃗_k⃗·n⃗_q⃗) [(k^2_e - (k⃗_e ·n⃗_q⃗)^2) + ω ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗) ×(n⃗_k⃗·n⃗_q⃗))] - 32 ωq_0 (E_e + ω + q_0)(E_e - k⃗_e ·n⃗_k⃗) [q_0(1 - ( n⃗_k⃗·n⃗_q⃗)^2) + ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) (n⃗_q⃗·n⃗_k⃗))] + 32 ωq_0 (E_e + ω + q_0) (1 - n⃗_k⃗·n⃗_k⃗) [(k^2_e - (k⃗_e ·n⃗_k⃗)^2) + q_0((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) (n⃗_q⃗·n⃗_k⃗))] - 16 ωq_0 (E_e + ω + q_0) (1 - n⃗_k⃗·n⃗_q⃗) [(k^2_e - (k⃗_e ·n⃗_k⃗)^2 - (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗)(k⃗_e ·n⃗_q⃗) (n⃗_k⃗·n⃗_q⃗)) - q_0 (n⃗_k⃗·n⃗_q⃗) × ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗))]- 16 ωq_0 (E_e + ω + q_0)(E_e - k⃗_e ·n⃗_q⃗) (n⃗_k⃗·n⃗_q⃗) [q_0(1 - (n⃗_k⃗·n⃗_q⃗)^2) + ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗) (n⃗_k⃗·n⃗_q⃗))]- 16 ωq_0 (E_e + ω + q_0) (n⃗_k⃗·n⃗_q⃗)((E_e - k⃗_e ·n⃗_q⃗) + ω(1 - n⃗_k⃗·n⃗_q⃗)) ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) (n⃗_q⃗·n⃗_k⃗))] - 16 ωq_0 (E_e + ω + q_0)(E_e - k⃗_e ·n⃗_q⃗) (n⃗_k⃗·n⃗_q⃗) [ω (1 - (n⃗_k⃗·n⃗_q⃗)^2) + ((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗) (n⃗_k⃗·n⃗_q⃗))]- 16 ωq_0 (E_e + ω + q_0) (1 - n⃗_k⃗·n⃗_q⃗) [(k^2_e - (k⃗_e ·n⃗_k⃗)^2 - (k⃗_e ·n⃗_q⃗)^2 + (k⃗_e ·n⃗_k⃗)(k⃗_e ·n⃗_q⃗)(n⃗_k⃗·n⃗_q⃗))- ω(n⃗_k⃗·n⃗_q⃗) ((k⃗_e ·n⃗_q⃗) - (k⃗_e ·n⃗_k⃗) (n⃗_q⃗·n⃗_k⃗))] - 32 ωq_0 (E_e + ω + q_0)(E_e - k⃗_e ·n⃗_q⃗) [((k⃗_e ·n⃗_k⃗) - (k⃗_e ·n⃗_q⃗) (n⃗_k⃗·n⃗_q⃗)) + ω (1 - (n⃗_k⃗·n⃗_q⃗)^2)].Thus, the rate of the neutron radiative β^-–decay with two photons in the final state for one of the photons from the energy region ω_ min≤ω≤ω_ max is given byλ^( Fig. <ref>)_βγγ(ω_ max, ω_ min) = (1 + 3λ^2) α^2/π^2 G^2_F|V_ud|^2/16π^3∫^ω_ max_ω_ min dω∫^E_0 - ω_m_e dE_ek_eF(E_e, Z = 1)∫^E_0 - E_e - ω_0 dq_0 (E_0 - E_e - ω - q_0)^2 ×∫dΩ_eγ/4π∫dΩ_eγ'/4π (ρ^(1)_eγγ'(E_e,k⃗_e, ω, n⃗_k⃗, q_0, n⃗_q⃗) + ρ^(2)_eγγ'(E_e,k⃗_e, ω, n⃗_k⃗, q_0, n⃗_q⃗) + ρ^(2)_eγγ'(E_e,k⃗_e, q_0, n⃗_q⃗, ω, n⃗_k⃗)).For the numerical calculation we use the following definitions ∫dΩ_eγ/4π … = 1/4 π∫^π_0dϑ_eγ sinϑ_eγ∫^2π_0dφ_eγ …, ∫dΩ_eγ'/4π … = 1/4 π∫^π_0dϑ_eγ' sinϑ_eγ'∫^2π_0dφ_eγ' …, k⃗_e ·n⃗_k⃗ = k_e cosϑ_eγ, k⃗_e ·n⃗_q⃗ = k_e cosϑ_eγ', n⃗_k⃗·n⃗_q⃗ = cosϑ_eγ cosϑ_eγ' + sinϑ_eγ sinϑ_eγ' cos(φ_eγ - φ_eγ').The integrals in Eq.(<ref>) we calculate for three photon energy regions 15keV≤ω≤ 340keV, 14keV≤ω≤ 782keV and 0.4keV≤ω≤ 14keV, respectively. 9 Berman1958 S. 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Michel, IL Nuovo Cimento 16, 560 (1960).Kim1973 Kwang Je Kim,Annals Phys. 79, 287 (1973).Johnson1998 A. S. Johnson, J. A. McNeil, and J. R. Shepard, Phys. Rev. D 58, 014001 (1998).Pisarski1984 R. D. Pisarski and F. Wilczek, Phys. Rev. D 29, 338 (1984).Grahl2013 M. Grahl and D. H. Rischke, Phys. Rev. D 88, 056014 (2013).Weinberg1967 S. Weinberg,Phys. Rev. Lett. 18 188 (1967).Gasiorowicz1969 S. Gasiorowicz and D. A.Geffen, Rev. Mod. Phys. 41, 531 (1969). | http://arxiv.org/abs/1706.08687v1 | {
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We point out that two of Milne's fourth-order integrators are well-suited to bit-reversible simulations. The fourth-order method improves on the accuracy of Levesque and Verlet's algorithm and simplifies the definition of the velocity v and energy e = (q^2 + v^2)/2 .( We use this one-dimensional oscillator problem as an illustration throughout this paper ). Milne's integrator is particularly useful for the analysis of Lyapunov ( exponential ) instability in dynamical systems, including manybody molecular dynamics.We include the details necessary to the implementation of Milne's Algorithms.Bit-Reversible Version of Milne's Fourth-Order Time-Reversible Integrator for Molecular Dynamics William Graham Hoover and Carol Griswold Hoover Ruby Valley Research InstituteHighway Contract 60, Box 601Ruby Valley, Nevada 89833December 30, 2023 =============================================================================================================================================================================================================§ INTRODUCTION William Milne's 1949 work Numerical Calculus<cit.> was republished by the Princeton University Press in 2015. The book is a particularly valuable source of clear and direct numerical methods. Research workers in statistical mechanics, molecular dynamics, and dynamical systems will find his approach to what is our own research interest, solving and analyzing differential equations for chaotic systems small and large, reliable and useful.Writing about a decade prior to the computer revolution Milne had no particular interest in “reversible computing” and the “bit-reversible” algorithms which make it possible to extend sequences of coordinates forward and backward in time stably and reversibly ad infinitum.Nevertheless his work is directly applicable to such finite-difference applications.In 1993 Dominique Levesque and Loup Verlet used an integer algorithm to solve problems in Newtonian mechanics with perfect time reversibility<cit.>. Loup had popularized Størmer and Newton's Leapfrog Algorithm a quarter century earlier, in the early days of molecular dynamics<cit.>,q_t+dt - 2q_t + q_t-dt = a_t(dt)^2.“Verlet's algorithm” appears on page 140 of Reference 1. If the righthand side of this finite-difference algorithm is truncated to an integer the resulting acceleration is precisely the same ( to the very last computational “bit” ) in either direction of time.Because this algorithm conserves phase volume when written in a “symplectic” centered-difference form :q_t+(dt/2) = q_t + v_t(dt/2);v_t+dt = v_t + a_t+(dt/2)dt;q_t+dt = q_t+(dt/2) + v_t+dt(dt/2),there is no tendency for energy drift. The errors in the velocity and energy in the leapfrog algorithm are unnecessarily large, so that two of Milne's algorithms ( both of them also on page 140 of Reference 1 ) can provide better accuracy for longer runs :q_t+2dt - q_t+dt - q_t-dt + q_t-2dt = [5a_t+dt + 2a_t + 5a_t-dt ](dt^2/4).The error, ≃(17/240)dt^6, is quite tolerable relative to the Størmer error, ≃ (1/12)dt^4. Milne also gives an even better corrector formula with an error ≃(-1/240)dt^6 .q_t+2dt - 2q_t+dt + q_t = [a_t+2dt + 10a_t+dt + a_t ](dt^2/12). § APPLICATIONS For several years now<cit.> we have been exploring the differences in Lyapunov spectra forward and backward in time in order to get insight into the Second Law of Thermodynamics.The fractal structures which arise in nonequilibrium deterministic and time-reversible steady-state problems provide explanations to both Loschmidt's Reversibility paradox and Zermélo's Recurrence paradox<cit.>.Levesque and Verlet's integer algorithm has proved to be a useful tool in these studies despite its relatively coarse description of particle trajectories.Integer algorithms are also useful in studies of the effects of finite precision ( single, double, quadruple, ... ) on phase-space distributions generated by flows and maps<cit.>. Mauricio Romero-Bastida<cit.> suggested the use of the integer-based leapfrog algorithm for generating a reversible reference trajectory of arbitrary length in his studies of the “covariant” Lyapunov exponents.In 2013 we were able to see a qualitative difference between the “important particles” ( those making above-average contributions to the Lyapunov instability ) forward and backward in time in the example inelastic-collision problem of Figure 1<cit.>.Continuing progress in low-cost computation caused us to revisit these problems in connection with a lecture course delivered at Kharagpur's Indian Institute of Technology in December 2016<cit.>.We were very pleased to find that Milne's work offers an improvement in the precision and accuracy of these Lyapunov studies and believe that others will find this approach useful to their own work.Although these improvements are not at all “new” we do expect that this work will accelerate progress in understanding the time-reversible simulation of irreversible processes.Figure 1 illustrates the important particles ( those making above-average contributions to the largest Lyapunov exponent ) forward and backward in time for the collision of two 400-particle balls. The 162 important particles forward in time are those blacked in along the interface between the balls while the 120 important particles backward in time are those blacked in in the necking regions where the plastic strain is greatest as the balls are separating.This simulation employed the Levesque-Verlet algorithm for the reference trajectory and a Runge-Kutta fourth-order algorithm for the two satellite trajectories ( one forward and another backward ) as is described in Reference 10. § NUMERICAL IMPLEMENTATION OF MILNE'S ALGORITHMS To illustrate the application of Milne's Algorithmswe consider an integer version of the simpler of his two fourth-order algorithms.We describe a harmonic oscillator with q̈ = -q . The preliminaries, which we give below, provide integer forms for five previous coordinates and the corresponding contributions to the acceleration, all of them multiplied by 10^15. We select an example timestep of (π/50) in the Fortran instructions so that an oscillator period corresponds to one hundred timesteps.We carried out two kinds of tests for the Milne integrator, reversibility, confirming that reversing the four prereversal coordinates exactly reverses the sequence of integers back to the initial value of 10^15. It is easy to show that the algorithm is exactly reversible in this way.Stability can be confirmed by solving for the dependence of the oscillation frequency on the timestep. Numerical work consistent with the linear analysis for the oscillator ( given in more detail in a Postscript ) shows that the dependence of the phase shift is quartic in the timestep for the range 0 < dt < 1.A direct simulation of the integer version of the algorithm, using two billion timesteps with dt = 0.2, showed no tendencytoward damping or instability.Similar results can be obtained by solving the floating-point version of the problem, where precise reversibility has to be abandoned ( because roundoff error will spoil it ). These results establish that the Milne algorithm is both reversible and stable for the oscillator. We recommend it to our colleagues for their use.The implementation of the algorithm is to some extent hardware dependent. On our various Mac computers using the free gnu compiler we had no trouble using 16-byte integers, giving roughly 15 digits for arithmetical operations.The following extract from the setup of the computation generates the inital data ( in this case four points from a cosine curve ) as well as the three integers, proportional to dt^2× 10^15, andneeded for the accelerations.On the following page we summarize the time-stepping loop where the three accelerations are expressed as integers { IAP,IA0,IAM }.We include at the end an indication of the coordinate reversal procedure needed to integrate backward.Here follows a bare-bones evolution loop for the integer coordinates. After ITMAX iterations the coordinate reversal steps make it possible to return precisely to, and beyond, the beginning. In the event that the velocities are to be calculated from Milne's fifth-order interpolation ( which is one order of overkill ) it is necessary to compute the integer coordinates to include IQPPP and IQMMM, getting IQPPP from the “step” IQPPP = IQPP + IQ0 - IQM - (IAPP + IAP + IA0).There is no difficulty in computing an accurate velocity with Milne's page 99 formula using six centered coordinates. This fifth-order interpolation gives not only good velocities, but also an accurate energy.Accurate values of these phase variables are a real advantage of the Milne algorithm over that of Levesque and Verlet.§ POSTSCRIPT ON THE STABILITY OF MILNE'S ALGORITHM The stability analysis for Milne's algorithm is straightforward. If we substitute the trial solution q ∝ e^iω t into the wholly linear algorithm the result is :cos(2ω dt) - cos(ω dt) + (dt^2/4)[5cos(ω dt) + 1] = 0.This simplifies to a quadratic equation in cos(ω dt) :2C^2 + [(5dt^2/4) - 1]C+ (dt^2/4) - 1= 0 where C ≡cos(ω dt) .Figure 2 shows that the dependence of the frequency error on the timestep is quartic, (1 - ω) ∝ dt^4, confirming the stability of the algorithm.§ ACKNOWLEDGEMENT The interest and support of Harald Posch ( Universität Wien ), Clint Sprott ( University of Wisconsin-Madison ), and Karl Travis ( University of Sheffield ) are gratefully acknowledged.99b1W. E. Milne, Numerical Calculus – Approximations, Interpolation, Finite Differences, Numerical Integration, and Curve Fitting (Princeton University Press, 2015).b2D. Levesque and L. Verlet, “Molecular Dynamics and Time Reversibility”, Journal of Statistical Physics 72, 519-537 (1993).b3L. Verlet, “ `Computer Experiments' on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules”, Physical Review 159, 98-103 (1967).b4H. A. Posch and W. G. Hoover, “Large-System Phase-Space Dimensionality Loss in Stationary Heat Flows”, Physica D 187, 281-293 (2004).b5O. Kum and Wm. G. Hoover, “Time-Reversible Continuum Mechanics”,Journal of Statistical Physics 76, 1075-1081 (1994).b6B. L. Holian, Wm. G. Hoover, and H. A. Posch, “Resolution of Loschmidt's Paradox: The Origin of Irreversible Behavior in Reversible Atomistic Dynamics”, Physical Review Letters 59, 10-13 (1987).b7C. Grebogi, E. Ott, and J. A. Yorke, “Roundoff-Induced Periodicity and the Correlation Dimension of Chaotic Attractors”, Physical Review A 38, 3688-3692 (1988).b8C. Dellago and Wm. G. Hoover, “Finite-Precision Stationary States At and Away from Equilibrium”, Physical Review E 62, 6275-6281 (2000).b9M. Romero-Bastida, D. Pazó, J. M. Lopéz, and M. A. Rodriguez, “Structure of Characteristic Lyapunov Vectors in Anharmonic Hamiltonian Lattices”, Physical Review E 82, 036205 (2010).b10Wm. G. Hoover and Carol G. Hoover, “Time-Symmetry Breaking in Hamiltonian Mechanics”, Computational Methods in Science and Technology 19, 77-87 (2013).b11Wm. G. Hoover and C. G. Hoover, The Kharagpur Lectures (World Scientific, Singapore, 2018, in preparation). | http://arxiv.org/abs/1706.08678v2 | {
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1Department of Astronomical Science, School of Physical Science, SOKENDAI (The Graduate University for Advanced Studies), Osawa, Mitaka, Tokyo 181-8588, Japan 2Nobeyama Radio Observatory, National Astronomical Observatory of Japan, Minamimaki, Minamisaku, Nagano 384-1305, Japan [email protected] [email protected] astrochemistry — ISM: individual objects (Taurus Molecular Cloud-1) — ISM: molecules LETTER First Detection of HC_5^15N in the Interstellar Medium Masao saito 1,2 Received: date / Accepted: date ================================================================ We report the first detection of HC_5^15N with the J=9-8 rotational line from the cyanopolyyne peak in Taurus Molecular Cloud-1 (TMC-1 CP) using the 45-m radio telescope of the Nobeyama Radio Observatory. The column density of HC_5^15N is derived to be (1.9 ± 0.5)× 10^11 cm^-2 (1σ). We apply the double isotope method to derive the ^14N/^15N ratios of HC_5N and HC_3N in TMC-1 CP. The ^14N/^15N ratios are calculated to be 344 ± 53 and 257 ± 54 for HC_5N and HC_3N, respectively. The ^14N/^15N ratio of HC_5N is lower than the elemental ratio in the local interstellar medium (∼ 440) and slightly higher than that of HC_3N in TMC-1 CP. Since HC_3N is formed via the neutral-neutral reaction between C_2H_2 and CN, the slightly higher ^14N/^15N ratio of HC_5N may support our previous suggestions that the main formation mechanism of HC_5N is the ion-molecule reactions between hydrocarbon ions (C_5H_n^+) and nitrogen atoms. § INTRODUCTIONCarbon-chain molecules are unique species in the interstellar medium, and the studies about their chemical mechanisms have been progressed mainly by radio astronomical observations. Survey observations showed that carbon-chain molecules are good chemical evolutional tracers <cit.>. Carbon-chain molecules such as CCS are abundant in young low-mass dark clouds and decrease in evolved star-forming cores. The tendency is explained by the chemical characteristics of carbon-chain species; they are formed from carbon cations (C^+) or carbon atoms (C) <cit.>, destroyed by the reactions with H^+, He^+, or O, and depleted onto dust grains <cit.>. However, since carbon-chain species have unsaturated bonds, they are unstable, and hence it is difficult to derive their main formation mechanisms from laboratory experiments. Consequently, the chemical network model calculations about carbon-chain molecules have large uncertainties despite many attempts.Recent development of the radio astronomical equipment allows us to detect weak lines, including isotopologues of carbon-chain molecules, within reasonable time. In order to investigate main formation mechanisms of carbon-chain molecules using their ^13C isotopic fractionation (the differences in abundance among the ^13C isotopologues), various observations deriving ^13C isotopic fractionation have been carried out in HC_3N <cit.>, HC_5N <cit.>, CCH <cit.>, CCS <cit.>, C_3S, and C_4H <cit.>toward the cyanopolyyne peak in Taurus Molecular Cloud-1 (TMC-1 CP; d = 140 pc), and in cyclic-C_3H_2 <cit.> toward the low-mass star-forming region L1527 (d = 140 pc). The main formation pathway of HC_3N in TMC-1 CP was suggested as the neutral-neutral reaction between C_2H_2 and CN, from the abundance ratios of [H^13CCCN]: [HC^13CCN]: [HCC^13CN] = 1.0 : 1.0 : 1.4 <cit.>. The observed abundance ratios can be explained by the reaction of C_2H_2 + CN, because C_2H_2 has two equivalent carbon atoms and ^13C tends to concentrate in CN via the exothermic reaction between ^13C^+ and CN <cit.>. <cit.> also carried out observations deriving ^13C isotopic fractionation of HC_3N toward the low-mass star-forming region L1527 and the high-mass star-forming region containing a hot core G28.28-0.36 (d=3 kpc).They suggested that the main formation pathways of HC_3N in the both star-forming regions are the same one as that in TMC-1 CP (C_2H_2 + CN).On the other hand, <cit.> proposed that the main formation mechanism of HC_5N in TMC-1 CP is the ion-molecule reactions between hydrocarbon ions (C_5H_n^+; n=3-5) and nitrogen atoms followed by the electron recombination reactions, based on the observational results showing the abundance ratios of [H^13CCCCCN]: [HC^13CCCCN]: [HCC^13CCCN]: [HCCC^13CCN]: [HCCCC^13CN] = 1.00:0.97:1.03:1.05:1.16 (± 0.19) (1σ). In the proposed ion-molecule reactions, all carbon atoms in HC_5N originate from the hydrocarbon ions.Such large hydrocarbon ions are produced through various processes, and there is no reason that ^13C is concentrated in a particular carbon atom in large hydrocarbon ions.In other words, there should be no clear ^13C isotopic fractionation. One difficulty using the ^13C isotopic fractionation method is that the differences in the ^12C/^13C ratio of each isotopologue are small, and we need long integration time to obtain spectra with sufficient signal-to-noise ratios.In the present letter, we report the first detection of HC_5^15N from TMC-1 CP. We derive its column density and the ^14N/^15N ratio of HC_5N. From the ^14N/^15N ratio, we suggest the ion-molecule reactions as the main formation mechanism of HC_5N in TMC-1 CP also suggested from our previous work of the ^13C isotopic fractionation of HC_5N. § OBSERVATIONSWe carried out observations of HC_5^15N (J=9-8; 23.37544 GHz <cit.>) with the Nobeyama 45-m radio telescope during 2014 December and 2015 January (2014-2015 season)[<cit.> described the observation date as 2014 March, April (2013-2014 season), December and 2015 January (2014-2015 season). In 2014 March and April, we carried out observations only in the 42 GHz band using the Z45 receiver. The observations in the 23 GHz band using the H22 receiver were conducted only in 2014 December and 2015 January; the data presented here were taken simultaneously with the normal species and the five ^13C isotopologues of HC_5N in the 23 GHz band in 2014 December and 2015 January.]. The observed position was (α_2000, δ_2000) = (04^ h41^ m4249, 2541270) for TMC-1 CP. The off-source position was set to be +30' away in the right ascension. We checked the telescope pointing every 1.5 hr by observing the SiO maser line (J=1-0) from NML Tau. The pointing error was less than 3".We used the H22 receiver, which enables us to obtain dual polarization data simultaneously. The H22 receiver is a single sideband (USB) receiver with its gain above 25 dB. The beam size and the main beam efficiency (η_B) were 72" and 0.8, respectively.The system temperatures were from 90 to 110 K, depending on the weather conditions and elevations. We used the SAM45 FX-type digital correlator in frequency setups whose bandwidths and frequency resolutions were 63 MHz and 15.26 kHz, respectively. The frequency resolution corresponds to the velocity resolution of 0.2 km s^-1.We used the chopper wheel method. We then estimated the absolute intensity calibration error at 10%, which is a typical value for the chopper wheel method.We employed the Smoothed Bandpass Calibration (SBC) method <cit.>. The SBC method allows us to reduce the time for observing off-source positions. The scan pattern was set as 20 seconds and 5 seconds for on-source and off-source positions, respectively. We applied 60 channel-smoothing only for off-source spectra.§ RESULTS AND ANALYSIS§.§ ResultsThe rotational line of HC_5^15N was clearly detected with the signal-to-noise ratio of 7, as shown in Figure <ref>. The on-source integration time is 45 hours 2 minutes, and the rms noise level in the line-free region is 2.4 mK in T_A^* with the velocity resolution of 0.2 km s^-1. We fitted the spectra with a Gaussian profile, and obtained the spectral line parameters. The value of V_LSR of the line is 5.7 ± 0.3 km s^-1, which is consistent with the systemic velocity of TMC-1 CP (5.85 km s^-1). The peak intensity (T_A^*), the line width (FWHM), and the integrated intensity (∫ T^∗_Adv) are 17 ± 2 mK, 0.42 ± 0.07 km s^-1, and 0.007 ± 0.002 K km s^-1 (1σ), respectively. The errors of the line parameters were derived from the Gaussian fitting.We verified the line identification from the line width and rest frequency. First, the derived line width (0.42 ± 0.07 km s^-1) is consistent with the typical value in TMC-1 CP (0.5 km s^-1, <cit.>) and the spectrum does not appear a spiky instrumental spurious. Second, there is no other detectable line in TMC-1 CP in the 23.35-23.4 GHz band according to the Splatalogue database for astronomical spectroscopy[http://www.cv.nrao.edu/php/splat/]. Thus, we concluded that the detected emission line should be identified as HC_5^15N. §.§ AnalysisWe calculated the column density assuming the local thermodynamic equilibrium (LTE). We used the following formulae <cit.>: τ = - ln[1- T^∗_ A/fη_ B{J(T_ex) - J(T_bg) }],whereJ(T) = hν/k{exp(hν/kT) -1} ^-1,andN = τ3hΔ v/8π ^3√(π/4ln2)Q1/μ ^21/J_lower+1exp(E_lower/kT_ex) ×{1-exp(-hν/kT_ex)} ^-1. In Equation (<ref>), T^∗_ A denotes the antenna temperature, f the beam filling factor, η_ B the main beam efficiency (0.8, Section <ref>), and τ the optical depth. Since a size of the emitting region of carbon-chain molecules in TMC-1 CP is approximately 2.5' according to the mapping observations by <cit.>, we used 0.8 for f.T_ex is the excitation temperature, T_bg is the cosmic microwave background temperature (≈ 2.7 K), and J(T) in Equation (<ref>) is the Planck function.In the calculation, we adopted the excitation temperature of 6.5 K of the normal species <cit.>. In Equation (<ref>), N denotes the column density, Δ v the line width (FWHM), Q the partition function, μ the permanent electric dipole moment of HC_5^15N (4.33 D; <cit.>), and E_lower the energy of the lower rotational energy level. The derived column density of HC_5^15N is (1.9 ± 0.5)× 10^11 cm^-2 (1σ). The ^14N/^15N ratio of HC_5N is determined to be 323 ± 80 (1σ), using the column density of the normal species ((6.2 ± 0.3) × 10^13 cm^-2, <cit.>). <cit.> derived the optical depth of the normal species to be 1.084 ± 0.014 (1σ), and the uncertainty of the column density of the normal species should be small. The errors contain the 1σ errors from the Gaussian fitting and the 10% absolute intensity uncertainty from the chopper wheel method.§ DISCUSSION§.§ Deriving the ^14N/^15N ratio of HC_5N using the double isotope method The ^14N/^15N ratios are often derived with the double isotope method, using the two isotopologues (e.g., <cit.>). In the case of HC_5N, we can calculate five cases independently, and reduce the uncertainty in the ^14N/^15N ratio statistically. We observed the normal species, the five ^13C isotopologues, and the ^15N isotopologue simultaneously. Therefore, we can exclude the pointing and calibration errors. We calculated the ^14N/^15N ratios of HC_5N with the double isotope method, using the following formula:HC_5 ^14N/HC_5 ^15N = < HC_m ^13CC_nN/HC_5 ^15N×HC_5 ^14N/HC_m ^13CC_nN>where m, n = 0-4 (m+n = 4).In Equation (<ref>), HC_5^14N/HC_m^13CC_nN represents the ^12C/^13C ratios (column of ^12C/^13C in Table <ref>, taken from <cit.>). We derived HC_m^13CC_nN/HC_5^15N using the integrated intensities (column of x/HC_5^15N in Table <ref>), because both ^15N and ^13C isotopologues are optically thin. < > denotes the mean value. We calculated the HC_5^14N/HC_5^15N ratio in the case of m = 0 and n=4 using Equation (<ref>). We repeated the calculations until m=4 and n=0, and finally we averaged the five HC_5^14N/HC_5^15N ratios. The calculated five ^14N/^15N ratios of HC_5N are summarized in Table <ref> (column of ^14N/^15N), and these values are consistent with each other. The averaged ^14N/^15N ratio of HC_5N was derived to be 344 ± 53 (1σ). The derived ^14N/^15N ratio of HC_5N using the double isotope method is consistent with 323 ± 80 (1σ) derived in Section <ref>. The ^14N/^15N ratio of HC_5N in TMC-1 CP is smaller than that of the elemental ratio of ∼ 440 <cit.>.§.§ Comparison of the ^14N/^15N ratios We also derived the ^14N/^15N ratio of HC_3N in TMC-1 CP using the double isotope method from the previous observational results with the Nobeyama 45-m telescope. We derived the integrated intensity ratios (column of x/HC_3^15N in Table <ref>) using the J=4-3 rotational transition at the 36 GHz band from <cit.>. <cit.> observed the normal species and the three ^13C isotopologues of HC_3N, and derived the three ^12C/^13C ratios.We assumed that the error in the result of <cit.> is 20%. The derived three ^14N/^15N ratios of HC_3N are listed in Table <ref> (column of ^14N/^15N). We estimated its ^14N/^15N ratio to be 257 ± 54 (1σ).We derived the column density of HC_3^15N using Equations (<ref>) - (<ref>) and the line parameters of its J=4-3 rotational line taken from <cit.>. We used the line, because we can assume that the filling factor is almost unity. We then used the beam filling factor of unity and the main beam efficiency of 0.8[http://www.nro.nao.ac.jp/ nro45mrt/html/prop/eff/eff_before2001.html#period5]. We used the excitation temperature of its normal species (7.1 K, <cit.>). We assumed that the uncertainty of the integrated intensity is 20%. The derived column density of HC_3^15N is (5.9 ± 0.5)× 10^11 cm^-2. The column density of the normal species is (1.6 ± 0.1)× 10^14 cm^-2 <cit.>. Therefore, the derived ^14N/^15N ratio of HC_3N is 270 ± 57 (1σ)[We divided the integrated intensity of HC_3^15N by a scaling factor of 1.3 to correct the difference between <cit.> and <cit.>. The scaling factor of 1.3 was derived by comparison of the integrated intensities of H^13CCCN (J=4-3) between <cit.> and <cit.>.]. The ^14N/^15N ratios derived by the two methods are well consistent with each other. §.§ Main formation mechanism of HC_5NThe elemental ^14N/^15N ratio in the local interstellar medium was estimated to be 441 ± 6 from the solar wind <cit.>. In addition, although the ^14N/^15N ratio of CN could not be derived due to a non-thermal intensity ratio of the hyperfine lines of the normal species of CN in TMC-1 <cit.>, ^15N generally tends to concentrate in CN molecules in cold environments <cit.>. The ^14N/^15N ratio of HC_3N in TMC-1 CP is smaller than that of the elemental ratio of 441 ± 6 <cit.>.The results suggest that N in HC_3N does not come from nitrogen atoms, but originates from CN, which agrees with the HC_3N formation pathway suggested from the ^13C isotopic fractionation <cit.>. In addition, the small ^14N/^15N ratio of HC_3N implies that ^15N is concentrated in CN molecules in TMC-1 CP, as suggested by the model calculation <cit.>. On the other hand, the derived ^14N/^15N ratio of HC_5N in TMC-1 CP is slightly higher than that of HC_3N in TMC-1 CP, if we take the ratio determined from the double isotope method.Hence, the results may suggest that the main formation mechanism of HC_5N is different from that of HC_3N.The possible formation pathways of HC_5N in TMC-1 CP were discussed in <cit.>, and they categorized the pathways into three mechanisms as following:Mechanism 1: the reactions of C_4H_2 + CN,Mechanism 2: the growth of the cyanopolyyne carbon chains via C_2H_2^+ + HC_3N, and Mechanism 3: the reactions between hydrocarbon ions and nitrogen atoms followed by electron recombination reactions. Only Mechanism 3 does not contain CN molecules intrinsically, and N in HC_5N originates from nitrogen atoms. The derived ^14N/^15N ratio of HC_5N in TMC-1 CP (344 ± 53 (1σ)) suggests that Mechanism 3 dominates the formation of HC_5N. The small difference in the ^14N/^15N ratio between HC_5N and the elemental ratio seems to imply that the reactions containing CN molecules (Mechanisms 1 and 2) partly contribute to the formation of HC_5N, as suggested by <cit.>. § CONCLUSIONS We have detected HC_5^15N from TMC-1 CP for the first time in the interstellar medium. Its column density is derived to be (1.9 ± 0.5)× 10^11 cm^-2 (1σ). The ^14N/^15N ratio of HC_5N is calculated to be 323 ± 80 (1σ). In addition, we evaluate its ^14N/^15N ratio using the double isotope method, and the value is determined at 344 ± 53 (1σ). The ^14N/^15N ratios derived from the column density and the double isotope method are consistent with each other. We also derived the ^14N/^15N ratio of HC_3N in TMC-1 CP using the previous observational results. The ratio is derived to be 257 ± 54 (1σ) from the double isotope method. The ^14N/^15N ratio of HC_3N is derived to be 270 ± 57 (1σ) using the column density. The double isotope method resulted in slightly higher ^14N/^15N ratio of HC_5N than HC_3N in TMC-1 CP. The results may support our previous work, proposing that the main formation mechanism of HC_5N is the ion-molecule reactions between hydrocarbon ions and nitrogen atoms followed by the electron recombination reactions, but other formation pathways including CN partly contribute to the formation of HC_5N.We deeply appreciate the anonymous referee for suggestions, which are very constructive to make our discussion. We are grateful to the staff of the Nobeyama Radio Observatory. Nobeyama Radio Observatory is a branch of the National Astronomical Observatory of Japan, National Institutes of Natural Sciences.[Alexander et al.(1976)]1976JMoSp..62..175A Alexander, A. J., Kroto, H. W., & Walton, D. R. M. 1976, Journal of Molecular Spectroscopy, 62, 175 [Hily-Blant et al. 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"authors": [
"Kotomi Taniguchi",
"Masao Saito"
],
"categories": [
"astro-ph.GA",
"astro-ph.SR"
],
"primary_category": "astro-ph.GA",
"published": "20170627033636",
"title": "First Detection of HC$_{5}$$^{15}$N in the Interstellar Medium"
} |
Laura N. Driessen [email protected] Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The University of Manchester, Manchester,M13 9PL, UKAnton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands GRAPPA, GRavitation and AstroParticle Physics Amsterdam, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, NetherlandsAnton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands GRAPPA, GRavitation and AstroParticle Physics Amsterdam, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands SRON, Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA, Utrecht, The NetherlandsAnton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands ASTRON, Netherlands Institute for Radio Astronomy, Postbus 2, 7990 AA, Dwingeloo, The NetherlandsAnton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The NetherlandsNYU Abu Dhabi, PO Box 129188, Abu Dhabi, UAE Affiliate Member, Center for Cosmology and Particle Physics, New York University, 726 Broadway, New York, NY 10003 We investigate six supernova remnant (SNR) candidates – G51.21+0.11, G52.37-0.70, G53.07+0.49, G53.41+0.03, G53.84-0.75, and the possible shell around G54.1+0.3 – in the Galactic Plane using newly acquired LOw-Frequency ARray (LOFAR) High-Band Antenna (HBA) observations, as well as archival Westerbork Synthesis Radio Telescope (WSRT) and Very Large Array Galactic Plane Survey (VGPS) mosaics. We find that G52.37-0.70, G53.84-0.75, and the possible shell around pulsar wind nebula G54.1+0.3 are unlikely to be SNRs, while G53.07+0.49 remains a candidate SNR. G51.21+0.11 has a spectral index of α=-0.7±0.21, but lacks X-ray observations and as such requires further investigation to confirm its nature.We confirm one candidate, G53.41+0.03, as a new SNR because it has a shell-like morphology, a radio spectral index of α=-0.6±0.2 and it has the X-ray spectral characteristics of a 1000-8000 year old SNR. The X-ray analysis was performed using archival XMM-Newton observations, which show that G53.41+0.03 has strong emission lines and is best characterized by a non-equilibrium ionization model, consistent with an SNR interpretation.Deep Arecibo radio telescope searches for a pulsar associated with G53.41+0.03 resulted in no detection, but place stringent upper limits on the flux density of such a source if it is beamed towards Earth. § INTRODUCTION There are many shell- and bubble-like objects in our Galaxy. For example, there are 295 supernova remnants (SNRs) in Green's SNR catalog <cit.>, 76 SNR candidates in a recent THOR+VGPS analysis <cit.>, and ∼1500 known HII regions (as well as ∼2500 probable and ∼4000 candidate HII regions) in the WISE HII region catalog[http://astro.phys.wvu.edu/wise/www.astro.phys.wvu.edu/wise] <cit.>. This means that observations and surveys of the Galactic Plane capable of investigating shell-like objects, particularly observations differentiating between candidate HII regions and SNRs, are extremely useful. As there are many sources and candidates, targeting individual objects with a single pointing per object is impractical. Interferometers that can observe large areas of the sky at low-frequencies with wide frequency bandwidth should prove to be excellent tools for Galactic Plane investigations.About 90% of SNRs and SNR candidates have been found in radio surveys <cit.>, but it is thought that there could be many missing SNRs <cit.>. Obtaining a morecomplete record of the SNR population, including confirming or rejecting the nature of SNR candidates, is important as it leads to better estimates of the Galactic supernova rate, the maximum ages of SNRs, and because SNRs are obvious locations for searching for young pulsars.Low-frequency (≲350 MHz) Galactic Plane observations are useful for investigating SNRs and SNR candidates, particularly for differentiating between SNR candidates and HII regions, due to the typically steeper radio spectral indices of SNRs <cit.> as compared to HII regions (α≳ 0); where S_ν ∝ ν^α for S_ν integrated flux density in Jy and ν frequency in Hz <cit.>. This means that SNRs are brighter at lower frequencies, while HII regions are brighter at higher frequencies. However, there have been relatively few low-frequency surveys of the Galactic Plane with high angular resolution and sensitivity. A good illustration of the capability of such surveys for SNR searches was demonstrated by a 333 MHz survey with the Very Large Array (VLA) of the Galactic Center region <cit.>. This survey resulted in the discovery of 35 new candidate SNRs, 31 of which are now confirmed. Multi-wavelength analysis is required to confirm (or reject) SNR candidates, such as X-ray observations or further radio observations at a different frequency to confirm the spectral index.The LOw-Frequency ARray <cit.> is an interferometer that observes at low-frequencies with a large field-of-view(FoV; e.g. ∼ 11 deg^2 using HBA Dual Inner mode), which means that it is ideal for observing and discovering steep-spectra objects and for differentiating between SNR candidates and HII regions. LOFAR consists of two arrays: the Low Band Antennas (LBA) and the High Band Antennas (HBA). The LBA observes between 10 and 80 MHz while the HBA observes between 110 and 250 MHz. The wide FoV also introduces many technical difficulties regarding calibration and imaging, particularly as the ionosphere can introduce significant phase and amplitude variations across the FoV. Here we discuss six SNR candidates in the FoV of proprietary LOFAR HBA observations (PI: J. D. Gelfand) that overlap with an archival Westerbork Synthesis Radio Telescope (WSRT) mosaic <cit.> and an archival VLA Galactic Plane Survey (VGPS) mosaic <cit.>. These SNR candidates were identified in a study of THOR+VGPS observations by <cit.> and are: G51.21+0.11, G52.37-0.70, G53.07+0.49, G53.41+0.03, G53.84-0.75, and G54.1+0.3. In particular, we present a multi-frequency analysis of SNR candidate G53.41+0.03. In Section <ref> we present the observations. In Section <ref> we present our results and in Section <ref> we discuss the SNR candidates.We conclude in Section <ref>.§ OBSERVATIONS AND ANALYSIS §.§ Radio observationsWe use radio observations at three different frequencies – 144 MHz (LOFAR HBA), 327 MHz (WSRT), and 1400 MHz (VGPS) – to investigate part of the Galactic Plane. Figure <ref> shows the FoV where our LOFAR HBA observations overlap archival WSRT and VGPS mosaics.We initially obtained and analyzed the LOFAR observations to investigate pulsar wind nebula (PWN) G54.1+0.3 but, due to the large FoV, we also investigated other promising SNR candidates. The LOFAR observations are centered on the PWN. The observations were taken on 2015 June 12 as part of project LC4_011 (ObsID: 345918) and were performed in HBA Dual Inner mode <cit.>. This means that the inner 24 tiles of the remote stations wereused resulting in a full-width half-maximum (FWHM) of the primary beam of 3.8^∘ and FoV of ∼11 deg^2 in this configuration. The LOFAR HBA target and calibrator scans cover the frequency range from 118.7 MHz to 169.5 MHz. The observing bandwidth was split into 260 subbands (SBs) with bandwidth of 195.3 kHz each. For these observations an 18 min calibrator scan of 3C380 was taken before and after the 3 hr target scan.The LOFAR observations were flagged, demixed, and averaged as part of standard LOFAR pre-processing. Demixing involves removing the effects of the very bright radio sources, Cassiopeia A and Cygnus A, that affect LOFAR images even when they are far from the phase center of the FoV. The data were averaged to 4 frequency channels per SB. The LOFAR synthesized beam size is 3.0'×2.2' with a position angle of 220.7 (with respect to the Galactic Plane) at 144 MHz using a Briggs weighting of 1.0 <cit.>. As this is a Galactic Plane observation the imaging calibration pipeline, prefactor <cit.>, was not successful. This is due to the significant extended emission in the Galactic Plane, across the FoV. Ionospheric variations during the observations were particularly pronounced. The observation was calibrated by transferring the time-independent, zero-phase gain solutions from the second calibrator scan to the target scan. The observations were then summed into 26 measurement sets of 10 SBs each. Two rounds of self-calibration were then performed on the target scan, the first using a model from the TIFR GMRT Sky Survey (TGSS) Alternative Data Release <cit.>. Multiscale imaging with Briggs 1.0 weighting was then performed using the WSClean tool <cit.>. The subband with a central frequency of 150 MHz was flux calibrated using the integrated flux density measurements of point sources from the TGSS ADR. It is important to note that the sensitivity of the LOFAR image drops significantly at the edge of the FWHM of the primary beam. This means that flux density values far from the phase center (PWN G54.1+0.3) are less reliable. Figure <ref> was produced by performing a multi-frequency (MFS) clean on all measurement sets.WSRT observations were obtained from a Galactic Plane point source survey at 327 MHz with a beam size of 60”×191” and a position angle of 61.3^∘ (with respect to the Galactic Plane) by <cit.>[<www.ras.ucalgary.ca/wsrt_survey.html>]. VLA observations from VGPS with a beam size of 1'×1' were also used <cit.>[<www.ras.ucalgary.ca/VGPS/VGPS_data.html>]. The VLA observations have the highest angular resolution of the available radio observations of this FoV. §.§ Radio pulse search observationsTo search for a pulsar towards G53.41+0.03, we observed the region using the 305-m Arecibo radio telescope and the 7-beam Arecibo L-band Feed Array (ALFA) receiver.On 2017 June 21, we made a 3-pointing grid of the region, where together the 21 observed beams were interleaved and cover a roughly 10^' region around the center of G53.41+0.03.The first pointing, where the central beam of ALFA was directly pointed towards the apparent center of G53.41+0.03, integrated for 2400 s.The other two interleaving pointings were integrated for 900 s.We recorded the resulting filterbank data using the Mock spectrometers, which provided two partially overlapping 172 MHz subbands centered at 1300 and 1450 MHz, respectively.Only total intensity was recorded, with 0.34-MHz spectral channels and 65.5 μs time resolution.We converted the raw samples from 16-bit to 4-bit values subsequent to the observation in order to reduce the data volume.At the start of the session, we observed PSR J1928+1746 in the central ALFA beam, in order to verify the configuration.We searched for radio pulsations in the direction of G53.41+0.03 using standard methods, as implemented in the PRESTO[<https://github.com/scottransom/presto>] software package.We chose to search the Mock subbands separately because the lower-frequency subband contains significantly more radio frequency interference (RFI).For each beam and subband we excised RFI using rfifind and then used multiple calls to prepsubband to generate dedispersed time series for dispersion measures in the range DM = 0 - 1019 pc cm^-3 in steps of1 pc cm^-3.The remaining dispersive smearing is ∼ 1 ms, even for the highest DMs in this range.Each dedispersed timeseries was then searched for periodicities using accelsearch with no additional search for linear acceleration (i.e. z_ max = 0).The cumulative set of candidates was then sifted and ranked using ACCEL_sift.py.We folded promising candidates — those with high signal-to-noise, high coherent power, and apparent peaks in signal-to-noise as a function of DM — using prepfold.Associated diagnostic plots for each candidate were then visually inspected.When this approach was applied to the test pulsar, J1928+1746, the expected signal was easily recovered in both subbands. §.§ Infrared observationsThe FoV coinciding with the radio observations was observed at 24.0 μ m as part of the Multiband Infrared Photometer for Spitzer GALactic Plane (MIPSGAL) survey <cit.>. MIPSGAL 24.0 μ m observations have a resolution of 6” and a 5σ root-mean-squared sensitivity of 1.3 mJy.§.§ X-ray observationsOf the six candidate SNRs that we investigate in this paper, only the possible shell around PWN G54.1+0.3 has been analysed previously in the X-ray band. It has been observed using Chandra <cit.>, Suzaku, and XMM-Newton <cit.>.The position of G53.84-0.75 has been observed in a ROSAT PSPC observation (ObsID: WG500209P.N1). Using the region size of 18.7 ' <cit.> we estimate the X-ray count rate with 2σ upper limit to be 1.5× 10^-2 counts/sec in the ROSAT 0.4 – 2.4 keV energy band. G53.41+0.03 is detected at the edge of the FoV of two ROSAT PSPC observations (ObsIDs: WG500042P.N2 and WG500209P.N1) and partially covered by an XMM-Newton observation taken on2008 Mar 29 (ObsID: 0503740101).The other three SNR candidates, G51.21+0.11, G52.37-0.70, and G53.07+0.49, have no complementary data available in the X-ray band.Although G53.41+0.03 lies at the edge of the detector in the XMM-Newton observation, the observation is important as it allows us to determine the nature of the X-ray emission through spectral analysis of the EPIC-MOS camera <cit.> data. We extracted thespectrum with the Science Analysis System (SAS) v14.0. Due to a failed CCD chip in MOS1 and the smaller FoV of the EPIC-PN detector only data from the MOS2 detector were used. The data were reduced using thetask and filtered for the background flaring. This resulted in 40.7 ks of cleaned exposure time. The source extraction region was a 1.8 ' radius circle centered on the extended X-ray source. The background was extracted using a region of the same size positioned in a nearby area of the detector devoid of X-ray sources.The source and background regions are shown in Figure <ref>. To perform the spectral analysis the SPEX fitting package version 3.04 (2017) together with SPEXACT 2.07 atomic tables were used <cit.>. The fitting statistics method employed was C-statistics <cit.>. Abundances were expressed with respect to Solar Abundance values of <cit.>. For the emission measure parameter (n_en_HV) we assumed a distance of 7.5 kpc (see Sec. <ref>). The analysis of the spectra was performed in the energy range between 0.7 – 3.0 keV, as this is the range in which the source spectrum dominates the background. Thecommand was used to obtain optimal binning of the spectra. After background subtraction the source spectrum consists of ∼ 2000 counts.The spectrum was fit with a non-equilibrium ionization (NEI) model with Galactic absorption. The Galactic background was represented by the modelin SPEX, with the temperature fixed to 0.5 eV to mimic absorption by neutral gas <cit.>. The NEI model was employed with the following free parameters: electron temperature T_2, ionization age τ = n_e t, normalization n_e n_H V, and abundances of elements Ne, Mg, Si, S, Fe. These elements have line emission in the energy band from 0.8 – 2.6keV, the band for which there was sufficient signal to noise.§.§ High-energy observations We searched the High Energy Stereoscopic System CATalog (HESSCAT[https://www.mpi-hd.mpg.de/hfm/HESS/pages/home/sources/www.mpi-hd.mpg.de/hfm/HESS/pages/home/sources/]) and Third Fermi LAT Catalog of High-Energy Sources <cit.> for high-energy sources associated with any of the SNR candidate shells. Fermi source 3FGL J1931.1+1659 is within the radius of SNR candidate G52.37-0.70. There are no other high-energy sources close to the other five SNR candidates.§ RESULTSThe VGPS, WSRT, and LOFAR HBA observations of the six SNR candidates in the FoV – G51.21+0.11, G52.37-0.70, G53.07+0.49, G53.41+0.03, G53.84-0.75, and G54.1+0.3 – are shown in Figures <ref> and <ref>. Only G53.41+0.03 and G54.1+0.3 have been observed in the X-ray band (see Sec. <ref>). As discussed by <cit.>, all six of the candidates have low thermal emission compared to the non-thermal emission, which we confirm using the MIPSGAL observations. §.§ Radio results The flux densities and spectral indices of SNR candidates G51.21+0.11, G52.37-0.70, G53.41+0.03, and G53.84-0.75, and the candidate shell around PWN G54.1+0.3 measured using the positions and radii reported by <cit.> are shown in Table <ref>. We subtracted the integrated flux density of the HII region overlapping G52.37-0.70 and the flux density of the bright point source within G53.84-0.75.Due to a drop-off in sensitivity away from the phase center of the HBA observation, we do not measure LOFAR integrated flux densities for G51.21+0.11 and G52.37-0.70.SNR candidate G51.21+0.11, shown in Figure. <ref> (top row), has a complex morphology with a bright radio filament type structure and a bright radio patch. It has an HII region, G051.010+00.060 <cit.>, on one side that appears to be coincident.52.37-0.70 is a faint radio shell visible most clearly in the VLA observation in the second row of Figure <ref>. There is a bright HII region, G052.174-00.567 <cit.>, on the upper right of this candidate and some smaller HII regions within the shell.G53.07+0.49 has a small angular size <cit.> and the location of the peak flux density is different for WSRT and LOFAR compared to the original VLA identification of the candidate. In Figure <ref> (bottom panel) we can also see that there is some extended emission around G53.07+0.49 that may or may not be associated with this candidate. As it is unclear which emission in the WSRT and LOFAR observations may or may not be associated with the candidate we do not measure WSRT or LOFAR flux densities for this candidate.There is diffuse emission and some radio point sources in the region where candidate G53.84-0.75 is located (Fig. <ref>, upper panel), but it is difficult to identify what emission is related to candidate G53.84-0.75 and whether there is a discrete object or if the extended emission is Galactic Plane dust.PWN G54.1+0.3 is shown in Figure <ref> (lower panel) where the bright spot in the center is the PWN and the partial loop around it is the known HII region G053.935+00.228 <cit.>. There is some faint, diffuse radio emission around the PWN in the VLA observation, which is the SNR-shell candidate. In the WSRT and LOFAR observations of PWN G54.1+0.3 shown in Figure <ref> it appears that the possible shell identified in the VLA observations <cit.> fades away or is part of the surrounding HII region. The large uncertainty in the spectral index in Table <ref> reflects that a powerlaw is not the best model; however, the flux density clearly decreases as the frequency decreases.In the LOFAR HBA and VLA observations G53.41+0.03 has a shell- or bubble-like morphology which is brighter on the upper edge, as shown in Figure <ref>. The radius of the shell at 144 MHz is ∼5'. As shown in Table <ref> G53.41+0.03 has a radio spectral index of α=-0.6±0.2.§.§ X-ray results As described in Sec. <ref>, G53.84-0.75 was observed by ROSAT. We used the PIMMS[<https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/w3pimms.pl>] tool with the optically thin plasma model APEC with temperature 0.3 keV and local Galactic absorption value of 2.4× 10^22 cm^-2 to obtain the 2σ upper limit for the flux. No X-ray feature coincident to the radio observations was detected. The 2σ upper limit for the absorbed/unabsorbed flux is F_0.4-2.4≈ 2.4× 10^-13 / 4.1× 10^-11 erg s^-1 cm^-2.The ROSAT and XMM-Newton X-ray observations of G53.41+0.03 confirm the existence of an extended X-ray source at the location of G53.41+0.03, particularly at the position of the radio-bright part of the shell[Since the spectral resolution of the ROSAT PSPC is poor and the images are noisy, we use only the XMM-Newton observation for further analysis.]. The XMM-Newton X-ray spectrum(Fig. <ref>) showsbright K-shell emission linesfrommagnesium, silicon, and sulfur and potential contributions from neon and iron around 1 keV. This is typical of thermal emission from an optically thin plasma. The absorbed/unabsorbed flux of the source measured using XMM-Newton in the 0.7 – 3.0 keV energy range is F=7.3 × 10^-13 / 3.1 × 10^-11 erg s^-1 cm^-2. The best-fit NEI model is represented by a C-stat / d.o.f. of 83.48/64. The parameters and 1σ errors are listed in Table <ref>, while the best fit model is shown in Figure <ref>.The ionization age informs us how far out of ionization equilibrium the plasmais, but given the narrow spectral range the parameter may correlate with the best-fit electron temperature T_2.To test the robustness of our best fit ionization age we calculate the error ellipse ofτand T_2, as shown in Figure <ref>.§.§ Radio pulsation search results After performing a pulsation search as described in Sec. <ref> we found no convincing astronomical signals in the data toward G53.41+0.03, and we ascribe the statistically significant signals that we did detect to RFI.Given the non-detection of radio pulsations toward G53.41+0.03, we can place an upper limit on the integrated flux density of any associated radio pulsar.We use the modified radiometer equation <cit.>, and assume that interstellar scattering does not have a significant effect on broadening the pulses through multi-path propagation.While the central ALFA beam has a gain of G ∼ 10 K Jy^-1, the 6 outer beams have G ∼ 8 K Jy^-1.We targeted the center of G53.41+0.03 (specifically, RA_ J2000 = 19^h29^m57.41^s, Dec_ J2000 = +18^∘09^'53.5^'') in a T = 2400-s pointing with the central ALFA beam, which covered a region of roughly 1.6^' in radius.Since G53.41+0.03 is roughly 10^' wide, we also gridded a much larger ∼ 10^' wide region around G53.41+0.03 in case the pulsar has moved from its birth site near the center of the SNR.In our sensitivity calculations we thus consider two scenarios: 1) where the pulsar is close to the center of G53.41+0.03, and where we should use G = 10 K Jy^-1 and T = 2400 s and 2) a scenario in which the pulsar is offset by several arcminutes, and where G = 8 K Jy^-1 and T = 900 s.Furthermore, if the pulsar is located towards the half-power sensitivity point of one of the beams, then the effective sensitivity is also half.We make this conservative assumption for scenario 2.The receiver temperature T_ rec = 25 K and the sky temperature in this direction of the Galactic plane is T_ sky = 5 K at 1400 MHz.We assume a W = 10% pulse duty cycle and a signal-to-noise S/N = 10 for detection.The two orthogonal linear polarizations, n_ p, of the receiver were summed, and the appropriate bandwidth is Δν = 172 MHz.Finally, using the modified radiometer equation, and assuming no additional losses due to digitization, we find for scenario 1: S^1_ max [mJy]S/N (T_ rec + T_ sky)/G √(T Δν n_ p)√(W/(1-W)) = 0.011mJy For scenario 2, where the putative pulsar is more offset from G53.41+0.03, S^2_ max = 0.045 mJy.These are deep upper-limits on the flux density of any pulsar associated with G53.41+0.03.Of the known young pulsars in the ATNF catalog, only a few have lower measured radio flux density <cit.>.However, because of beaming and the possibility of significant interstellar scattering, these limits do not definitively exclude a young pulsar associated with G53.41+0.03.§ DISCUSSIONHere we will discuss the characteristics and nature of each SNR candidate. We will focus on G53.41+0.03, including calculating its approximate distance and age.G51.21+0.11: SNR candidate G51.21+0.11 has a negative spectral index, α=-0.7±0.21, and a complex morphology coincident with a known HII region. There are no XMM-Newton or Chandra observations in the direction of the candidate to confirm its nature. We find G51.21+0.11 to be an interesting object that is possibly an SNR, but further investigation using X-ray observations is required.G52.37-0.70: Although G52.37-0.70 has a shell-like morphology in the VLA observations, it has a spectral index of α=0.3±0.3 fitted using the VLA and WSRT integrated flux densities. The spectral index indicates that this candidate is unlikely to be an SNR, and as such the Fermi source within the radius of the candidate (see Sec. <ref>) is unlikely to be associated.G53.07+0.49: Candidate G53.07+0.49 has a small angular size in the VLA observations, but the peak flux density in the WSRT and LOFAR observations is offset from the SNR candidate location suggested by <cit.>. As such we do not measure WSRT or LOFAR flux densities for this candidate, and as there are no X-ray observations available, further investigation using X-ray or higher resolution low-frequency observations is required to comment on the nature of this candidate.G53.84-0.75: It is not clear what emission is SNR candidate G53.84-0.75 and there are large errors on the VLA integrated flux density from <cit.>. This, as well as the strange spectral shape, suggests that there is no discrete, extended object at this position. This is supported by the ROSAT X-ray non-detection. For this reason we find it unlikely that G53.84-0.75 is an SNR.G54.1+0.3: Whether PWN G54.1+0.3 has an SNR shell has been in question since <cit.> found faint radio emission around the PWN, which is just visible in the VLA observation (Fig. <ref>). <cit.> found no evidence of a shell in their Chandra observations, while <cit.> found hints of a very faint, diffuse shell using Suzaku and XMM-Newton. <cit.> find that the shell suggested by <cit.> is more likely to be part of the surrounding HII region. Alternatively, <cit.> suggest a slightly smaller radius shell (7.2') as a possible shell around PWN G54.1+0.3 with an integrated flux density of 1.46 Jy at 1.4 GHz. There is no evidence for extended emission around PWN G54.1+0.3 in our LOFAR HBA observation, as can be seen in Figure <ref> (bottom panel), aside from the known HII region G053.935+0.228 <cit.>. This is supported by the low flux-densities measured by WSRT and LOFAR (shown in Table <ref>) using a region of radius 7.2' and subtracting the flux density of the PWN. We find it unlikely that there is a shell around PWN G54.1+0.3.G53.41+0.03: G53.41+0.03 has a morphology common to SNRs. Using the flux densities shown in Table <ref> we find that the G53.41+0.03 has a steep negative radio spectral index, α=-0.6±0.2, as expected for an SNR. X-ray analysis indicates that the plasma of G53.41+0.03 has a relatively high temperature of T_2∼ 0.8 keV. The ionization age τ∼10^10.6 s·cm^-3 is much lower than needed for ionization/recombination balance (τ≥10^12 s·cm^-3). The fact that the spectrum is far out of ionization equilibrium is a clear signature that the source is an SNR <cit.>, as no other known source class has gas tenuous enough and/or is young enough to be far out of ionization equilibrium. We therefore confirm that G53.41+0.03 is an SNR, and further investigate it by calculating its approximate distance and age. §.§ The distance to G53.41+0.03 Estimating the distance to Galactic SNRs is notoriously difficult. There are few methodsthat give reliable results, such as kinematic methods, based onoptical Doppler shifts combined withproper motion of optical filaments <cit.>, or, less reliably,21cm line absorption combined with a Galacticrotation model <cit.>. In contrast, SNRs located in the Magellanic Clouds can be reliably placed at the distance of these satellite galaxies.By using reliable distance estimates some secondary distance indicators have been developed, such as the X-ray Galactic absorption column <cit.> and the Σ-D relation <cit.>.A first indication of the distance of an SNR can be its positional association witha spiral arm.However, the reason that the investigated fieldis so rich in sources is that the line of sight crosses the Sagittarius-Carina arm tangentially as well as regions of the Perseus arm. Taking the Galactic spiral arm model of <cit.>, wefind that thel=53.4^∘ line of sight intercepts the Sagittarius arm <cit.> between ∼ 4 kpc and 7.5 kpc, and the Perseus arm at 9.6 kpc. Given that the Sagittarius arm istangential along the line of sight, this suggests a probable distance between 4.5 and 7.5 kpc.<cit.> derived a relation between column density and distance of N_H=8.4× 10^21 d^1.58 cm^-2. The measuredcolumn density ofN_H=2.4× 10^22 cm^-2 (Table <ref>), therefore, suggests a distance of ∼ 8.4 kpc. However, one should be cautious here, because the line of sight crosses the arm tangentially, whichis likely to lead to a column density that is higher than averagefor a given distance.The surface brightness of G53.41+0.03 normalized to 1 GHz isΣ = 8.3× 10^-21 W m^-2 Hz^-1sr^-1. The 1 GHz surface brightness was obtained using the 1.4 GHz flux density measured by <cit.> and a spectral index of α=-0.6 (see Tab. <ref>). Using the relation between diameter and surface brightness (the Σ-D relation) in <cit.> gives yet another distance estimate of 8 kpc. However, we know that the Σ-D relation is controversial, as there is large scatter which may relate to the SNR environments, and there is debate on the statistical validity of the relation <cit.>.The distance estimates based on the X-ray absorption and Σ-D relation, although uncertain, are consistent with the idea that the SNR is located in the Sagittarius-Carina arm, but suggest that the SNR is on the far-side of the arm. We therefore adopt a distance of 7.5 kpc for G53.41+0.03. The angular radius of ∼ 5^' translates then into a physical radius of 10.7d_7.5 pc, with d_7.5 the distance in units of 7.5 kpc. §.§ The age of G53.41+0.03The spectrum of G53.41+0.03 allows us to put some constraints on the density and age of the SNR. To do this we need a volume estimate. Given a typical volume filling fraction of 25%[A strong shock has a compression factor of 4. This means that roughly 25% of the volume, approximated by a sphere, will emit.] and assuming a spherical morphology, we estimate the volume to be V_SNR=3.3× 10^58d_7.5^3 cm^3. The X-ray spectrum was obtained for only ∼20% of the shell, so we take V_X≈ 6.7× 10^57d_7.5^3 cm^3 to be the volume pertaining to the X-ray spectrum. Taking n_e≈ 1.2n_H in the emission measure n_e n_H V, we obtain the density n_H≈ 0.8 d_7.5^-3/2 cm^-3.Using this number together with the best-fit ionization age of n_et=4× 10^10 cm^-3s we find an approximate age of 1600d_7.5^3/2 yr.The measured electron temperature corresponds to a shockvelocity of V_s≈ 800 km s^-1 or higher if the electron temperature is lower than the ion temperature <cit.>.For the Sedov-Taylor self-similar evolution model we have V_s=0.4 R/t.Using R=10.7 d_7.5 pc,gives then an approximate ageof ∼ 5300d_7.5 yr. Using the Sedov-Taylorevolution model of R^5=2.026 Et^2/ρ, with E=10^51 erg gives yet another estimate of the age of∼ 7800d_7.5^7/4 yr. The two estimates based on the Sedov-Taylor model give roughly similar results for the canonical explosion energy of 10^51 erg (t≈ 6500 ± 1500 yr), whereas the estimate based on the ionization age suggests a younger age. This discrepancy may be due to non-standard evolution scenarios, for example evolution in a wind-blow cavity. This needs to be addressed in follow-up studies. However, these estimates agree that G53.41+0.03 is an SNR with an age somewhere between 1000 and 8000 yr. X-ray observations centered on and covering the whole SNR are needed to fully characterize the properties of G53.41+0.03.§ CONCLUSIONWe confirm that SNR candidate, G53.41+0.03, is in fact an SNR using XMM-Newton observations, and LOFAR observations targeting PWN G54.1+0.3. G53.41+0.03 has a shell-like morphology in the radio, with a radius of ∼ 5 '. Using LOFAR HBA observations, as well as archival WSRT and VGPS mosaics, we confirm that G53.41+0.03 has a steep spectral index (α=-0.6±0.2), typical of synchrotron radiation from SNRs. MIPSGAL observations show that G53.41+0.03 has no IR component. Archival XMM-Newton observations show that G53.41+0.03 has an associated X-ray component with a coincident morphology to the radio shell. Furthermore, analysis and fitting of the XMM-Newton observation show that G53.41+0.03 has strong emission lines and is best characterized by a non-equilibrium ionization model, with an ionization age and normalization typical for an SNR with an age between 1000 and 8000 yr and a density of n_H≈ 0.8 d_7.5^-3/2 cm^-3. Given the X-ray, IR, and radio characteristics of G53.41+0.03, we confirm that it is a new Galactic Plane SNR. We do not find a pulsar associated with G53.41+0.03, but the upper-limits on the flux density do not exclude the possibility of a young pulsar that is exceptionally weak or not beamed towards Earth.We also investigate five other SNR candidates from <cit.> in the same LOFAR FoV. We show that three of these candidates (G52.37-0.70, G53.84-0.75 and the shell around PWN G54.1+0.3) are unlikely to be SNRs and one, G51.21+0.11, is a good SNR candidate that requires further investigation. This demonstrates that it is important to further investigate SNR candidates using low-frequency observations with telescopes such as WSRT and LOFAR.We would like to thank Vincent Morello, George Heald, Raymond Oonk, Andre Offringa, Jess Broderick, Pedro Salas, Alex Mechev, and Irene Polderman for useful discussions and assistance with LOFAR imaging and calibration. LND and JWTH acknowledge support from the European Research Council under the European Union's Seventh Network Framework Programme (FP/2007-2013) / ERC Grant Agreement nr. 337062. JWTH is an NWO Vidi fellow. This paper is based (in part) on data obtained with the International LOFAR Telescope (ILT) under project code LC4_011. LOFAR <cit.> is the Low Frequency Array designed and constructed by ASTRON. It has observing, data processing, and data storage facilities in several countries, that are owned by various parties (each with their own funding sources), and that are collectively operated by the ILT foundation under a joint scientific policy. The ILT resources have benefited from the following recent major funding sources: CNRS-INSU, Observatoire de Paris and Université d'Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI), Ireland; NWO, The Netherlands; The Science and Technology Facilities Council, UK. 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C., & Varoquaux, G. 2011, ArXiv e-prints, http://arxiv.org/abs/1102.1523arXiv:1102.1523 [cs.MS][van Haarlem et al.(2013)van Haarlem, Wise, Gunst, Heald, McKean, Hessels, de Bruyn, Nijboer, Swinbank, Fallows, Brentjens, Nelles, Beck, Falcke, Fender, Hörandel, Koopmans, Mann, Miley, Röttgering, Stappers, Wijers, Zaroubi, van den Akker, Alexov, Anderson, Anderson, van Ardenne, Arts, Asgekar, Avruch, Batejat, Bähren, Bell, Bell, van Bemmel, Bennema, Bentum, Bernardi, Best, Bîrzan, Bonafede, Boonstra, Braun, Bregman, Breitling, van de Brink, Broderick, Broekema, Brouw, Brüggen, Butcher, van Cappellen, Ciardi, Coenen, Conway, Coolen, Corstanje, Damstra, Davies, Deller, Dettmar, van Diepen, Dijkstra, Donker, Doorduin, Dromer, Drost, van Duin, Eislöffel, van Enst, Ferrari, Frieswijk, Gankema, Garrett, de Gasperin, Gerbers, de Geus, Grießmeier, Grit, Gruppen, Hamaker, Hassall, Hoeft, Holties, Horneffer, van der Horst, van Houwelingen, Huijgen, Iacobelli, Intema, Jackson, Jelic, de Jong, Juette, Kant, Karastergiou, Koers, Kollen, Kondratiev, Kooistra, Koopman, Koster, Kuniyoshi, Kramer, Kuper, Lambropoulos, Law, van Leeuwen, Lemaitre, Loose, Maat, Macario, Markoff, Masters, McFadden, McKay-Bukowski, Meijering, Meulman, Mevius, Middelberg, Millenaar, Miller-Jones, Mohan, Mol, Morawietz, Morganti, Mulcahy, Mulder, Munk, Nieuwenhuis, van Nieuwpoort, Noordam, Norden, Noutsos, Offringa, Olofsson, Omar, Orrú, Overeem, Paas, Pandey-Pommier, Pandey, Pizzo, Polatidis, Rafferty, Rawlings, Reich, de Reijer, Reitsma, Renting, Riemers, Rol, Romein, Roosjen, Ruiter, Scaife, van der Schaaf, Scheers, Schellart, Schoenmakers, Schoonderbeek, Serylak, Shulevski, Sluman, Smirnov, Sobey, Spreeuw, Steinmetz, Sterks, Stiepel, Stuurwold, Tagger, Tang, Tasse, Thomas, Thoudam, Toribio, van der Tol, Usov, van Veelen, van der Veen, ter Veen, Verbiest, Vermeulen, Vermaas, Vocks, Vogt, de Vos, van der Wal, van Weeren, Weggemans, Weltevrede, White, Wijnholds, Wilhelmsson, Wucknitz, Yatawatta, Zarka, Zensus, & van Zwieten]2013A A...556A...2V van Haarlem, M. P., Wise, M. W., Gunst, A. W., et al. 2013, http://dx.doi.org/10.1051/0004-6361/201220873, 556, A2[van Weeren et al.(2016)van Weeren, Williams, Hardcastle, Shimwell, Rafferty, Sabater, Heald, Sridhar, Dijkema, Brunetti, Brüggen, Andrade-Santos, Ogrean, Röttgering, Dawson, Forman, de Gasperin, Jones, Miley, Rudnick, Sarazin, Bonafede, Best, Bîrzan, Cassano, Chyży, Croston, Ensslin, Ferrari, Hoeft, Horellou, Jarvis, Kraft, Mevius, Intema, Murray, Orrú, Pizzo, Simionescu, Stroe, van der Tol, & White]2016ApJS..223....2V van Weeren, R. J., Williams, W. L., Hardcastle, M. J., et al. 2016, http://dx.doi.org/10.3847/0067-0049/223/1/2The Astrophysical Journal Supplement Series, 223, 2[Vink(2012)]Vink2012 Vink, J. 2012, http://dx.doi.org/10.1007/s00159-011-0049-1The Astronomy and Astrophysics Review, 20, 49 | http://arxiv.org/abs/1706.08826v2 | {
"authors": [
"Laura N. Driessen",
"Vladimír Domček",
"Jacco Vink",
"Maria Arias",
"Jason W. T. Hessels",
"Joseph D. Gelfand"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170627130357",
"title": "Investigating Galactic supernova remnant candidates with LOFAR"
} |
^1 Section de mathématiques, Université de Genève, 2-4 rue du Lièvre,Genève [email protected] ^2 Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy ^3 INFN, Sezione di Torino, Via P. Giura 1, I-10125 Torino, Italy^4 Malaysia Italy Centre of Excellence for Mathematical Sciences University Putra Malaysia, 43400 Serdang, Selangor, Malaysia [email protected] A one-dimensional quantum system with off diagonal disorder, consisting of a sample of conducting regionsrandomly interspersed within potential barriers is considered. Results mainly concerning the large N limit are presented. In particular, the effect of compression on the transmission coefficient is investigated. A numericalmethod to simulate such a system, for a physically relevant number of barriers, is proposed. It is shown that the disordered model converges to the periodic case as N increases, with a rate of convergence which depends on the disorder degree. Compression always leads to a decrease of the transmission coefficient which may be exploitedto design nano-technological sensors. Effective choices for the physical parameters to improve the sensitivity are provided. Eventually large fluctuations and rate functions are analysed. disordered systems, compression, fluctuations, Kronig-Penny model, transfer matrix technique.§ INTRODUCTIONEquilibrium and nonequilibrium thermodynamics <cit.> are based on the vast separation between thespace and time scales of the microscopic, mesoscopic and macroscopic physical realms. Such a separation of scalesrequires the systems of interest to be made of very large numbers of microscopic constituents and it allows the state of local thermodynamic equilibrium. In that state, microscopicfluctuations of physical quantities are negligible, so that the thermodynamic fields are definedand they are described by the thermodynamic laws. In certain small systems, pertaining e.g.to modern bio- and nano-technologies, the separation of scales is not realized, and thephysical properties of interest are characterized by fluctuations of size comparable to thatof the average signals. In this work, the investigation of Refs.<cit.>, concerning a variation of the Anderson model <cit.> of disordered solids, is developed in order to account for the effects of compression on the electron transmission coefficient. Indeed, since Anderson's paper, the study of electron transport has played a key role in the development of modern solid state physic, see for instance Refs. <cit.> and references therein. The systems of interest, here, are nanostructured devices made of an insulating matrix embedding randomly distributeddrops of conducting material. Such complex objects can be represented by 1-dimensional modelsconsisting of conducting regions delimited by N randomly placed potential barriers, in whichelectrons are injected from one electrode at a given temperature T <cit.>. The large N limit is taken under the constraint that the sum of the N barrier widthsand the total length of the system remain constant as N grows. This is at variance with models that grow in size with N.Unlike usual models foundin the literature <cit.>, the one of Refs.<cit.> enjoys a purely off-diagonal disorder <cit.>that affects the tunnelling couplings among the wells, but not the energies of the bound states within the wells. This is not the case of the original tight-binding model introduced by Anderson to describelocalization phenomena in disordered solids <cit.>, in which random fluctuationsonly concern the energy of a bound state. Furthermore, increasing the number of barriersleads, in the Anderson model, to the infinitely large system limit, while increasingN in the model investigated here, produces finer and finer distributions of the same amountof conductor dispersed within the same amount of insulating material. Therefore, the two large N limits do not describe the same situation: Anderson's limit views the system of interest asmacroscopic, i.e. very large compared to its microscopic constituents, while the limit ofRefs.<cit.> refers to system sizes that can be small compared to macroscopic objects.The relevant different mathematical constructions imply substantial differences, describing such different physical situations. While Anderson's limit suits macroscopic objects, the limit of Refs.<cit.> better describes systems atthe mesoscopic scale.In <cit.>, the N →∞ limit led to the conclusion that a large deviation principleholds for the fluctuations of the transmission coefficient, with a proper scaling for the rate function. In the present article, we focus on the behaviour of the transmission coefficient forphysically relevant numbers of potential barriers, and we study the effects of compression, that can be realized in practice in numerous nanostructured devices.Our findings are the following: * We have extended the continuum limit results proposed in <cit.>, observing that the rate of convergence of our model to the Kronig-Penny case <cit.> averaged over the energy strongly depends on the disorder degree. * Unlike the Anderson model, our large N limit implies no localization. Nevertheless, increasing the disorder degree at fixed N leads to a substantial reduction of the transmission coefficient, which may be viewed as a phenomenon in some sense analogous to localization.* A mathematical framework of compression has been introduced and two different situations have been simulated and compared. In both cases, compression induces a decrease of the transmission coefficient. * Analysing the relative percentage change of the transmission coefficient, an optimal configuration has been identified to design an effective sensor. It is found that a moderate number of barriers and strong disorder imply high sensitivity to compression.* Fluctuations and rate function have been investigated, obtaining that they may be exploited to reveal the compression state of the system.* A numerical scheme which does not suffer from overflow and Ω problem has been developed.This article is organized as follows: Section 2 describes the mathematical model to be used for disorderand compression. Section 3 introduces our numerical results and it is divided in subsections concerning linear compression model and a generalized version. Section 4 deals with fluctuationsand rate functions for systems under compression. Section 5 recapitulates the contents of the article and in the appendix the numerical scheme developed to tackle the issues raised by the range of energies and lengths of physical interest is explained. § THE MODEL Consider a 1-dimensional system of length L, consisting of an array of N potential barriersseparating N-1 potential wells, in equilibrium with one electrode that acts as an external thermostatat temperature T, cf. Fig.<ref>. This means that the mean energy of the plane waves enteringfrom the left boundary is k_BT/2. Let the wells have same width δ_N, so that the totallength of the N-1 wells is L_cond = (N-1) δ_N = α L, where α∈ (0,1), and letthe widths of the N potential barriers be picked at random with uniform distribution, to reach thetotal length (1-α) L (cf. section <ref> for details).Let all potential barriers have same constant height V (x) = V, and let their boundary points bedenoted by x=x_0,...,x_2N-1. For fixed barrier width, we would have a variation of the Kronig-Penney model<cit.>.In a steady state, the microscopic behavior of the electrons in this environment is given by thetime independent Schrödinger equation:d^2/dx^2ψ=2m/ħ^2(V-E)ψ,x∈[0,L]where m is the mass of an electron, and ħ is the reduced Planck constant. Denoting by U_l the l-th region, for l∈{ 0,2,...,2N }, the solutions of eq.(<ref>) for E<V have the form:ψ_l(x)=A_2le^ikx+A_2l+1e^-ikx V(x)=0)A_2le^-zx+A_2l+1e^zx V(x)=V)with k=√(2mE)/ħ and z=√(2m(V-E))/ħ. The boundary conditions prescribe A_0>0 for the amplitude of the plane wave entering from theleft boundary, and A_4N+1=0 since no wave enters or is reflected from the right boundary. The steady state current is defined by <cit.>,j_l(x)=ħ/2mi[ψ_l(x)^*(d/dxψ_l(x))- (d/dxψ_l(x)^*)ψ_l(x)] =j^tr_l(A_2l)-j^ref_l(A_2l+1),where j^tr_l(A_2l)=ħ k|A_2l|^2 /m denotes the current transmitted from the (l-1)-th barrieron the left and j^ref_l(A_2l+1)=ħ k|A_2l+1|^2/m denotes the current reflected from the (l+1)-thbarrier.Considering eqs.(<ref>) and (<ref>), we get the following definitionfor the transmission coefficient S across the system:S(N)=j^tr_2N(A_4N)/j^tr_0(A_0)=|A_4N|^2/|A_0|^2. To numerically compute the coefficient S as a function of the various parameters of the model,it is convenient to rewrite eq.(<ref>) in terms of the characteristic quantities,introducing x̂=x/L, ψ̂=ψ√(L), Ê=E/E_T and V̂=V/E_T,with E_T=K_bT, which is twice the mean kinetic energy of the plane waves entering from the leftthermostat. Further, introducing the scalar parameter γ=ħ^2 /(2mL^2E_T), the expressionfor the dimensionless wave vectors takes the form: k̂=√(Ê)/√(γ)and ẑ=√(V̂-Ê)/√(γ). In the following, we refer only to dimensionless quantities, but for sake of simplicity, we omit thehat over the corresponding symbols. Hence, the dimensionless form of eq.(<ref>) reads:d^2/dx^2ψ(x)=1/γ(V-E)ψ(x),x∈ [0,1]. §.§ Mathematical treatment of disorder We introduce disorder in our systems by picking the dimensionless potential barrier widths, λ̂_i,i=1,...,N, from a given probability distribution ρ(λ)dλ. We begin with a uniform distribution: ρ(λ) = 11 - 2 η , λ∈ [η,1-η] , η∈ (0, 1/2),where, for a given L, η is chosen in order to avoid physical nuisances, such as barriers widths smallerthan single atoms. The smaller is η, the larger is the support of the probability density function ρ(λ), thus a measure of the disorder degree is given by the value of η. The empirical mean width for a single realization of the disorder is a random variable denoted by:λ̂_N = 1/N∑_i=1^Nλ̂_i.The weak law of large numbers implies that λ̂_N converges in probability to the mean⟨λ̂⟩, in the large N limit.After the N widths have been generated, the total length of the sample may exceed or be smaller than the desired value, therefore we rescale all lengths introducing the parameter c_N=L(1-α)/Nλ̂_Nso that λ_i=c_Nλ̂_i and∑_i=1^Nλ_i=∑_i=1^NL(1-α)/Nλ̂_Nλ̂_i=L(1-α)=L_insLet us denote by Λ_N={λ_1,...,λ_N} the set of barrier widths. Among the possible realizations of Λ_N, the regular barrier distributionΛ_B={λ_B,...,λ_B} plays a crucial role, since it correspondsto the Kronig-Penney model, the continuum limit of which has been considered in Ref.<cit.>. We call periodic the case of Λ_B.Considering an observable A, defined as a function of a given realization of barriers, and denoting by Ω={Λ_N^(1),Λ_N^(2),....,Λ_N^(ℓ)} a set ofrealizations, the corresponding ensemble average is given by:⟨ A⟩_Ω=1/ℓ∑_i=1^ℓ A(Λ_N^(i)).We are interested in the observable S, which is also a function of the energy E of the incoming particle, of the potential height V and of the temperature T that determines the distribution of the particles energies: S=S(Λ_N;V,E,T).Averaging over the particles energy gives the coefficientS(Λ_N;V,T)=∫_0^∞S(Λ_N;E,V,T) f_eq(E)dE,where the Maxwellian probability densityf_eq(E)=√(1/π E)e^-Eis used to represent the electrode on the left as a classical heat reservoir. §.§ Sample compressionBecause of externally exerted pressure, the sample length may be reduced by an amount Γ, so that its length is given by L_compr = L - Γ. If the insulator is e.g. polymeric and the conductoris e.g. metallic, we may in first approximation assume that the length reduction only concerns the potential barrier widths. In any event, introducing the ratio r for the effect of compressionon the two materials, we may write:L_ins,compr=L_ins-Γ· rL_cond,compr=L_cond-Γ· (1-r) L_ins,compr+L_cond,compr=L_comprwhere the index compr denotes the lengths regarding the compressed state. For instance, the case r=1 describes the situation in which only the insulator is affected by the compression. Introducing the parameter α_compr=L_cond,compr/L_compr, the compressed state canbe described by the function f_Γ,r: (L,α,Λ_N,V)→ (L_compr,α_compr,Λ_N,compr,V_compr)that associates the old system, characterized by (L,α,Λ_N,V) with the compressed systemcharacterized by (L_compr,α_compr,Λ_N,compr,V_compr), where the notation indicates that the compression modifies the realization of the barrier widths and, consequently, that it mayaffect the potential height. One possibility for the variation of the potential under compression is that the area under a barrier,i.e. barrier width times barrier height, is constant.The idea is that the compression leads to higher insulator density, hence to an increase of the potential.The specific form of the increase is irrelevant here, since other rules may be simply implemented in our framework.One may ask whether the compression introduces disorder also in the potential strength, because of different increments in barriers of different widths. Using our rule, this does not happen.Indeed, consider a system composed by two barriers of width λ_1 and λ_2divided by a conduction region whose length is δ. One hasL=λ_1+λ_2+δ , L_ins=λ_1+λ_2 ,L_cond=δ ,α=δ/λ_1+λ_2+δCompressing the system by a quantity d, and distributing the compression with ratio r, one gets:L_compr=λ_1+λ_2+δ-d ,L_ins,compr=λ_1+δ-d· r , L_cond,compr=δ-d·(1-r) , α_compr=δ-d·(1-r)/λ_1+λ_2+δ-dObserve that the widths λ_1 and λ_2 arise from the normalization of realizationsλ̂_1 and λ̂_2 picked at random from the chosen distribution of widths. Then, we may writeλ_i = L(1-α)/Nλ_Nλ̂_i ,i=1,2If the area under each barrier is kept constant under compression, we haveλ_i V = λ_i,compr V_i,compr ,i=1,2which, thanks to eq.(<ref>) can be rewritten as:L(1-α)/Nλ_Nλ̂_i V = L_compr(1-α_compr)/Nλ_Nλ̂_i V_i, compr ,i=1,2This implies:V_1,compr=V_2,compr=V L(1-α)/L_compr(1-α_compr)The reasoning can be easily extended to any numbers of barriers. It follows that the heights of the potential barriers depend only on the compression level and on the ratio r, not on the realization of the microscopic disorder.§ NUMERICAL RESULTSThe solution (<ref>) of eq.(<ref>) must be subjected to the classical BenDaniel-Dukeboundary conditions on the generic l-th node, with l∈{0,1,...,2N-1}, which require the continuity both of the wave function and of its first derivative at each node: ψ_l(x_l)=ψ_l+1(x_l) ψ^'_l(x_l)=ψ^'_l+1(x_l)where x_l= ∑_i=1^(l/2)λ_i+δl/2, if l is even, and x_l=∑_i=1^(l+1)/2λ_i + δl-1/2 if l is odd, where λ_i denotes the random width of the i-th barrier. With this notation, eq.(<ref>) may be written as:𝐌_0(x_0) ·[ A_0; A_1 ] = 𝐌_1(x_0) [ A_2; A_3 ] 𝐌_2(x_1) ·[ A_2; A_3 ] = 𝐌_3(x_1) [ A_4; A_5 ] 𝐌_4(x_0) ·[ A_4; A_5 ] = 𝐌_5(x_0) [ A_6; A_7 ]where the support matrices 𝐌_2l and 𝐌_2l+1 have been introduced, and𝐌_4N-2(x_2N-1) ·[ A_4N-2; A_4N-1 ] = 𝐌_4N-1(x_2N-1) [ A_4N; A_4N+1 ]For E<V, these 2x2 matrices of coefficients 𝐌_2l(x_l) and 𝐌_2l+1(x_l) read:𝐌_2l(x_l)=[ e^ikx_le^-ikl; ike^ikx_l -ike^-ikl ]𝐌_2l+1(x_l)=[ e^-zx_le^zx_l; -ze^-zx_l ze^zx_l ]for even l, and 𝐌_2l(x_l)=[ e^-zx_le^zx_l; -ze^-zx_l ze^zx_l ]𝐌_2l+1(x_l)=[ e^ikx_le^-ikx_l; ike^ikx_l -ike^-ikx_l ]for odd l. Assuming that the amplitude of the incoming wave A_0 is known, and imposing A_4N+1=0,since there is no reflection at the right boundary, these equations constitute a set of 4N equationsin 4N variables, for which the support matrices 𝐌_2l and 𝐌_2l+1 allow us to write:[ A_0; A_1 ]=𝐌_0^-1·𝐌_1·𝐌_2^-1·𝐌_3···𝐌_4N-2^-1𝐌_4N-1·[ A_4N;0 ]=𝐌[ A_4N;0 ]where 𝐌 denotes the product of the 𝐌_i. It follows that A_0=M_11A_4Nwhere M_11 is the first entry of M. Consequently, eq.(<ref>) may be written as:S=A^*_4NA_4N/A^*_0A_0=1/|M_11|^2which is, in principle, a simple and efficient expression for the transmission coefficient.In practice, however, the range of energies and lengths of nanotechnological interest makeeq.(<ref>) hardly of any use for numerical calculations.For instance, L=500nm and energy of the order of E_T at room temperature imply that the dimensional variable z ranges between100 and 1000, which make overflow the entries of the matrices 𝐌_i, see e.g. Ref.<cit.> for overflow and Ω problems. To overcome these difficulties, we have developed a numerical scheme which relies uniquely upon the scattering matrix, and that is described in the Appendix.For our numerical results, if not otherwise stated, we refer to L=500nm, which is a length suitable forpresent nanotechnology, to V=3 for the dimensionless potential, and to α=10/11, meaning that theinsulator length amounts to the fraction 1/11 of the total sample length.Figure <ref> shows the common behaviour of the ensemble average⟨ S(Λ_N; V, T) ⟩_Ω as a function of the number of barriers N, computed overdifferent realizations of the microscopic disorder. The maximum value N=400 is determined by the fact that forL=500nm, one obtains barrier widths of the order of 10^-10m, below which the physical significance is lost. The right panel of Figure <ref> concerns the behaviour of ⟨ S(Λ_N;V,T)⟩_Ω for small values of N.Let us understand as greater disorder the situation in which the support of the uniform distribution of widhs ρis wider, i.e. the case in which η is smaller. Then, Fig.<ref> shows that theperiodic case enjoys the highest transmission coefficient, and that growing disorder implies adecay of ⟨ S ⟩_Ω.At the same time, the growth of N at fixed disorder degree makes ⟨ S ⟩_Ω increase, apart from a minimal decrease at small N. The periodic case, in particular, reaches a plateau at N≈ 200; in other words the periodic case attains within physically relevant scales the maximum transmissioncoefficient that the model allows and that remains throughout the physically relevant range. The disordered cases, onthe other hand, may also reach a plateau, but presumably at scales thatexceed the physically relevant ones. Therefore, in their cases, larger N, i.e. finer structures, correspond to higher ⟨ S ⟩_Ω.This statement agrees with Ref.<cit.>, in which a closed formula for the asymptotic behavior of the transmission coefficient in the periodic case has been given:Ŝ=lim_N→∞ S_N= [ 1+Ẽ^2/4E(sin(L√(E-Ẽ))/√(E-Ẽ))^2 ]^-1As we are interested in the average with respect to the energy distribution, we numerically computedS_B = ∫_Ef_eq(E) [ 1+Ẽ^2/4E(sin(L√(E-Ẽ))/√(E-Ẽ))^2 ]^-1for different disorder intensities. Setting the parameters given at the beginning of this section we getS_B=0.4178, while ⟨ S(Λ_4300;V,T)⟩_Ω = 0.4076, and⟨ S(Λ_10^5;,V,T)⟩_Ω = 0.4150, with λ∈ [0.4,0.6].For λ∈ [0.1,0.9], we get instead ⟨ S(Λ_7· 10^4;V,T)⟩_Ω = 0.4073We conclude that in the large N limit our model tends to the Kronig Penney model, with a rate of convergence thatdepends on the disorder degree. This confirms the results of Ref.<cit.>, although for highly disordered cases the asymptotic properties do not suit the nanotechnological interests.These observations mean that there are no localization effects in our model, unlike the case ofthe Anderson model. The origins of this discrepancy may be traced back to the fact that Anderson's model is based on a discrete tight binding Hamiltonian, that we do not have, and to the inapplicabilityin our model of Furstenberg's theorem, from which localization depending on thefirst Lyapunov exponent follows <cit.>,<cit.>. While the sequence of barriers of Anderson's modelincreases by adding new barriers without modifying the previous ones, adding a barrier in our construction alters the preceding barriers in order to keep unchanged the insulator amount, cf.eq.(<ref>). The hypothesis of Furstenberg's theorem are thus violated and we are in a framework that has been little investigated so far.§.§ Linear compression model and design optimization for sensor devicesSuppose now that our samples have been compressed according to the model described in section <ref>.Figure <ref> shows that increasing the compression percentage leads in our model to a mild decrease of S,for large N, and to an equally mild increase for small N. The cross-over between the two regimes grows with the disorder. Note that the growth of the disorder also seems to move forward, away form the physically interesting region, the asymptotic regime.Figure <ref> shows the dependence of ⟨ S(Λ_400;V,T)⟩ onthe compression percentage. More precisely, Fig <ref>(a) and <ref>(b) corresponds to Fig <ref>(a) and Fig <ref>(b), while Fig <ref>(c) and Fig <ref>(d) refer to the same setting, but r=1. It is evident that the reduction of the transmission coefficient is linear asa function of the compression factor. Furthermore table <ref> allows us to conclude that for given disorder, the absolute value of the rate of decrease, a, increases with r, the fraction of compression attributed to the insulator. For fixed r,the absolute value of a decreases if the disorder is higher. The increment of a for growing r means that the increment of the potential height is more significant than thereduction of L_ins. Figure. <ref> shows the behaviour of the probability current defined by eq. (<ref>) for different temperatures. Being the problem time independent, the current is constant along the system and it is sufficient to compute it at one of the extreme. For the sake of simplicity, we have fixed A_0=1 in the simulations. We next focus on possible optimal choices for the design of effective sensor devices. From this point of view, it is convenient to examine the relative percentage change of the transmission coefficient under compression, rather than the absolute variation investigated previously. In the following, the relative percentage change is defined as Δ(β)=|⟨ S(Λ_N,β)⟩ -⟨ S(Λ_N,0)⟩|/⟨S(Λ_N,0)⟩, where β∈ [0,100] is the compression percentage. In particular, we look for good choices for the number of barriers N and for the disorder degree, in order to have a high sensitivity to compression, i.e. large relative percentage change Δ(β) under compression.Guided by the behaviour of ⟨ S(N)⟩ described by figures <ref>, we have considered three possible optimal choices for the variable N, that are a)small number of barriers, N≈ 10; b) high number of barriers, N=400;c) intermediate number of barriers, N≈ 150. Table <ref> (see appendix <ref>)summarizes the most interesting values of Δ(β) for these different N, disorder degree and ratio r. The results show that even tough for N=400 we have the maximum absolute drop of the transmission coefficient under compression, the maximum relative drop is attained for smaller numbers of barriers. In fact, especially for strong disorder degree, we have that in the range N≈ 100∼ 200, Δ(β) is significantly larger than for N=400. For very low N, we have a large relative drop of S which might be in theory exploited. Nevertheless since S is very small in absolute value, there might be difficulties to measure the corresponding low currents. Considering the ratio, the higher is r, the greater is Δ(β) as we would expect. For r=1 and β=10 results are not shown, because that corresponds to a negative insulator length, cf. (<ref>).We observe that the higher the disorder, the higher the relative percentage change Δ(β), if the other parameters are fixed. Therefore in spite of all the other possible choices, randomness enhances the sensitivity to compression. For this reason, we have also simulated a system in which not only the barriers but also the wells are random. With λ∈ [0.1,0.9], and a weak disorder for the wells δ∈ [0.4,0.6]. Fig <ref> shows the behaviour of the transmission coefficient for this system. We observe that the behaviour changes since ⟨ S⟩is flat and almost vanishing for N<100 and then it grows quickly suggesting a faster rate of convergence to the periodic case than the fixed wells width case. Nevertheless the relative percentage change remains similar, even tough the crossover zone restricts, as well as the interval of moderate N values for which the relative percentage change is significant. Therefore, introducing randomness in the wells widths does not appear to improve the sensitivity.All things considered, the optimal design choice for a compression sensor whose barriers height grows linearly with compression, requires a number of barriers N≈ 100∼ 200, strong disorder only for the barriers width and ratio r close to one. Nevertheless the absolute variation of the transmission coefficient is small.§.§ Generalized compression modelThe numerical results of section <ref> show that a model of compression that preserves the area of the potential barriers producesa limited decrease of the transmission coefficient under compression. We therefore propose and numerically test another possibility. In particular we consider the following rule for the potential height: V_compr,a=V(L(1-α)/L_compr(1-α_compr))^2=V(L(1-α)/L_ins,compr)^2=V(L(1-α)/L_ins-L· r·β/100)^2where β is the compression percentage. In this case, the potential increases as1/(C-β/100)^2 for β/100→ C, where C=L_ins/L· r.Again, it is to be remarked that for every power p, the ruleV_compr=V(L(1-α)/L_ins-L· r·β/100)^p does not introduce any disorder in the potential heights.Taking p >1, compression makes the potential increase significantly more than in the case analysed in the previous section, thus we expect the transmissioncoefficient to drop much faster as a function of disorder. This is confirmed by Fig.<ref>. In Fig.<ref> a polynomial regression is shown, to find the decay rate of the transmission coefficient under compression, with the new potential barriers. We observe a linear decrease of S with the compression factor for a wide compression range, followed by a nonlinear, milder decay regime at high compressions. Clearly the absolute value of a is larger than the counterpart for the linear compression model. This indicates that the selection of the material plays a determinant role for the physical properties of the system, and that a non-linear behaviours of the potential height with compression are to be preferred. § FLUCTUATIONS AND RATE FUNCTIONS In <cit.>, <cit.>, the authors studied the decay of fluctuations as N increases, they identified micro,meso and macroscales and checked the validity of a large deviation principle(LDP). It is therefore interesting tocheck how the fluctuations are affected by compression. In Fig. <ref>, we plot √(⟨ (S-⟨ S⟩_Ω)^2)⟩/⟨ S⟩_Ω for different percentages of compression. We observe that compression enhances the relative size of fluctuations, and it does so more efficiently at small N. On the contrary, growing N implies smaller fluctuations relative size.Introducing the variable X_N=S_N/⟨ S_N ⟩, which is the transmission coefficient normalized toits expected value, approximated by the empirical mean, and denoting by ρ_N(X) the probability distribution of X_N, we can write⟨ S ⟩_Ω = ∫ S ρ_N(X) dX and we may consider now the behaviour under compression of the rate function Ξ (x), <cit.> <cit.> defined by:lim_N→∞-logρ_N(x)/N=Ξ(x)Figure <ref>(a) shows that, in accord with Fig.<ref>(b),the probability distribution covers a wider range of valuesunder larger compression rates. Therefore, also the properties of the fluctuations can be used to reveal the compression state. Furthermore, this can be done more efficiently for higher disorder. § CONCLUSION AND FUTURE DEVELOPMENTSIn the present article we have investigated the behaviour of a thermostatted disordered system under compression.Our results indicate that for physically relevant N, the randomness of the barrier widths leads to a decreaseof the transmission coefficient, which is more significant for stronger microscopic disorder.Considering the large N limit, we have shown numerically that our model behaves similarly tothe large N limit of the Kronig-Penny model recently studied in <cit.> and we have expanded that work considering energy averages.We have then shifted our attention to compressed systems, providing a mathematical framework suitable for real cases and amenable to experimental tests. For two compression models we find that compression causes a decrease of S. Modifying the degree of freedom p, which represents the power law followed by the potential heights under compression,our numerical simulations show that for quitea large interval of compression percentages, the decrease can be assumed to be linear.Furthermore for the linear compression model we have extensively investigated the relative percentage change of the transmission coefficient, identifying the best possible configuration for effective sensors. Eventually we have noticed that compression increases the fluctuations of S, as shown by a probability density and rate function estimation. This effect may be used to reveal the compression state of the sample. § ACKNOWLEDGMENTSThe authors are grateful to M. Colangeli for very useful remarks. Computational resources were provided by HPC@POLITO (http://hpc.polito.it)§ REFERENCES§ APPENDIX §.§ Numerical scheme: the transfer matrix and scattering matrix techniquesTo overcome the numerical difficulties described in section 2, we have developed a numerical scheme that relies uniquely upon the scattering matrix. If the transfer matrix relates linearly the wave amplitudes on the left side with the wave amplitudes on the right side, the scattering matrix relates linearly the amplitudes of wave exiting the barrier potential with the amplitudes of the wave entering the barrier potential. Therefore, considering a single barrier, the following relations hold:[ A_4; A_5 ]=M_3^-1M_2M_1^-1M_0[ A_0; A_1 ]=T[ A_0; A_1 ]=[ T_11 T_12; T_21 T_22 ][ A_0; A_1 ] [ A_4; A_1 ]=S[ A_0; A_5 ]=[ S_11 S_12; S_21 S_22 ][ A_0; A_5 ]It is straightforward to verify that:𝐒=[ S_11 S_12; S_21 S_22 ]=[ T_11T_22-T_21T_12/T_22T_12/T_22; -T_21/T_22 1/T_22 ]Since all the components of 𝐓 scale at most as e^zd, S_12, S_21, S_22 are bounded. S_11 might instead explode, because the numerator scales as e^2zd. Nevertheless, introducing a_1=(1-z/ik)(1-ik/z)a_2=(1+z/ik)(1+ik/z)a_3=(1-z/ik)(1+ik/z)a_2=(1+z/ik)(1-ik/z)one finds that the leading term of the numerator of S_1,1 is (a_1a_2-a_3a_4)e^2zd.Since a_1a_2-a_3a_4=0, we conclude that all the components of 𝐒 are bounded. Suppose now that the scattering matrix 𝐒̂ links linearly the wave amplitudes that enter and exit a sequence of N barriers, while 𝐒 describes the scattering process trough the (N+1)th that is added to the system. The following relations allow us to construct a unique scattering matrix for the whole system.A_4N=Ŝ_11S_11/1-S_12Ŝ_21 A_0 + (Ŝ_11S_12Ŝ_22/1-S_12Ŝ_21+Ŝ_12) A_4N+1A_1=( S_21+S_22Ŝ_21S_11/1-S_12Ŝ_21) A_0 +S_22Ŝ_22/1-S_12Ŝ_21 A_4N+1Once we have the total scattering matrix, it is easy to compute the transmission coefficient throughS=|A_4N|^2/|A_0|^2=|S_11|^2The scheme illustrated here has the advantage of being numerically stable and not subjected to overflow problems.Nevertheless, this advantage comes at the cost of having to deal with non linear relations, which require agreater computational effort than the simpler matrix multiplications of Eq (<ref>).Given the present day computer facilities, this is not a serious hinderance.§.§ Relative percentangechange under compression | http://arxiv.org/abs/1706.08284v2 | {
"authors": [
"Tommaso Vanzan",
"Lamberto Rondoni"
],
"categories": [
"cond-mat.dis-nn",
"quant-ph"
],
"primary_category": "cond-mat.dis-nn",
"published": "20170626085010",
"title": "Quantum thermostatted disordered systems and sensitivity under compression"
} |
Independent Motion Detection with Event-driven Cameras V. Vasco1, A. Glover1, E. Mueggler2, D. Scaramuzza2, L. Natale1 and C. Bartolozzi11iCub Facility, Istituto Italiano di Tecnologia, Genova, Italy{valentina.vasco, arren.glover, lorenzo.natale, chiara.bartolozzi}@iit.it2Robotics and Perception Group, University of Zurich, Zurich, Switzerland{mueggler, sdavide}@ifi.uzh.ch December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================Unlike standard cameras that send intensity images at a constant frame rate, event-driven cameras asynchronously report pixel-level brightness changes, offering low latency and high temporal resolution (both in the order of micro-seconds). As such, they have great potential for fast and low power vision algorithms for robots.Visual tracking, for example, is easily achieved even for very fast stimuli, as only moving objects cause brightness changes. However, cameras mounted on a moving robot are typically non-stationary and the same tracking problem becomes confounded by background clutter events due to the robot ego-motion. In this paper, we propose a method for segmenting the motion of an independently moving object for event-driven cameras. Our method detects and tracks corners in the event stream and learns the statistics of their motion as a function of the robot's joint velocities when no independently moving objects are present. During robot operation, independently moving objects are identified by discrepancies between the predicted corner velocities from ego-motion and the measured corner velocities. We validate the algorithm on data collected from the neuromorphic iCub robot. We achieve a precision of ∼90% and show that the method is robust to changes in speed of both the head and the target.Event-driven cameras, neuromorphic processing, independent motion. § INTRODUCTIONThe direct interaction between a robot and its surroundings is one of the major challenges in robotics. The iCub, designed to be anthropomorphic with a three-and-a-half year old child, primarily uses vision to measure the state of the external environment and, as such, visual motion estimation is fundamental. Unlike standard cameras that read the full sensor array to produce images at a fixed frame rate, event cameras only report change in pixel-level brightness above a threshold. The “events” are produced independently and asynchronously for each pixel sensor. They offer a high dynamic range (140 as compared to 60 of standard cameras), together with low latency and high temporal resolution (both in the order of micro-seconds). Processing of redundant information is avoided as pixels that do not experience a change simply do not activate.As such, event cameras are a promising technology for fast, accurate and low power vision algorithms for robots in dynamic environments. Since the output of event cameras is fundamentally different from standard cameras (a continuous, asynchronous stream of events instead of a sequence of images), new algorithms are required to deal with these data. The iCub <cit.> is a humanoid robot designed with sensory and actuation capabilities to interact with a dynamic environment, in which objects and the robot itself move simultaneously. The neuromorphic iCub is equipped with a stereo pair of ATIS event cameras <cit.>. Solving a problem such as visual tracking can be performed almost trivially using a stationary event camera: only the motion of the target causes events to be produced. The problem of segmenting the background from the target is inherently solved by the sensor. However, the movement of an event camera mounted on a robot causes events to be generated due to all contrast in the field-of-view and the problem of tracking or recognising the motion of a target object becomes more difficult. Differentiating the event camera signal caused by ego-motion from the signal caused by independent motion has many potential uses in event-driven robotics. The problem of Independent Motion Detection (IMD) was first studied in <cit.>. However, as noted by <cit.>, the apparent motion induced by the ego-motion has “some degree of statistical regularity” and “the structure of the environment [..] is far from arbitrary” for many applications (in their case, autonomous ground vehicles). Also constraints on the vehicle motion were suggest to be exploited <cit.>.Besides visual sensing, most robots are also equipped with proprioceptive sensors such as inertial measurement units and joint encoders. Thus, a typical strategy is to use the knowledge of the robot kinematics to predict the apparent motion induced by ego-motion <cit.>. However, due to imprecision in the mechanical structure and sensor acquisition errors, significant noise is present in these predictions. Therefore, <cit.> proposed to learn the correlation joint velocities and flow statistics, without requiring a kinematic model of the robot. Independent motion is found by predicting a probability distribution of the optical flow and identifying flow vectors that do not belong to this distribution. This approach estimates ego-motion correctly, even if the majority of the image plane is independent motion (e.g., a large object in front of the camera), which is not possible by using methods relying on vision alone.In this paper we present a method to segment events caused by ego-motion from those caused by independent object motion. Such a technique has wide applicability for event-driven algorithms in robotics. For example, identifying independent object motion makes detection and tracking of moving objects simple. In a dynamic environment moving objects are usually very relevant to behaviour (e.g. for avoidance or attentional interaction with human collaborators handling objects).Alternatively, segmentation of ego-motion events can be used to remove outliers to improve event-based visual odometry methods (e.g. <cit.>) in dynamic scenes. The use of event cameras for this task is driven by the strong potential for low-latency, low-power robotic vision. Our algorithm takes advantage of previous work in event-based corner detection <cit.> as well as traditional approaches to ego-motion segmentation <cit.>.§ RELATED WORK§.§ Independent Motion Detection for Robot Applications In <cit.>, the authors proposed the MotionCUT framework, based on Lucas-Kanade tracker failures. They observed that Lucas-Kanade algorithm typically fails around rotations and occlusions, which are likely caused by objects independently moving. This approach assumes visual motion to be dominated by ego-motion and only a small portion of it to be produced by independent motion. In case of large objects or objects close to the camera, the camera translation component is not negligible and it can produce a failure in the Lucas-Kanade tracking, as well as an independent moving object. The assumption of the ego-motion being the dominant motion of the scene is often violated in dynamic environments where many objects move at the same time. In <cit.>, the kinematics of the robots were assumed to be known, but with input-dependent noise due to mechanical imperfections and sensor noise. To address these issues, the correlation between predicted and tracked features was learned. For the prediction, the depth of the features was estimated using stereo. The statistics were learned in situations without independently moving objects, i.e., the apparent motion was only due to the ego-motion. In the testing phase, these statistics are used to detect anomalies that correspond to independent motion. In <cit.>, the detection of independent motion was used for object segmentation. It builds upon <cit.>, but additionally includes inertial measurements.§.§ Event-based Motion Estimation Much work on motion estimation with event cameras has focused on SLAM and visual odometry algorithms.Estimating only the rotational component of a camera has been performed using various methods <cit.>, however, the motion of the iCub's eyes includes a non-negligible amount of translational motion, and these algorithms are not suitable, especially when objects are close to the camera. Event-based 6-DOF SLAM and visual odometry methods <cit.> assume static scenes and sufficiently large independently moving objects introduce errors. In this paper, we consider environments in which objects are also moving and are interested in their motion, rather than solely the robot's ego-motion. It was shown that specific algorithmic adjustments were required to perform tracking in a highly-dynamic environment, which had simultaneous camera and object motion <cit.>. When using an event-based camera, the problem difficulty increases dramatically. In addition, the experiments relied solely on visual data as they did not have access to the robot kinematics. Previously, combining event camera vision with external sensors has been performed using a gyroscope for event-stabilisation <cit.>.Again, this method can only estimate the apparent motion due to camera rotation, but not translation.To segment ego-motion induced events from those induced from an independently moving object, first a measure of the optical flow is required. Optical flow can be easily calculated using event cameras, by fitting planes in the three-dimensional spatio-temporal space in which events exist <cit.>. However, similarly to frame-based cameras, when considering only small spatial windows of events, aperture problems cause incorrect flow estimation along long uniform edges. With event-based cameras, the locally spatial plane fitting method, in combination with the temporally asynchronous nature of events, makes it difficult to apply global correction methods.Unique features are typically used to avoid the aperture problem, from which the true optical flow can be calculated. Event-based corner detection methods have been proposed <cit.>. In this paper, we use <cit.>, that adapted the Harris method to the event data stream. It has also been shown that tracking corners in event-space is possible, resulting in a faster than frame-rate update of corner positions. § METHODOLOGYThe pipeline for detecting independent motion consists of visual corner detection, tracking and velocity estimation in parallel with the estimation of the joint velocities from motor encoders, as shown in Fig. <ref>. In a learning phase using a completely static scene, a model of the correlation between motor velocities and the resulting visual motion is developed (following <cit.>). During operation, computed visual motion is compared to the expected visual motion given the model and the movement of the robot. Large discrepancies between estimated motion and computed motion can be classified as being caused by an independently moving object. §.§ Corner Detection The dynamic vision circuitry of the ATIS camera provides a stream of events, in the form of {x_i, y_i, p_i, ts_i} (pixel position in (x, y), polarity, and timestamp). The stream of events is first reduced to the subspace of corner events, detected using <cit.>, with the following implementation changes. Previously a fixed event window was used that stored a fixed number of the most-recent events globally across the sensor space in the form of a “surface” <cit.>. Good results were achieved with the size of the window tuned according to the scene complexity. However this can be problematic when motion is not uniform over the visual scene (e.g. in our case of an object moving with motion independent of the camera motion), as higher velocities produce a higher number of events per second, manifesting in the event window as “thicker edges”. Instead of a single global event window, we implemented the same fixed event surface on a local scale, one for each pixel location. The window size becomes no-longer dependent on the particular scene and object motion, but on the feature type used: in our case, corners. As the features are constant, the window size can also be constant and independent of the scene. For corner detection we set the window size to be 2 × l, where l is the radius of the window used. §.§ Corner clustering and tracking Events that are labelled as corners are clustered into corner tracks, from which the velocity of each corner can be estimated. The first cluster is initialized with the first corner event and following corner events are added based on the spatial distance to the clusters: events are added if the distance is less than a threshold D; if no cluster is less than D pixels to the current event, a new cluster is created.Such a greedy corner allocation method can be applied as corner detection error is typically limited to 2 pixels <cit.> and we can assume to observe the full trajectory (pixel by pixel) of objects in the event space (i.e. an observed object cannot jump more than one pixel when using an event-based camera, whereas it is a common occurrence for objects that move faster than the frame-rate of a traditional camera).Each cluster is updated according to a first-in first-out rule: when the maximum size S is reached, the oldest corner event is removed from the cluster. To avoid tracking corner events that do not reflect the current motion, clusters are deleted if they are not updated for a time higher than t_refresh. Corners positions are tracked over time using regression to fit a line in the (x, y, t) space. Given a set of corner events _i = {x_i, y_i, p_i, ts_i} that belong to the current cluster _k, we find the function f that minimizes the sum of the squared deviations: f : min∑_i = 1^n (ts_i - f(x_i, y_i, a, b, c))^2. (a, b, c) defines the direction of the line and provides the components of the flow in both directions = (v_xi, v_yi) = 1/c(a, b). To minimise the error on velocity estimation, we define a minimum number of events m for the clusters to be informative. The corner event _i is augmented with the additional information from the velocity calculation, generating a flow event: _i {x_i,y_i,p_i,ts_i,v_xi,v_yi}.The algorithm is detailed in Algorithm <ref>.Corner clusters are updated and an estimation of visual velocity is calculated asynchronously as events occur, with a higher than microsecond resolution. The flow of the entire scene can be found by querying all event clusters that are active at any point in time. The most recent flow event within the cluster holds the most up-to-date velocity calculation for each corner cluster. §.§ Model Learning A model of average visual motion given motor encoder velocities is learned from data. Robot joint velocities, _e, can be estimated by differentiating encoder positions and applying a filter <cit.>. Supervised learning is performed every time joint velocities are read such that the input is _e(t) and the learning signal is the optical flow statistics _v(t), S_v(t) computed from corner tracking. _v(t) and S_v(t) are the mean scene velocities (μ_v_x, μ_v_y) and the covariance matrix of all active clusters queried at time t. Importantly, the model must be learned (once) in a completely static environment such that ego-motion alone contributes to the visual flow. Joint velocities are updated every 10 ms, while the estimated velocity from the corner clusters is asynchronously updated for each event, with an initial latency of m events. When at least half of the clusters reach m events, we read the encoder velocities and update the statistics, considering only clusters that satisfy the requirement. To avoid using just one cluster which would not be informative to learn scene statistics, we also define a minimum number of clusters n that need to be active at the same time.For fast speed motion, the speed of the algorithm is limited by the encoder update frequency, but for slower motion, the update is tailored to the dynamics of the scene, and the update is done only when sufficient information is gathered, saving computation and power.Joint velocities and associated cluster velocities are then used as training examples for ν-SVM <cit.> with an RBF kernel, to learn five regressors, namely (μ̂_̂v̂x̂, μ̂_̂v̂ŷ, σ̂_̂v̂x̂, σ̂_̂v̂x̂ ̂v̂ŷ, σ̂_̂v̂ŷ).The algorithm is detailed in <ref>.§.§ Independent Motion Classification During robot operation we compare the computed velocity of corner events _i = (v_x, v_y) with the expected distribution _̂v̂ = (μ̂_̂v̂x̂, μ̂_̂v̂ŷ), Ŝ_̂v̂ = (σ̂_̂v̂x̂, σ̂_̂v̂x̂ ̂v̂ŷ; σ̂_̂v̂x̂ ̂v̂ŷ, σ̂_̂v̂ŷ) predicted using the model, given the joint velocities. The Mahalanobis distance is used as a metric of how likely the calculated velocities belong to the ego-motion distribution: d = √(( - )^T Ŝ^-1 ( - )).Classification is performed using a distance threshold T, such that flow events _i that are below the threshold can be assumed to be created from the motion of the robot itself, while those that exceed the threshold can be labelled as independent motion events.§ EXPERIMENTS AND RESULTS§.§ Experiments We characterized and tested the algorithm on data collected from the event camera (ATIS) mounted on the iCub robot. iCub head has a total of 6 Degrees of Freedom (DoFs): 3 for the neck (pitch, roll and yaw), and 3 for the eyes (tilt, version and vergence). The iCub GazeController <cit.> was used to move the robot head position, controlling both neck and eyes independently. Ego-motion was generated by defining 3D target positions in the environment to gaze at, and different speeds were achieved by specifying the time for the trajectory to be completed.During the learning phase, in order to get a representative dataset for training, we sampled a static environment selecting targets for the controller distributed in a rectangular area, while changing the times for gazing. The data collected were used to train the ego-motion prediction model.For the testing phase, we performed two sets of experiments. In the first, we moved the head randomly in the environment at a fixed speed and simultaneously the hand around the yaw axis, while holding an object (a tea box). This method was used to control the velocity of the object and the ground truth. The hand moved at 4 different speeds, in order to evaluate the detection response with different velocities of the independently moving object.In the second experiment, we changed the velocity of the head, while maintaining fixed the velocity of the hand, in order to evaluate the detection response with different velocities of the ego-motion. Fig. <ref> shows the experimental setup. We controlled the hand and the head velocities, respectively at 120, 130, 140 and 150^∘/s and 3, 5 and 10^∘/s.For all testing datasets, the region-of-interest defining the position of the object that underwent independent motion waslabelled by hand. The corner events falling within the region-of-interest formed the ground-truth true-positive detections.We empirically selected the following parameters: l = 5 px, D = 5 px, n = 5, S = 50, m = 15, t_refresh = 1 s. §.§ Results Sparse corner flow events were compared to the learned model using the metric defined in Eq. <ref>. Fig. <ref> shows the distance between the velocity vectors and the predicted motion computed according to Eq. <ref>, grouped into background and independent motion (blue and red line) according to the ground truth. The average distance over ∼10 ms for each group is shown. In general, velocity vectors that belong to independent motion exhibit higher distances to the model than background vectors. This indicates the potential for separating ego-motion and independent motion using the proposed method.At some points in the dataset the distance for the independent motion corners drops to a similar level as the ego-motion corners (red and blue lines overlap, for example, between 0-1 s, 7-8 s, 15-16 s and in the last 5 s). This happens as the object stops moving and becomes indistinguishable from the background motion. These points do not correspond to failure in the detection algorithm, but represent an intrinsic limitation that originates from the use of motion to detect the target.The performance of the algorithm, on an event-by-event basis, as the detection threshold T changes, was evaluated in terms of precision and recall, shown in Fig. <ref>. At low thresholds (i.e. low recall), the precision is ∼90 %, which indicates that a strong “independent motion” response was always present in the system. Such a response is caused by noise in the detection algorithm, however a precision of 100 % may not be required for many robotic applications. The precision is stable over a wide range of thresholds, until ∼40 % recall rate is achieved. We can therefore select a threshold T to achieve a precision of ∼90 %. Performances are consistent with different speeds of the target and iCub head. The algorithm is therefore robust to changes in velocities and a valid threshold can be chosen that should be robust to speed variation.Example snapshots of events (accumulated over 350 ms and labelled according to the selected threshold T = 4), are shown in Fig. <ref> along with the corresponding motion distributions (in orientation and magnitude). In Fig. <ref>, corner events on the target fall within the ego-motion distribution, as the target is not moving. Coherently with Fig. <ref>, we can select a proper threshold to separate the independent motion from the background, both in conditions in which motion magnitudes and orientations are separable, as shown in Fig. <ref> (fig:frame2fig:mag2fig:ori2), and when only one of the two components is separable (a different magnitude but orientation falling in the same distribution is shown in Fig. <ref> (fig:frame3fig:mag3fig:ori3). Some corner events are not labelled as independent motion even though they belong to the moving object (Fig <ref> fig:frame2fig:frame3). This happens mainly when motion direction changes, in this case, the relative motion between the object and the background approaches zero and independent motion is similar to the ego-motion generated flow. Additionally, during sharp changes in motion, there is a small latency in recovering the clusters and re-evaluating the new velocity.We finally analysed the trajectories traced by corner events labelled as independent motion, grouped according to the ground truth (Fig. <ref>). Only trajectories along the x sensor plane are shown for clarity. Ideally, all corner events within the ground truth region should be classified as independent motion (i.e. in Fig. <ref>), and all corner events from the background should be ego-motion (i.e. in Fig. <ref>). Despite the recall of ∼ 40 % a consistent detection is still achieved over time, indicating a segmentation algorithm could potentially achieve a consistent result. Importantly, false positives are sparse and don't form a coherent pattern, such that a simple filter could easily reject such detections of independent motion.§ CONCLUSIONSIn this work, we have presented an event-based independent motion detector using the event camera, which disentangles the independent motion that occurs in the visual scene from the robot ego-motion. As background clutter can induce many additional events (but irrelevant for certain tasks), this task is crucial for event-driven scenarios where cameras are non stationary (on a robot). The use of cluster events reduces the data flow to the most informative events, enabling efficient, real-time implementation of many different event-driven vision algorithms for robotics.We detect and track corners in the space of events and learn the correlation between their motion and robot's joint velocities, when there is no moving object in the scene. We then label as belonging to independent motion corner events whose motion does not agree with the predicted velocity.We model ego-motion with first-order statistics, relying on the assumption of negligible motion parallax (which depends on the structure of the scene) and motion induced by rotation around the optical axis of the cameras (the head roll), as in <cit.>. This assumption did not affect the result as the algorithm was able to detect independent motion, with a precision of ∼ 90%, consistently with changing speed of both the target and the head. However we plan to model and learn the ego-motion using an affine motion model to be robust to head rotations. The detection can be problematic when the object changes direction, as the relative motion with the background approaches zero. However we do not need dense detection in time as sparse detections of independent motion can be used as triggering locations for visual tracking. Finally we show that sparse optical flow can be effectively used to address independent motion detection, reducing therefore the amount of data to process.We plan to use these sparse detections to segment a moving object in a cluttered scene on the iCub, implementing an event-based attention mechanism driven by the motion of the target, which would facilitate visual tracking.§ ACKNOWLEDGMENTThis research was supported by the Swiss National Science Foundation through the National Center of Competence in Research Robotics.IEEEtran | http://arxiv.org/abs/1706.08713v2 | {
"authors": [
"Valentina Vasco",
"Arren Glover",
"Elias Mueggler",
"Davide Scaramuzza",
"Lorenzo Natale",
"Chiara Bartolozzi"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170627081750",
"title": "Independent Motion Detection with Event-driven Cameras"
} |
Accretion disc around PSR B1259-63]A new approach to the GeV flare of PSR B1259-63/LS2883Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong†:[email protected] ⋆:[email protected] PSR B1259-63/LS2883 is a binary system composed of a pulsar and a Be star. The Be star has an equatorial circumstellar disk (CD). The Fermi satellite discovered unexpected gamma-ray flares around 30 days after the last two periastron passages. The origin of the flares remain puzzling. In this work, we explore the possibility that, the GeV flares are consequences of inverse Compton-scattering of soft photons by the pulsar wind. The soft photons are from an accretion disk around the pulsar, which is composed by the matter from CD captured by the pulsar's gravity at disk-crossing before the periastron. At the other disk-crossing after the periastron, the density of the CD is not high enough so that accretion is prevented by the pulsar wind shock. This model can reproduce the observed SEDs and light curves satisfactorily.§ INTRODUCTIONPSR B1259-63/LS 2883 is a binary system composed of a radio pulsar and a massive main sequence Be star <cit.>. The pulsar PSR B1259-63 is a non-recycled, spin-down powered radio pulsar, with a spin period of 47.76 ms, and a period derivative of 2.27×10^-15 <cit.>. The companion star is a Be star with the mass of ∼31 M_⊙, and rotating at a near-break-up rate <cit.>. The binary orbit is highly eccentric, with the eccentricity e=0.87, orbital period P=1237 days and semimajor axis a=7.2 AU. The stellar wind of the Be star interacts with the pulsar wind fiercely, forming a termination shock front, which is believed to be the origin of un-pulsed, orbital modulated high energy radiations of X-rays <cit.> and TeV gamma-rays <cit.>. The ejected matter at the equator of the Be star builds up a Keplerian decretion disk, which transports the angular momentum of the star outwardly <cit.>. The obstruction of the circumstellar disk (CD) leads to eclipse of the pulse radio emission near the periastron <cit.>, and the interaction between the pulsar and the CD gives rise to the double peaks structure of the X-ray light curve <cit.>.During the periastron passage in 2010/2011, the first passage after the launch of the Fermi satellite, <cit.> and <cit.> discovered the gamma-rays emission (100 MeV-100 GeV) from the system with the Fermi-LAT detector. Marginal gamma-rays were detected within period around the time of the periastron t_p, before it faded at ∼ t_p+15 days. Then at ∼ t_p+30 days, an sudden re-emergence of the gamma-ray emission surprisingly occurred, with the flux fast rising up to 20-30 times of its previous value over a few days, followed by a slow decay about two weeks before its disappear after t_p+60 days. The GeV flare reappeared during the following periastron passage in 2014 <cit.> at approximate the same orbital phase. The similarities of the flux, on-set phases, duration and sub-scale structures in the light curves between GeV flares in two periastron passages <cit.> suggest its repeating nature and the origin due to the interaction between the pulsar and the circumstellar matter. The cause of the GeV flare is puzzling: as the first-considered radiation mechanism, inverse Compton scattering (IC) of the cool pulsar wind electrons with seed photons from either the Be star or the CD gives the maximum of GeV flux at the periastron, where the external photons field is densest[Considered the anisotropy of IC, the GeV light curve peaks slightly before the periastron.]. Whereas the flare occurs ∼30 days after the periastron <cit.> ; on the other hand, if we trust the geometry of the CD implied by the X-ray light curve <cit.>, then the flare appears when the pulsar has well left the CD. Therefore it is difficult to attribute the flare to the immediate interaction between the pulsar and the CD.The mystery of the GeV flare attracts many explaining attempts: <cit.> still tried to explain the flare as a consequence of cool pulsar wind IC with the soft photons from the CD, but they also took into account of the time variance of the pulsar wind zone length (PWZ) towards the observer. PWZ is the place where the IC process take place. They argued that the rapid rising of the gamma-ray flux corresponds to the fast growth of the PWZ immediately after the pulsar's exiting from the CD. The main difficulty of this model is insufficient soft photons from CD for IC. As mentioned by the authors, the required target photon luminosity is ∼40% of LS 2883. Significant heating of the CD by the passage of the pulsar is supposed by the authors to reconcile this problem. Using also IC, <cit.> sought for seed photons in X-rays from the synchrotron radiation of the shock-heated electrons. Since in this model the seed photons have higher energy of a few keV, the bulk Lorentz factor of the pulsar wind should be ∼500 to give the observed gamma-ray emission. The Lorentz factor is much less than the typical values of pre-shock pulsar wind (seeand references therein). It is difficult for this model to explain the delay between the GeV flare and the X-ray lightcurve maximum.Another approach to the GeV flare, proposed by <cit.> and modeled in detail by <cit.>, is the Doopler-boosted synchrotron radiation in the shock tail: The stellar wind from the Be star collides with the pulsar wind to form a termination shock. Electrons (and positrons) in the pulsar wind are shock-heated and flow along the shock-tail at a mildly relativistic velocity. The hot electrons radiate away their energy through the synchrotron radiation, which is in X-ray in the flow co-moving frame and is Doopler-boosted to gamma-ray at the rest frame. This model is favored by the observational fact that the GeV flare appears when the pulsar is around the inferior conjunction, where the shock tail is close to the line of sight. However this model predicts that the light curves of the X-ray and gamma-ray peak at the same orbital phase, which violates the observation.<cit.> explored the possibility that, the flare is the result of a transit between the driven reconnection and the electromagnetic precursor scenarios of the termination shock. The above mentioned scenarios are both ways in which the oscillating magnetic field of the rotating pulsar dissipates its energy. The radiation efficiency is low in the driven reconnection scenario, and is high in the electromagnetic precursor scenario. When the condition of transition is satisfied, the radiation efficiency jumps thus gives the observed flare. This model also predicts another GeV flare at a different orbital phase when the transition condition is satisfied again (the peak is not necessary in the GeV band as mentioned by the authors, depending on the wind conditions.) The prediction is not supported yet by observations.A recent work of <cit.> studied numerically the impact of stellar wind clumps or inhomogeneities on the high-energy non-thermal radiation in binary system like this. They applied their study to PSR B1259-63/LS2883, and explored the scenario that a dense matter clump impacts on the two-wind interaction region. Although enlightening, their simulation is not able to reproduce the GeV flare satisfactorily.We explore the possibility in this paper that, when the pulsar passes across the CD, some of the matter in the CD is captured by the gravity of the neutron star. After the pulsar exits the CD, the captured matter forms an accretion disk. Soft photons emitted from the accretion disk are up-scattered by the pulsar wind, results in the observed gamma-ray flares. The gravity capture happens only when the termination shock front are inside the capture radius, inside which the velocity of the CD matter in the pulsar rest frame is less than the escape velocity from the gravity field of the pulsar. If the radius of the termination shock front is larger than the capture radius, the matter flow will be redirected before it enters the capture radius. We should also check whether the captured matter has enough specific angular momenta to form an accretion disk. The detailed calculations in section 2.1 show that the formation of an accretion disk by this scenario is likely.This paper is organized as follows. In section 2 we describe the model: In section 2.1 we study how the CD matter is captured by the gravity of the pulsar; Then in section 2.2 we describe how an accretion disk is formed from the captured matter, and how the accretion disk evolves with time; In section 2.3 we study the IC process, in which the pulsar wind up-scatters the accretion disk soft photons to gamma-rays. In section 3 we compare our model calculations with observations. We conclude and discuss in section 4.§ MODEL §.§ Accretion from the stellar discTo describe the motion of the pulsar and the CD matter, we construct the Cartesian coordinates frame as follows: the origin is on the barycenter of Be star and the pulsar, the x-axis is towards the periastron, z-axis is along the angular momentum of the orbit. The norm vector of the Be star's CD is𝐧_𝐜𝐝=(sinθ_ncosϕ_n,sinθ_nsinϕ_n,cosθ_n).Where θ_n and ϕ_n are the polar angle and the azimuthal angle of the norm vector respectively. The pulsar intersects with the mid-plane of the CD at the true anomaly ϕ_cd,±=ϕ_n±π/2. Based on the Kepler equations, the velocity of the pulsar as a function of the true anomaly ϕ is:[ v_p,x=e√(pμ)sinϕcosϕ/p-sinϕ√(pμ)/r; v_p,y=e√(pμ)sin^2ϕ/p+cosϕ√(pμ)/r;]where r=p/(1+ecosϕ) is the pulsar-star distance, e is the eccentricity of the orbit, μ≡ G(M_⋆+M_p) is the gravity constant times the sum of the total mass of the Be star and the pulsar, p≡ a(1-e^2) and a is the semi-major axis. The CD is thought to be Keplerian, therefore we assume the velocity of the disk matter is pure tangential and ignore the outward velocity. As a result, at any point 𝐫 the velocity of the disk matter is:𝐯_𝐜𝐝=√(GM_⋆/r)𝐧_cd×𝐫/r.Here we presume that the rotation of CD is prograde with the orbit of the pulsar. The retrograde case is discussed later. The velocity of the disk matter viewed at the pulsar rest frame is 𝐯_𝐫𝐞𝐥=𝐯_𝐜𝐝-𝐯_p. Any matter with the impact parameter less than the radiusr_BH=2GM_p/v^2_rel,is thought to be captured by the gravity of the neutron star. r_BH is the Bondi-Hoyle value <cit.> when v_rel is much larger than the sound speed. Mass transfer via gravity-capture is only possible when the shock front is within r_BH, i.e., r_s<r_BH as discussed in above section, wherer_s=√(L_spin/4πρ_cdv^2_relc),and L_spin is the spin down power of the pulsar, ρ_cd is the local density of the CD.We adopt the following parameters: * e=0.87, a=7.2 AU* M_⋆=31 M_⊙ <cit.>, M_p=1.4 M_⊙* θ_cd=45^∘, ϕ_cd=19^∘ <cit.>* L_spin=8×10^35 ergs/s, The density distribution of the decretion disk of the Be stellar has been modeled by previous researchers <cit.> as:ρ_cd = ρ_0(R_⋆/R)^nexp(-z^2/2H^2)= ρ_0(R_⋆/R)^nexp(-(ϕ-ϕ_cd)^2/2Δϕ^2),where n is in the range 3∼3.5, with 3.5 corresponds to an steady state isothermal outflow, H is the scale height of the disk, z is the cylindrical coordinate in the vertical direction, Δϕ is the half-opening angle of the disk projected on the orbital plane, R_⋆ is the radius of the Be star, which is ∼10 R_⊙ <cit.>. The second part of above equation is the disk density profile projected to the orbital plane, and Δϕ=18.5^∘ as modeled by <cit.>.<cit.> argued that a large base density (ρ_0∼1×10^-9 g/cm^3) is needed for this system to account for the double peak structure of the X-ray light curve.In the upper panel of figure <ref>, we plot the r_s and r_BH as function of ϕ under different ρ_cd profile index n, with ρ_0=10^-9 g/cm^3. ϕ is the true anomaly and the zero point is at the periastron. We ignore the perturbation of the density profile of the disk by the pulsar. We can see that the scenario of mass transfer is sensitive to n: when n=3, mass transfer occurs in both CD crossing; When n=3.3, mass transfer only happens during a part of the CD crossing before the periastron (“–" crossing hereafter, the other CD crossing is denoted as “+"); When n=3.5, no mass transfer from the CD during both crossings. Changing the n from 3 to 3.5 is equivalent to changing ρ_0 from ∼0.5 to ∼2 times the current value. The illustration of the orbit and the position of the disk (grey shade region) is plotted in the bottom panel of figure <ref> with n=3.3, with the mass transfer region shaded in red. When the condition that r_s<r_BH is met, the Bondi-Hoyle like process transfers matter into the gravity-bounded sphere at a rate:Ṁ_trans, BH=π r^2_BHρ_cdv_rel.Since v_rel and ρ_cd are not uniform and have gradient across the radius of the CD, the gravity-captured matter possess the specific angular momentum of <cit.>:l(t)=(GM_p)^2/v^3_rel(|∇ v_rel|/v_rel+|∇ρ_cd|/ρ_cd). During the mass transfer process, the total accreted mass is:M_tot=η∫Ṁ_trans,BHdt,where η accounts for the inefficiency of the Bondi-Hoyle accretion. According to numerical study by <cit.>, under the influence of the bow shock of the pulsar η can be as small as ∼1%. Here we leave η to be determined by fitting to the observation.After ramming into the capture radius, the gases dissipate their energy and shift to the orbit of lowest energy for a given angular momentum, i.e., a torus of gas at the so called circular radius: R_circ≡l̅^2/(GM_p), wherel̅=η∫ l(t)Ṁ_trans,BHdt/M_tot.After that, the matter losses angular momentum via viscosity and spread from R_circ to form an accretion disk. In table <ref> we list different mass transfer scenarios under variance of disk parameters. We see from this table that the mass transfer scenarios are not sensitive to slight changes of disk orientation and inclination (see the definition of three CD configurations in the caption of table <ref>). However, whether or not the mass transfer occurs depends strongly on the rotation direction of the CD: mass transfer is expected to occur when the CD is prograde, but not retrograde with the orbit. By study the spin-orbit coupling of this system, <cit.> suggested the misalignment of the spin axis of LS 2883 and orbit axis is less than 90^∘, which supports a prograde CD.For the purpose of further modeling, we work with θ_n=45^∘ ,ϕ_n=19^∘ and the assumption that the CD is prograde and n=3.3. In this case, M_tot=η7×10^23 g and R_circ=1.4×10^10 cm. R_circ is much larger than the radius of the light cylinder (7.6×10^7 cm), therefore the angular momentum of the accreted matter is enough to form an accretion disk. The mass transfer process lasts ∼5 days, while the flare lasts more than 60 days after the disk passage. Therefore the accretion rate of the disk should be much less than the mass transfer rate. Thus, as an approximation, we consider the gravity-captured matter accumulated around the pulsar within r_BH during the mass transfer process. When the matter transfer ends, the accumulated matter begins to form an accretion disk. §.§ Evolution of the accretion diskAs described above, after being captured by the gravity of the neutron star, complex physics processes (including shocks) will convert much of the kinetic energy of the gas into radiation and redistribute the remaining kinetic energy and angular momenta, before a torus of gas is formed at R_circ. This phase is much alike the situation where a tidal disrupted star's debris is accreted onto a black hole <cit.>. The torus is the predecessor of an accretion disk. After the accretion disk is developed, the inner edge decreases until it is disrupted by the torque of the magnetic field of the pulsar. The inner most radius r_M≈ r_A <cit.>, wherer_A=5.1×10^8Ṁ_acc,16^-2/7m_p^-1/7μ_30^4/7 cmis the Alfven radius, Ṁ_acc,16 is the accretion rate of the accretion disk in units of 10^16 g/s, μ_30 is the magnetic dipole in units of 10^30 G cm^3, m_p≡ M_p/M_⊙. After the accretion disk reaches the r_M, the propeller effect ejects the accreting matter thus the mass of the accretion disk declines steadily. This is much alike the situation where a pulsar is accreting from a fossil disk. Therefore, we follow the strategy of <cit.>, to assume the accretion rate in two phases:Ṁ_acc=Ṁ_acc,0,(r_in>r_M)Ṁ_acc=Ṁ_acc,0(t/τ)^-β,(r_in=r_M)where t=0 is the onset of disk formation and t=τ is the moment when the disk descends to r_M, β is the index to describe the decreasing of the accretion rate, β=19/16 for an electron scattering dominated disk opacity, and β=1.25 for a Kramer opacity. β=5/3 is a well-known value given by <cit.>, when considering a tidal disrupted star's debris accreting onto a black hole. r_in evolves with time as:r_in=R_circ-v_rt,(t<τ) r_in=r_M,(t≥τ), The constant accretion rate Ṁ_acc,0 is normalized to the total bounded mass:M_tot=∫^∞_0Ṁ_accdt,which gives:Ṁ_acc,0=(β-1)M_tot/βτ.On the other hand, if we assume the accretion disk to be a Shakura-Sunyaev disk <cit.>, thenτ ≈ R_circ/v_r≈ 1/2.7α^-4/5Ṁ_acc,16^-3/10m_p^1/4R_circ^5/4×10^6 s.where α is the viscosity index.Combining equations (<ref>, <ref>), the accretion rate can be solved as:Ṁ_acc,0/10^16 g/s=[2.7α^4/5β-1/βM_tot,22m_p^-1/4R_circ,10^-5/4]^10/7where R_circ,10 is R_circ in units of 10^10 cm.Since the temperature of the accretion disk increase inwardly, the inner most region of the accretion disk dominates the radiation, where the temperature is:T=1.4×10^4α^-1/5Ṁ_acc,16^3/10m_p^1/4r_in,10^-3/4 K,where r_in,10 is the inner radius of the accretion disk in units of 10^10 cm. With the time evolution of Ṁ_acc and r_in from equations (<ref>, <ref>), the temperature as a function of t is shown in figures <ref> and <ref>. §.§ Inverse Compton scattering from the pulsar windThe soft photons from the accretion disk are inverse-Compton scattered to high energy by electrons in the pulsar wind. If we assume a monochromatic electron energy of the pulsar wind, with the Lorentz factor Γ, then the characteristic energy of the scattered photon is <cit.>: E_γ≈300(kT/1 eV)Γ_0,4^2 MeV,in the Thomson regime (kTΓ/0.511 MeV≪1), andE_γ≈5.11Γ_0,4 GeV,in the Klein-Nishina regime (kTΓ_0/0.511 MeV≳1), where Γ_4≡Γ/10^4. For kT∼10 eV as we calculated above, a pulsar wind with Γ∼10^3-10^4 will produce inverse Compton scattered photons in the Fermi-LAT's energy range (100 MeV–100 GeV). Assuming an isotropic soft photons field, the energy differential power distribution (photons per energy range, per unit time) of inverse Compton scattering in Thomson limit is <cit.>:Ṅ_γ=∫ dN_e(l)∫^∞_02/Γ^2ϵ_phπ r_0^2cn(ϵ_ph,l)f(x)dϵ_ph,where dN_e(l) is the number of electrons at distance l along the line of sight, within range dl. In figure <ref>, we illustrate the IC process. The accretion disk is not necessarily face on as plotted; If we suppose nearly all the spin down energy of the pulsar is converted into the kinetic energy of electrons in the pulsar wind at l (as adopted by <cit.>, ∼96% of the pulsar wind energy is in the kinetic energy of electrons for this system. For a discussion on the role of pulsar wind in spin braking, seeand references therein), then we have:dN_e(l)=L_spin/Γ m_ec^2dl/c.ϵ_ph in equation (<ref>) is the energy of the soft photons, r_0 is the classic radius of the electron, n(ϵ_ph,l) is energy differential number density of the soft photons at l, f(x)=2xln x+x+1-2x^2 for 0<x<1, f(x)=0 for x>1 and x≡ E_γ/(4Γ^2ϵ_ph).For simplicity, we consider the case where the accretion disk is viewed face-on. The soft photon field at l is:n(ϵ_ph,l)=4π l/h^3c^3∫^R_out_R_inϵ_ph^2RdR/(R^2+l^2)^3/2(expϵ_ph/kT(R)-1).A change of the inclination will slightly reduce the density of photon field. This will eventually lead to larger fitted η. Combining equations (<ref>,<ref>,<ref>), and work out the integrate over l,Ṅ_γ=8π^2r_0^2/h^3c^3L_spin/Γ^3m_ec^2∫ϵ_phf(x)∫_R_in^R_outdR/expϵ_ph/kT(R)-1dϵ_ph, Since the surface brightness of a black body ∝ T^4, and T∝ R^-3/4, the soft photon field from the accretion disk can be represented by a ring with width of r_in and temperature of T(r_in). Therefore the integration over R in Equation (<ref>) can be simplified as:Ṅ_γ=8π^2r_0^2/h^3c^3L_spinr_in/Γ^3m_ec^2∫ϵ_phf(x)/expϵ_ph/kT(r_in)-1dϵ_ph,Assuming the pulsar wind is isotropic, the spectrum of the inverse Compton process is:F_γ=Ṅ_γ/4π D^2, With above equations, and the known temporal function of T and r_in in equations (<ref>) and (<ref>), we know how the spectrum energy distribution (SED) evolves with time, as shown in figure <ref>. For each time, we show the SED corresponding for (β, η)=(19/16, 0.059), (1.25, 0.047), (5/3, 0.024) in black dashed, red dash-dotted and blue solid curves respectively. α is 0.4 as the best fitted value found in the next section. The numbers labeled besides the apexes of each group of curves indicates the corresponding days after the periastron.§ COMPARING WITH OBSERVATIONS In order to evaluate this model, we compare it against observations. <cit.> presented the light curves and SEDs of the gamma-ray flares in 2010/2011 and 2014 passages. They divided the light curves of the flares into two phases based on their fluxes level: the peak phase and the tail phase. The peak phase of the first/second passage is from the 31 st day to 40 th/42 nd day after the periastron, and the tail phase is from the 40 th/42 nd to 71st day after the periastron. They obtained the SEDs by integrating the photons over the peak phases, tail phases and all the flares duration. To compare with them, we also average our calculated SEDs over the corresponding time ranges. In figures <ref> and <ref>, sub-panels titled with “All", “Peak" and “Tail" correspond to SEDs averaged over the whole durations, peak and tail phases. The black dashed, red dash-dotted and blue solid curves are SEDs calculated with different pairs of (β, η) (see the captions of the figures for details). The green points with error bars are observed SED data from <cit.>, those points with downward arrows are observational upper limits. Integrating equation (<ref>) over the Fermi-LAT energy range (100 MeV-100 GeV) at each instance gives the flux as function with time, i.e., the light curves. We compare the calculated light curves with the observed ones in the panels titled with “Light curve" of figures <ref> and <ref>. § CONCLUSION AND DISCUSSIONIn this paper, we study a new explanation of the observed post-periastron GeV flares of the binary system PSR B1259-63/LS2883. This phenomenon could be results as the pulsar wind inverse Compton-scattering of the soft photons, which are emitted from an accretion disk around the pulsar. The accretion disk composed by the matter from the circumstellar disk (CD) captured by the pulsar's gravity at disk-crossing. The pulsar crosses the CD twice in each orbital period, one pre-periastron and another post-periastron. In the post-periastron crossing, the density of the CD is not enough so that the pulsar wind prevents the matter from being accreted. While in the other crossing the density is sufficient. That explain the GeV flares are only observed once in each orbit.With certain parameters of the accretion disk, this model can reproduce the observed SEDs and light curves satisfactorily (see figures <ref> and <ref>).§.§ Sub-structures in the light curves and the disk instabilityIt is worth mentioning that the light curves in <cit.> show some significant sub-structures other than a pure decay. The disk instability might produce the above-mentioned small-scaled outbursts (seeand references therein; seefor a review). The recurrence time of those outbursts is of the viscous timescale near the inner edge of the disc, which is ∼0.1 days estimated with the best fitted parameters. However the variance time scale of the sub-structures in the light curves is of several days. Therefore, if the disk instability accounts for the sub-structures in the light curves, a different α≲0.05 is needed in the decaying phase than that in the rising phase. It is possible that the α of an accretion disk is not a constant and is time-dependent via other disk properties <cit.>. In this work, we only consider the quasi-stable disk as a simplification and do not try to reproduce the sub-structures.§.§ UV excess from the accretion diskOne critical evidence to prove or to disprove the formation of the accretion disk is the expected UV emission excess. The frequency differential energy flux at frequency ν from the accretion disk is:F_ν=4π hν^3cos i/c^2D^2∫^R_out_R_inRdR/exp hν/kT -1,where i is the inclination of the disk and D is the distance to the system. We assume i=45^∘ and use the simplification that the integration over the whole accretion disk can be approximated by a isothermal ring with width r_in and temperature T(r_in). While the flux from the Be star is:F_ν,⋆=2hν^3π R^2_⋆/c^2D^21/exp hν/kT_⋆-1,where T_⋆=30200 K is the temperature of the Be star, and the flux from the CD F_ν,sd can be represented by an isothermal disk with a temperature of 0.6 T_⋆ and an effective radius of the pseudophotosphere R_eff(ν) <cit.>.The SED in the optical band is shown in figure <ref>. An excess of the SED in UV band is a natural prediction. The energy flux (ν F_ν) from 2×10^16 Hz (82.7 eV) to 3×10^16 Hz (165 eV) is from 3.2×10^-11 ergs/cm^2/s to ∼1×10^-12 ergs/cm^2/s at the peak of the flare. It is higher than the low-energy tail of the soft X-ray emission from the shock, extrapolating to the same energy range from the figure 3 of <cit.>. Therefore, an extreme-UV observation covering 2×10^16-3×10^17 Hz (10-15 nm) during the GeV flare period can be used to test the proposed model. §.§ The dependence on the CD density profile of the modelWe see from table <ref> that the mass transfer scenario dependents on the density profile of the CD. If n=3, or equivalently ρ_0 halves from the current assumed value of 1×10^9 g/cm^3, the mass transfer condition will not be satisfied at disk-crossing; on the other hand, if n=3.5 or equivalently ρ_0 doubles from the current value, the mass transfer can occur at both disk-crossings. Thus people expect a smaller GeV peak ∼20 days after the main flare with ∼1/4 of the flux. The fluxes uncertainties of the current light curves are too large to confirm or to exclude it.§.§ The accretion of angular momentumIn our treatment, the accretion of angular momentum is due to the velocity and density gradient along the CD. Early studies suggested the specific angular momentum captured in this way is equation (<ref>). However <cit.> and <cit.> argued analytically and numerically respectively that, very little (if not none) angular momentum can be accreted by the center mass in this way. Later more sophisticated numerical calculations <cit.> showed that up to 70% of the total available angular momentum can be accreted. In our paper, as long as the actual specific angular momentum is enough, so that the circular radius R_circ is well outside the light cylinder, the validity of our model is not damaged. In our model, R_circ influences the gamma-ray flux in the combination α^4/5R^-5/4_circ, thus any change of R^-5/4_circ will be absorbed by a change of α during the fitting.In fact, as α varies in a reasonable range, the time delay between the disk-crossing and the flare can be adjusted accordingly in a range of a couple of weeks. As a result, this model has a weak predictive power over the position of the CD. As we mentioned in above paragraphs, that this model is potentially testable with observation of the UV excess. Besides, we expect that the spin period derivative of PSR B1259-63 increases during the GeV flare, due to the additional torque acting on the pulsar magnetic fields by the accretion disk.SXY thanks Prof. J. Takata for helpful instructions. The authors thanks anonymous referee for his/her careful reviewing and helpful comments, which brings much improvement of the manuscript. 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"authors": [
"Shu-Xu Yi",
"K. S. Cheng"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170627082133",
"title": "A new approach to the GeV flare of PSR B1259-63/LS2883"
} |
cfaHarvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138, USA; mailto:[email protected]@cfa.harvard.edu northwesternCenter for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208 ohioAstrophysical Institute, Department of Physics and Astronomy, 251B Clippinger Lab, Ohio University, Athens, OH 45701, USAMatt Nichollcfa, Edo Bergercfa, Raffaella Marguttinorthwestern, Peter K. Blanchardcfa, James Guillochoncfa, Joel Lejacfa and Ryan ChornockohioAt redshift z=0.03, the recently-discovered SN 2017egm is the nearest Type I superluminous supernova (SLSN) to date, and first near the center of a massive spiral galaxy (NGC 3191). Using SDSS spectra of NGC 3191, we find a metallicity ∼2at the nucleus and ∼1.3for a star forming region at a radial offset similar to SN 2017egm. Archival radio-to-UV photometry reveals a star-formation rate ∼15yr^-1 (with ∼70% dust-obscured), which can account for a Swift X-ray detection, and stellar mass ∼10^10.7 . We model the early UV-optical light curves with a magnetar central-engine model, using the Bayesian light curve fitting tool . The fits indicate ejecta mass 2-4 , spin period 4-6 ms, magnetic field (0.7-1.7)×10^14G, and kinetic energy 1-2×10^51 erg. These parameters are consistent with the overall distributions for SLSNe, modeled by <cit.>, although the derived mass and spin are towards the low end, possibly indicating enhanced loss of mass and angular momentum before explosion. This has two implications: (i) SLSNe can occur at solar metallicity, although with a low fraction ∼10%; and (ii) metallicity has at most a modest effect on their properties. Both conclusions are in line with results for long gamma-ray bursts. Assuming a monotonic rise gives an explosion date MJD 57889±1. However, a short-lived excess in the data relative to the best-fitting models may indicate an early-time `bump'. If confirmed, SN 2017egm would be the first SLSN with a spectrum during the bump-phase; this shows the same O2 lines seen at maximum light, which may be an important clue for explaining these bumps. § INTRODUCTION Modern time-domain surveys have become adept at uncovering new and rare types of transients. One of the most important findings has been the discovery of hydrogen-poor stellar explosions that reach peak absolute magnitudes M≲ -21 mag <cit.> – the so-called Type I superluminous supernovae (here referred to simply as superluminous supernovae or SLSNe).To date, these have been found almost exclusively in metal-poor dwarf galaxies at redshifts z≈ 0.1-2 <cit.>. SLSNe span a wide range of luminosities and timescales, but are uniquely defined by a blue spectrum at early time (with blackbody temperature ≳ 15000 K), which later cools to resemble more typical Type Ic SNe <cit.>.While the power source of SLSNe was initially a great mystery, a wide range of recent studies point to a `central engine', namely the rotational power of a central compact remnant.Given that the luminosity of SLSNe remains high for weeks to months, spin-down power from a millisecond magnetar <cit.> is more plausible than black hole accretion <cit.>. <cit.> recently presented magnetar model fits to the multicolor light curves of 38 SLSNe to uniformly determine the parameter space occupied by these explosions and their magnetar engines, using our new public Bayesian code . Key findings of this work include a continuous, and relatively narrow distribution of the SN and magnetar parameters, and a lack of dependence on metallicity.Here, we use the same framework to study the early evolution of SN 2017egm in the first 50 days after explosion. SN 2017egm is the nearest SLSN to date (z=0.0307, d_L=135 Mpc) and the first to be robustly associated with a nearby massive spiral galaxy, NGC 3191 <cit.>. Our goals are threefold: (i) determine the time of explosion and the SN and engine properties; (ii) determine the properties of the host, NGC 3191; and (iii) explore how SN 2017egm fits into the broader SLSN class in light of its unusual host environment.The former will help to focus searches for any pre-peak `bumps' as seen in several previous SLSNe <cit.>, and to inform on-going follow-up strategies. The latter two points will shed light on the nature of the engine and SN and their relation (if any) to the environment's metallicity, as well as on the comparison of SLSNe to long gamma-ray bursts (LGRBs).§ SN 2017EGM: A LOW-REDSHIFT SLSN§.§ Discovery SN 2017egm was discovered by the Gaia Satellite on 2017 May 23 UT and given the internal designation Gaia17biu. It was not detected in previous observations on 2017 Apr 20. It was reported to the Transient Name Server[https://wis-tns.weizmann.ac.il/] by the Gaia Photometric Science Alerts team <cit.>. The SN is coincident with the star-forming galaxy NGC 3191, with a radial offset of 5.2” from the nucleus (see <ref>).An optical spectrum was obtained by <cit.> on 2017 May 26.6 using the 2.16 m telescope at Xinglong Station of the National Astronomical Observatories of China. The spectrum was blue with only weak features, and the authors found a best match to a young Type II SN using the SuperNova Identification Code <cit.>; however they also noted that the absolute magnitude, ≈ -19 mag, was unusually high for this spectral type.<cit.> re-observed SN 2017egm at higher signal-to-noise on 2017 May 30 with the Nordic Optical Telescope as part of the NOT Unbiased Transient Survey.They found broad absorption features matching the O2 series common in SLSNe.Photometry subsequently obtained with the Swift UV Optical Telescope (UVOT) showed an absolute magnitude of M_V≈ -20.6 and a colour temperature of ≳ 15,000 K, confirming that SN 2017egm is a SLSN at a distance roughly half that of the previous nearest event <cit.>. <cit.> also noted that the massive spiral host galaxy is very unusual for SLSNe. §.§ Early Spectra We observed SN 2017egm with the FAST spectrograph <cit.> on the 60-inch telescope at Fred Lawrence Whipple Observatory (FLWO) on 2017 June 18 and with MMT using the Blue Channel Spectrograph and 300 grating on 2017 June 29 UT. The data were reduced using standard procedures in ; see Figure <ref>.We also show the original classification spectrum, available from the Transient Name Server <cit.>.For comparison, we plot the pre-maximum spectra of 5 well-studied SLSNe from the literature. The striking similarity in spectral slope and the characteristic O2 lines at 3500-4500 Å supports the classification by <cit.>, and confirms that SN 2017egm belongs to this class.We arrange the spectra in Figure <ref> in order of rest-frame time since explosion, using the best-fit explosion dates from the models of <cit.>. The spectra were obtained up to 2 weeks before maximum light, but this corresponds to a range of times since explosion given the diversity in rise times; we discuss the estimated explosion date for SN 2017egm in <ref>.The first spectrum of SN 2017egm is one of the earliest spectra ever obtained for a SLSN (≈ 10 d after explosion). After smoothing the data with a Savitsky-Golay filter, we recover the strongest O2 lines even at this early phase.The centroids of the O2 absorption lines exhibit a clear trend towards redder wavelengths with increasing time since explosion. In the case of SN 2017egm, the implied velocity decreases from ≈ 19,000to ≈ 13,000over a period of 24 d. This is comparable to the rate of decrease in Fe2 velocity shortly after maximum light <cit.>, but it is the first time this has been observed at such an early phase, using O2 lines, when few spectra are typically available.By the time of our MMT spectrum, obtained close to optical maximum light, the velocity of the O2 blends has decreased sufficiently that individual components can be resolved (see inset). This spectrum likely also shows blends of Fe2 and Fe3 <cit.>, and is more developed in the red, with overall similar features to SN 2015bn in particular <cit.>.We will show in <ref> that the earliest spectrum of SN 2017egm may have coincided with an initial bump in the light curve. This would be the first time that a spectrum has been obtained for a SLSN in this phase. The similarity of the early spectrum to those at maximum light is an important clue to the mechanism powering the bumps.§ HOST GALAXY: NGC 3191Integrated photometry of NGC 3191 is available from radio to UV. Using the near-UV flux density measured with the Galaxy Evolution Explorer <cit.>, we infer a star formation rate of SFR=1.4× 10^-28L_ν,UV≈ 5 M_⊙ yr^-1.On the other hand, infrared measurements at 25, 70, and 100 μm from the Infrared Astronomy Satellite <cit.> yield SFR≈ 11 M_⊙ yr^-1 (using the relation of ).NGC 3191 also exhibits resolved emission at 1.4 GHz in data from the Faint Images of the Radio Sky at Twenty centimeters survey <cit.>, pointing to a star formation origin, with an inferred value of SFR≈ 13 M_⊙ yr^-1 (using the relation of ).<cit.> recently detected X-ray emission from NGC 3191 using the X-ray Telescope (XRT) on Swift. We analyze the XRT data spanning 2017 June 2–20(exposure time of 19.2 ks) and find a source with a count-rate of (1.6± 0.4)× 10^-3 s^-1 at position RA=101904.44, Dec=+462718.7. This is about 13” away from the SN position, making a direct association unlikely. We fit the extracted counts with a power law spectral model using a Galactic column of N_H, MW=9.6× 10^19 cm^-2 <cit.>. We find no evidence for intrinsic absorption (N_H, int≲ 0.7× 10^22 cm^-2; 3σ). The power law index is Γ=2.2± 0.4, leading to an unabsorbed flux of 6.2^+1.8_-1.2× 10^-14 erg cm^-2 s^-1 (0.3-10 keV) or L_X≈ 1.3× 10^41 erg s^-1.Interpreting this luminosity as due to star formation yields a value of SFR≈ 10 M_⊙ yr^-1 (converting to the 2-10 keV band and using the relation of ), in excellent agreement with the IR and radio inferred values.We therefore conclude that NGC 3191 has a total star formation rate of ≈ 15 M_⊙ yr^-1, of which about 70% is dust obscured, and that the X-ray emission detected with XRT is likely not dominated by SN 2017egm itself. We further fit the broad-band spectral energy distribution of NGC 3191 using prospector <cit.> to determine additional galaxy parameters (Figure <ref>). The results indicate a star-forming galaxy with a moderate amount of dust and no evidence for AGN activity. The stellar mass is log M_* = 10.7 ± 0.1 , and SFR =14.8 ± 1.2yr^-1, in excellent agreement with our earlier estimates. The integrated metallicity is 0.7± 0.3 [Note that the metallicity grid inextends only to 1.5 ].The half-light radius of NGC 3191 as determined from Sloan Digital Sky Survey <cit.> u-band imaging is 6.9± 0.2”. The host-normalized offset of SN 2017egm from the nucleus is therefore R/R_ half≈ 0.75. This is close to the median for the SLSN sample <cit.> and the LGRB sample <cit.>.Finally, we use archival SDSS spectra taken both at the host nucleus and at the location of a bright star forming region about 7” away from the center to infer the gas-phase metallicity (Figure <ref>). The line ratios log ([N2]/Hα) = -0.40 and log ([O3]/Hβ) = -0.35 are typical of SDSS star-forming galaxies with minimal contribution from an AGN <cit.>. For the purpose of comparison with previous metallicity measurements for SLSN hosts, we use the calibration of <cit.>, utilizing both the [N2]/Hα ratio and the R_23 diagnostic. We find that at the galaxy center the metallicity is 12+ log(O/H)≈ 9.0 (≈ 2 Z_⊙), while at the location of the star forming region it is 12+ log(O/H)≈ 8.8 (≈ 1.3 Z_⊙), indicative of a mild gradient.While we do not have a metallicity measurement directly at the SN position, its radial offset from the center is comparable to that of the star forming region (see Figure <ref>) and we therefore conclude that it is most likely ≳ Z_⊙. In Figure <ref> we show NGC 3191 on the mass-metallicity diagram, along with previous SLSN, LGRB, and core-collapse SN host galaxies.Clearly, NGC 3191 has a higher metallicity and stellar mass than most previous SLSN hosts, comparable to those of regular core-collapse SN hosts.However, similar cases exist in the LGRB sample, and two other SLSNe appear to have metal-rich hosts: MLS121104 <cit.> and PTF10uhf <cit.>. Both were at z≈ 0.3 and occurred in the outskirts of their host galaxies, and therefore it is not possible to exclude significant metallicity differences between the integrated host and the explosion site, or even the presence of undetected dwarf satellites as the true hosts (e.g., ). <cit.> found that PTF10uhf was coincident with a merger between two star-forming galaxies, and may be more likely associated with the less massive of the pair. However, the metallicity they inferred from a spectrum extracted at the SN location did not indicate a significantly lower metallicity there. For SN 2017egm, due to its low redshift and small offset from the host nucleus, a Solar metallicity is more robust.This in turn lends support to the possibility that MLS121104 and PTF10uhf also occurred in Solar metallicity environments.We therefore conclude that ∼ 10% of SLSNe may occur in galaxies with Z≳ Z_⊙, while the bulk of the sample remains skewed to lower metallicities (e.g., ); this is similar to the case for LGRBs <cit.>.While the precise fraction of SLSNe observed in metal-rich galaxies will be sensitive to selection effects and detection efficiencies, our results indicate that there is no hard upper bound on the metallicity of SLSN environments. § LIGHT CURVE ANALYSISUpon the classification of SN 2017egm as a SLSN, Swift UV-Optical Telescope (UVOT) imaging was obtained starting on UT 2017 June 2 <cit.>. We downloaded the data from the public Swift archive and extract the UVOT light curves in the UVW2, UVM2, UVW1, U, B, and V filters following the procedures outlined in <cit.>, using a 3” aperture to minimise contamination from the underlying host galaxy light.The magnitudes are calibrated in the Swift photometric system <cit.>. We estimate the host contamination by extracting the flux in a 3” aperture centered on a bright part of the galaxy far from the SN position, and subtract this contribution from the UVOT photometry. Given the resulting surface brightness of m_u≈ 23 AB mag arsec^-2 the host contribution is only a few percent and it therefore has a minimal effect on the light curves and our models. We also obtained g, r, i imaging with the FLWO 48-inch telescope and the 1.3-m McGraw-Hill telescope at MDM observatory.We fit the multicolour UVOT light curves and the Gaia data point using the Modular Open Source Fitter for Transients (). Specifically, we use the magnetar-powered model recently fit to a sample of 38 SLSNe by <cit.>, with identical distributions of priors.The model and implementation in , and results for the SLSN population, are described in detail by <cit.>; a fuller description of the code and its usage will be provided by Guillochon et al. (in preparation).The earliest light curve data for SLSNe are often complicated by the presence of fast initial peaks, or `bumps' around the time of explosion <cit.>. These have been variously interpreted as shock-cooling of extended material <cit.>, a secondary shock driven by the engine <cit.>, or most recently a shocked `cocoon' surrounding a jet breakout <cit.>. However, our magnetar model for the main light curve peak does not accommodate these bumps explicitly. Hence we test for a bump in SN 2017egm by fitting the models to three versions of the light curves: with all data included (i.e., no bump; Model 1); with the Gaia data point excluded assuming that it represents a rapid bump phase (Model 2); and with the Gaia data point and first two UVOT epochs excluded assuming that they represent an extended bump (Model 3). All of these models and their associated posteriors are shown in Figure <ref>.§.§ Observational properties SN 2017egm reached maximum light in U-band on MJD 57924 with m_U ≈ 13.5 mag, or M_U≈ -22.2 mag. The apparent magnitude is ≳ 2 mag brighter than any previous SLSN due to the proximity of SN 2017egm, but the absolute magnitude is typical for SLSNe.We use our light curve models to estimate the explosion date and rise time to maximum light. The explosion date varies between our models depending on the assumption of a pre-explosion bump. If we assume the rise is smooth (Model 1), we find an explosion date of MJD 57889± 1 (2017 May 16).In Model 2 we find MJD 57895± 1 (2017 May 22). In Model 3, we find MJD 57901± 1 (2017 May 28), which is after the Gaia detection – in this case a precise estimate of the explosion date would require a detailed model for the bump itself (since the rise in our model is always monotonic).Overall, the range of rise times corresponding to the various model fits is ≈ 27-34 days, consistent with typical values for SLSNe <cit.>.Comparing the different models in Figure <ref>, we find that models that exclude points potentially contaminated by a bump (Models 2 and 3) provide somewhat better fits to the rise and maximum light behaviour. This is quantified in terms of a variance parameter <cit.>. We find median values of σ = 0.14, 0.11 and 0.08 mag for Models 1, 2 and 3, respectively. In the case of Model 3, this is similar to the observed photometric errors. Therefore there is modest evidence favouring the fits in which we assume a bump. Future data during the light curve decline will place further constraints on the model fits, which may help to emphasise the differences with and without a bump.In the latter models, the light curve fits are well below the earliest points from Gaia and UVOT, and the first points in the UV bands appear to show little evolution or even a slight dimming. This is fully consistent with the properties of the early bumps in SLSNe, and suggests that the Gaia discovery mayhave occurred during this phase.We therefore encourage all surveys to search their archives for more data around this time.It is therefore possible that the spectrum from <cit.>, obtained only 3 days after discovery (Figure <ref>), is the first spectrum of a SLSN ever taken during the bump phase. It is therefore noteworthy that it is remarkably similar to the spectrum at maximum light, with high velocity lines and a smooth velocity evolution.Thiscan inform models for the underlying mechanism of these bumps.SN 2017egm will soon enter Solar conjunction and will be unobservable between 2017 July 4 and September 16.While this prevents monitoring over several months of post-maximum evolution, SN 2017egm will remain bright enough for detailed study long after it becomes visible again. The late-time decline rate is not well constrained by our data, but our models provide probabilistic estimates. Assuming a limit of ≲ 23 mag for spectroscopy, our ensemble of fits suggests that SN 2017egm will remain brighter than this limit for ≳ 2 years, enabling unprecedented deep observations at phases that have not been possible to probe for more distant SLSNe.§.§ Physical parameters: is SN 2017egm unique? Given themetal-rich environment of SN 2017egm, it is imperative to explore whether this nearby event is representative of the general SLSN population, or if it differs in some fundamental properties. We can address this question with our light curve fit and the broad comparison sample of <cit.>.In Figure <ref> we plot the medians and error bars for the ejecta mass (M_ ej), kinetic energy (E_K), magnetar spin period (P), and magnetic field (B) estimated from our three model fits to the existing data for SN 2017egm, in comparison to the full SLSN sample (see Figure 5 of ).We find P≈ 4-6 ms, B≈ (0.7-1.7)× 10^14 G, M_ ej≈ 2-4 , and E_K≈ 1-2× 10^51 erg.These values are well within the distribution for the overall SLSN sample (all are within ≈ 1σ of the population medians), indicating that SN 2017egm isnot an atypical event. Thus, despite its occurrence in a metal-rich environment, SN 2017egm appears to be a typical member of the SLSN population.We do however note that the relatively low ejecta mass and modest spin period we infer could in principle be a reflection of the environment, in the sense that higher metallicity may lead to greater loss of mass and angular momentum before explosion.<cit.> found a possible correlation between host metallicity and magnetar spin period, although <cit.> did not find such a trend with a larger sample of events.A larger number of SLSNe at the high-metallicity end will be needed to test if there does exist a dependence of spin period or ejecta mass on metallicity.Given that it overlaps the rest of the population at the ≈ 1σ level in all parameters, SN 2017egm does demonstrate that any such dependence is weak. Thus, the main effect of metallicity is a reduction in the rate of occurrence of SLSNe, rather than a change in the explosion properties.§ CONCLUSIONS We have analysed the host galaxy and early evolution of SN 2017egm — the nearest SLSN to date and the most securely associated with a Solar-metallicity environment.Our analysis makes predictions for the explosion time and future light curve evolution of SN 2017egm, suggests that the early data may represent a `bump' phase, and demonstrates that despite the higher metallicity the properties of SN 2017egm are typical of the overall SLSN sample.Using archival photometry and spectroscopy of NGC 3191 we find M_*≈ 10^10.7 M_⊙, SFR≈ 15yr^-1, and a metallicity at a comparable radial offset to that of SN 2017egm of ≈ 1.3 (≈ 2 at the nucleus). Together with two other SLSNe from the literature that appear to be in similar galaxies (but at z≈ 0.3), we estimate that up to ≈ 10% of SLSNe may occur at Solar metallicity, similar to the findings for LGRBs.While the SLSN rate is clearly suppressed at high metallicity <cit.>, there does not seem to be a strong upper bound on the metallicity.We model the pre-maximum UV and optical photometry with , assuming a magnetar central engine, and find parameters typical of the general SLSN population <cit.>. We use the ensemble of fits to make probabilistic estimates for observables such as the explosion date and the time for which SN 2017egm will be observable; SN 2017egm should be observable spectroscopically for ≳ 2 years, allowing for future detailed studies of unprecedented detail.If we assume a monotonic rise, we estimate an explosion date of MJD 57889± 1 d. This will inform archival searches for any pre-maximum `bumps' around the time of explosion <cit.>.In fact, our modeling already shows some discrepancies with the data at the earliest epochs that likely indicates a bump has occurred. If confirmed, this makes the spectrum obtained by <cit.> the first ever obtained during the bump phase of a SLSN. We found that that this spectrum matches typical SLSN spectra at maximum light, though with O2 lines that are more strongly blue-shifted.The most important conclusion of our study is that metallicity has at most a modest effect on the physical parameters of SLSNe and their engines, and primarily impacts on the overall rate. This supports the analysis of <cit.>, who found no correlations between metallicity and any model parameters for their SLSN sample. A similar conclusion has been reached for LGRBs <cit.>. However, it is also possible that the relatively slow spin period (while still overlapping the rest of the spin distribution for SLSNe) is a reflection of the metal-rich environment <cit.>.Continued multi-wavelength observations of SN 2017egm will constrain the parameters more tightly.We will continue to model the data and distribute results via the Open Supernova Catalog <cit.>, providing a test for whether appropriately calibrated physical models with can provide robust predictions to aid in follow-up observations of SN 2017egm and future SLSNe and other transients. If SN 2017egm continues to evolve in a similar fashion to other SLSNe, it will support a picture where fairly normal SLSNe can sometimes occur in unexpectedly metal-rich environments. Future data will also tell us whether the closest SLSN to date has any more surprises in store. We thank Iair Arcavi for important comments that improved this work. The Berger Time-Domain Group at Harvard is supported in part by NSF grant AST-1411763 and NASA ADA grant NNX15AE50G. This paper uses data products produced by the OIR Telescope Data Center, supported by the Smithsonian Astrophysical Observatory. This work is based in part on observations obtained at the MDM Observatory, operated by Dartmouth College, Columbia University, Ohio State University, Ohio University, and the University of Michigan. natexlab#1#1[Albareti et al.(2016)Albareti, Prieto, Almeida, Anders, Anderson, Andrews, Aragon-Salamanca, Argudo-Fernandez, Armengaud, Aubourg, et al.]alb2016 Albareti, F. D., Prieto, C. A., Almeida, A., et al. 2016, arXiv preprint arXiv:1608.02013[Angus et al.(2016)Angus, Levan, Perley, Tanvir, Lyman, Stanway, & Fruchter]ang2016 Angus, C. R., Levan, A. J., Perley, D. 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"authors": [
"Matt Nicholl",
"Edo Berger",
"Raffaella Margutti",
"Peter K. Blanchard",
"James Guillochon",
"Joel Leja",
"Ryan Chornock"
],
"categories": [
"astro-ph.HE",
"astro-ph.CO",
"astro-ph.SR"
],
"primary_category": "astro-ph.HE",
"published": "20170627161146",
"title": "The superluminous supernova SN 2017egm in the nearby galaxy NGC 3191: a metal-rich environment can support a typical SLSN evolution"
} |
firstpage–lastpage 2002Thermal and energetic processing of astrophysical ice analogues rich in SO_2 Z. Kaňuchová corresponding author1,3 Ph. Boduch 2 A. Domaracka 2 M.E. Palumbo3 H. Rothard2G. Strazzulla 3 ==========================================================================================================================================================================================================================================ξ^1 CMa is a monoperiodically pulsating, magnetic β Cep star with magnetospheric X-ray emission which, uniquely amongst magnetic stars, is clearly modulated with the star's pulsation period. The rotational period P_ rot has yet to be identified, with multiple competing claims in the literature. We present an analysis of a large ESPaDOnS dataset with a 9-year baseline. The longitudinal magnetic field shows a significant annual variation, suggesting that P_ rot is at least on the order of decades. The possibility that the star's Hα emission originates around a classical Be companion star is explored and rejected based upon VLTI AMBER and PIONIER interferometry, indicating that the emission must instead originate in the star's magnetosphere and should therefore also be modulated with P_ rot. Period analysis of Hα EWs measured from ESPaDOnS and CORALIE spectra indicates P_ rot > 30 yr. All evidence thus supports that ξ^1 CMa is a very slowly rotating magnetic star hosting a dynamical magnetosphere. Hα also shows evidence for modulation with the pulsation period, a phenomenon which we show cannot be explained by variability of the underlying photospheric line profile, i.e. it may reflect changes in the quantity and distribution of magnetically confined plasma in the circumstellar environment. In comparison to other magnetic stars with similar stellar properties, ξ^1 CMa is by far the most slowly rotating magnetic B-type star, is the only slowly rotating B-type star with a magnetosphere detectable in Hα (and thus, the coolest star with an optically detectable dynamical magnetosphere), and is the only known early-type magnetic star with Hα emission modulated by both pulsation and rotation.Stars : individual : ξ^1 CMa – Stars: magnetic field – Stars: early-type – Stars: oscillations – Stars: mass-loss. § INTRODUCTION The bright (V = 4.3 mag), sharp-lined β Cep pulsator ξ^1 CMa (HD 46328, B0.5 IV) was first reported to be magnetic by <cit.>, based upon FORS observations at the Very Large Telescope (VLT). <cit.> confirmed the detection using ESPaDOnS at the Canada-France-Hawaii Telescope (CFHT). Despite intensive observation campaigns with different spectropolarimeters, there are conflicting claims regarding the star's rotational period: <cit.> derived a rotational period of ∼2.18 d based on FORS1/2 and SOFIN data, while <cit.> found a longer period of ∼4.27 d based on ESPaDOnS measurements. Bothandagreed that the star was likely being viewed with a rotational pole close to the line of sight. However,noted that given the small variation in they could not rule out intrinsically slow rotation. <cit.> was the first to note ξ^1 CMa's variable radial velocity. <cit.> found a pulsation period of ∼0.2 d, identifying the star as a β Cephei variable. While many β Cep stars are multi-periodic, ξ^1 CMa appears to be a monoperiodic pulsator <cit.> with an essentially constant period <cit.>. In their analysis of CORALIE high-resolution spectroscopy, <cit.> found a pulsation period of 0.2095764(4) d (where the number in brackets refers to the uncertainty in the final digit).also reported that ξ^1 CMa is one of the few β Cep stars that achieves supersonic pulsation velocities. Several spectroscopic mode identification methods revealed that the oscillation frequency most likely corresponds to either a radial (l,m)=(0,0) mode, or a dipolar (l,m)=(1,0) mode, the latter viewed at small inclination. An l=2 mode was ruled out via modelling of the velocity moments. Photometric mode identification indicates that only the radial mode agrees with the frequency <cit.>.Like many magnetic early-type stars, ξ^1 CMa displays the emission signatures of a magnetized stellar wind in optical, ultraviolet, and X-ray spectra. The star's X-ray emission spectrum is very hard, and the brightest amongst all magnetic β Cep stars, as demonstrated with Einstein <cit.>, ROSAT <cit.>, and XMM-Newton <cit.>. The star's X-ray emission has recently been shown to be modulated with the pulsation period <cit.>, a so-far unique discovery amongst massive magnetic pulsators <cit.>[Short-term X-ray variability has also been reported for the magnetic β Cep star β Cen <cit.>, and for the non-magnetic (Jason Grunhut, priv. comm.) β Cep star β Cru <cit.>. However, while subsequent analyses of the X-ray light curves of these stars have confirmed variability, they have not confirmed those variations to be coherent with the primary pulsation frequencies <cit.>, thus ξ^1 CMa is the only β Cep star for which X-ray variation is unambiguously coherent with the pulsation period.]. Various wind-sensitive ultraviolet lines show strong emission <cit.>, which is typical for magnetic stars. ξ^1 CMa has also been reported to display Hα emission <cit.>, although the properties of this emission have not yet been investigated in detail. Our goals in this paper are, first, to determine the rotational period based upon an expanded ESPaDOnS dataset, and second, to examine the star's Hα emission properties. The observations are presented in 2. In 3 we refine the pulsation period using radial velocity measurements. In 4 we analyze the line broadening, determine the stellar parameters, and investigate the relationship between the star's pulsations and its effective temperature. In 5 we examine the possibility that the Hα emission may be a consequence of an undetected binary companion, constraining the brightness of such a companion using both interferometry and radial velocity measurements. The magnetic measurements and magnetic period analysis are presented in 6. We examine the long- and short-term variability of the Hα emission in 7, and investigate the influence of pulsation on wind-sensitive UV resonance lines. Magnetic and magnetospheric parameters are determined in 8, 9 presents a discussion of the paper's results, and the conclusions are summarized in 10. § OBSERVATIONS §.§ ESPaDOnS spectropolarimetry Under the auspices of the Magnetism in Massive Stars (MiMeS) CFHT Large Program <cit.>, 29 ESPaDOnS Stokes V spectra were acquired between 2008/01 and 2013/02. One additional observation was already published by <cit.>. A further 4 ESPaDOnS observations were acquired by a PI program in 2014, and another 22 by a separate PI program in 2017[Program codes 14AC010 and 17AC16, PI M. Shultz.]. ESPaDOnS is a fibre-fed echelle spectropolarimeter, with a spectral resolution λ/Δλ∼ 65,000, and a spectral range from 370 to 1050 nm over 40 spectral orders. Each observation consists of 4 polarimetric sub-exposures, between which the orientation of the instrument's Fresnel rhombs are changed, yielding 4 intensity (Stokes I) spectra, 1 circularly polarized (Stokes V) spectrum, and 2 null polarization (N) spectra, the latter obtained in such a way as to cancel out the intrinsic polarization of the source. The majority of the data were acquired using a sub-exposure time of 60 s, with the exception of the first observation (75 s), and the data acquired in 2017, for which a 72 s sub-exposure time was used to compensate for degradation of the coating of CFHT's mirror. The data were reduced using CFHT's Upena reduction pipeline, which incorporates Libre-ESPRIT, a descendent of the ESPRIT code described by <cit.>. <cit.> describe the reduction and analysis of MiMeS ESPaDOnS data in detail. The log of ESPaDOnS observations is provided in an online Appendix in Table <ref>. The quality of the data is excellent, with a median peak signal-to-noise ratio (S/N) per spectral pixel of 828 in the combined Stokes V spectrum. The 2 observations acquired on 2017/02/11 had a peak S/N below 500, and were discarded from the magnetic analysis. The log of sub-exposures, which were used for spectroscopic analysis, is provided in an online Appendix in Table <ref>. Note that while there are 56 full polarization sequences, due to one incomplete polarization sequence there are 227 rather than 224 sub-exposures listed in Table <ref>.§.§ MuSiCoS spectropolarimetry Three Stokes V spectra were obtained in 2000/02 with the MuSiCoS spectropolarimeter on the Bernard Lyot Telescope (TBL) at the Pic du Midi Observatory. This instrument, one of several similar fibre-fed échelle multi-site continuous spectroscopy (hence MuSiCoS) instruments <cit.> constructed at various observatories was uniquely coupled to a polarimeter <cit.>. It had a spectral resolution of 35,000 and covered the wavelength range 450–660 nm, across 40 spectral orders. As with ESPaDOnS, each polarimetric sequence consisted of 4 polarized subexposures, from which Stokes I and V spectra, as well as a diagnostic null N spectrum, were extracted. The log of MuSiCoS data is provided in an online Appendix in Table <ref>. One of the MuSiCoS observations has a low S/N (below 100), and was discarded from the analysis. The data were reduced using ESPRIT <cit.>.Normal operation of the instrument was verified by observation of magnetic standard stars in the context of other observing programs <cit.>. In addition to this, we utilize the MuSiCoS spectra of 36 Lyn presented by <cit.>, one of which was obtained during the same observing run as those of ξ^1 CMa, in order to compare them to observations of the same star acquired recently with Narval, a clone of ESPaDOnS which replaced MuSiCoS at TBL, and which achieves essentially identical results <cit.>. This comparison is provided in Appendix <ref>.§.§ CORALIE optical spectroscopy A large dataset (401 spectra) was obtained between 2000/02 and 2004/10 with the CORALIE fibre-fed echelle spectrograph installed at the Nasmyth focus of the Swiss 1.2 m Leonard Euler telescope at the European Southern Observatory's (ESO) La Silla facility <cit.>. The spectrograph has a spectral resolving power of ∼100,000, and covers the wavelength range 387–680 nm across 68 spectral orders. The data were reduced with TACOS <cit.>.The first analysis of these data was presented by <cit.>.The log of CORALIE observations is provided in an online Appendix in Table <ref>. The median peak S/N is 205. One observation, acquired on 10/12/2003, was discarded as it had a S/N of 16, too low for useful measurements.§.§ IUE ultraviolet spectroscopy ξ^1 CMa was observed numerous times with the International Ultraviolet Explorer (IUE). The IUE could operate in two modes, high-dispersion (R∼2000) or low-dispersion (R∼300), with two cameras, the Short Wavelength (SW) from 115 to 200 nm and the Long Wavelength (LW) from 185 to 330 nm. We retrieved the data from the MAST archive[Available at https://archive.stsci.edu/iue/]. The data were reduced with the New Spectral Image Processing System (NEWSIPS). There are two simultaneous low-resolution spectra obtained with the Short Wavelength Prime (SWP) and Long Wavelength Redundant (LWR) cameras, along with 13 high-resolution spectra obtained with the SWP camera. The high-resolution data are of uniform quality as measured by the S/N, which is approximately 20 in all cases. Twelve of the spectra were obtained in close temporal proximity, covering a single pulsation cycle in approximately even phase intervals. The first observation was obtained 154 days previously. We used the absolute calibrated flux, discarded all pixels flagged as anomalous, and merged the various spectral orders. §.§ VLTI near infrared interferometry While the angular radii of ξ^1 CMa and its circumstellar material are certainly too small to be resolved interferometrically, the star is bright enough that the data can be used to search for a high-contrast binary companion.We have acquired four Very Large Telescope Interferometer (VLTI) observations: low-resolution AMBER H and K photometry, a high-resolution AMBER NIR spectro-interferogram, and one low-resolution H band PIONIER observation. AMBER (Astronomical Multi-Beam Recombiner) offers three baselines and can operate in either low-resolution photometric mode or high-resolution (R∼12000) spectro-interferometric mode <cit.>. PIONIER (Precision Integrated-Optics Near-infrared Imaging ExpeRiment) combines light from 4 telescopes, offering visibilities across 6 baselines together with 4 closure phase measuremnts <cit.>. All data were obtained using the 1.8 m Auxilliary Telescopes (ATs), which have longer available baselines, and hence better angular resolution than the 8 m Unit Telescopes (UTs). AMBER observations were obtained with baselines ranging from 80 to 129 m. PIONIER was configured in the large quadruplet, with a longest baseline of 140 m. The nearby star ξ^2 CMa (A0 III, H = 4.63) was used as a standard star for calibration. The observing log is given in Table <ref>. As the execution time for a full measurement (∼1 hr) is a significant fraction of the pulsation period, pulsation phases are not given. Owing to the relatively faint magnitude, the wavelength edges of the AMBER H and K profiles were particularly noisy. These data points were edited out by hand before commencing the analysis. No such procedure was necessary for the PIONIER observation.§ PULSATION PERIOD§.§ Radial Velocities We measured radial velocities (RVs) from the ESPaDOnS and CORALIE spectra using the centre-of-gravity method <cit.>. In addition to using the same line (Si iii 455.3 nm) used by <cit.>, we have used: N ii 404.4 and 422.8 nm; N iii 463.4 nm; O ii 407.2, 407.9, 418.5, and 445.2 nm; Ne ii 439.2 nm; Al iii 451.3 nm; and Si iv 411.6 nm. For ESPaDOnS data we used individual sub-exposures rather than the Stokes I profiles corresponding to the full polarization sequences: each polarization sequence encompasses about 2.3% of a pulsation cycle (4×60 s sub-exposures + 3×60 s chip readouts), as compared to about 0.33% for the sub-exposures (although the 2017 data had slightly longer subexposure times, this was compensated for by shorter readout times). The CORALIE data are not as uniform as the ESPaDOnS data in this regard: the median exposure time corresponds to about 2.8% of a pulsation cycle, and some are up to about 10% (all CORALIE measurements were retained, however HJDs were calculated at the middle rather than the beginning of each exposure). Since centre-of-gravity measurements can be biased by inclusion of too much continuum, it is important to choose integration limits with care. Thus RVs were measured iteratively. A first set of measurements was conducted using a wide integration range (± 50 about the systemic velocity), chosen to encompass the full range of variation. These initial RVs were then used as the central velocities for a second set of measurements, with an integration range of ± 30 . These were in turn used a final time as the central velocities to refine the RVs with a third iteration using the same integration range. The error bar weighted mean RV across all lines was then taken as the final RV measurement for each observation, with the standard deviation across all lines as the uncertainty in the final RV.ESPaDOnS and CORALIE RV measurements are tabulated in an online Appendix in Tables <ref> and <ref>, respectively. We determined the systemic velocity from the mean RV across all observations to be 22.5 ± 1 .§.§ Frequency Analysis We analyed the RVs using period04 <cit.>. Frequency spectra are shown in Fig. <ref>, where the top panels show the frequency spectra for the original dataset, and the panels below show frequency spectra after pre-whitening with the most significant frequencies from previous frequency spectra. A S/N threshold of 4 was adopted as the minimum S/N for significance <cit.>. The uncertainty in each frequency was determined using the formula from <cit.>, σ_ F=√(6)σ_ obs/(π√(N_ obs)AΔ T), where σ_ obs is the mean uncertainty in the RV measurements, N_ obs is the number of measurements, A is the amplitude of the RV curve, and Δ T is the timespan of observations.Period analysis of the CORALIE dataset (left panels in Fig. <ref>) yielded the same results as those reported by <cit.>, with significant frequencies at f_ pul,COR = 4.7715297(5) d^-1 with an amplitude of 16.6 , and at the harmonics 2f_ pul,COR and 3f_ pul,COR with amplitudes of 0.6 and 0.3 , respectively. After pre-whitening with these frequencies, no significant frequencies remain (bottom left panel of Fig. <ref>), and all peaks are at the 1d^-1 aliases of the spectral window. The same analysis of the ESPaDOnS data (middle panels in Fig. <ref>) finds maximum power at f_ pul,ESP = 4.7715007(4) d^-1, and at 2f_ pul,ESP. The combined dataset (right panels in Fig. <ref>) yields the strongest signal at f_ pul,Comb = 4.7715121(3) d^-1, along with significant peaks at 2f_ pul,Comb and 3f_ pul,Comb. In this case pre-whitening does not completely remove the peak corresponding to f_ pul,Comb. This could indicate that the pulsation frequency is not the same between the datasets. The difference in frequencies between the CORALIE and ESPaDOnS datasets, 2.9× 10^-5 d^-1, is about 30 times larger than the formal uncertainties in either frequency. This difference corresponds to an increase in the pulsation period of 0.1 s, approximately compatible with the increase of 0.063 ± 0.009 s expected from the constant period change of +0.0037(5) s yr^-1 reported by <cit.>. To explore this hypothesis, phases were calculated assuming a constant rate of period change -0.02 < Ṗ < +0.02 s yr^-1, with f_0 = 4.771529(7) d^-1 taken as the frequency inferred from the CORALIE data acquired in 2000, and allowed to vary within the uncertainty in this frequency. The phases ϕ were calculated asϕ =HJD - (T_0 + NP_0 + 0.5ṖN^2)/P_0 + NṖ, where P_0 is the initial period, and N is the number of pulsation cycles elapsed between the observation and the reference epoch T_0 = 2451591.42576, defined as the time of the first RV maximum one cycle before the first observation in the dataset. The goodness-of-fit χ^2 statistic was calculated for each combination of P_0 and Ṗ using a 3^rd-order least-squares sinusoidal fit in order to account for the pulsation frequency and its first two harmonics. The minimum χ^2 solution was found for P_0 = 0.2095763(1) d and Ṗ=+0.0096(5) s yr^-1, where the uncertainties were determined from the range over which χ^2 does not change appreciably. The RVs are shown phased with Eqn. <ref> using these parameters in Fig. <ref>. The bottom panel of Fig. <ref> shows the residual RVs after subtraction of the 3^rd-order sinusoidal fit. The standard deviation of the residuals is 0.48 , an improvement over the standard deviation of 0.94 obtained using the constant period determined from the full dataset. The bottom right panel of Fig. <ref> shows the frequency spectrum obtained for the residual RVs in Fig. <ref>: in contrast to the results obtained via pre-whitening using a constant pulsation frequency, all power at f_ pul and its harmonics is removed, with no significant frequencies remainining.§.§ Comparison to previous results Our analysis of the CORALIE dataset recovers the same pulsation frequency as that found by the analysis performed by <cit.> of the same data. However, we find Ṗ to be almost 3 times larger than the rate found by <cit.>. Whether this reflects an acceleration in the rate of period change will need to be explored in the future when larger datasets with a longer temporal baseline are available. <cit.> noted that ξ^1 CMa is one of the few β Cep stars for which Ṗ is low enough to be consistent with stellar evolutionary models. The higher Ṗ suggested by our results brings ξ^1 CMa closer to the range observed for other β Cep stars, which is to say, slightly above the Ṗ predicted by evolutionary models for a star of ξ^1 CMa's luminosity. § STELLAR PARAMETERS§.§ Line Broadening The high S/N, high spectral resolution, and wide wavelength coverage of the ESPaDOnS data, combined with the sharp spectral lines of this star, present numerous opportunities for constraining the line broadening mechanisms. As with measurement of RVs, in order to minimize the impact of RV variation, Stokes I spectra from individual sub-exposures were used rather than the spectra computed from the combined exposures. We selected two lines for analysis: the Si iii 455.3 nm line, which is sensitive to the pulsational properties of β Cep stars <cit.>, and for comparison N ii 404.4 nm, a weaker line with lower sensitivity to pulsation.We applied a model incorporating the projected rotation velocity , radial/tangential macroturbulence v_ mac, and assumed radial pulsations, performing a goodness-of-fit (GOF) test on a grid spanning 0–20 in and 0–30 in v_ mac, in 1 increments. The disk integration model is essentially as described by <cit.>, with the exception of radial pulsations. Pulsations were modeled as a uniform velocity component normal to the photosphere, with the local line profiles in each surface area element shifted by the projected line-of-sight component of the pulsation velocity. A limb darkening coefficient of ϵ=0.36 was used, obtained from the tables calculated by <cit.> for a star with =27 kK and logg=3.75. Local profiles were broadened with a thermal velocity component of 4 for Si iii, and 5.6 for N ii, and the disk integrated profile was convolved according to the resolving power of ESPaDOnS. Line strength was set by normalizing the EWs of the synthetic line profiles to match the EWs of the observed line profiles. In the course of this analysis we found a Baade-Wesselink projection factor of measured-to-disk-centre radial velocity of p=1.45±0.02, which is consistent with determinations for other β Cep variables (e.g., ).Fig. <ref> shows the resulting best-fit models. Taking the mean and standard deviation across all observations, both Si iii 455.3 nm and N ii 404.4 nm yield vsini=5±3 . The two lines give different values for v_ mac, however: 19±1 and 8±2 , respectively. The much lower value of v_ mac determined from N ii 404.4 nm may indicate that the extended wings of Si iii 455.3 nm, which require a higher value of v_ mac to fit, are at least partly a consequence of pulsations. As demonstrated in Fig. <ref>, there is essentially no difference in quality of fit between models with nonzero and v_ mac, and zero ; conversely, a much worse fit is obtained by setting v_ mac=0 . For Si iii 455.3 nm, the χ^2 of the model without v_ mac is 13 times higher than the χ^2 of the model without , while the χ^2 of the model without is only 1.4 times higher than the model with both turbulent and rotational broadening. The difference is not as great for N ii 404.4 nm, although the χ^2 of the model with rotational broadening only is still higher than either the model with turbulent broadening only, or both rotational and turbulent broadening. <cit.> used magnetic O stars with rotational periods known to be extremely long (i.e. with ∼ 0 ) and found that both the goodness-of-fit test and the Fourier transform methods severely over-estimate when v_ mac > vsini, as is the case for both of the lines we have examined. We conclude that the star's line profiles are consistent with = 0 . §.§ Surface Gravity The surface gravity logg was determined by fitting tlusty BSTAR2006 synthetic spectra <cit.> to the Hβ and Hγ lines of the ESPaDOnS spectrum acquired at pulsation phases at which the RV was closest to the systemic velocity of 22.5 (i.e. at pulsation phases 0.25 or 0.75). As both of these lines are close to the edges of their respective orders, in order to avoid warping the line the orders were merged from un-normalized spectra, then normalized using a linear fit to nearby continuum regions. Five surface gravities were tested between logg=3.25 and logg=4.25, and a low-order polynomial was fit to the resulting χ^2 in order to identify the lowest χ^2 solution. Uncertainties were determined by fitting models with of 26, 27, and 28 kK, spanning the approximate range in the uncertainty in (see below). Fig. <ref> shows the best-fit models, logg = 3.78 ± 0.07, compared to the observed Hβ line. Fitting to Hγ yielded identical results.§.§ Effective Temperature The effective temperature was determined using photometry, spectrophotometry, and spectroscopy. Using the Strömgren uvbyβ photometric indices obtained by <cit.>, and the idl program uvbybeta.pro[Available at <http://idlastro.gsfc.nasa.gov/ftp/pro/astro/uvbybeta.pro>.] which implements the calibrations determined by <cit.>, yields =26.2 kK. Using the Johnson UBVRIJHK photometry collected by <cit.> and the colour-temperature calibration presented by <cit.> yields =25±3 kK, where the uncertainty was determined from the range of across the (U-B), (V-K), (B-V), (V-R), and (J-K) colours.The was measured using spectrophotometric data by fitting synthetic BSTAR2006 spectra to the IUE low-dispersion spectra, Johnson UBVRI photometry, and Strömgren uvbyβ photometry. The Johnson photometry was converted into absolute flux units using the calibration provided by <cit.>, and the Strömgren photometry converted using the calibration determined by <cit.>. The synthetic spectra were scaled by 1/d^2 and 4π R_*^2, where the distance d was fixed by the Hipparcos parallax and the stellar radius R_* was left as a free parameter. The resulting fit is shown in Fig. <ref>. The best-fit model yields =26.8 ± 0.4 kK and R_*=7.8 R_⊙. The uncertainty comes from the spread in over the range of logg=3.78±0.07, and for two values of E(B-V), 0 and 0.02. Fluxes were dereddened using the idl routine fmunred[Available at<http://idlastro.gsfc.nasa.gov/ftp/pro/astro/fm_unred.pro>.], which utilizes the <cit.> extinction curve parameterization. As demonstrated by the inset in Fig. <ref>, there is a clear absence of interstellar silicate absorption near 220 nm. The de-reddened spectra in the inset in Fig. <ref> assume the strength of the absorption bump is c_3=2.1, the minimum Galactic value <cit.>. Even with this low value of c_3, using E(B-V)=0.04 results in an increase of the flux near the silicate absorption bump which is not observed. Using the average Galactic value of c_3=3.23 instead would require that E(B-V) be even lower.§.§ Comparison to previous results Previous measurements of have in general yielded somewhat higher, non-zero results than those obtained here (15± 1.5 , ; 9±2 , ; 14±5 , ).did not include v_ mac in their profile fit, but did include pulsation and thermal broadening;did not consider pulsations; andincluded v_ mac, but neither pulsation velocity nor thermal broadening. Our inclusion of thermal, macroturbulent, pulsational, and rotational velocity fields likely explains why our best-fit value of , 5 ± 3 , is somewhat lower than the values found previously, since all four sources of line broadening are of a similar magnitude in this star. Stellar parameters for ξ^1 CMa have been determined numerous times using a variety of different spectral modelling methodologies, e.g., DETAIL/SURFACE <cit.>, FASTWIND <cit.>, and PoWR <cit.>. <cit.> employed an algorithmic method to simultaneously determine reddening, , metallicities, and surface gravities for β Cep stars observed with IUE, and for ξ^1 CMa found =24 ± 1 kK, logg=3.89, and E(B-V) < 0.01, i.e. their analysis favoured a cooler, slightly less evolved star, but with the same very low level of reddening.derived logg from photometric calibrations, rather than line profile fitting, which likely explains the different values of logg. <cit.> found =27.5 kK and logg=3.75, <cit.> derived =27 ± 1 kK and logg=3.80 ± 0.15, and <cit.> found =27.5 ± 0.5 kK: all are in agreement with the results found here. The slightly higher found in these works is likely a consequence of the higher value of reddening which, as is shown in <ref> and Fig. <ref>, is not consistent with the absence of silicate absorption. While reddening has an effect on the spectrophotometric method applied in <ref>, we obtained essentially identical results using line strength ratios and ionization balances below in <ref>, which should not be affected by reddening.§.§ Effective Temperature VariationThe photospheric temperature of a β Cephei star is variable, a property which can be explored both photometrically and spectroscopically. The only public two-colour photometry for ξ^1 CMa are the Tycho BV_ T observations. While pulsational modulation is visible in both passbands, the precision of the data is not sufficient to detect a coherent signal in the colours. This is not surprising, as B-V colours are poorly suited to determining for such hot stars. We therefore attempted to constrain the change via the ionization balances of various spectral lines, under the assumption that all equivalent width (EW) changes are due to temperature variation. We first measured EWs from the ESPaDOnS observations, and from appropriate synthetic spectra obtained from the tlusty BSTAR2006 library <cit.>. We used the ESPaDOnS Stokes I spectra obtained from the combined sub-exposures, in order to maximize the S/N. The following spectral lines were used: He i 587.6 nm, He ii 468.6 nm, C ii 426.7 nm, C iii 406.8 nm, N ii 404.4 nm, N iii 463.4 nm, O i 777.4 nm, O ii 407.9 nm, Ne i 640.2 nm, Ne ii 439.2 nm, Si ii 413.1 nm, Si iii 455.3 nm, and Si iv 411.6 nm. Since the EW can depend on logg as well as , the model EW ratios were linearly interpolated to logg=3.78 between the results for logg=3.75 and logg=4.0. We then determined by interpolating through the resulting grid to the EW ratios measured from the observed spectra. The results are illustrated in Fig. <ref>. While there is a considerable spread in values, from approximately 25 to 28 kK, the mean value of 26.8 kK agrees well with the spectrophotometric determination. A coherent variation with the pulsation period is seen for all ion strength ratios. Individual atomic species yield variations with semi-amplitudes up to Δ T_ eff = ± 500 K. Taking the weighted mean across the effective temperatures from all ion strength ratios at each phase yields a variation of Δ T_ eff = ± 310 ± 30 K. If the zero-point of each variation is first subtracted, the resulting mean semi-amplitude is 330 ± 30 K, slightly larger but overlapping within the uncertainty.To check the validity of the temperature variation, we used it to model the Hipparcos light curve, as demonstrated in Fig. <ref>. With the uncertainty in the pulsation period and rate of period change determined in <ref>, the accumulated uncertainty in pulsation phase for the Hipparcos photometry is between 0.009 and 0.012 cycles. We began by determining the mean radius R_* from the mean and the luminosity logL. logL was obtained from the Hipparcos parallax distance d=424±35 pc, the apparent V magnitude 4.06, and the bolometric correction BC=-2.61 ± 0.09 mag obtained by linear interpolation through the theoretical BSTAR2006 grid according to and logg <cit.>. The absolute V magnitude m_V = V - A_V - μ = -3.86 ± 0.18 mag, where μ = 5logd - 5 = 8.13 ± 0.18 mag is the distance modulus and A_V = E(B-V)/3.1 < 0.06 mag is the extinction. The bolometric magnitude is then M_ bol = m_V + BC = -6.47 ± 0.27 mag, yielding log(L/L_⊙) = (M_ bol,⊙-M_ bol)/2.5=4.49 ± 0.11 where M_ bol,⊙ = 4.74 mag. This yields R_* = √((L/L_⊙)/(T_ eff/T_ eff,⊙)^4) = 7.9 ± 0.6 R_⊙, where T_ eff,⊙=5.78 kK. This radius agrees well with the value found via spectrophotometric modelling, R_*=7.8 R_⊙, in which the radius was left as a free parameter (Fig. <ref>).Integrating the radial velocity curve (Fig. <ref>) in order to obtain the absolute change in radius, and assuming radial pulsation, yields a relatively small change in radius of ± 6.58 × 10^4 km, corresponding to between 1.0 and 1.5% of the stellar radius. The radius variation was then combined with the variation from Fig. <ref> to obtain logL and the BC at each phase. The apparent V magnitude was then obtained by reversing the calculations in the previous paragraph. The solid red line in Fig. <ref> shows the resulting model light curve compared to the Hipparcos photometry, where we made the assumption that V and H_ p are approximately equivalent. The larger 500 K semi-amplitude measured from individual pairs of ions is not consistent with the light curve, yielding a photometric variation larger than observed, with a semi-amplitude of 0.037 mag as compared to the observed semi-amplitude of ∼0.021 mag. The apparent phase offset of ∼0.05 cycles between the predicted and observed photometric extrema may suggest that Ṗ may not in fact be constant; alternatively, there may simply be too few measurements to constrain the variation well enough to obtain a close fit to the photometric data, as the latter is clearly not a perfect sinusoid.§.§ Mass and Age Fig. <ref> shows ξ^1 CMa's position on the Hertzsprung-Russell diagram (HRD) (top) and the -logg diagram (bottom), where the mean was used. Comparison to the evolutionary models calculated by <cit.> (which assume an initial rotational velocity of 40% of the critical velocity) indicates that the stellar mass M_*=14.2±0.4 M_⊙, and that the absolute stellar age is t=11±0.7 Myr. The <cit.> models do not include the effects of magnetic fields, however, grids of self-consistent evolutionary models including these effects in a realistic fashion are not yet available.The positions of the star on the two diagrams are mutually consistent. The other known magnetic B-type stars with similar stellar parameters are also shown in Fig. <ref>. The stellar parameters of the majority of the other stars were obtained from the catalogue of magnetic hot stars published by <cit.> and references therein; those of β CMa and ϵ CMa were obtained from <cit.>. ξ^1 CMa is one of the most evolved stars in the ensemble, and is the most evolved star with a mass above about 12 . Its position on the -logg diagram indicates it may be a more evolved analogue of NU Ori, HD 63425, and HD 66665. As an additional check on the stellar mass and radius, we utilized the pulsation period and the relation P_ fun = Q/√(ρ), where P_ fun is the fundamental pulsation period, ρ is the mean density, and Q is the pulsation constant. With M_* = 14.2±0.4 M_⊙, R_* = 7.9 ± 0.6 R_⊙, and the theoretical value Q = 0.035 <cit.>, this yields P_ fun = 0.18 ± 0.02 d, very close to the true pulsation period P ∼ 0.21 d. If the actual pulsation period is used to calculate Q, we obtain Q = P_ fun√(ρ) = 0.041 ± 0.003. § BINARITY§.§ Interferometry <cit.> reported that ξ^1 CMa hosts weak Hα emission, which is atypical for a star of ξ^1 CMa's stellar properties. There are two possibilities to explain this: first, that it originates in a stellar magnetosphere, and second, that it originates in the decretion disk of a heretofore undetected Be companion star. It seems reasonable to expect that ξ^1 CMa has a binary companion, as the binary fraction is ∼65% for B0 stars <cit.>. Furthermore, early claims that the Hα emission of β Cep, a similar magnetic early-B type pulsator, originated in its magnetosphere <cit.>, proved unfounded following the detection of a Be companion star by <cit.>. The peak Hα emission has a maximum strength of 28% of the continuum (see <ref>). Be star emission lines can range up to ∼ 10× of the continuum, although the emission is typically much less than this. We therefore expect that the putative companion must have a luminosity of ≥ 2.8% of that of ξ^1 CMa, yielding log(L/L_⊙)≥ 2.9 or V < 7.6. Assuming the pair to be coeval, and therefore locating the star on the same isochrone as ξ^1 CMa, the companion's mass should be near 5 M_⊙, compatible with a classical Be star. ξ^1 CMa is listed in the Washington Double Star Catalogue as possessing a binary companion <cit.>. However, the reported companion is both too dim to account for the Hα emission (V = 14) and too far away, located approximately 28" from the primary, i.e. well outside the 1.6" ESPaDOnS aperture.We acquired H and K low-resolution VLTI-AMBER data, together with VLTI-PIONIER data, in order to search for the presence of a binary star. The squared visibilities V^2 and closure phases ϕ_ C of these data are shown as functions of spatial frequency in Fig. <ref>. Both the AMBER K band and PIONIER V^2 measurements are entirely flat, compatible with a single unresolved source. There is furthermore no signal in the PIONIER ϕ_ C measurements, which are more precise than those available from AMBER. We analyzed the data using the standard litpro package[litpro software is available at http://www.jmmc.fr/litpro] <cit.>, fitting a two-point model, with one point fixed in the centre of the map and the (x,y) coordinates of the second free to move. The overall upper limit on the flux of a secondary component is 1.7% of the primary star's flux. In order to constrain the maximum flux of a binary companion at different distances from the primary, we repeated the two-point model fit in successively wider boxes (as the current version of litpro does not support polar coordinates, a Cartesian approximation to annuli was used). As these flux ratios are in the H and K bands, but our estimated minimum flux ratio is in the V band, we used synthetic tlusty SEDs <cit.> to convert the H and K band flux ratios to V band flux ratios. In this step we assumed that the secondary's is near 20 kK, appropriate to a 5-6 M_⊙ star near the main sequence. A companion of the required brightness is ruled out beyond ∼ 40 AU. Close binaries containing magnetic, hot stars are extremely rare, with <2% of close binary systems containing a magnetic companion earlier than F0 <cit.>. This does not mean that a close companion can be ruled out a priori. Such a companion may be detectable via radial velocities. As found in <ref>, the RV curve of ξ^1 CMa is extremely stable, with a standard deviation of the residual RVs of ∼0.5 . To determine if a Be companion should have been detected, we computed radial velocities across a grid of models with secondary masses 1 M_⊙ < M_ S < 10 M_⊙, semi-major axes 0.05 AU < a < 500 AU, eccentricities 0 < e < 0.9, and inclinations of the orbital axis from the line of sight 1^∘ < i < 90^∘. We then phased the JDs of the observations with the orbital periods P_ orb, using a single zero-point if P_ orb was less than the time-span of the observations and multiple, evenly-spaced zero-points if P_ orb was greater than this span. We then calculated the expected RV of the primary at each orbital phase, and compared the standard deviation σ_ orb of these RVs with the observed standard deviation σ_ obs (RV amplitudes were not used as, for orbits with periods longer than the timespan of observations, the full RV variation would not have been sampled). A given orbit was considered detectable if σ_ orb > 3σ_ obs. Numerical experiments with synthetic RV curves including gaussian noise with a standard deviation of 0.5 indicate that this criterion is likely conservative: with 624 RV measurements, input periods can generally be recovered even when the semi-amplitude of the RV curve is similar to the noise level. Each orbit was assigned a probability P(i) assuming a random distribution of i over 4π steradians (i.e., P(i) = 1-cosi), and a flat probability distribution for M_ S, e, and a. From this we obtained the 1, 2, and 3σ upper limits on a companion's mass as a function of a. This mass upper limit was then transformed into a V magnitude lower limit by interpolating along the 10 Myr isochrone in Fig. <ref>. Within the 40 AU inner boundary of the interferometric constraints, a binary companion of sufficient brightness to host the Hα emission can be ruled out entirely at 1σ confidence, and almost entirely at 2σ. We have proceeded under the very conservative assumption of a Be star with Hα emission of 10× its continuum level, however the majority of Be stars have emission of at most a few times the continuum, and many have much less than this. Thus, the upper limits on companion mass and brightness determined above rule out all but the most exceptional of classical Be stars as a possible origin for the Hα emission. We conclude that there is unlikely to be an undetected binary classical Be companion star that is sufficiently bright to be the source of the star's Hα emission. §.§ Spectro-interferometry The formation of the line emission around the β Cep star itself is supported by the absence of any spectrointerferometric signal across the Brγ line. Such observations were taken with AMBER, and no signature in phase was detected on the level of 2^∘ (Fig. <ref>). This means that the photometric position as a function of wavelength remained stable on the level of 60 μas <cit.>. An emission component of about 30% of the strength of the continuum (Brγ emission typically being comparable in strength to that of Hα in Be stars) must therefore have its photocenter within 60 μas from the photocenter of the nearby continuum, as otherwise it would have produced a detectable offset of the phase signal. This not only excludes a general offset from the central star, i.e. formation around a companion, but also an extended orbiting structure, in which the blue emission would be formed at an offset opposite tothe red emission.§ MAGNETIC FIELD§.§ Least Squares Deconvolution While ξ^1 CMa's sharp spectral lines and the high S/N of the ESPaDOnS observations mean that Zeeman signatures are visible in numerous individual spectral lines, in order to maximize the S/N we employed the usual Least Squares Deconvolution (LSD; ) multiline analysis procedure. In particular we utilized the `improved LSD' (iLSD) package described by <cit.>. iLSD enables LSD profiles to be extracted with two complementary line masks, improving the reproduction of Stokes I and V obtained from masks limited to a select number of lines. We used a line mask obtained from the Vienna Atomic Line Database (VALD3; ) for a solar metallicity star with = 27 kK. While magnetic early-type stars are often chemically peculiar, ξ^1 CMa is of essentially solar composition <cit.>, albeit with a mild N enhancement <cit.>, thus a solar metallicity mask is appropriate. The mask was cleaned and tweaked as per the usual procedure, described in detail by <cit.>, such that only metallic lines unblended with H, He, interstellar, or telluric lines remained, with the strengths of the remaining lines adjusted to match as closely as possible the observed line depths. Of the initial 578 lines in the mask, 338 remained following the cleaning/tweaking procedure. The resulting LSD profiles are shown in Fig. <ref> (top panel). Note that, due to intrinsic line profile variability and the longer subexposure times used for the 2017 data, the line profiles of the most recent data are slightly broader than thedata acquired previously.In order to directly compare MuSiCoS and ESPaDOnS results, a second line mask was employed with all lines outside of the MuSiCoS spectral range removed, leaving 139 lines. The bottom panel of Fig. <ref> shows a comparison between the highest S/N MuSiCoS LSD profile and the ESPaDOnS LSD profile with the closest pulsation phase (0.4377 vs. 0.4425). The Stokes I profiles from the 2 instruments agree well, considering both the lower spectral resolution and the much longer exposure times of MuSiCoS (corresponding to ∼15% of a pulsation period, as compared to ∼2.3% of a pulsation period for ESPaDOnS). The key point of interest is in Stokes V, which is clearly negative in the MuSiCoS LSD profile, but positive in all ESPaDOnS LSD profiles. The second MuSiCoS observation also yields a negative Stokes V profile. The reliability of these MuSiCoS observations is evaluated in detailin Appendix <ref>. False Alarm Probabilities (FAPs) were calculated inside and outside of the line profile, and classified as Definite Detections (DDs), Marginal Detections (MDs), or Non-Detections (NDs) according the methodology and criteria described by <cit.>. Detection flags for Stokes V and diagnostic null N are given in Table <ref>. All MuSiCoS observations are formal non-detections. All ESPaDOnS observations are DDs(with the exception of the two discarded measurements from 2017/02/11, both of which yielded NDs due to their low S/N). However, in numerous ESPaDOnS observations, N also yields a DD. This phenomenon is considered in greater detail in <ref>.§.§ Longitudinal Magnetic Field The longitudinal magnetic field was measured from the LSD profiles by taking the first-order moment of the Stokes V profile normalized by the equivalent width of Stokes I <cit.>. The same measurement using N yields the null measurement , which should be consistent with 0 G. In order to ensure a homogeneous analysis, the LSD profiles were first shifted by their measured RVs to zero velocity, and an identical integration range of ± 30 around line centre employed. and measurements are reported in Table <ref>. As expected from the LSD profiles, < 0 for both MuSiCoS measurements, while > 0 for all ESPaDOnS measurements. The ESPaDOnS data are of much higher quality, with a median σ_B = 6 G, as compared to 57 G for the MuSiCoS data. In contrast to the FAPs, in which many ESPaDOnS N profiles yield definite detections, /σ_N is typically very low, with a maximum of 2.4 and a median of 0.6, i.e. statistically identical to zero. Because is expected to be modulated by stellar rotation, it can be used to determine the rotation period. The periodograms for and are shown in Fig. <ref>, where the latter shows the variation arising from noise. There are numerous peaks at periods of a few tens to a few hundreds of days which appear in both the and the periodograms. Phasing with the periods corresponding to these peaks does not produce a coherent variation (e.g., phasing the data with the highest peak in this range, at 177 d, and fitting a first-order sinusoid, yields a reduced χ^2 of 1162). The strongest peak in the periodogram is at 5100 ± 300 d. This peak does not appear in the periodogram. This is similar to the timespan of the ESPaDOnS dataset, and so the formal uncertainty is certainly under-estimated, with the 5100 d period representing a lower limit. However, the S/N of this peak is 26, above the threshold for statistical significance, indicating that the long-term variation is probably real.Fig. <ref> shows the measurements, both individual (filled symbols) and in annual bins (open circles), as a function of time. There is an obvious long-term modulation, with steadily declining from a peak of ∼330 G in 2010 (HJD 2455200) to value of ∼80 G in 2017 (HJD 2457800). The annual mean measurements are provided in Table <ref>. The MuSiCoS observations, both of which are of negative polarity, indicate (in combination with the ESPaDOnS data) that the rotational period must be longer than 5100 d. The time difference between the MuSiCoS measurements and the maximum ESPaDOnS observations is ∼3600 d. If these two epochs sample the curve at its positive and negative extrema, then the rotational period must be at least 7200 d (∼20 years), or a half-integer multiple if there is more than one cycle between the observed extrema. The curvature of the ESPaDOnS suggests that _ max occurred at ∼HJD 2455200, however as it cannot be ruled out that _ min < -127 ± 40 G, it is possible that P_ rot > 7200 d. Indeed, a longer period seems likely: phasing with a 20-year period, and fitting a least-squares 1^st-order sinusoid (as expected for a dipolar magnetic field), produces a very poor fit as compared to longer periods. In Fig. <ref>, the illustrative sinusoudal fit was performed using a 30-year period, which achieves a reasonable fit to the data (the reduced χ^2 is 2.6). Note that with this fit the MuSiCoS data do not define _ min. Longer periods can also be accommodated, however at the expense of |⟨ B_z⟩_ min| > |⟨ B_z⟩_ max|. We thus adopt P_ rot = 30 yr as the most conservative option allowed by the data.§.§ Comparison to previous results There are two competing claims for rotational periods in the literature. The first, based on spectropolarimetry collected with FORS1, FORS2, and SOFIN, is approximately 2.18 d <cit.>. The second period, based on a preliminary analysis of an earlier, smaller ESPaDOnS dataset, is ∼ 4.27 d <cit.>. Period analysis of the ESPaDOnS measurements using Lomb-Scargle statistics rules out both of these periods. The rotational periods provided by <cit.> and <cit.> are respectively indicated with black and red arrows in the top panel of Fig. <ref>. There is no significant power in the periodogram at either period. The peak corresponding to the <cit.> period appears in both the and periodograms, suggesting it to be a consequence of noise. Phasing the data with the periods given by <cit.> or <cit.> does not produce a coherent variation, as is demonstrated in Fig. <ref>. It impossible for a period on the order of days to account for the systematic decline of ∼200 G between the earliest data and the most recent data. We also note that the ESPaDOnS dataset presented here includes two epochs with superior time-sampling to that of the FORS1/2 datasets: 14 observations over 10 d in 2010, and 20 observations over 38 d in 2017, as compared to 13 FORS1 observations over 1075 d and 11 FORS2 observations over 60 d. The ESPaDOnS dataset thus enables a much better probe than the FORS1/2 dataset of short-term as well as long-term variability.All FORS1/2 and SOFIN measurements are of positive polarity <cit.>, in agreement with ESPaDOnS data. Furthermore, the magnitude, ∼300 G, is similar. The two SOFIN measurements also agree well with the ESPaDOnS data. The much shorter 2.18 d period determined by <cit.> is due to an apparent variation with a semi-amplitude that is significant at 1.8σ in comparison with the median uncertainties in these measurements. <cit.> showed that systematic sources of uncertainty such as instrumental flextures must be taken into account in the evaluation of uncertainties from FORS1 data, that the uncertainties in FORS1 measurements should thus be about 50% higher, and therefore that FORS1/2 detections are only reliable at a significance of >5σ, a much higher threshold than is satisfied by the variation the 2.18 d period is based upon. The catalogue of FORS1 measurements published by <cit.> additionally reveals differences of up to 150 G for observations of ξ^1 CMa between results from different pipelines. Using the published uncertainties, the FORS1/2 measurements presented by <cit.> are generally within 2σ of the sinusoidal fit to the ESPaDOnS and MuSiCoS data, and only 3 differ by greater than 5σ. Curiously, the FORS2 data published by <cit.> are systematically 80 G lower than the FORS1 measurements. We conclude that the previously published low-resolution magnetic data are not in contradiction with the long-term modulation inferred from high-resolution measurements. § EMISSION LINES Having rejected the possibility that ξ^1 CMa's Hα emission originates in the decretion disk of a classical Be companion star ( <ref>), we proceed under the assumption that it is formed within the stellar magnetosphere. In this section we explore the star's UV and Hα emission properties. We evaluate short-term variability with the stellar pulsations, as well as long-term variability consistent with the slow rotation inferred from the magnetic data.§.§ Hα emission §.§.§ Long-term modulation Fig. <ref> shows Hα in 2000, 2004, 2010, 2014, and 2017, selected so as to share the same pulsation phase (ϕ = 0.75 ± 0.03). A synthetic line profile calculated using the physical parameters obtained in <ref> is overplotted as a thick line. Comparison of observed to synthetic line profiles demonstrates that emission is present at all epochs, although it is substantially weaker in the earlier CORALIE data. The agreement of Hβ with the model is reasonable at all epochs, although there is also weak emission present in the line core (Fig. <ref>). Close analysis of the line core of Hβ shows evidence of evolution, with the same trend as in Hα, although this evolution is of a much lower amplitude. Hα EWs were measured with an integration range of ±0.4 nm of the rest wavelength in order to include only the region with emission (Fig. <ref>). The spectra were first shifted to a rest velocity of 0 by subtracting the RVs measured in <ref>. The periodogram for these measurements is shown in the middle panel of Fig. <ref>. It shows a peak at 6700 d or 18 yr. The S/N of this peak is 29, indicating that it is statistically significant. This is consistent with the very long timescale of variation inferred from the ESPaDOnS periodogram (Fig. <ref>), and with the minimum 7200 d period inferred from the positive and negative extrema in the ESPaDOnS and MuSiCoS data. The bottom panel of Fig. <ref> shows the Hα EWs as a function of time. Maximum emission and maximum occur at approximately the same phase, as do the minimum and the minimum emission strength. The variation in EW is seen more clearly in the annual mean EWs (open circles), which are tabulated in Table <ref>. The relative phasing of and EW is consistent with a co-rotating magnetosphere, and supports the accuracy of the adopted period. Since is both positive and negative, two local emission maxima are expected at the positive and negative extrema of the curve, as at these rotational phases the magnetospheric plasma is seen closest to face on. The solid curve in the bottom panel of Fig. <ref> shows a 2^nd-order sinusoidal fit to the annual mean EWs, which indeed yields two local maxima, the strongest corresponding to =_ max, and the second maximum predicted to occur at =_ min. Note that, due to the incomplete phase coverage (less than half of a rotational cycle), in the event that the Hα variation is indeed a double-wave a Lomb-Scargle periodogram should show maximum power at close to half of the rotational period. The periodogram peak at 18-yr would then indicate P_ rot = 36 yr, consistent with the rotational period inferred from . §.§.§ Short-term modulation As a first step to investigating whether Hα is affected by pulsation, residual EWs were obtained by subtracting the least-squares 2^nd-order sinusoidal fit to the annual mean EWs. The bottom panel of Fig. <ref> shows the period spectrum of the residual EWs. The strongest peak is at the stellar pulsation period, with a S/N of 12. After prewhitening the EWs with both the rotational and pulsational frequencies, the S/N of the highest peak in the periodogram is close to 4, suggesting that all significant variation is accounted for by these two frequencies.While the semi-amplitude of the residual EW variation is similar to the median error bar, binning the residual EWs by pulsation phase does not change the semi-amplitude, but increases the significance of the variation to ∼ 8σ with respect to the mean error bar. The phase-binned residual EWs are shown phased with the pulsation period in the top panel of Fig. <ref>. There is a coherent variation of the phase-binned residual EWs with the pulsation period, with a semi-amplitude of approximately 0.003 nm. One obvious candidate mechanism for producing this pulsational modulation is the change in the EW of the underlying photospheric profile due to the changing , as explored in <ref>. To investigate this hypothesis we calculated synthetic spectra via linear interpolation between the grid of tlusty BSTAR2006 models <cit.>. We used the physical parameters from <ref>, including the ±300 K variation found in <ref>. Deformation of the line profile due to pulsation was accounted for using the RV curve from <ref>, and modelling the (assumed radial) pulsations as described in <ref>. Spectra were calculated at 20 pulsation phases, and the EW was measured at each phase using the same integration range as used for the observed data, and after moving the synthetic spectra to zero RV. The black line in the top panel of Fig. <ref> shows the resulting variation, normalized by subtracting the mean EW across all models. The semi-amplitude of the EW variation expected due to changes in the photospheric profile due to pulsation is about 0.001 nm, much smaller than observed. Furthermore, it is out of phase with the observed variation by about 0.5 pulsation cycles. Note that the maximum and minimum EWs of the predicted variation correspond to the minimum and maximum of the variation (Fig. <ref>), as expected for H lines, which grow weaker with increasing in this range. Radial pulsation introduces asymmetry into the line profile, which can be quantified by V/R, defined as the ratio of the EW in the blue half to the EW in the red half of the line. We measured V/R from EWs calculated from -0.4 nm to line centre (defined at the laboratory rest wavelength shifted by the RV measured from metallic lines in <ref>), and line centre to +0.4 nm. The bottom panel of Fig. <ref> shows the V/R variation. The semi-amplitude of the variation is about 0.25. This is much higher than the semi-amplitude of 0.0007 predicted by synthetic spectra calculated using a radially pulsating photospheric model; since this is essentially flat on the scale of Fig. <ref>, the photospheric variation is not shown. The low level of line asymmetry in the synthetic spectra is a consequence of the large Doppler broadening of the Hα line, approximately 30 or about twice the semi-amplitude of the RV curve. For narrower lines, e.g. the C ii 656.3 nm line for which the Doppler velocity is similar to the RV semi-amplitude, the predicted and observed V/R variations are in reasonable agreement. The semi-amplitude of the Hα V/R variation increases steadily from 0.16 in 2000-2001, to 0.20 in 2002-2004, to 0.35 in 2008-2010, and then declines to 0.24 in 2012-2014 and 0.13 in 2017. This is the same pattern as the change in total emission strength, thus, the amplitude of the V/R variation correlates to the total emission strength and, therefore, to . For a more detailed view of the pulsational modulation of Hα we calculated dynamic spectra phased with the pulsation period. These are shown in Fig. <ref>, using the synthetic line profile from Fig. <ref> as a reference spectrum. Individual line profiles were moved to 0 by subtracting their RVs, and then binned by pulsation phase using phase bins of 0.05 cycles. The two panels of Fig. <ref> show dynamic spectra in the epochs of minimum emission (2000-2001) and maximum emission (2008-2010). The emission line morphology and pattern of variability is essentially identical in other epochs. In both cases, the emission peaks near the centre of the line. In the CORALIE data, the Hα emission peak is ∼20% of the continuum. This rises to about 28% of the continuum in the ESPaDOnS data. The emission peak anticorrelates slightly with the RV, as shown by the overplotted white lines. The strong red and blue emission variability revealed by the V/R variation in the bottom panel of Fig. <ref> is due to secondary emission peaks which occur at phases 0.0 and 0.5, respectively blue- and red-shifted with respect to the line centre. As with the central emission peak, these are stronger in the ESPaDOnS data, reflecting the change in V/R amplitude over time. Note that, in the data acquired at earlier epochs, these secondary emission bumps are apparently separated from the main emission peak, while in later epochs they are connected (although this depends on the choice of the reference spectrum).Both the emission strength (as measured by the residual EWs after pre-whitening with the 2^nd-order fit in Fig. <ref>) and the line asymmetry (as quantified by V/R) vary coherently with the pulsation phase. Synthetic photospheric spectra calculated using a radially pulsating model are unable to reproduce the residual EW variation, which is both 3× larger than predicted, and 0.5 cycles out of phase. The amplitude of the observed V/R variation is about 300× larger than predicted by the model, which does not reproduce the prominent blue- and red-shifted secondary emission bumps. The amplitude of V/R is furthermore variable with time, increasing and decreasing in strength in the same fashion as the total emission strength. As the star's pulsation amplitude is extremely regular, a change in the amplitude of V/R with epoch cannot be explained by photospheric pulsation. These discrepancies between model and observation suggest either that the origin of the pulsational modulation of Hα is either not photospheric, or that it cannot be explained due to variation alone. Exotic processes, such as temperature inversions related to shockwaves produced by the star's supersonic pulsations, may be one explanation. Alternatively, the origin of the pulsational modulation may reside within the magnetosphere. §.§ UV emission lines Emission is present in four of the UV doublets often used to diagnose the wind properties of early-type stars: N v 1239, 1243 Å; Si iv 1394, 1403 Å; C iv 1548, 1551 Å; and Al iii 1854, 1863 Å. The emission profiles of these doublets are similar to those of the magnetic β Cep pulsator β Cep at maximum emission. <cit.> performed this comparison for the C iv doublet. Fig. <ref> compares the mean line profiles for ξ^1 CMa's N v and Al iii lines to those of β Cep, and demonstrates that these lines are also similar to those of β Cep at maximum emission. Such emission is unique to magnetic stars, and as an indirect diagnostic of stellar magnetism has historically prompted the search for magnetic fields in such stars (e.g., ; indeed, the detection of ξ^1 CMa's UV emission motivated the collection of the MuSiCoS data). The presence of emission in the N v line is particularly interesting. This line is not seen in normal early B-type stars <cit.>, nor is it present in most normal late O-type stars (e.g. ). In their study of IUE data for 4 magnetic B-type stars, including β Cep, <cit.> concluded that this doublet must be formed at ∼30 kK, somewhat higher than the photospheric . The presence of N v lines in the UV spectra of relatively cool stars is thought to be a consequence of Auger ionization due to the presence of X-rays <cit.>. Since ξ^1 CMa has a strong magnetic field, it is expected to be overluminous in X-rays due to magnetically confined wind shocks, and is indeed observed to be overluminous in X-rays to the degree predicted by models <cit.>.β Cep's wind lines show clear variability synchronized with its rotation period <cit.>. In contrast, ξ^1 CMa's wind lines show only a very low level of variability, more similar to that seen in a normal (magnetically unconfined) stellar wind <cit.>. The bottom panels of Fig. <ref> show Temporal Variance Spectra (TVS), which compare the variance within spectral lines to the variance in the continuum <cit.>. To minimize variation due to pulsation, spectra were moved to 0 by subtracting the RV computed on the basis of the RV curve and ephemeris determined in Section <ref>. With the uncertainty in P_0 and in Ṗ ( <ref>), the uncertainty in pulsation phase is about 0.028, corresponding to a maximum uncertainty in RV of 3 , less than the ∼7.8 velocity pixel of the IUE data. RV correction reduces the TVS pseudocontinuum level by a factor of about 2 to 3.Comparison of EW measurements of these lines to EWs of synthetic spectra, using the same radially pulsating model described above in <ref>, yielded ambiguous results, due to the small number of high-dispersion IUE observations (13 spectra), and formal uncertainties similar to the maximum level of variability, ∼0.02 nm. This low level of variability is consistent with the weak variability of the residual Hα EWs. The lack of variability in ξ^1 CMa's wind lines is consistent with a rotational pole aligned with the line of sight, an aligned dipole, a long rotation period, or some combination of these. The similarity to β Cep's emission lines at maximum emission suggests that the magnetic pole was close to being aligned with the line of sight when the UV data were acquired. The long-term modulations of both Hα and favour an oblique dipole with a long rotational period, in which case the UV data should have been acquired at a rotational phase corresponding to one of the extrema of the curve. Phasing the UV data with the same 30 yr rotational period as in Fig. <ref> yields a phase close to 1.0, i.e. they would indeed have been obtained close to magnetic maximum. Assuming that the data must have been acquired near an extremum of the curve (i.e. at a phase close to 0.5 or 1.0) would require a period of 10, 15, 20, 30, or 60 years, of which only 30 and 60 years are not excluded by the magnetic data. This assumes that the UV data were, in fact, acquired close to an extremum. Given that the UV emission is only slightly stronger than that of β Cep, which has a weaker magnetic field and no Hα emission, it may be the case that the true maximum UV emission strength is significantly in excess of observations, in which case the UV data cannot be used to infer P_ rot.§ MAGNETIC AND MAGNETOSPHERIC PARAMETERS In this section, the ∼ 30 yr rotation period inferred from magnetic and spectroscopic data is used with the star's physical parameters and measurements to establish constaints on the properties of ξ^1 CMa's surface magnetic field, circumstellar magnetosphere, and spindown timescales, using the self-consistent Monte Carlo method described by <cit.>. These results are summarized in Table <ref>. The variation of a rotating star with a dipolar magnetic field can be reproduced with a three-parameter model: the inclination angle i between the rotational axis and the line of sight, the obliquity angle β between the magnetic and rotational axes, and the polar strength of the magnetic dipole at the stellar surface B_ d. The two angular parameters are related via tanβ = (1-r)/(1+r)i, where r is the ratio defined by <cit.> as r=(|B_0|-B_1)/(|B_0|+B_1), with B_0 and B_1 the mean and semi-amplitude of the sinusoidal fit to when phased with P_ rot. We determined r=-0.78± 0.02 from the sinusoidal fit to the annual mean measurements shown in the top panel of Fig. <ref>. In consequence these results rely on the 30-year period. The degeneracy between i and β is usually broken by determining i independently, e.g. from , P_ rot, and R_*. Using this method <cit.> found i ∼ 3^∘, however, this was based on their rotation period of 2.18 d, which was shown in <ref> to be incorrect. Indeed, the equatorial rotational velocity implied by a 30-year period, v_ eq<0.04 , is much less than the upper limit on <8 found in <ref> (and indeed, much less than the 1.8 velocity resolution of ESPaDOnS data). Since i cannot be constrained, we applied a probabilistic prior, requiring that i be drawn from a random distribution over 4π steradians such that P(i)=1-cosi (e.g., ). The corresponding probability density function (PDF) is shown in the bottom right panel of Fig. <ref>. The bottom middle panel shows the resulting PDF for β, which peaks at 83^∘. Despite the inverse relationship of i and β obtained for r=-0.78 (middle panel of Fig. <ref>), and the bias towards large i in the prior, large β are favoured overall. We note that we obtain a similar β to that given by <cit.>, however this is simply a coincidence as their value was obtained with an incorrect rotation period and a very different value of the Preston r parameter (0.56, as compared to -0.78).For each (i, β) pair, B_ d was determined using Eqn. 1 from <cit.>. This also requires knowledge of the limb darkening coefficient ϵ, which we obtained from the tables calculated by <cit.> as ϵ=0.36±0.02 (<ref>). The resulting PDF peaks sharply at the minimum value permitted by Preston's equation, B_ d=1.1 kG. As demonstrated in the middle and upper left panels of Fig. <ref>, B_ d < 2 kG over the range 20^∘<i<85^∘, with a low-probability tail extending out to several kG for very large and very small i and β. This is much lower than the value of ∼5.3 kG given by <cit.>, which arose from their very small i.If instead the ESPaDOnS and MuSiCoS measurements define the extrema of , we obtain r=-0.33 ± 0.02. This does not affected B_ d, although a slightly smaller β∼ 75^∘ is favoured.Magnetic wind confinement is governed by the balance of kinetic energy density in the radiative wind to the magnetic energy density. This is expressed by the dimensionless wind magnetic confinement parameter η_* <cit.>. If η_* > 1, the star possesses a magnetosphere. <cit.> analyzed IUE observations of ξ^1 CMa using the Potsdam Wolf-Rayet (PoWR) code in order to determine the wind parameters, obtaining log(Ṁ/M_⊙ yr^-1)=-10 and =700 . However, magnetic confinement reduces the net mass-loss rate and, more seriously, strongly affects line diagnostics due to the departure from spherical symmetry in the circumstellar environment. Indeed,were unable to achieve a simultaneous fit to the Si iv and C iv doublets. Comparison of magnetohydrodynamic (MHD) simulations and spherically symmetric models to the magnetospheric emission of Of?p stars has demonstrated that MHD simulations yield a superior fit, and require mass-loss rates comparable to those obtained from the <cit.> recipe, whereas spherically symmetric models in general require much lower mass-loss rates to achieve a relatively poor match to the observations <cit.>. In addition to this, magnetic wind confinement is not expected to modify the surface mass flux, which is the relevant quantity in calculating the strength of magnetic confinement <cit.>. Thus, η_* should be determined using the wind parameters as they would be in the absence of a magnetic field, rather than those measured via spectral modelling (e.g., ). Using the mass-loss recipe of <cit.> with , logL, and M_* from Table <ref>, assuming the metallicity Z/Z_⊙=1, and determining the wind terminal velocity =2070±60 via scaling the star's escape velocity by 2.6 as suggested by Vink et al., yields a mass-loss rate log(Ṁ/M_⊙ yr^-1)=-8.0±0.1. From Eqn. 7 of <cit.>, η_* > 340, so the wind is magnetically confined. The physical extent of the magnetosphere is given by the Alfvén radius , defined as the maximum extent of closed magnetic field lines in the circumstellar environment. can be calculated heuristically from η_* (Eqn. 7 in ): we find >4.6 R_*. Unless i is particularly large or small, in which case B_ d is significantly in excess of 2 kG, the magnetic and magnetospheric parameters will be close to the derived lower limits. is almost certainly below 20 R_*.ξ^1 CMa has the highest X-ray luminosity and the hardest X-ray spectrum of any of the magnetic β Cep stars <cit.>. The X-ray luminosity, corrected for interstellar absorption corresponding to E(B-V) = 0.015, is 2.4 ×10^31 erg s^-1 <cit.>. Using 2D MHD simulations <cit.> calculated X-ray emission from magnetically confined wind shocks and developed an X-ray Analytic Dynamical Magnetosphere (XADM) scaling for with , R_*, , and . Comparison to available X-ray data for magnetic early-type stars has indicated that predicted by XADM should be scaled by ∼5-20% to match the observed <cit.>. This efficiency factor accounts for dynamical infall of the plasma. XADM successfully predicts the X-ray luminosity of magnetic, hot stars across 3 decades in logL, 2 decades in B_ d, and 5 decades in <cit.>, i.e. the range of the model's successful application is much larger than the uncertainty introduced by the efficiency factor. Assuming an efficiency of 10%, B_ d=1.1 kG, and taking into account the uncertainties in the stellar parameters in Table <ref>, XADM predicts =31.5 ± 0.1, in excellent agreement with the observed X-ray luminosity =31.47. Adopting the higher value of B_ d=5.3 kG suggested by <cit.> yields =31.7, slightly higher than observed (although this can be reconciled by lowering the efficiency factor to 5%). If the lower and determined by <cit.> are used instead, XADM predicts =28.9 with an efficiency factor of 100%, 2.6 dex lower than observed. Since the efficiency factor can only lower , it is impossible for the XADM model to match ξ^1 CMa's observed X-ray luminosity with a mass-loss rate significantly lower than the <cit.> prediction. <cit.> divided magnetic, massive stars into two classes, those with dynamical magnetospheres (DMs) only, and those also possessing centrifugal magnetospheres (CMs). CMs appear when < , where is the Kepler radius, defined as the radius at which the centrifugal force due to corotation compensates for the gravitational force <cit.>. Solving for using Eqn. 12 from <cit.> with M_* from Table <ref> and P_ rot>30 yr yields >580 R_*. As ≫ , the magnetosphere does not include a CM. The Kepler radius is related to the dimensionless rotation parameter W≡ v_ eq/v_ orb=R_ K^-3/2, where v_ orb is the velocity required to maintain a Keplerian orbit at the stellar surface <cit.>. Critical rotation corresponds to W=1, and no rotation to W=0. For ξ^1 CMa, W<7 × 10^-5.Magnetic wind confinement leads to rapid spindown due to angular momentum loss via the extended moment arm of the magnetized wind <cit.>. The rotation period will decrease exponetially, with a characteristic angular momentum loss timescale τ_ J of <cit.>: τ_ J = 3/2f τ_ M(R_*/R_ A)^2, where τ_ M≡ M_*/Ṁ is the mass-loss timescale, and f is the moment of inertia factor, which can be evaluated from the star's radius of gyration r_ gyr as f = r_ gyr^2. Consulting the internal structure models calculated by <cit.>, f ≃ 0.06 for a star of ξ^1 CMa's mass and age. Solving Eqn. <ref> then yields τ_ J=4±1 Myr. The rotation parameter at a time t after the birth of the star is W(t) = W_0 e^-t/τ_ J, where W_0 is the initial rotation parameter. Assuming W_0=1 yields the maximum spindown age t_ S,max=42±12 Myr <cit.>. This is about 4 times longer than the age inferred from evolutionary tracks. Solving Eqn. <ref> for W_0 yields W_0 < 0.02. Thus, either magnetic braking must have been much more rapid than predicted, or almost all of the star's angular momentum loss must have occurred before it began its main sequence evolution, i.e. the star was already a slow rotator at the ZAMS.Given the important role played by the mass-loss rate in determining and τ_ J, it is of interest to explore the sensitivity of these results to different mass-loss prescriptions. <cit.> found that they were unable to reproduce the observed mass-loss rates of stars with logL < 5.2, suggesting that may be lower than predicted by the <cit.> recipe. We first note that, since τ_ J increases with decreasing Ṁ, a lower mass-loss rate cannot resolve the discrepancy between t_ S,max and the age inferred from the HRD. Second, we note that satisfying the condition for a CM (>) requires log(Ṁ/M_⊙ yr^-1)≤ -16.5. If the lower and found by <cit.> are used to calculate the magnetospheric and spindown parameters, none of the above conclusions are fundamentally changed ( ∼ 14 R_* ≪ R_ K , and t_ S,max = 250^+80_-100 Myr). <cit.> computed mass-loss rates for late O-type stars in the weak-wind domain using the theory of reversing layers <cit.>, and for a star with ξ^1 CMa's and logg predict log(Ṁ/M_⊙ yr^-1)∼ -8.8. <cit.> provided an alternate calculation for , intended specifically for B-type stars, and for ξ^1 CMa predicted log(Ṁ/M_⊙ yr^-1) = -8.9 (see Table 5 in ), similar to the <cit.> prediction. Thus, while there are systematic differences between theoretical mass-loss rates, these are much smaller than would be required to change the basic conclusions that the star lacks a CM and has a spindown age much less than its evolutionary age. § DISCUSSION§.§ Impact of pulsation on the magnetometry While exhibits a clear long-term modulation, it also shows evidence for short-term variability. In particular in 2010 (HJD∼2455200), the most densely time-sampled epoch, there is substantial apparent scatter in the measurements: the mean error bar is 5 G, but the standard deviation of in this epoch is 15 G. We calculated residual measurements by subtracting the mean at each annual epoch in order to remove the long-term trend. These are shown phased with the pulsation period in Fig. <ref>. The reduced χ^2 of a least-squares sinusoidal fit to all residual measurements is 1.2, as compared to 2.8 for the null hypothesis of no variation. In 2010 there is an apparently coherent variation with a semi-amplitude of 16±2 G, consistent with the standard deviation in of 15 G, and 3 times larger than the median error bar of 5 G. The reduced χ^2 of the least-squares sinusoidal fit to the 2010 residual measurements is 0.9, as compared to 6.1 for the null hypothesis. In 2017, the epoch with the largest number of observations and the most systematic sampling of the phase curve, the same fit yields a semi-amplitude of 7±2 G, which is identical to both the standard deviation of, and the median uncertainty in, during this epoch. The reduced χ^2 of a sinusoidal fit and the null hypothesis to the 2017 data is close to 1 in both cases, indicating that in 2017 there is no evidence for a coherent variation with pulsation phase. In 2013, the only other epoch with sufficient observations to constrain a sinusoidal fit, the semi-amplitude is 10±5 G, the reduced χ^2 of the fit is 1.3, and the reduced χ^2 of the null hypothesis is 2.3, i.e. 2013 yields results intermediate between 2010 and 2017. This suggests that the observed variation of may be explained as the superposition of a short-term variation with the stellar pulsation period on top of the long-term variation due to rotation, with the short-term variation declining in amplitude with . We next turn our attention to the LSD profiles. It is clear from the numerous `definite' detections in LSD N profiles that pulsation has affected our data to some degree. Signatures in N profiles are sometimes seen in stars in which the stellar lines vary during acquisition of the polarized sub-exposures used to construct the final Stokes V spectrum. Following the analysis performed by <cit.>, we modelled the N profiles by creating disk-integrated Stokes I and V profiles at the radial velocities corresponding to the pulsation phases of each sub-exposure, and then combining the individual model profiles in such a way as to simulate the double-ratio method's combination of I ± V beams to yield Stokes I, Stokes V, and N <cit.>. Each model profile was created with =0 , a macroturbulent velocity of 8 , the measured RVs, a projection factor of 1.45, and a constant dipolar magnetic field with i = β = 90^∘ and =1.1 kG, with the positive magnetic pole at the centre of the stellar disk. The model profiles were normalized to the observed Stokes I EW.Illustrative results are shown for Stokes I, N, and Stokes V in Fig. <ref>. The model reproduces the variation in N reasonably well. We conclude from this that the N profile signatures are principally a consequence of line profile variability between sub-exposures. There is no change in measured from model LSD profiles created with or without introducing RV shifts between sub-exposures. Furthermore, remains null, as expected. This result is in agreement with the observational results of <cit.>, who found that while shifting sub-exposures by their pulsation velocities greatly reduced the N profile signatures, remained unchanged within error bars. If the modulation of with pulsation phase is not a consequence of an instrumental effect, it might be due to a real change in the intrinsic surface magnetic field strength of the star. An obvious mechanism that might produce such an effect is conservation of magnetic flux throughout radial pulsation cycles. Magnetic flux is conserved as 1/R_*^2, and the semi-amplitude of the change in stellar radius is Δ r/R_* ∼ 1-1.5%, thus could vary by about ±2–3% or ± 7-10 G at _ max=328 G, which is close to the observed semi-amplitude in 2010 of 16±2 G. In this scenario, maximum should occur at phase 0.25, when the star is at its most contracted, and minimum at phase 0.75, when the star is at its most extended; this is indeed what is observed. Thus, a modulation of arising due to an actual change in the surface magnetic field strength with the pulsation period is consistent with both the magnitude and the phasing of the observed effect. The apparent proportionality of the semi-amplitude of the pulsational modulation of to the magnitude of is also consistent with this hypothesis, as a change in due to a change in B_ d should be largest when there is the least amount of magnetic flux cancellation across the stellar disk (i.e. at magnetic maximum). While our model does not predict a change in due to RV shifts between subexposures, we cannot entirely rule out the possibility that the modulation shown in Fig. <ref> is due to line profile variability. Whatever its origin, the influence of pulsation on is much smaller than the long-term modulation of , thus the fundamental conclusion that P_ rot > 30 yr should not be affected by pulsation. §.§ Does extremely slow rotation in an evolved magnetic B-type star make sense? Due to magnetic braking evolved magnetic hot stars are expected to rotate more slowly than either similar, younger magnetic stars or non-magnetic stars of comparable age and mass. However, as was shown in <ref>, the spindown age inferred for ξ^1 CMa is difficult to reconcile with the age inferred from its position on the HRD. The <cit.> evolutionary models imply an age of 11±0.7 Myr, while its spindown timescale is τ_ J=4±1 Myr. Thus, at most two or three e-foldings of the initial rotational period can have occurred since the ZAMS. Assuming initially critical rotation, ξ^1 CMa should now have a rotational period of about 6 d; conversely, calculating from its actual rotation period, its initial rotation fraction W_0 must have already been very close to 0, implying a rotational period on the ZAMS already on the order of years. <cit.> calculated spindown ages for all stars in Fig. <ref> except β CMa and ϵ CMa, finding 40 < t_ S,max < 130 Myr for these stars, i.e. their maximum spindown ages are 4 to 10 times greater than their main-sequence ages, which for stars in this mass range are typically 8 to 15 Myr. We note that the rotational periods of τ Sco and β Cep are both known to high precision <cit.>. For τ Sco, t_ S,max=126^+16_-32 Myr, while its position on the HRD indicates t=3.3^+0.5_-1.1 Myr, a discrepancy of 1.6 dex; similarly for β Cep, t_ S,max=204^+9_-19 Myr, differing by 1.2 dex with its age of t=13.2^+0.4_-0.7 Myr <cit.>. These discrepancies between the stellar age and the maximum spindown age cannot be reconciled by utilizing a different set of evolutionary models: while the absolute ages inferred from different prescriptions vary depending on assumptions about rotation, mixing, or mass-loss, these uncertainties in the models are typically on the order of 1 to 2 Myr for a 15 star, much less than the ∼1 dex difference between gyrochronological and evolutionary ages. It should be noted that the <cit.> models do not include magnetic fields. Fully self-consistent evolutionary models of magnetic stars are not yet available; however, it seems unlikely that such models will resolve the discrepancy via modification of the age inferred from a star's position on the HRD. <cit.> also found that spindown timescales are typically much longer than stellar ages for magnetic, massive stars, which they interpreted as evidence for rapid magnetic flux decay. At first order, the surface magnetic field should weaken as 1/R_*^2 due to conservation of fossil magnetic flux. As an evolved star, ξ^1 CMa's radius has increased by a factor of 1.8±0.1 since the ZAMS, thus its minimum surface magnetic field strength on the ZAMS should have been B_ d,ZAMS > 3.2 kG. Recalculating τ_ J on the ZAMS using Vink mass-loss rates suggests that τ_ J should not have been very different from its present value, as the lower mass-loss rate on the ZAMS compensates for the stronger magnetic field and greater Alfvén radius. Thus, magnetic flux conservation alone cannot account for ξ^1 CMa's slow rotation, and some other mechanism or combination of mechanisms – magnetic flux decay, rapid angular momentum loss on the PMS, or modification of the internal structure of the star due to the magnetic field – is necessary. While it is of course possible that the theoretical mass-loss rates are simply incorrect, resolving the discrepancy by changing alone would require an increase of ∼1 dex, well outside the range of uncertainty of current models which typically differ a factor of 2 to 3, and are in addition usually lower than the <cit.> mass-loss rates (e.g., ).The irreconcilability of gyrochronological and evolutionary timescales amongst magnetic early B-type stars is worthy of future investigation. Here, it is pointed out primarily to emphasize that the difficulty in explaining ξ^1 CMa's extremely slow rotation should not count as an argument against a long rotational period, as this problem is shared by all similar stars.Fig. <ref> shows P_ rot as a function of fractional main sequence age τ_ MS for those stars from Fig. <ref> for which both P_ rot and B_ d are known. The majority of the dipolar field strengths and rotational periods were obtained from the catalogue published by <cit.>, and references therein. β CMa's magnetic, rotational, and stellar properties were obtained from <cit.>. The rotational periods of NU Ori and HD 63425 were determined using ESPaDOnS data <cit.>. Masses and fractional main sequence ages were determined via linear interpolation between the <cit.> evolutionary models on the basis of the stars' positions on Fig. <ref>. For HD 63425 and HD 66665, we used the -logg diagram to infer stellar parameters, as the uncertainties in logL are very large for these stars. In addition to having by far the longest rotational period of any magnetic B-type star, ξ^1 CMa has a stronger magnetic field than any star in Fig. <ref> but NU Ori. It is also amongst the most evolved stars in the sample: the only stars of a comparable fractional age are β Cep and β CMa, both of which are less massive (∼12 ), and have much weaker magnetic fields (∼0.1 kG). Since mass-loss rates increase with M_*, more massive stars should spin down more rapidly than less massive stars. With the exception of NU Ori, stars more massive than ξ^1 CMa all have rotational periods of 10-100 d, although they are less evolved. NU Ori, which has a stronger magnetic field, a much shorter rotational period, and a slightly higher stellar mass than ξ^1 CMa, is also the youngest star on the diagram, likely accounting for its rapid rotation. This comparison shows that, viewed in the context of stars with similar magnetic and stellar parameters, ξ^1 CMa's very slow rotation is not anomalous given its age.§.§ A B-type star with an optically detectable dynamical magnetosphere? In DMs, plasma deposited in the magnetosphere by the stellar wind falls back due to gravity, while in CMs centrifugal support due to the magnetically enforced corotation of the plasma with the star prevents infall above . <cit.> noted that stars without CMs only display emission if their mass-loss rates are high enough to replenish the magnetosphere on dynamical timescales, thus enabling the circumstellar environment to remain optically thick despite continuous depletion as material returns to the photosphere. As a consequence, the only stars with optical emission lines originating in their DMs are the magnetic O-type stars, whereas the only magnetic B-type stars listed byas hosting Hα emission possess CMs. While they were aware of ξ^1 CMa's Hα emission,assumed the 4.26 d rotation period found by <cit.>, and thus that the emission originated in the star's CM. However, as ≫ , ξ^1 CMa does not have a CM.The Hα emission profile shown in Fig. <ref> is furthermore entirely distinct from the double-horned profile characteristic of a CM, which generally produces two emission peaks located at approximately ± (e.g. ). The strong central peak is much more similar to the typical profile of a magnetic O-type star with a DM (e.g. ), although the emission peaks of such stars tend to be slightly red-shifted, whereas ξ^1 CMa's is closer to line centre. ξ^1 CMa's ultraviolet emission line morphology is also consistent with an origin in a DM, at least insofar as it is very similar to the emission lines of β Cep, which does not have a CM. This leads to the question of how a B-type star can have an optically detectable DM. The quantity of plasma in a DM is a function of two parameters: the size of the magnetosphere, and the mass-loss rate which feeds it. As is clear from Fig. <ref>, ξ^1 CMa is one of the most luminous magnetic B-type stars known: while τ Sco and NU Ori have similar logL, no stars are known to have a higher luminosity. In Fig. <ref> is plotted as a function of logL for the stars in Fig. <ref>. Not only does ξ^1 CMa have a higher luminosity than most of the B-type stars, it also has one of the largest Alfvén radii. Also plotted on Fig. <ref> are the magnetic O-type stars from the <cit.> catalogue. All of these stars have emission, despite the very small of the stars with higher luminosities. The dashed diagonal line indicates a possible division between stars with and without emission line DMs. Under the assumption that a star with a smaller requires a higher mass-loss rate in order to show emission, stars above the line should show emission, while those below should not. Alternatively, the presence or absence of emission could be related purely to the mass-loss rate, as indicated by the vertical dotted line: in this case, stars with logL/L_⊙≳ 4.5 should show emission. ξ^1 CMa's position in either case is consistent with the presence of Hα emission. The presence or absence of magnetospheric emission in magnetic OB stars with 4.5 ≤logL/L_⊙≤ 5.0, but ≥ 3 R_*, would help to distinguish between these scenarios.Only one other magnetic B-type star, NU Ori, is in the same part of the diagram as ξ^1 CMa. As noted above, NU Ori is a rapid rotator, and possesses a CM. This star shows neither Hα nor UV emission <cit.>. The absence of emission, given its high luminosity and comparable magnetic confinement strength to ξ^1 CMa, is a distinct puzzle deserving of future investigation. §.§ A pulsating magnetosphere? Due to the limited number and quality of IUE observations, the evidence for pulsational modulation of UV resonance doublets is inconclusive. Evidence for pulsational modulation of the star's Hα emission is much stronger. The residual Hα EWs, after subtraction of a sinusoidal fit to the annual mean EWs phased with the presumed rotation period, yield a clear peak in the periodogram at the pulsation period. When phased with the pulsation period, the amplitude of the residual EW variation is 3 times higher than the amplitude expected from a photospheric modulation of the absorption line's EW, and is furthermore 0.5 pulsation cycles out of phase with the predicted photospheric variation. Finally, Hα shows a much stronger V/R variation than is predicted by the photospheric model. Stochastic changes in the EW of the Hα line have been observed for some magnetic O-type stars with dynamical magnetospheres, e.g. the well-studied object θ^1 Ori C <cit.>. 3D MHD simulations are able to reproduce this short-term variability as a consequence of turbulent plasma infall <cit.>. As is constant for such stars, their short-term variability is wholly a consequence of the complex behaviour of MHD plasmas. Since ξ^1 CMa's mass-loss rate should change with the pulsation phase, it is conceivable that this could force a periodicity in the outflow and infall of the magnetically confined plasma. The observation of prominent blue- and red-shifted secondary peaks in the Hα emission structure occuring at distinct pulsation phases, together with the stronger modulation in Hα V/R as compared to residual Hα EWs, further suggests that, throughout the course of a pulsation cycle, there are larger local changes in the distribution of plasma within the magnetosphere, as compared to the total mass of magnetically confined plasma. The majority of the Hα emission is expected to be produced in the innermost regions <cit.>. The wind flow timescale, calculated using a standard β = 1 velocity law[While confinement in a dipolar magnetic field does modify the wind-flow timescale, in practice this is a second-order effect <cit.>.], ranges from ∼0.13 d at a distance of 2 R_*, to ∼0.19 d at the minimum Alfvén radius of 4.6 R_*. The free-fall timescale is somewhat longer, ranging from 0.2 to 1.2 d over this same distance. These timescales are similar to the pulsation period, suggesting that periodic density waves arising due to the varying mass-loss rate might lead to a periodicity in the plasma infall. The strongest blue-shifted secondary emission peak occurs at about phase 1.0, and the strongest secondary red-shifted emission peak at phase 0.5. If the blue- and red-shifted emission peaks are respectively consequences of outflowing and infalling plasma, and assuming the majority of the emission is formed relatively close to the star, a lag of approximately 0.7 cycles between the peak mass-loss rate (at phase 0.3) and the strongest blue-shifted emission (at phase 1.0) would make sense.While no other magnetic, massive star's Hα emission has ever been found to be modulated by pulsation as well as rotation, <cit.> reported that ξ^1 CMa's X-ray light curve exhibits a coherent variation with the pulsation period, with both hard and soft X-ray emission peaking ∼0.1 cycles after the maximum of the visible light curve. Since ξ^1 CMa is a pulsating star, is not constant. The changes in and logL_* should lead to a variation in with a semi-amplitude of 0.05 dex. Changes in should be negligible. Calculation of η_* and as a function of pulsation phase suggests the overall changes in these parameters to be small (∼20% in η_* and 4% in ). Similar X-ray pulsations have also been reported for the β Cep star β Cru <cit.>. As β Cru does not have a detected magnetic field (J. Grunhut, priv. comm.), this argues against a purely magnetic explanation for the X-ray variability. <cit.> considered a variation in as a possible mechanism behind the star's X-ray pulsations, and rejected it based upon the absence of X-ray pulsations in other β Cep stars, either magnetic or not. However, they did not consider their results in the context of the XADM model. The XADM model predicts a modulation in of approximately ±10% to arise from a variation in of ±0.05 dex, with the maximum and minimum of the predicted X-ray light curve corresponding to the maximum and minimum of the visible light curve. This is the same amplitude as that reported by <cit.>. As a sanity check, we performed similar calculations for β Cep and β Cen, the only other magnetic β Cep stars with X-ray light curves. β Cep is slightly cooler (∼26 kK), less strongly magnetized (B_ d∼260 G; ), and has somewhat lower amplitude pulsations leading to a smaller change in luminosity (±0.015 mag and ±0.006 dex, respectively, as evaluated by phasing the star's Hipparcos observations with its primary pulsation period of 0.19 d; ) and, hence, a smaller change in the mass-loss rate (±0.03 dex). The XADM model predicts that β Cep should have X-ray pulsations with a semi-amplitude of ≤ 6%, close to the upper limit determined by <cit.>. For β Cen, it is not clear which of the many pulsation frequencies identified by <cit.> belongs to the magnetic secondary. However, even the highest amplitude frequency (±0.0015 mag) leads to a luminosity modulation of just 0.006 dex, hence a change in of less than ±0.01 dex, and, with B_ d=200 G <cit.>, a variation in the X-ray luminosity of <3%, below the upper limit established by <cit.>. It is probably not the case that simple calculation of global parameters at each pulsation phase provides an accurate picture. Strong local density variations due to the changing may lead to correspondingly strong local changes in the magnetic confinement strength. Magnetohydynamic simulations will be necessary to properly explore the impact of stellar pulsation on magnetospheric dynamics. § CONCLUSIONS AND FUTURE WORK The principal result of this paper is that both magnetic and spectroscopic data indicate that the rotation period of ξ^1 CMa is very long, on the order of decades. Rotational periods on the order of days, as previously suggested in the literature, are conclusively ruled out by the ESPaDOnS measurements alone, which require a period of at least 5100 d. Inclusion of the MuSiCoS results requires that P_ rot be about 30 years, a conclusion supported by the modulation of the Hα emission strength seen in the CORALIE and ESPaDOnS datasets. A period of approximately 30 years is consistent with the timing of the star's strong UV emission, which should correspond to an extremum of the curve. Given that phase coverage of the rotational cycle is incomplete to an unknown degree, the constraints that can be placed upon the geometry and strength of the magnetic field are necessarily somewhat loose. Further spectropolarimetric observation over decades is essential to characterization of the magnetic field. If the 30-year period is correct, magnetic crossover (=0) will occur in 2018, and magnetic minimum in 2025. We have conducted the first detailed examination of the star's weak Hα emission, which we detect in all available spectroscopic data. There is no evidence in the interferometric data of a binary companion, and combined constraints from interferometry and RV residuals rule out a binary companion of sufficient luminosity to host the weak emission. Furthermore, there is no indication of a Keplerian disk in the high-resolution AMBER spectro-interferometry. Thus, the interferometric data does not support the hypothesis that the emission originates in the Keplerian disk of a classical Be companion star, indicating that this emission is produced in ξ^1 CMa's circumstellar environment. This makes ξ^1 CMa the first magnetic β Cep star with a magnetosphere detectable in visible light. Given that its extremely slow rotation implies a dynamical rather than a centrifugal magnetosphere, ξ^1 CMa is also the coolest star with a dynamical magnetosphere detectable in Hα.Finally, in addition to the already reported modulation of X-ray flux with the pulsation period, the Hα emission also appears to correlate with the pulsations in a fashion that seems likely to be intrinsic to the circumstellar plasma. The best evidence for this is in the strong Hα V/R variations, which are much greater than expected from a radially pulsating photospheric model. The total Hα EW in any given epoch also varies in antiphase with the expected EW variation due to photospheric variation, and furthermore has a larger amplitude than predicted by a photospheric model. However, the EW is only weakly variable compared to V/R, suggesting that the distribution of plasma within the magnetosphere, rather than the total quantity of magnetically confined material, is the primary origin of the variability. Given the complex velocity fields within dynamical magnetospheres, magnetohydrodynamic (MHD) simulations will be necessary to understand the interplay between pulsation, mass-loss, and magnetic confinement. As the magnetic wind confinement parameter η_* is on the order of 10^2, modeling ξ^1 CMa's magnetosphere should be within the capabilities of the current generation of 2D MHD codes.We also find a weak modulation of with the pulsation period. Further modeling is needed to understand the origin of this phenomenon. Signatures in the diagnostic null profile are accurately reproduced as a consequence of RV variations between subexposures, however, this model does not predict any modulation of . Polarized radiative transfer in a moving atmosphere will be essential to answering the question of whether these modulations are an artefact of the measurement methodology, or intrinsic to the stellar magnetic field.Acknowledgements The authors acknowledge the assistance of Conny Aerts, who provided the CORALIE dataset. This work has made use of the VALD database, operated at Uppsala University, the Institute of Astronomy RAS in Moscow, and the University of Vienna. MS acknowledges the financial support provided by the European Southern Observatory studentship program in Santiago, Chile. MS and GAW acknowledge support from the Natural Science and Engineering Research Council of Canada (NSERC). CN and the MiMeS collaboration acknowledge financial support from the Programme National de Physique Stellaire (PNPS) of INSU/CNRS. We acknowledge the Canadian Astronomy Data Centre (CADC). The IUE data presented in this paper were obtained from the Multimission Archive at the Space Telescope Science Institute (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NAG5-7584 and by other grants and contracts. This project used the facilities of SIMBAD and Hipparcos. This research has made use of the Jean-Marie Mariotti litpro service co-developed by CRAL, IPAG, and LAGRANGE. § RELIABILITY OF MUSICOS DATA<cit.> objected to a long rotation period, first suggested by <cit.>, on the grounds that the period is largely constrained by the MuSiCoS measurements, and further suggesting that as these data were collected at high airmass the measurements were unreliable. As we have shown, it is not the case that a long rotation period is implied primarily by the MuSiCoS data: to the contrary, a gradual decline in is clearly evident in the ESPaDOnS data alone, and the conclusion that the star is slowly rotating is also supported by the long-term modulation of the Hα emission strength. However, the MuSiCoS measurements are an important constraint, and in this appendix we address the question of their reliability. While we are aware of no reason why the sign of the Stokes V signature should be flipped due to the star being observed at high airmass, the fibers were plugged in differently each observing run, which could result in an apparent polarity change. To check this possibility, we acquired the MuSiCoS dataset of the Bp star 36 Lyn reported by <cit.>, of which one measurement was acquired on 02/10/2000, during the same observing run as the ξ^1 CMa data. We compare these data with Narval observations of 36 Lyn. Narval, a clone of ESPaDOnS, obtains essentially identical results to its sibling instrument <cit.>. LSD profiles were extracted from the two datasets using the same method described in <ref>, with a line mask appropriate to 36 Lyn's . Fig. <ref> shows the resulting measurements, phased with the rotational ephemeris determined by <cit.>. The two datasets agree well at all phases. While the measurement obtained on 02/10/2000 is in its expected position, =-40±170 G is close to 0, thus a sign flip would not change its position on the curve. As an additional check, Fig. <ref> compares the LSD profile of this MuSiCoS observation to the Narval LSD profile with the closest rotational phase. The Stokes V profiles of the two observations are a good match to one another. Moreover, a change in polarity of the MuSiCoS signature would render it in conflict with the Narval profiles. We conclude that there is no compelling reason to doubt the reliability of MuSiCoS, which obtained results fully consistent with those of modern high-resolution spectropolarimeters, and thus that the negative polarity of ξ^1 CMa's Stokes V profile during this epoch is real.§ LOG OF MAGNETIC MEASUREMENTS § RADIAL VELOCITY AND HΑ EW MEASUREMENTS § LOG OF INTERFEROMETRIC OBSERVATIONS | http://arxiv.org/abs/1706.08820v1 | {
"authors": [
"M. Shultz",
"G. A. Wade",
"Th. Rivinius",
"C. Neiner",
"H. Henrichs",
"W. Marcolino",
"the MiMeS Collaboration"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170627124225",
"title": "The pulsating magnetosphere of the extremely slowly rotating magnetic $β$ Cep star $ξ^1$ CMa"
} |
Multilevel Monte Carlo Method for StatisticalModel Checking of Hybrid Systems Sadegh Esmaeil Zadeh Soudjani1 Rupak Majumdar1 Tigran Nagapetyan2 December 30, 2023 =============================================================================== Insertion Sequences (ISs) are small DNA segments that have the ability of moving themselves into genomes. These types of mobile genetic elements (MGEs) seem to play an essential role in genomes rearrangements and evolution of prokaryotic genomes, but the tools that deal with discovering ISs in an efficient and accurate way are still too few and not totally precise. Two main factors have big effects on IS discovery, namely: genes annotation and functionality prediction. Indeed, some specific genes called “transposases” are enzymes that are responsible of the production and catalysis for such transposition, but there is currently no fully accurate method that could decide whether a given predicted gene is either a real transposase or not. This is why authors of this article aim at designing a novel pipeline for ISs detection and classification, which embeds the most recently available tools developed in this field of research, namely OASIS (Optimized Annotation System for Insertion Sequence) and ISFinder database (an up-to-date and accurate repository of known insertion sequences). As this latter depend on predicted coding sequences, the proposed pipeline will encompass too various kinds of bacterial genes annotation tools (that is, Prokka, BASys, and Prodigal). A complete IS detection and classification pipeline is then proposed and tested on a set of 23 complete genomes of Pseudomonas aeruginosa. This pipeline can also be used as an investigator of annotation tools performance, which has led us to conclude that Prodigal is the best software for IS prediction. A deepen study regarding IS elements in P.aeruginosa has then been conducted, leading to the conclusion that close genomes inside this species have also a close numbers of IS families and groups. § INTRODUCTION The number of completely sequenced bacterial and archaeal genomes are rising steadily,such an increasing makes it possible to develop novel kind of large scale approachesto understand genomes structure and evolution over time. Gene content prediction and genome comparison have both provided new important information anddeciphering keys to understand evolution of prokaryotes <cit.>.Important sequences in understanding rearrangement of genomes during evolution are so-calledtransposable elements (TEs), which are DNA fragments or segments thathave the ability to insert themselves into new chromosomal locations, and oftenmake duplicate copies of themselves during transposition process <cit.>. Remark that, inbacterial reign, only cut-and-paste mechanism of transposition can be found, thetransposable elements involved in such a move being the insertion sequences (ISs).Insertion sequences range in size from 600 to more than 3000bp.They are divided into 26 main different families in prokaryotes, as described in ISFinder[<www-is.biotoul.fr>] <cit.>, an internationalreference database for bacterial and archaeal ISs that includes background information on transposons.The main function of ISFinder is to assign IS names and to produce a focalpoint for a coherent nomenclature for all discovered insertion sequences.This database includes over than 3500 bacterial ISs <cit.>.Data come from a detection of repeated patterns, which can beeasily found by using homology-based techniques <cit.>.Classification process of families, for its part, depends on transposases homologyand overall genetic organization. Indeed, most ISs consist of short inverted repeatsequences that flank one or more open reading frames (ORFs, see Figure <ref>),whose products encode the transposase proteins necessary for transposition process. The main problem with such approaches for ISs detection and classification is that they are obviously highly dependent on the annotations, and existing tools evoked above only usethe NCBI ones, whose quality is limited and very variable.In this research work, the authors' intention is to find an accurate method for discovering insertion sequences in prokaryotic genomes.To achieve this goal, we propose to use one of the most recent computational toolfor automated annotation of insertion sequences, namely OASIS, together with theinternational database for all known IS sequences (ISFinder).More precisely, OASIS works with genbank files that have fullydescribed genes functionality: this tool identifies ISs ineach genome by finding conserved regions surrounding already-annotatedtransposases. Such technique makes it possible to discover new insertion sequences, evenif they are not in ISFinder database. A novel pipeline that solves the dependence on NCBI annotations,and that works with any annotation tool (with or without description ofgene functionality) is then proposed.The output of our pipeline contains all detected IS sequences supported with other important information like inverted repeats (IRs) sequences, lengths, positions, names of family and group, and other details that help in studying IS structures.The contributions of this article can be summarized as follows. (1)A pipeline for insertion sequences discovery and classification is proposed, which does not dependon NCBI annotations. It uses unannotated genomes and embeds various annotation toolsspecific to Bacteria (such as Prokka, BASys, and Prodigal) in its process. (2) Overlapping and consensus problems that naturally appear aftermerging annotation methods recalled above are solved, in order to obtain large and accurate number of ISs with their names of families and groups. And finally (3) the pipeline is testedon a set of 23 complete genomes of Pseudomonas aeruginosa, and biological consequences are outlined. The remainder of this article is organized as follows. In Section <ref>, various toolsfor discovering IS elements in different species of Bacteria and Archaea are presented.The suggested methodology for increasing both the number and accuracy of detectingIS elements is explained in Section <ref>. The pipeline is detailed inSection <ref>, while an application example using 23 completed genomes ofP. aeurigonsa is provided in Section <ref>.This article ends by a conclusion section, in which the contributions are summarized andintended future work is detailed.§ STATE OF THE ART IN ISS DETECTION OR ANNOTATION The study on the plant-pathogenic prokaryote Xanthomonas oryzae pv. oryzae (Xoo), which causesbacterial blight (one of the most important diseases of rice) was published in 2005 by Ochiai et al. <cit.>. They used GeneHacker <cit.>, GenomeGambler version 1.51, and Glimmer program <cit.> for coding sequence prediction.Insertion sequences were finally classified by a BLAST analysis using ISFinder database evoked previously. IScan, developed by Wagner et al. <cit.>, has thenbeen proposed in 2007. Inverted repeats are found using smith waterman localalignments on transposase references found with BLAST and usedas a local database. This tool has beenapplied on 438 completely sequenced bacterial genomes by using BLAST with referenced transposases, to determine which transposases are related to insertion sequences. Touchon et al., for their parts, have analyzed262 differentbacterial and archaeal genomes downloaded from GenBank NCBI in 2007 <cit.>. A coding sequence has then been considered as an IS element if itsBLASTP best hit in ISFinder databasehas an e-value lower than 10^-10. ISA has been created by Zhou et al. in 2008 <cit.>.This annotation program depends on both NCBI annotations and ISFinder. More precisely, authors manually collected 1,356 IS elements with both sequences and terminal signals from the ISFinder database, which have been used as templates for identification of all IS elements and mapconstruction in the targeted genomes. ISA, which is not publicly available, has finally been used for an analysis of 19 cyanobacterial and 31 archaeal annotated genomes downloaded from NCBI.In 2010, Plague et al. analyzed the neighboring gene orientations (NGOs) of all ISs in 326 fully sequenced bacterial chromosomes. They obtained primary annotations from the Comprehensive Microbial Resource database (release 1.0-20.0) at the Institute for GenomicResearch[<http://cmr.tigr.org/tigr-scripts/CMR/CmrHomePage.cgi>].Their approach for extracting IS elements from these genomes was to considerthat a coding sequence with a best BLASTX hit e-value lower than 10^-10 isan insertion sequence <cit.>. ISsage, for its part, has been developed in 2011 by Varaniet al. <cit.>. They used eight different bacterial genomes downloaded from NCBI, and produced a web application pipeline that allows semi-automated annotationbased on BLAST against the ISFinder database. However ISsage cannotautomatically identify new insertion sequences which are not already present inISFinder database.A new computational tool for automated annotation of ISs has then beenreleased in 2012 by Robinson et al. <cit.>. This tool has been called OASIS, which stands for “OptimizedAnnotation System for Insertion Sequences”. They worked with 1,737 bacterial and archaeal genomes downloaded from NCBI.OASIS identifies ISs in each genome by finding conserved regions surrounding already-annotated transposase genes. OASIS uses a maximum likelihood algorithmto determine the edges of multicopy ISs based on conservation between their surrounding regions. For defining inverted repeats, the same strategy as IScanwas used (Smith-Waterman alignment). Authors also used hierarchicalagglomerative clustering to identify groups of IS lengths. The ISs set is then classifiedaccording to the family and group after a BLASTP best hit in ISFinder databasewith an e-value lower than 10^-12. When a clustercannot match with any entry of the database, the IS set isconsidered as new. Thus OASIS has the ability to discover newinsertion sequences, that is, which cannot be found in ISFinder. Finally, in 2014, the analysis of the NGOs for all IS elements within 155 fully sequenced Archaea genomes was presented by Floreket al. <cit.>. To do so, they have launched a BLASTP in the ISFinder, with an e-value less than or equal to 10^-10, forall protein coding sequences downloaded from NCBI which are related to ISelements. Two major concerns with the state of the art detailed above can be emphasized. Firstly, most of them cannot detect new insertion sequences. Secondly, all these tools are based on NCBI annotations of very relative and variable qualities – except ISsaga, which could work with other annotation tools (but it depends only on transposase ORFs that have been already defined in ISFinder). Our objective in the next section is to propose a pipelinethat solves these two issues, being able to deal with unannotated genomes and to detect unknown ISs. § PREDICTION AND MODULES BASED ON OASIS For illustration purpose, the proposed pipeline system for IS elements prediction will be presented using 23 complete genomes of P. aeruginosa available on theNCBI website, RefSeq and INCDS/Genebank databases, see Table <ref> (RefSeq genomes were prefered when available). The prediction of IS elements in the proposed pipeline depends on both OASIS <cit.> and ISFinder <cit.>.§.§ Prediction of IS elements from Pseudomonas aeruginosa OASIS is used in this pipeline for predicting insertion sequences in prokaryotic genomes.This latter detects ISs in each genome by finding conserved regionssurrounding already-annotated transposase genes, which are identified by theword ¨transposase¨ in the “product” field of the GenBank file. Obviously OASIS highly depends on the quality of annotations <cit.>, while to determine whether a given gene is a transposase or notis a very difficult task (indeed transposases are amongthe most abundant and ubiquitous genes in nature <cit.>, and they are widely separated in Prokaryote genomes).OASIS deals with files having genbank format. It takesthem as input and then produces two output files for each provided genome. The first one is a fasta file that contains all IS nucleotide sequences,with start and end positions. It also contains the amino acid sequence for each ORF. The second file is a summary table providing attributesthat describe the insertion sequence: set-id, family, group,IS positions, inverted repeat left (IRL) and right (IRR),and orientation. Remark that most of these information arein the ISFinder database too. IndeedOASIS find them alone but it extracts family names and group from ISFinder. The main problem found in OASIS is solved in the proposed pipeline byusing different types of annotations: NCBI will not be used alone,and gene functionality taken from annotation tools will either or not be used depending on the situation. Finally, transposaseswithin IS will be verified using ISFinder database. OASIS can thus be used in two different ways in our pipeline, depending on theprovided genbank file. These two modules have been named NOASIS,whichuses the original input genbank genome file provided by the NCBI (as it is, without any modification), and DOASIS, which deals with modified genbank files that have been updated to obtain more accurate results than NOASIS. These modules are described thereafter. §.§ Normal OASIS (NOASIS) For finding predicted IS in NOASIS module, we simply applied OASIS on the inputset of genomes with their NCBI annotations, that is, with the original downloaded genbank file.Using the reference genome named PAO1, the summary outputted by the pipeline isgiven in Tables <ref>and <ref>. In these NOASIS tables, the summary produced by OASIS is enriched with new features described below: * Real IS IS sequences that have best match (first hit) when using BLASTN with ISFinder database, an e-value equal to 0.0, and with a functionality of each ORF within the IS recognized as a transposase. * Partial IS Sequences that match part of known IS from ISFinder (i.e., have e-value lowerthan 10^-10) and havealso a transposase gene functionality for the ORFs.* Putative New IS Sequences with bad score after making a BLASTN with ISFinder, but with a transposase. They may be real insertion sequences not already added in ISFinder database or false positives, requiring human curation. Applying this slightly improved version of OASIS in the 23 genomes of Pseudomonas leads to a major issue: surprisingly,NOASIS found no real insertion sequences in some genomeslike PACS2 or SCV20265.The problem is that OASIS findmultiple copies of IS elements in each genome by identifyingconserved regions surrounding transposase genes. However someof the considered genomes either have no information about transposase geneinto their feature genbank tables or have simply no featuretable in their genbank format files. This issue is at thebasis of our improved module called DOASIS, which is explained below.For the sake of comparison, Figure <ref> contains similar results for Mycobacterium tuberculosis genus.§.§ Developed OASIS (DOASIS)The main idea for DOASIS module is that information about transposases within genbank files are potentially incorrect (i.e., may all be false positives). Sowesimply decide to remove all transposase words in the product fieldsfrom all inputted genomes. We thus update these information as follows. Step 1: genbank update. Inputted genbank files are modified following one of the three methodsbelow. * All-Tpase: we consider that all the genes may potentially be a transposase. So all product fields are set to “transposase”.* Zigzag Odd: we suggest that genes in odd positions are putative transposases and we update the genbank file adequately. Oddly, this new path will produce new candidates whichare not detected during All-Tpase.* Zigzag Even: similar to Zigzag Odd, but on even positions. We checked also a randomized method (i.e., by putting “transposase” in randomly picked genes). However we found poorernumber of predictive real ISs or new real ISs compared with the three methods previously presented. For these reasons, we will not further investigate the randomized method. Step 2. We apply OASIS three times (i.e., one time per method)on all genomes, and then we take the output fasta file thatcontains both nucleotides and amino acids sequences for each IS element.Step 3.A BLASTN with ISFinder is applied on each IS sequence.If the e-value of the first hit is 0.0, thenthe ORF within thisIS belongs to known (Real) IS already existing in the ISFinder database. Else, if the e-value is lower than 10^-10, then we found a Partial IS.Step 4. Collect all Real IS from previous three methods (ALL_Tpase, Zigzagodd, and Zigzag even) and then remove overlaps among them. Finally, produce bestReal IS with all information. Remark that the problem of finding consensusand overlaps can be treated as a lexical parsing problem. § THE PROPOSED PIPELINE It is now possible to describe the proposed pipeline that can use the two modulesdetailed in the previous section. This pipeline, depicted in Figure <ref>,will increase the number of Real IS detected on the set ofP.aeruginosa genomes under consideration(indeed, the detection is improved in all categories of insertion sequences, but we only focus on Real IS in the remainder of this article, for the sake of concision). Its steps are detailed in what follows. Step 1: ORF identification. Our pipeline is currently compatible with any type of annotationtools, having either functionality capability or not, but for comparison we only focus in this article on the following tools: BASys, Prokka, and Prodigal. BASys (Bacterial Annotation System) is a web server that performsautomated, in-depth annotation of bacterial genomic (chromosomal and plasmid)sequences.It uses more than 30 programs to determine nearly 60 annotation subfields for each gene. Remark that genomes must be sent online manually, and that somecuration stage may be required to remove some DNA ambiguity onreturned genbank files.Prokka (rapid prokaryotic genome annotation), for its part, isa classical command line software for fully annotating draft bacterial genomes,producing standards-compliant output files for further analysis <cit.>.Finally, Prodigal (Prokaryotic Dynamic Programming Genefinding Algorithm) is an accurate bacterial and archaeal genes findingsoftwareprovided by the Oak Ridge National Laboratory <cit.>. Step 2: IS Prediction. The second stage of the pipeline consists in usingeither NOASIS or DOASIS for predicting IS elements. Notice that NOASIScannot be used with Prodigal, as this module requires information about gene functionality (both NOASIS and DOASIS can be use withProkka and BASys annotations).Step 3: IS Validation. This step is realized by launching BLASTN on eachpredicted IS sequence with ISFinder. The e-value of the first hit is then checked: if it is 0.0, then the ORF within this sequence is a Real IS known by ISFinder. Asdescribed previously, it will be considered as Partial IS ifits e-value is lower than 10^-10. Both IS names of family and group are returned too. § RESULTS AND DISCUSSION We can firstly remark in Figure <ref> that,using either Prokka or BASys for genes detection and functionality prediction is better than taking directly the annotated genomes from NCBI: a larger number of Real IS can be found. Additionally, this comparison shows that Prokka outperforms BASys in 3 families of ISs (namely: IS3, IS30, and ISNCY), while BASys seems better for detecting insertion sequences belonging in the IS5, IS1182, and TN3 families. This variability may be explained by the fact that functionality annotations of these tools depend probably onIS families that where known when these tools have been released.The effects of DOASIS module compared to single OASIS on annotated NCBI genomes are depicted in Figure <ref>. The improvement in real IS discovery is obvious, illustrating the low quality and inadequacy ofNCBI annotations for studying insertion sequences in bacterial genomes, and the improvements when using our pipeline. This chart shows too that a zigzag path in the annotation can oddly improve the detection of insertion sequences. The prediction of real ISs is based on finding conserved regions (i.e., inverted repeats (IRs)) surrounded by transposase genes.Some ISs have been lost in All_Tpase, for the following reason: when we suggested that all genes are transposases, OASIS found predicted ISs that consist of large sets of transposases surrounded by IR in their left and right boundaries. But when these predicted ISs have been verified using ISFinder database, we did not find any good match. Contrarily, in Zigzag methods, good matches have been found (real ISs), because many of these elements consist of one or two transposase genes flanked by IRs. These results are listed with detail in Table <ref> using BASys annotation tools. We can thus wonder if the source of a wrong prediction of real IS is due to a wrong coding sequence prediction, or tofunctionality errors. Switching between NOASIS and DOASIS allows us to answer this question. We can concludefrom Table <ref> that (1) annotation errors are more frequent on NCBI, while Prokka annotates well the sequences related to ISs (see NOASIS columns), and that (2) both NCBI and Prokka have a better coding sequence prediction than BASys, at least when considering sequences involved in IS elements (see DOASIS columns and the correlation line). More precisely, the correlation is based on the number of predicted real IS elements between NOASIS and DOASIS. Prodigal has been studied separately, as it does not provide genes functionality. The number of Real ISs per genome returned by our pipeline using prodigal is given in Figure <ref>. As shown in Table <ref>, the quality ofcoding sequences predicted with prodigal compared with other annotation tools allows us to discover the best number of real ISs. In particular, we have improved a lot of results produced by OASIS and ISFinder on NCBI annotations, which is usually used in the literature that focuses on bacterial insertion sequences. Furthermore, this table illustrates a certain sensitivity ofcoding sequence prediction tools with functionality annotation capabilities to detect ISs in some specific genomes like PA7. Indeed we discovered,during other studies we realized on this set of Pseudomonas strains, that PA7 has a lot of specific genes, that is, which are not in the core genome of all Pseudomonases, which may explain such a sensitivity. § CONCLUSIONInsertion sequences of bacterial genomes are usually studied using OASIS and ISFinder on NCBI annotations. We have shown in thisarticle that a pipeline can be designed to improve the accuracy of IS detection and classification by improving the coding sequence prediction stage, and by considering a priori each sequence as a transposase. The source code for this pipeline can be download from the link [<http://members.femto-st.fr/christophe-guyeux/en/insertion-sequences>].A comparison has been conducted on a set of Pseudomonas aeruginosa, showing an obvious improvement in the detection of insertion sequences for some particular configurations of our pipeline. In future work, we intend to enlarge the number of coding sequence and functionality prediction tools and to merge all the Real IS results in order to improve again the accuracy of our pipeline. We will then focus on the impact of IS elements in P.aeruginosa evolution, comparing the phylogenetic tree of strains of this species witha phylogeny of their insertion sequences. Insertion events will then be investigated, and related to genomes rearrangements found in this collection of strains. We will finally enlarge our pipeline toeukariotic genomes and to other kind of transposable elements.plain | http://arxiv.org/abs/1706.08267v1 | {
"authors": [
"Huda Al-Nayyef",
"Christophe Guyeux",
"Jacques M. Bahi"
],
"categories": [
"q-bio.GN"
],
"primary_category": "q-bio.GN",
"published": "20170626075950",
"title": "A Pipeline for Insertion Sequence Detection and Study for Bacterial Genome"
} |
These authors contributed equally to this work. Department of Electrical and Computer Engineering, University of Rochester, Rochester, NY 14627These authors contributed equally to this work. Institute of Optics, University of Rochester, Rochester, NY 14627Department of Electrical and Computer Engineering, University of Rochester, Rochester, NY 14627Department of Electrical and Computer Engineering, University of Rochester, Rochester, NY 14627 School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China [email protected] Institute of Optics, University of Rochester, Rochester, NY 14627 Department of Electrical and Computer Engineering, University of Rochester, Rochester, NY 14627 Lithium niobate (LN) exhibits unique material characteristics that have found many important applications. Scaling LN devices down to a nanoscopic scale can dramatically enhance light-matter interaction that would enable nonlinear and quantum photonic functionalities beyond the reach of conventional means. However, developing LN-based nanophotonic devices turns out to be nontrivial. Although significant efforts have been devoted in recent years, LN photonic crystal structures developed to date exhibit fairly low quality. Here we demonstrate LN photonic crystal nanobeam resonators with optical Q as high as 10^5, more than two orders of magnitude higher than other LN nanocavities reported to date. The high optical quality together with tight mode confinement leads to extremely strong nonlinear photorefractive effect, with a resonance tuning rate of ∼0.64 GHz/aJ, or equivalently ∼84 MHz/photon, three orders of magnitude greater than other LN resonators. In particular, we observed intriguing quenching of photorefraction that has never been reported before. The devices also exhibit strong optomechanical coupling with gigahertz nanomechanical mode with a significant f · Q product of 1.47× 10^12 Hz. The demonstration of high-Q LN photonic crystal nanoresonators paves a crucial step towards LN nanophotonics that could integrate the outstanding material properties with versatile nanoscale device engineering for diverse intriguing functionalities. High-quality lithium niobate photonic crystal nanocavities Qiang Lin December 30, 2023 ==========================================================Lithium niobate (LN) exhibits outstanding electro-optic, nonlinear optical, acousto-optic, piezoelectric, photorefractive, pyroelectric, and photoconductive properties <cit.> that have found very broad applications in telecommunication <cit.>, nonlinear/quantum photonics <cit.>, microelectromechanics <cit.>, information storage <cit.>, sensing <cit.>, among many others <cit.>. Recently, significant interest has been attracted to develop LN photonic devices on chip-scale platforms <cit.>, which have shown significant advantage in device engineering compared with conventional approaches. Miniaturization of device dimensions dramatically enhances optical field in the devices which enables a variety of nonlinear optical, quantum optical, and optomechanical functionalities. Among various approaches developed to date, photonic crystal is probably one of the most efficient ones for light confinement <cit.>, which has been demonstrated on a variety of material platforms <cit.>. For lithium niobate, however, it remains an open challenge to achieve high optical quality, primarily due to significant challenge in device fabrication <cit.>. The LN photonic crystal nanocavities demonstrated to date generally exhibit low optical Q in the order of ∼ 100 <cit.>, which seriously limits their potential applications.An alternative approach to get around the fabrication challenge is to fabricate waveguide structures on a different material deposited on top of a LN substrate to provide wave guidance while using LN as a cladding material <cit.>. This approach, however, limits the extent of optical mode overlap with the LN layer as well as the design flexibility of waveguide structure, due to the limitation of index contrast required between the waveguide material and the LN substrate.In this paper, we demonstrate LN photonic crystal nanobeam resonators with optical Q up to 1.09× 10^5, more than two orders of magnitude higher than any other LN photonic crystal nanocavities reported to date <cit.>. The high optical Q together with the tiny effective mode volume (∼ 1.03(λ/n)^3) leads to extremely strong nonlinear photorefractive effect, with a resonance tuning rate of ∼0.64 GHz/aJ, corresponding to ∼84 MHz/photon, three orders of magnitude greater than other LN resonators <cit.>. In particular, it enables us to observe the intriguing quenching of photorefraction that has never been reported before. It also results in strong coupling between the optical cavity mode and the mechanical motion of the device structure, which allows us to sensitively probe the rich nanomechanical properties of the LN photonic crystal nanobeams up to ∼1 GHz. The demonstration of high-Q LN photonic crystal nanocavities paves the foundation towards LN nanophotonics that would combine elegantly the unique material properties of lithium niobate and versatile nanophotonic device design/fabrication, for broad nonlinear photonic, quantum photonic, optoelectronic, and optomechanical applications.§ DEVICE DESIGN AND FABRICATIONCurrent plasma etching approaches to fabricate high-quality LN photonic devices generally produce a slant angle on the device sidewall <cit.>. Although it might help improve the optical quality of LN microresonators, it impacts seriously on LN photonic crystals which have stringent requirement on the precision of device fine structures. To achieve high optical Q, we tailored our design to incorporate such slant angle into the structure of photonic crystals. The insets of Fig. <ref>(a) show the rectangular-shaped unit cell of the designed photonic crystal nanobeam (Fig. <ref>(c), inset), where the angles of inside and outside sidewalls (Fig. <ref>(b)), θ_in = 45^∘ and θ_out =75^∘, are determined by the plasma etching process. The width W of the nanobeam, the layer thickness H, and the lattice constant a are the free parameters which we optimized to produce an optimal bandgap. Figure <ref>(c) shows the band diagram simulated by the finite element method, where a LN photonic crystal nanobeam with dimensions of W=750 nm, H=250 nm, and a lattice constant of a=600 nm exhibits a bandgap of 28 THz covering optical frequency from 203 to 231 THz, for the transverse-electric-like (TE-like) polarization with the electric field dominantly lying in the device plane.To produce a defect cavity, we gradually decreased the lattice constant from 600 nm to 540 nm around the center of the nanobeam. We optimized the nanobeam with a pattern of lattice constants as shown in Fig. <ref>(d), which results in a localized defect cavity at the center of the nanobeam whose fundamental cavity mode exhibits a resonance frequency close to the center of the photonic bandgap, as indicated by the blue dot in Fig. <ref>(c). Figure <ref>(e) and (f) show the optical mode field profiles of the fundamental (TE0) and second-order (TE1) TE-like cavity modes, simulated by the finite element method. The simulations show that the two modes exhibit radiation-limited optical Qs of 6.0× 10^6 and 5.2× 10^5, respectively, with effective mode volumes as small as 1.03(λ/n)^3 and 1.80(λ/n)^3 (where λ is the optical resonance wavelength and n is the refractive index). Our devices were fabricated on a 300-nm-thick x-cut congruent single-crystalline LN thin film sitting on a 2-μ m-thick buried oxide layer. The structure was patterned with ZEP-520A positive resist as a mask via electron beam lithography (Fig. <ref>(a)) and was etched with the Ar-ion milling process <cit.>. We developed an over-etching process to produce desired fine structures and sidewall smoothness, as schematically shown in Fig. <ref>(b)-(d). During the beginning stage of etching, the Ar-ion milling process produces slant angles on the device sidewall, leading to a trapezoid-shaped cross section (Fig. <ref>(b)). Further Ar-ion milling etched the ZEP-520A mask away and thinned the thickness of the LN layer down to ∼250 nm, eventually forming a triangularly-shaped cross section (Fig. <ref>(c)). Finally, the buried oxide layer was undercut by diluted hydrofluoric acid to form a suspended photonic crystal nanobeam (Fig. <ref>(d)). § LINEAR OPTICAL PROPERTIESFigure <ref>(a) and (b) show a fabricated device, which clearly show smooth and well defined fine features of the device structure. To characterize the optical property of the device, we launched a continuous-wave tunable laser into the device via evanescent coupling with a tapered optical fiber. Figure <ref>(c) shows the schematic of the experimental testing setup, where the optical wave transmitted out from the device is detected by a high-speed detector with a 3-dB bandwidth of 1.3 GHz whose output is characterized by an oscilloscope or an electrical spectrum analyzer, depending on the measured contents. The laser wavelength is calibrated by a Mach-Zehnder interferometer.By scanning the laser wavelength over a broad telecom band and monitoring the power transmission from the device, we obtained the transmission spectrum of the device shown in Fig. <ref>(a). Figure <ref>(a) shows that the device exhibits two high-Q optical resonances at 1452 and 1511 nm, respectively, which correspond to the fundamental and second-order cavity modes (Fig. <ref>(e) and (f)). Detailed characterization of these two modes (Fig. <ref>(b) and (c)) shows that the TE0 and TE1 modes exhibit optical Q as high as 1.09 × 10^5 and 1.08 × 10^5, respectively. These values are more than two orders of magnitude higher than other LN photonic crystal nanocavities that have ever been reported to date <cit.>. As discussed in the previous section, the TE0 mode has a radiation-limited optical Q about one order of magnitude higher than the TE1 mode. Therefore, the similarity of optical Qs for these two modes in our devices infers that the optical quality of the devices are still limited by the scattering loss from the sidewall roughness, which can be improved by further optimization of device fabrication.We are able to precisely control the device dimensions to tune the cavity resonance, as shown in Fig. <ref>(d). On one hand, the cavity resonance depends nearly linearly on the lattice constant. By tuning the lattice constants by an amount between -20 nm and 20 nm in a step of 5 nm from the nominal values shown in Fig. <ref>(d), we are able to shift the cavity resonance wavelength in a linear fashion from 1480 nm to 1560 nm, by a step of about 10 nm (Fig. <ref>(d), black dots). On the other hand, the cavity resonance is sensitive to the width and the thickness of the photonic crystal nanobeam. As shown in Fig. <ref>(d), a similar broadband tuning range of cavity resonance can be obtained by varying simultaneously the width and the thickness of the photonic crystal nanobeam while keeping the ratio of W/H constant. § PHOTOREFRACTION AND ITS SATURATION AND QUENCHING The high quality of the LN photonic crystal nanobeams enables us to observe intriguing nonlinear optical phenomena. Figure <ref> shows an example. We scanned the laser wavelength across a cavity resonance back and forth in a periodic triangular fashion, and monitored the transmission of the device. When the input optical power increases from 330 nW to 8 μW, the transmission spectrum changes from a Lorentzian shape to a bistability-type shape while the overall resonance wavelength shifts towards blue by about 55 pm (Fig. <ref>(a), Region I). The bistability-type behavior is simply due to the thermo-optic nonlinearity that responds fairly rapidly to photothermal heating <cit.>, which does not affect the overall position of the cavity resonance. The overall blue shift is a typical feature of the photorefractive effect that originates from the electro-optic effect introduced by the space-charge electric field produced via photovoltaic drift current <cit.>. The slow relaxation of space charge distribution leads to a net decrease of refractive index which results in an overall blue shift of the cavity resonance <cit.>.As the linewidth of the loaded cavity resonance is about 15 pm with a coupling depth of 30 % while the laser continuously scans over a tuning range of 280 pm, we estimate the average optical power coupled into the cavity is ∼133 nW, which corresponds to an averaged energy of ∼11.5 aJ and an averaged photon number of only ∼87 inside the cavity. This results in a blue tuning rate of ∼0.64 GHz/aJ, corresponding to ∼84 MHz/photon or ∼55 GHz/μ W, which is 3 orders of magnitude larger than those observed in millimeter-size LN resonators <cit.>, clearly showing the dramatically enhanced nonlinear optical effect in LN photonic crystal nanobeam. Such an energy-efficient resonance tuning is of great potential for applications such as all-optical wavelength routing and photonic circuit reconfiguration that are essential for photonic interconnect and optical data communication.When the input power increases further from 8 μW to 41 μW (Fig. <ref>(a), Region II), although the thermo-optic bistability becomes more profound, as expected, the left edge of the cavity resonance stays at a same wavelength location, as indicated by the red dashed line in Fig. <ref>(a). This infers that the overall cavity resonance wavelength remains unchanged, implying that the photorefraction saturates completely with increased power, in contrast to the photorefraction phenomena observed in other devices<cit.>. The underlying mechanism is likely due to the saturation of the generation of space charges responsible for photorefraction, since the extremely tiny physical size of the LN photonic crystal nanocavity leads to a limited number of donors/acceptors that can be excited by optical absorption to produce space charge carriers. Of particular surprise is that, when we maintained the periodic laser scanning of the cavity mode at an input power of 41 μW, the cavity resonance wavelength moves gradually by itself back to its original value of the passive cavity in the absence of optical power, as indicated by the arrows in Fig. <ref>. After this stage, the overall resonance remains unchanged at its passive value no matter how much optical power is launched into the device, as indicated by the blue dashed line in Fig. <ref>(b) showing the left edge of the cavity resonance. This indicates that the photorefraction is completely quenched by the optical wave launched into the device, which has never been observed before. At this state, no matter if we decreased or increased optical power, the phenomena remain same as Fig. <ref>(b), with the overall resonance wavelength nearly intact, except that the extent of thermo-optic bistability varies with optical power. Interestingly, the whole process is reversible. For example, after the photorefraction is quenched, if the device stays at rest for a few hours in the absence of optical wave, it will recover to its original state and all the phenomena shown in Fig. <ref>, such as resonance blue shifting, saturation and quenching of photorefraction, re-appear. The physical nature underlying the observed quenching phenomena is not clear at this moment, which requires further exploration. The quenching of photorefraction would be of great importance for nonlinear optical applications of LN nanophotonic devices, since photorefraction has been shown to be potentially detrimental to nonlinear optical processes <cit.>.§ NANO-OPTOMECHANICAL PROPERTIES The high quality of the LN photonic crystal nanobeams together with tight optical mode confinement results in strong coupling between the optical field inside the cavity and the mechanical motion of the device structure <cit.>, which would enable us to probe the optomechanical properties of the device. To do so, we locked the laser wavelength half way into the cavity resonance at the blue detuned side, and monitored the power spectrum of the cavity transmission. The device was tested in the atmospheric environment at room temperature.Figure <ref>(a) and (b) show recorded power spectra of a device, which shows rich mechanical mode families extending over a broad frequency range. As shown in Fig. <ref>(a), the device exhibits a mechanical mode with a frequency at Ω_m/2π = 1.003 GHz. Detailed characterization (Fig. <ref>(c)) shows that this mode exhibits an intrinsic mechanical Q of 1465, corresponding to a f · Q product of 1.47 × 10^12 Hz, which is comparable to state-of-the-art LN micromechanical resonators <cit.>. We believe that the mechanical damping is dominated by clamping loss, as the device has not been engineered to isolate the mechanical mode from environment. Numerical simulations show that this mechanical mode corresponds to a breathing mode (Fig. <ref>(a), inset) with an effective motion mass of m_ eff = 0.81 picograms and a theoretical frequency of 1.099 GHz. Detailed comparison of the experimental spectrum with theory shows that this mode exhibits an optomechanical coupling coefficient of |g_ OM|/2π = 22 GH/nm, which corresponds to a single-photon/single-phonon optomechanical coupling rate of |g_o|/2π = |g_ OM|/2π√(ħ/2 m_ effΩ_m) = 71 kHz. This value is comparable to those observed in most other optomechanical crystals <cit.>, although our devices are not specifically designed for optomechanical applications. It is lower than those in optimized optomechanical crystals reported in <cit.> that were optimized to enhance the photoelastic contribution. As LN exhibits outstanding acousto-optic property <cit.>, we expect that future optimization of device design would be able to significantly improve the optomechanical properties of the LN photonic crystal nanobeams.On the other hand, detailed characterization of low-frequency modes (Fig. <ref>(b)) shows that a majority of them exhibit low mechanical qualities in the order of ∼100, which is primarily due to air damping since low-frequency mechanical modes exhibit large amplitudes of thermal mechanical motion, sensitive to air damping. Two examples are given in Fig. <ref>(d) and (e), where the modes at 1.71 MHz and 4.68 MHz exhibits mechanical Qs ∼80. Numerical simulations show that these two modes correspond to the first-order and second-order flexural modes (Fig. <ref>(b), inset I and II), respectively, with effective motional masses of 7.2 and 7.9 picograms. Comparison of the experimental spectra with theory shows that these two modes exhibit |g_ OM|/2π= 0.35 and 0.45 GHz/nm, respectively, corresponding to |g_o|/2π= 9.1 and 6.8 kHz. The small values of optomechanical coupling are primarily due to the nature of the mechanical modes (Fig. <ref>(b), inset I and II) which do not couple well with the optical cavity mode localized at the beam center. Figure <ref>(f) shows that a mechanical mode at 11.18 MHz shows a high mechanical Q of 6142, which is likely to be a high-order flexural mode (Fig. <ref>(b), inset III) that is not as sensitive to air damping as other modes.§ CONCLUSION AND DISCUSSION In summary, we have demonstrated LN photonic crystal nanobeam resonators with optical Q up to 10^5 that is more than two orders of magnitude higher than other LN photonic crystal nanocavities reported to date <cit.>. The devices exhibit an effective mode volume as small as ∼ 1.03(λ/n)^3. The high optical Q together with tight optical mode confinement results in intriguing nonlinear optical phenomena. We have observed significant cavity resonance tuning induced by the photorefractive effect, with a tuning rate of ∼0.64 GHz/aJ, corresponding to ∼84 MHz/photon, three orders of magnitude greater than other LN resonators <cit.>. In particular, the devices exhibit strong saturation and quenching of photorefraction that has never been observed before. Photorefraction-induced optical damage is known to be detrimental to nonlinear optical processes in LN crystals <cit.>, which has become a major obstacle to LN nonlinear photonics. Conventional approaches to mitigate photorefraction is to dope LN crystal with certain ions to increase the photorefraction threshold <cit.>. The strong saturation and quenching of photorefraction observed in our devices might offer an elegant solution to this problem, making LN nanophotonic devices particularly promising for nonlinear photonic applications.On the other hand, the demonstrated devices exhibit strong coupling between the optical cavity mode and the mechanical motion of the device structures, with which we were able to characterize the rich nanomechanical motions of the device. We observed mechanical modes with frequency up to 1.003 GHz with a f · Q product of 1.47 × 10^12 Hz that is comparable to state-of-the-art LN micromechanical devices <cit.>. The devices exhibit a single-photon/single-phonon optomechanical coupling rate of |g_o|/2π = 71 kHz that is comparable to most other optomechanical crystals <cit.>, although our devices are not specifically designed for optomechanical applications. LN exhibits strong piezoelectric effect, electro-optic effect, and electromechanical coupling, significantly larger than other materials such as aluminum nitride and gallium arsenide <cit.>. 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"authors": [
"Hanxiao Liang",
"Rui Luo",
"Yang He",
"Haowei Jiang",
"Qiang Lin"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170627151900",
"title": "High-quality lithium niobate photonic crystal nanocavities"
} |
e1e-mail: [email protected] Max-Planck-Gesellschaft Halbleiterlabor, Otto Hahn Ring 6, D-81739 München, Germany Institut für Hochenergiephysik der Österreichischen Akademie der Wissenschaften, Nikolsdorfer Gasse 18, A-1050 Wien, Austria Atominstitut, Technische Universität Wien, Stadionallee 2,A-1020 Wien, AustriaDEPFET detectors for direct detection of MeV Dark Matter particles A. Bähraddr1 H. Kluckaddr2,addr3 J. Ninkovicaddr1 J. Schiecke1,addr2,addr3 J. Treisaddr1 Received: date / Accepted: date ============================================================================================ The existence of dark matter is undisputed, while the nature of it is still unknown.Explaining dark matter with the existence of a new unobserved particle is among the most promising possible solutions. Recently dark matter candidates in theMeV mass region received more and more interest. In comparison to the mass region between a few GeV to several TeV, this region is experimentally largelyunexplored. We discuss the application of a RNDR DEPFET semiconductor detectorfor direct searches for dark matter in the MeV mass region. We present the working principle of the RNDR DEPFET devices and review the performance obtainedby previously performed prototype measurements. The future potential of the technology as dark matter detector is discussed and the sensitivity for MeV dark matter detectionwithRNDR DEPFET sensors is presented.Under the assumption of six background events in the region of interest and anexposure of one kg·y a sensitivity of about σ_e = 10^-41 cm^2 fordark matter particles with a mass of 10 MeV can be reached.§ INTRODUCTIONSeveral independent measurements clearly point towards the existence of dark matter. The nature of dark matter is still not understood and is among the biggest outstanding problems of modernphysics <cit.>. A well motivated solution to this problem is the existence of anew particle candidate, which interacts at most weakly with standard model particles. The possible mass range of this particle candidate, as well as the possibleinteraction strength with ordinary matter, spans several orders of magnitude <cit.>. Recentlyseveral theoretical studies focus on possible dark matter candidates in the MeV mass region,below the mass scale of weakly interacting massiveparticles <cit.>. This mass region is experimentally less explored and opens a large space for undiscovered dark matter candidates. Direct detection experiments search for relic dark matter particles by looking forelastic scatterings between a dark matter candidate and a nucleus.The energy deposited in the scattering processes, the nuclear recoil-energy, can be measured by the experiment. By using simple kinematic relations the mass of the dark matter particle can be inferred from the recoil energy.The sensitivity towards low mass dark matter particles is determined by the detection thresholdfor nuclear recoils. Dark matter candidates with masses below 100MeV lead to a nuclear recoilas low as a few eV and are therefore below the threshold of direct dark matter detection experiments.The search for light dark matter particles via scattering with an electron opens the opportunity to extend the reach towards even smaller masses, down to a few MeV. However, the theoretical prediction of the dark matter-electron scattering rate is more complex, compared to the nuclear scattering. In this paper we study the possibility to measure dark matter-electron scattering using a silicon based detector. In solid state detectors electrons are bound to thenucleus and can no longer being considered as free particles. Electrons are not at rest and the typical speed is greater compared to the average speed of the dark matter particle, leadingto a different kinematics of the process. In addition the complicatedelectronic structure of the semiconductor makes the calculation of the scattering rate more complicated. This topic has been discussed in detail in the literature, e.g.in <cit.>, and is only summarised here. A semiconductor detector based on the DEPFET principle (DEpleted P-channel Field Effect Transistor) with repetitive non-destructive readout (RNDR) <cit.> offers the possibility to perform a low-noise measurementof the ionisation signal originating from a dark matter-electron inelastic scattering process, downto a single electron. The excellent noise performance for the ionisation signal is reached by repeating the measurement in a statistically independent way. With the average ionisation energy for a single electron of a few eV the detector performancecan be transformed to a sensitivity for dark matter masses down to a few MeV. We briefly review the detection of MeV dark matter with semiconductor targets in section <ref>, the RNDR DEPFET detector principle and the expected detector performance is discussed in section <ref> and in section <ref> we present the expected sensitivity for MeV direct dark matter detection. In section <ref> we summarise the potential of RNDR DEPFET detectors for direct dark matter detection. § DETECTION OF MEV DARK MATTER BY DARK MATTER-ELECTRON SCATTERING The process of dark matter-electron scattering is derived and discussed in references <cit.>. In this section we summarise the key findings of <cit.>, which are necessary to discuss the expected sensitivity for RNDR DEPFET dark matter detectors in section <ref>. Thereader is referred to <cit.> for a the complete derivation, in particular about the crystal form factor of silicon, which contains relevant information about the electron binding in the corresponding material.The measurement of the recoil energy distribution from the dark matter scattering process, togetherwith the expected velocity distribution of the dark matter gives an estimate of the mass of theincoming dark matter particle. For dark matter-nucleus scattering the mass can bederived by simple kinematic calculations and the deposited recoil energy is proportional to 1/m_N, with m_N being the mass of the target nucleus. A lighter target material therefore returns an increased average recoil energy, which is experimentally easier to measure.The scattering between a dark matter particle and an electron is more complicated andrequires a careful discussion. Compared to the dark matter-nucleus scattering no simple interpretation of the scattering rate in terms of cross-section and dark matter mass is possible. Two points are discussed in order to understand the relation between the dark matter scattering rate and the underlying dark matter parameters: the kinematic relation of the scattering process of MeV dark matter particlesand the relevant binding effects of electrons in silicon.In a solid state device made of silicon electrons are bound and cannot be considered to be at rest. The energy transferred to the electron E_e can be derived from a simple energy conservation relation E_e = - Δ E_χ- E_N <cit.>, with Δ E_χ being the energy loss of the dark matter particle and E_N being the recoil energy of the whole atom. Please note that the energy E_e is the total energy and only parts of the energy is finally transferred as the kinetic energy of the electron, while therest is needed to move the electron from the valence band to the conduction band. We consider small energy transfers only and thereforethe recoil energy of the atom, E_N, can be safely set to zero. The average velocity of the electron can be related to its binding energy,v_e∼ Z_eff α, with α≈ 1/137 being the fine-structure constant and the effective charge of the nucleus Z_eff being one for outer electrons. The velocity v_e is large compared tothe velocity of the incomingdark matter particle, v/c ∼ 10^-3.The average momentum transfer of the scattering process is therefore dominatedby the momentum of the bound electron. This information, together with the energyconservation relation, can be used to show that the typical available momentum transfer q in MeV dark matter scatterings is enough to move electrons from the valence band to the conduction band of silicon,with a band gap in the order of a few eV. Parts of the energy transferred from the dark matter particle to the electron E_eis needed to move the electron from the valence band to the conductance band. To predict the electron scattering rate the relevant electronbinding effects for silicon need to be calculated. The calculation of a dimensionless crystal form factor f_crystal(q,E_e) was performedfor the first time in <cit.> and can be considered as akey input to the prediction of the dark matter-electron scattering rate in silicon.The form factor calculation implies that the scattering processes with larger q-values are suppressed compared to processes withlow q, leading to a sensitivity increase towards low energy recoils.The differential recoil rate can be written as <cit.>:dR/d ln E_e=ρ_χ/m_χN_cell σ_e α m_e^2/μ^2_χ e ×∫ d ln q (E_e/qη( v_min(q,E_e) )) F_DM(q)^2|f_crystal(q,E_e)|^2,with ρ_χ being the local dark matter density, m_χ the mass of the dark matter particle, N_cell the number of unit cells in the target, σ_e parametrizing the strength of the interaction, m_ethe mass of the electron,μ_χ e the reduced mass of the dark matter-electron system andη( v_min(q,E_e)) parametrizing the dark matter density profile.The dark matter form factor F_DM(q) parametrises the momentum dependence of the interaction.For F_DM(q)=1 the interaction strength σ_e is reduced to a simple point like interaction. F_DM(q)=(αm_e/q) corresponds to an electric dipole moment and F_DM(q)=(αm_e/q)^2 corresponds to the exchange of a massless (or ultra-light) vector mediator. For our studies we choose the simplest momentum dependency and we set F_DM(q)=1, as expected for a point-like interaction. The energy deposited via thedark matter scattering process E_e is converted toan average number of produced electrons Q, by setting the average ionisation energy to E_ion=3.6 eV and the band-gap energy to E_gap=1.11 eV. The ionization Q is given by Q(E_e)= 1 + Int[(E_e-E_gap)/E_ion], <cit.>.The expected recoil rate as a function of deposited energy E_e is shown in figure <ref>. For a 10 MeV dark matter particle about 65% of all events generateat least two electrons in the detector.The rate is calculated by using the publicly available QEdarkcode <cit.> [http://ddldm.physics.sunysb.edu/ddlDM/].The expected sensitivity is presented after discussing the expected performance of the RNDR DEPFET device insection <ref> in terms of detected electrons. § RNDR DEPFET SENSORS FOR DIRECT DARK MATTER DETECTION§.§ Concept of RNDR DEPFET devicesThe basic idea behind repetitive non-destructive readout(RNDR) is to apply one of the most important implications of the central limit theorem on the field of detectors.Any charge generated in the sensitive detector bulk is collected in the internal gates.Due to the excellent charge carrier lifetime, charge loss can be virtually excluded. TheRMS noise of a single measurement is determined by the electronic noise of the transistor current measurement.By repetitively measuring the identicalsignal charge in a statistically independent way, the value resulting from the average of the individualmeasurements has a standard deviation of σ_eff=σ/√(n), with σ being the RMSnoise of a single measurement, and n being the number of readings. In this way, the standard deviation of the mean can be considered to be the effective noise of the measurement. Devices based on the combined detector-amplifier structure DEPFET are applied for a varietyof particle physics and astrophysical experiments <cit.>. In their mostsimple form, they provide an active pixel sensor with pixel-individual charge storage and readout at high speedwith very good signal-to-noise ratio (SNR). In addition, however, they providean ideal platform to realise the RNDR principle for radiation detectors. The simplest DEPFET cell <cit.> consists of a P-channel FET integrated on a silicon bulk, which is fully depleted by means of sidewards depletion(see figure <ref>). By an additional deep-n implant directly below the gate, a potential minimum for electrons is created, which all bulk-generatedelectrons will drift to. In case a transistor current is present, their presence modulates the conductivity of the transistor channel, and this modulation isdetected by appropriate subsequent electronics. Hereby, the potential minimum has the same effect on the channel as the external gate, and it is therefore alsoreferred to as internal gate. High-accuracy measurements rely on correlated double-sampling (CDS) to determine the amount of charge. After an initialmeasurement of the transistor state, the charge is removed from the internal gate by an attached n-channel MOSFET, the ClearFET, and the transistor state ismeasured again with empty internal gate. The actual amount of charge can be precisely determined by the difference. In this way, standard DEPFET cells in circulargeometry (see figure <ref>) have been operated with an equivalent noise charge (ENC) of 4-5 e^- RMS for a readout time of 4 μ s <cit.>. The fact, however, that the quantity of charge is sensed indirectly via the channel conductivity enables an efficient implementation of a DEPFET devicecapable of RNDR. In case the charge is not cleared away during a CDS cycle, but transferred to an adjacent storage node, where the charge is still preserved and where it has also no influence on the DEPFET channel conductivity, mimics a clear process and a nondestructive CDS cycle can be implemented. Transferring the chargeback to the DEPFETs internal gate again after the CDS cycle has been finished starts a new CDS cycle for the identical signal charge. In case of the DEPFET, thesecond storage node can even be the internal gate of a second DEPFET adjacent to the first one, and the transfer can be conducted by means of an additional so-calledtransfer gate interposed between the two DEPFETs. The second DEPFET can also be used to conduct a CDS measurement, where the clear is replaced by thetransfer back to the original DEPFET. This process can be repeated arbitrary times,until the charge is removed by the ClearFET after the final acquisition.In this way, one device pixel can be considered to be a superpixelbeing composed of two DEPFET subpixels, whose internal gates are connected bythe transfer gate. An example for a circuit representation and the respective layout is shown in figure <ref>.§.§ Performance model and prototype results In practice, however, the RNDR process is disturbed by the advent of additional signal- and leakage electrons from thebulk during the signal evaluation process.This leads to a deviation from the ideal behaviour, which has been described by Bähr's equation <cit.>: σ_eff^2 = σ^2/n+Δσ^2·(1/2+1/3· n - 5/6·1/n)where Δσ is the expected increase in noise during one CDS acquisition in the RNDR cycle. For givenΔσ and σ, an optimum number of transfer cycles can be derived:n^opt=√(3·σ^2/Δσ^2-5/2)resulting in an optimum achievable effective noise of:σ^opt_eff = √(σ^2/n^opt +Δσ^2·(1/2+1/3· n^opt - 5/6·1/n^opt))An example for the dependence can be seen in figure <ref>.The second summand under the square-root in Eq. <ref> describes the deviation from the expected 1/√(n) behaviour due tothe influence of the increase in noise Δσoriginating from the leakage current. Inside asilicon detector, this contribution can be efficiently suppressed, but not eliminated, by cooling. For low temperatures, the performance curvewill approximate the ideal1/√(n) behaviour.A DEPFET based RNDR device optimised for the detection of the extremely weak signals (i.e. σ < 2-3e^- ENC)can be operated with an optimum number of readout cycles, which allows to lowerσ^opt_eff down to a level, where the minimum detectable signal (i.e. one electron) can be only generated by noisefluctuations with 5 sigma probability or lower.The application of cumulative measurement techniques (i.e. using the non-destructive readoutwithout clearing of the pixel charge) helps to reduce this source of background(i.e. seeming single electron signals due to noise fluctuations) even further. Here, suppression of the Δσ-contribution in the perturbation term of Bähr's equation to 10^-4, and even lower, helps to achieve a σ^opt_eff of 0.2 e^- and below.This is achieved by adopting either the electronic shutter option or the Infinipix topology. Nevertheless, even in case the detector is operated with an effective threshold of one electron, volume leakage current collected during the sensors integration time is a source ofirreducible background. To maintain the sensitivity for the WIMP interaction signature as lowas 2-3 e^-, the aim must be to lower the probability of two leakagecurrent electrons within one pixel and frame to as low a level as possible. This can be achieved by operating the device at lowest possible temperatures to decrease the absolute magnitude of the leakage current, or by increasing the readout rate to limit the integration time, or by a combination of both methods. Again, cumulative measurements can help to preserve the statistical significance by preventing performance deterioration due to recombination noise. Standard mode RNDR DEPFET in circular geometry furnished with compact subpixels sharing the clear contact (see figure <ref>) have been operated in single-pixel and small matrix environments for proof-of-principle measurements and verification of the performance model. Results have beenreported in <cit.> and <cit.>,some results are shown in figure <ref>. The predictions of Bähr's equation are nicely confirmed by both measurementsand Monte Carlo simulations modelling the extended weightingfunction for the RNDR cycle. The value forσ_eff^opt of 0.18 e^- RMS corresponds to the prediction for a device with a value for of 3 e^- at- 50 ^∘C and 256 transfer cycles. The single electron resolving capability was verified for amounts of charge of up to 10^3 e^-, the peaks are nicely separated. The high resistivity float zone silicon used for the fabrication of the sensors has a charge carrier lifetime at room temperature at the order of one second. This has to be seen in relation to the drift time in the depleting field, which, depending on the bias voltage, is at the order of 10 - 20 ns. Coolingof the sensor increases the charge carrier lifetime to levels of minutes, so that an efficiency of 100 % for bulk generated electrons can be assumed with an accuracy of 10^-9. Theprobability to create an electron-hole pair by the dark matter-electron scattering is fully described by Eq. <ref>.Highly doped regions on the front- and backside of the sensor, however, can be considered as dead material, reducing the effective mass of thedetector and therefore the exposure. The exposure quoted in the sensitivity studies described in section <ref> does not include the dead material. The amount of dead material at sensor front- and backside, which is expected to be in the range of a few percent,needs to be determined by simulation using the final sensor layout. §.§ Planned improvements for future devices In addition to the leakage current, a more serious perturbation of the RNDR process arises from the DEPFET's permanent sensitivity. In case signal charge arrivesduring the RNDR cycle, the signal charge is altered and the resulting mean value of the n measurements does not represent the original signal charge. This is mainly aproblem for applications were the incoming radiation is not synchronized with the readout cycle and for the background events for applications where it is. Althoughrunning average techniques can be applied during the RNDR process to detect the occurrence of these so-called misfit events, it is better to reduce their overallinfluence or even to completely avoid it. In this respect, two different approaches have been pursued to optimize RNDR-based detectors for future applications:* A substantial reduction of the initial noise figure σ for a single reading decreases not only σ_eff^opt (see Eq. <ref>), but also n^opt and,accordingly, the required time for the RNDR cycle. Constant signal rate provided, this in proportion reduces the probability for misfit events.* In addition, the introduction of a global electronic shutter to the pixel array decouples the DEFET superpixels from the detector bulk.Charge generated in the silicon bulk while the shutter is active will be extracted from the detector volume without being detected.Although this approach introduces some degree of dead-time, it provides a reduction of misfit background by at least two ordersof magnitude in addition to the improvements achieved by reducing the noise.Both options have been evaluated with respect to feasibility.Concerning the first option, an optimization of the DEPFET response by adapting geometry and standard process technology parameters is expected to lower the initial noise figure down to values of 2-1.5 e^- ENC, depending on the shaping time. This lowers both n^opt by an order of magnitude and, accordingly, σ^opt_eff to levels far below the single electron threshold <cit.>. More advanced modifications of the process technology, which are currently under investigation,have the potential to improve the performance even further. The implementation of an electronic shutter has been evaluated via simulations and on a prototype level, and its functionality has been verified.The introduction of additional blind and blind-gatecontacts surrounding the pixel structure allow to extract electrons on demand, providing a charge suppression factor of 10^-3 and higher for the superpixels, while maintaining fullretention of charge already stored in the internal gates. The shutter speed is below 100 ns. Figure <ref> shows layout and circuit representationof a typical RNDR pixel with shutter functionality. One of the biggest drawbacks of DEPFET based devices is the pixel size. Current RNDR DEPFET prototype devices exhibit pixel sizes atthe order of 75 × 75 μ m^2. This relatively large pixel size is partiallycounterbalanced by the full depletion in combination with the relatively large device thickness of 450 μ m. Nevertheless, this large pixel sizelimits the capability for background suppression on the base of cluster analysisespecially for events in a very shallow depth beneath the pixel structure.For this reason, compact devices have been designed, which providefor a pixel size of 36 × 36 μ m^2. This very compact design (seefigure <ref>) has been realized by combining clear and shutter contacts. The design implements global clear and shutter functionality and allows for incrementalas well as absolute chargemeasurements. In combination with the large bulk thickness, cluster analysis is possible to some extent. For dark matter detection, arrays of 1k × 1k of these pixels are proposed covering an area of ≈3.7 ×3.7 cm^2, on a fully depleted detectorbulk of 1 mm thickness. Detector mass is at the order of 3.2 g. Initial noise is expected to be 1.5 e^- ENC, target noise is < 0.2 e^- ENC.§.§ Planned prototype measurementsThe base for the development of such devices will be the data gathered from the upcoming prototype measurements. Here, RNDR DEPFET devices with standard topology with and without global shutter functionality as shown infigure <ref> and <ref> respectively will for the first time be operated on a larger matrix scale in a low background environment. The devices consist of an array of 64 × 64 pixels integrated on a 0.45 mm thick silicon bulk. Goal of the measurement is the complete parametrization of the devices in terms of operational parameters, operating temperature for lowest leakage current and optimizedreadout for optimum noise performance. The readout setup is optimized for background shielding and low noise rather than high speed readout, as theframe rate is at the order of mHz or even lower.§ DARK MATTER SENSITIVITY STUDIES The low-noise measurement performance of the ionisation signal from an inelastic dark matter-electron scattering measured with the RNDR DEPFET sensor described in section <ref> can be translated into a sensitivity for the detection of MeV dark matter. Like in section <ref>we use the publicly available QEdark code <cit.> for the estimate.We analyse the impact of three key-parameters on the experimental sensitivity to low mass dark matter.Besides the threshold for the ionisation measurement, we study the exposure and the impact ofbackground events on the sensitivity. While the DEPFET devices describedin section <ref> have a mass of 3.1 g only, we will discuss our results with a default exposure of one kg·y and presents results with 0.1 kg · y as an alternative scenario. We investigate two main background sources, which could influence the sensitivity: background events caused by the energy depositions from radioactive decays from inside or outside of the experiment and background events generatedby the leakage current present during the operation of the silicon sensor. The two background sources have a different impact on the operation of the DEPFET device.§.§ Background events from the leakage currentFor the operation of the RNDR DEPFET detector a bias voltage is applied to the sensor. A very small leakage current is generated in the sensor, which can lead to the collection of electrons in theinternal gate. These electrons from the leakage current generate background events. The size of the leakage current, and therefore the number of background events, can be reduced by operating the device at lowertemperatures.Even the smallest known leakage current in silicon devices generate a significantbackground event rate for single pixel hits. The total number of background eventsfrom single electron events originating from the leakage current grows proportional to the total exposure time. Any increase of the readout rate of the device will not change the picture.The situation changes for background events with two electrons collected ina single pixel, assuming theprobability to generate a single electron from the leakage current is uncorrelated. The probability to collect two electrons from the leakage currentin the same pixel is significantly reduced and, in addition, the increase of thereadout rate with a regular clear of the internal gates will further reducethe probability to collect two electrons from leakage current in the same pixel.Alternatively, RNDR DEPFET devices allow for cumulative measurements, as the charge withinone superpixel does not necessarily have to be cleared, but may remain within the superpixelfor later comparative measurements. This can help to detect the presence of leakage current electronswithin a pixelduring a“reference” acquisition, whose presence may be confirmed or disproved duringsubsequent reference acquisitions and can later be subtracted from the “final” acquisition data, thuscombining the benefits of a fast readout rate without the drawback of increased noise hit rate.This feature, however, is mainly interesting in the case of relatively high initial noise values. For this sensitivitystudies we assume a default threshold of Q=2 e^-; in addition we also study the expected sensitivity for a threshold of Q=1 e^- and Q=3 e^-. The impact of background events from the leakage current is crucial and issubject to detailed device studies planned for the future.§.§ Simulation of background contributions from intrinsic radioactivity In <cit.> a limit is derived for a background free experiment, while for thisstudy we will discuss in addition the influence of background on the sensitivity of the experiment. Background from radioactive decays can be subdivided in two different categories, intrinsic background and background from external sources. We assume the shielding from external background sources to be very efficient so that remaining external backgrounds create surface events which can be rejected to a large extend, similar to the procedureused for detecting dark matter with semiconductor devices <cit.>. An irreducible background from internal radioactive decays is expected. The sensitivedetector elements consist mainly of silicon and previous studies indicate, that the decay of ^32Si is expected to be the leading contribution to the internal background <cit.>. Cosmogenic activation of Si can produce the unstable isotope ^32Si, which decays via β^--decay with a half-life of t_1/2=153 y and an energy release of 227.2 keV. The decay leads to an energy deposition in the sensor and generates background events in the region of interest. The decay product ^32P is also unstable and decays with a half-life of t_1/2=14.268 d and an energy release of 1.711 MeV to the stable isotope ^32S. A further cosmogenic background is ^3H produced via muon spallation and inelastic scattering of neutrons on silicon <cit.>. It undergoes a β^--decay into the stable ^3Hewith an energy release of 18.592 keV.We simulate the energy deposition in silicon of the ^32Si and the subsequent ^32P decay with the GEANT4 simulation package in version 10.2p1 using mostly the default processes described by the “Low Energy Electromagnetic Physics Working Group” <cit.>. Only the size of the sampling bins of the β^- spectra are decreased by a factor 100 relative to the default settings to increase the precision atlowest energies. We model a silicon only device with the geometry similar to the device to be used for initial dark matter searches. We set the activity of ^32Si in the sensor to 80 kg^-1d^-1 <cit.>. The decaying ^32Si isotopes are randomly distributed in the sensor. We explicitly note, since ^32Si is generated via cosmogenic activation, that the activity strongly depends on the time the silicon device is exposed to cosmic rays and the activity might vary for other devices.To our knowledge no measurement of the cosmogenic ^3H production rate R_3H in Si exists. Therefore, we rely on the simulation study <cit.> which found a strong dependence of R_3H on the used simulation code, resulting in values ranging from27.29 to 108.74 kg^-1d^-1. In a conservative approach we use the upper limit and set R_3H=108.74 kg^-1d^-1. The cosmogenic induced activity A_3H is then given by <cit.>A_3H=R_3H·( 1-e^-ln 2 · t_exp/t_1/2) · e^-ln 2 · t_cool/t_1/2,where t_1/2=12.32 y is the half-life of ^3H, t_exp is the period of exposure to cosmic rays, and t_cool is the cooling time, i.e. the duration at an underground location. Assuming a duration of t_exp=2 y between growth of the Si crystal and movement of the assembled detector to an underground laboratory, and afterwards an immediate start of operation, i.e.t_cool=0 y, results in A_3H=11.57 kg^-1d^-1. The simulated spectrum of the deposited energy in silicon from ^32Si, ^32P, and ^3H decays is shown in figure <ref>. The simulation returns a flat spectrum with an activity of ≤ 2.26 kg^-1 d^-1 keV^-1,corresponding to0.825 kg^-1 y^-1 eV^-1, for energy depositions below 1 keV. The RNDR DEPFET detector is able to detect single electrons with a resolution of 0.2 e^-.To estimate the total background rate we follow the conversion from the total deposited energyto ionization as used in <cit.> and summarised in Eq. <ref>.We define as the signal region the energy range between the band gap energy of silicon (1.1 eV)and the minimum energy needed to generate three electrons (8.3 eV), corresponding to the first two bins in figure. <ref>. By defining the first Q-bin as part of the signal regionwe follow a conservative approach and allow upward fluctuations of Q=1 to Q=2 hits, generated by the leakage current. With the given background activity of 0.825 kg^-1 y^-1 eV^-1 for energy depositions below 1 keV, as reported above, weexpect a background rate of 5.94 kg^-1 y^-1 in the region of Q=1 to Q=2.For thesensitivity studies we use a background rate of 6 kg^-1 y^-1.§.§ Expected sensitivity for detecting MeV dark matter with DETFET-RNDR detectors We use the number of predicted background events from ^32Si, ^32P, and ^3H decays together with the code QEdark code to calculate the expected sensitivity of the experiment <cit.>.We consider a constant form factor of F_DM(q)=1 only. We determine the expected sensitivity assuming six background events, an exposure of one kg·y and a threshold of Q=2 e^-. We use the statistical approach described in <cit.> to determine the expected sensitivity. We assign nouncertainty to the number of expected background events and we take the number of observed events to be equal to the number of background events. The upperlimit for the number of signal events for six background events is 6.75 events (95 % C.L.). The expected sensitivity for different assumptions is shown in figure <ref> and figure <ref>. Please note, that we assume in all cases no background events fromleakage current events. With the default assumption of an energy threshold of two electrons, six background events and an exposure of one kg·ywe can reach a sensitivity of about σ_e = 10^-41 cm^2 fordark matter particles with a mass of 10 MeV. Assuming six backgroundevents the maximal sensitivity can be reached with an exposure of about one kg·y, an exposure of three years improves the sensitivity only marginally. Increasing the threshold from two electrons to three electrons reduces the sensitivity in theMeV mass region by almost one order of magnitude. A reduction of the exposure to 0.1 kg · y will lead to a similar loss in sensitivity.§ CONCLUSION The quest for particle dark matter is among the most urgent open topics of modern physics. Themass range for dark matter candidates, as well as the interaction rate with ordinary matter, is unpredicted.The parameter space for light dark matter in the MeV mass rangestill has some experimentally unexplored regions. We discuss the possibility touse a silicon detector operated as a RNDR DEPFET device to detect single electronsbeing produced by possible dark matter-electron scatterings. Measurements using a RNDR DEPFET prototype return an effective noise of 0.18 e^- RMS, allowing to resolve single electrons.Assuming six background events in the signal region, a threshold of two electrons and an exposure of one kg·y, we determinethe expected sensitivity to be about σ_e = 10^-41 cm^2 for dark matter particles with a mass of 10 MeV.Bertone:2004pz G. Bertone, D. Hooper and J. Silk, Phys. 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"authors": [
"Alexander Baehr",
"Holger Kluck",
"Jelena Ninkovic",
"Jochen Schieck",
"Johannes Treis"
],
"categories": [
"physics.ins-det",
"astro-ph.CO",
"astro-ph.IM"
],
"primary_category": "physics.ins-det",
"published": "20170627041520",
"title": "DEPFET detectors for direct detection of MeV Dark Matter particles"
} |
UTP,QCMD,UMD,NIST]L. [email protected] QCMD]C. de la Cruz UTMT]M. R. Koehler MSTD]M. A. McGuire UTMT]V. Keppens UTP,UTMT,MSTD]D. Mandrus QCMD,UTP]A. D. Christianson [cor1]Corresponding author[UTP]Department of Physics & Astronomy, University of Tennessee, Knoxville, TN-37996, USA [QCMD]Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, TN-37831, USA [UMD]Department of Materials Science & Engineering, University of Maryland, College Park, MD 20742 [NIST]NIST Center of Neutron Research, Gaithersburg, MD-20899 [UTMT]Department of Material Science & Engineering, University of Tennessee, Knoxville, TN-37996, USA [MSTD]Materials Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN-37831, USAStructural properties ofhave been investigated using neutron and x-ray diffraction, and resonant ultrasound spectroscopy (RUS) measurements. Diffraction measurements indicate a continuous structural transition from orthorhombic (Pnma) to monoclinic (P2_1/c) structure. RUS measurements show softening of natural frequencies at the structural transition, consistent with the elastic nature of the structural ground state. The structural transition temperatures indecrease with Ag composition until the monoclinic phase is completely suppressed at x_c = 0.225. All of the evidence is consistent with the presence of an elastic quantum critical point in .Elastic quantum critical pointQuantum phase transition § INTRODUCTION Quantum criticality continues to be a key pillar of research in condensed matter physics. Prototype examples include magnetic quantum critical points (QCP) in which quantum fluctuations of spins melt an ordered state of matter <cit.>. However, in recent years, interest is beginning to shift toward new paradigms of quantum criticality <cit.>. For example, SrTiO_3 and KTaO_3 have been identified as being close to a ferroelectric QCP <cit.>, and the superconducting state in (Ca,Sr)_3Rh_4Sn_13 is associated with the nearby structural QCP <cit.>. Collectively these new examples of quantum criticality provide an opportunity for fresh perspectives and the discovery of new unifying insights. Arising out of this emerging interest, an elastic QCP has been theoretically proposed <cit.>. This type of critical phenomenon is expected to occur when the quantum zero point motion of atoms generates a residual strain in the lattice suppressing the structural ground state. In this sense, the elastic QCP is fundamentally different from the magnetic and other structural counterparts <cit.>. Furthermore, recent experimental work demonstrates that LaCu_6-xAu_x shows the promise of hosting an elastic QCP <cit.>. To provide a more comprehensive understanding of the tunability of the structural phase transition leading to an elastic QCP, here we present a study of the related seriesas a potential candidate of elastic QCP. In this paper, we present structural properties ofas a function of Ag composition and temperature. The monoclinic phase ofis gradually suppressed with Ag substitution. The structural transition is accompanied by a gradual softening of some natural frequencies, as is expected for a continuous elastic phase transition. Linear extrapolation of T_S with x shows that a complete suppression of the monoclinic structure occurs at the critical composition x_QCP = 0.225. All of the measurements are consistent with the presence of an elastic QCP in .§ EXPERIMENTAL DETAILS Polycrystalline samples of(x = 0, 0.075, 0.1, 0.125, 0.135, 0.15, 0.155, 0.175, 0.2, 0.225, 0.25, 0.3) were synthesized by arc melting the elements La, Cu and Ag in stoichiometric proportions. The phase purity of the sample was characterized by laboratory x-ray measurements at room temperature. Samples with compositions x = 0.075, 0.1, and 0.125 were also measured on a PANalytical X'Pert Pro MPD powder x-ray diffractometer using CuK_α,1 radiation (λ =1.5406). For the characterization of the structural transition, diffraction measurements were performed at room temperature and 20 K, and temperature dependence of selected Bragg peaks was obtained in 10 K steps. Resonant ultrasound spectroscopy (RUS) measurements were performed on the polycrystalline samples with compositions , x = 0.135, 0.155, and 0.175 using a set-up as described in the Ref. <cit.>. For the measurement, a rectangular parallelepiped sample was held between two transducers. Mechanical resonances within the range 10 - 1000 kHz were collected as a function of temperature.Neutron diffraction measurements were performed on the samples of (x = 0, 0.15, 0.2, 0.225, 0.25) with the HB-2A powder diffractometer at the High Flux Isotope Reactor (HFIR) of Oak Ridge National Laboratory (ORNL). Neutrons of wavelength 1.54were used for the measurement. Collimators containing parallel blades of cadmium coated steel were positioned before the monochromator, sample, and detector with divergence of 12^'-21^'-6^' respectively. The sample of mass ≈ 5 g was finely ground in a glove box and placed inside a vanadium can with helium as an exchange gas. The can containing the sample was loaded in a closed cycle refrigerator system. Diffraction patterns were collected at room temperature and at 4 K. For x = 0.15 and 0.2, diffraction patterns at several temperatures were obtained near T_S. For x = 0.225 and 0.25, diffraction patterns were obtained at room temperature and at 4 K only. The structural parameters were obtained using Rietveld refinement with the FullProf Suite software <cit.>.High resolution x-ray diffraction measurements of LaCu_5.7Ag_0.3 were performed using transmission geometry with 11-BM at the Advanced Photon Source at Argonne National Laboratory <cit.>. A monochromatic x-ray of wavelength λ = 0.413 was used for the measurement, which provides a Q-resolution of Δ Q/Q = 2× 10^-4 . The sample was finely ground inside a glove box filled with argon, which was then mixed with amorphous SiO_2 in the molar ratio of 1:3 to minimize x-ray absorption. The mixture was packed inside a Kapton tube of 0.8 mm diameter. The Kapton tube containing the sample was spun at 60 Hz to achieve an efficient powder averaging during the measurement.§ RESULTS AND DISCUSSION Analysis of the room temperature laboratory x-ray diffraction pattern shows that the samples are of high purity and are consistent with either the orthorhombic (space group: Pnma) (x ≥ 0.075) or monoclinic (space group: P2_1/c) structure (x = 0). The structural phase transition inwas characterized by different methods: neutron and x-ray diffraction, and RUS measurements. Using x-ray diffraction, the temperature dependence of selected Bragg peaks was measured. At T_S, some of the structural Bragg peaks pertaining to the orthorhombic structure split into two monoclinic peaks, which is shown in Fig. <ref>(a). The splitting occurs only for the Bijovet pairs (H K L) with H≠0 and K≠ 0, which, due to the monoclinic distortion, acquire different d-spacing at T_S. Neutron diffraction measurements were used to determine the temperature dependence of monoclinic angle β. Near T_S, β gradually increases with lowering temperature as expected for a second order phase transition, and consequently, there is a continuous evolution of shear strain (e_12∝cos(β)) in the monoclinic phase. Therefore, cos^2(β) is linearly extrapolated to zero for the estimation of T_S, as shown in Fig. <ref>(b). RUS measures the resonances that occur when the frequency of an ultrasonic wave matches with the natural frequency of the sample. The RUS measurements show that some of the natural frequencies gradually become soft as T_S is approached. As the square of a natural frequency (F^2) is directly proportional to the combination of elastic constants, we have used F^2 as an indirect probe of elastic behavior in . T_S is characterized by the change in the slope of F^2 versus temperature. An example is shown in <ref>(c), where the lines representing the slope of F^2 in the orthorhombic and monoclinic phases intersect at T_S. The change in the slope of F^2 versus temperature can be attributed to a complete softening of the C_66 = C_1212 elastic constant <cit.>, which is expected as the phase transition takes place with softening of corresponding acoustic phonon Γ-X and evolution of shear strain e_12 <cit.>. A phase diagram summarizing T_S obtained from diffraction and RUS measurements is presented in Fig. <ref>. T_S inlinearly decreases with Ag composition. A linear extrapolation of T_S with Ag composition shows that the monoclinic phase is completely suppressed near x_c = 0.225. For the compositions at and above x_c, no structural phase transition is observed above 4 K (marked as squares in the phase diagram).For a detailed understanding of the orthorhombic structure inand in particular, to investigate the distribution of Ag atoms in the orthorhombic unit cell, high resolution synchrotron x-ray measurements were performed for the composition LaCu_5.7Ag_0.3. As expected from the phase diagram, the diffraction pattern is consistent with the orthorhombic structure. In addition to the reflections from the main phase, small impurity peaks consistent with elemental copper appear in the diffraction pattern. The intensity of the impurity peaks corresponds to only 0.39% of copper by weight. No additional impurities were detected. The measured pattern with the fit of orthorhombic crystal structure is shown in Fig. <ref>(a). The orthorhombic unit cell consists of four formula units, in which La and Cu/Ag are distributed in one general and five special sites. The crystal structure after the Rietveld refinement is shown in Fig. <ref>(b). Details of the Rietveld refinements are presented in Table <ref>. The result presented here illustrates that the structural properties ofare microscopically similar to the related series LaCu_6-xAu_x, which has recently been identified as a host of an elastic QCP, and also to other members of the CeCu_6-xT_x family <cit.>. In particular, the Rietveld analysis of the x-ray diffraction measurements shows that the crystal structure ofis virtually identical to that of LaCu_6-xAu_x <cit.>. The substituent Ag inexclusively occupies the special copper position Cu2, and T_S decreases linearly with chemical substitution as in the case of LaCu_6-xAu_x. Furthermore, RUS measurements indicate that the elastic properties ofare similar to the related compound CeCu_6, indicating a similarity in structural properties. The structural resemblance ofwith the LaCu_6-xAu_x and CeCu_6-xT_x family indicates that the suppression of the monoclinic phase inresults in an elastic QCP. § CONCLUSIONIn conclusion, a structural phase diagram ofis constructed using neutron and x-ray diffraction and RUS measurements. High resolution synchrotron x-ray diffraction measurement demonstrates thatbears a structural resemblance to the related series LaCu_6-xAu_x and CeCu_6-xT_x. The phase transition inis driven by elastic instabilities, with the RUS measurement showing softening of natural frequencies at T_S. T_S incan be suppressed with Ag substitution, and the monoclinic phase is completely terminated at the critical composition x_QCP = 0.225. The evidence taken together suggests thatis a promising host of an elastic QCP. § ACKNOWLEDGEMENTWe acknowledge D. Singh for useful discussions and M. Suchomel for assistance with the synchrotron x-ray measurements. The research at the High Flux Isotope Reactor at Oak Ridge National Laboratory is supported by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy (DOE). MAM and DM acknowledge support from the U. S. DOE, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. Use of the Advanced Photon Source at Argonne National Laboratory was supported by the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). § REFERENCES10 url<#>1urlprefixURL href#1#2#2 #1#1sachdev2007quantum S. Sachdev, Quantum phase transitions, Wiley Online Library, 2007.Loh_REV H. v. Löhneysen, A. Rosch, M. Vojta, P. Wölfle, Fermi-liquid instabilities at magnetic quantum phase transitions, Rev. Mod. Phys. 79 (2007) 1015–1075.senthil2004deconfined T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, M. P. 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Grosche, Pressure- and composition-induced structural quantum phase transition in the cubic superconductor (Sr,Ca)_3Ir_4Sn_13, Phys. Rev. Lett. 109 (2012) 237008.scf3 S. U. Handunkanda, E. B. Curry, V. Voronov, A. H. Said, G. G. Guzmán-Verri, R. T. Brierley, P. B. Littlewood, J. N. Hancock, Large isotropic negative thermal expansion above a structural quantum phase transition, Phys. Rev. B 92 (2015) 134101.zacharias M. Zacharias, I. Paul, M. Garst, Quantum critical elasticity, Phys. Rev. Lett. 115 (2015) 025703.elastic L. Poudel, A. F. May, M. R. Koehler, M. A. McGuire, S. Mukhopadhyay, S. Calder, R. E. Baumbach, R. Mukherjee, D. Sapkota, C. de la Cruz, D. J. Singh, D. Mandrus, A. D. Christianson, Candidate elastic quantum critical point in LaCu_6-xAu_x, Phys. Rev. Lett. 117 (2016) 235701.migliori2005implementation A. Migliori, J. 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Komatsubara, Elastic soft mode and crystalline field effect of Kondo lattice substance: CeCu_6, J. Phys. Soc. Jpn. 54 (7) (1985) 2367–2370.poudel L. Poudel, C. de la Cruz, E. A. Payzant, A. F. May, M. Koehler, V. O. Garlea, A. E. Taylor, D. S. Parker, H. B. Cao, M. A. McGuire, W. Tian, M. Matsuda, H. Jeen, H. N. Lee, T. Hong, S. Calder, H. D. Zhou, M. D. Lumsden, V. Keppens, D. Mandrus, A. D. Christianson, Structural and magnetic phase transitions in CeCu_6-xT_x ( = Ag, Pd), Phys. Rev. B 92 (2015) 214421.grube1999suppression K. Grube, W. H. Fietz, U. Tutsch, O. Stockert, H. v. Löhneysen, Suppression of the structural phase transition in CeCu_6-xAu_x by pressure and Au doping, Phys. Rev. B 60 (1999) 11947–11953. | http://arxiv.org/abs/1706.08433v1 | {
"authors": [
"L. Poudel",
"C. de la Cruz",
"M. R. Koehler",
"M. A. McGuire",
"V. Keppens",
"D. Mandrus",
"A. D. Christianson"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170626151422",
"title": "LaCu$_{6-x}$Ag$_{x}$: A promising host of an elastic quantum critical point"
} |
Double crystallographic groups and their representations on the Bilbao Crystallographic Server Mois I. Aroyo December 30, 2023 ==============================================================================================In this paper, we present a novel massively parallel algorithm for accelerating the decision tree building procedure on GPUs (Graphics Processing Units), which is a crucial step in Gradient Boosted Decision Tree (GBDT) and random forests training. Previous GPU based tree building algorithms are based on parallel multi-scan or radix sort to find the exact tree split, and thus suffer from scalability and performance issues. We show that using a histogram based algorithm to approximately find the best split is more efficient and scalable on GPU.By identifying the difference between classical GPU-based image histogram construction and the feature histogram construction in decision tree training, we develop a fast feature histogram building kernel on GPU with carefully designed computational and memory access sequence to reduce atomic update conflict and maximize GPU utilization. Our algorithm can be used as a drop-in replacement for histogram construction in popular tree boosting systems to improve their scalability. As an example, to train GBDT on dataset, our method using a main-stream GPU is 7-8 times faster than histogram based algorithm on CPU in LightGBM and 25 times faster than the exact-split finding algorithm in XGBoost on a dual-socket 28-core Xeon server, while achieving similar prediction accuracy.§ INTRODUCTION Decision tree ensemble algorithms are increasingly adopted as a crucial solution to modern machine learning applications such as ranking and classification. The major computation cost of training decision tree ensemble comes from training a single decision tree, and the key challenge of decision tree building process is the high cost in finding the best split for each leaf, which requires scanning through all the training data in the current sub-tree.Since a tree ensemble algorithm typically has more than a hundred trees, while each tree has around 10 layers, the computation may require thousands of data passes. As a result, training tree ensemble algorithms is time consuming for datasets with millions of data points and thousands of features.Several parallel algorithms have been proposed to solve the scalability issue of building decision trees in multi-core or distributed settings. For example,XGBoost <cit.> has a high-quality multi-core implementation for gradient boosted decision tree (GBDT) training which partitions the work by features, and it has been extended to distributed settings like Hadoop or Spark. Recent works <cit.> proposed several different approaches for parallelizing decision tree building on distributed systems, andLightGBM[https://github.com/Microsoft/LightGBM], a popular tree boosting software package considers both data-partitioning and feature partitioning in their distributed version. As an important resource of parallel computing, GPU is much cheaper than building distributed systems and becomes a standard computing unit for big data analytics.However, the use of GPU is seldom exploited in decision tree building process and tree ensemble algorithms.To the best of our knowledge, among popular decision tree implementation packages, only XGBoost implements a GPU accelerated algorithm, but we find that it is slower than running on a 28-core CPU in our benchmarks because it implements the exact-split algorithm on GPU. Furthermore, due to GPU's strict memory constraint this method does not scale to large dataset. In this paper, we propose a novel GPU-based algorithm for building decision tree using a histogram-based method, and observe a significant speedup over all the existing multi-core CPU and GPU implementations for GBDT training.Unlike previous GPU approaches that parallelize sorting and scanning to find the exact split, our algorithm constructs histograms for all features on GPU, and then approximately finds the best split using these histograms. Our main contribution is in three folds: * We show that histogram based methods for decision tree construction on GPU is more efficient than existing approaches,which are based on multi-scan and radix sort to find the exact split. The exact-split based GPU tree builder in XGBoost actually cannot compete with a server with 28 cores. We design a very efficient algorithm for building feature histograms on GPU and integrate it into a popular GBDT learning system, LightGBM.* We show significant speedup on large-scale experiments. For dataset, XGBoost (with exact-split tree builder) takes over 4,100 seconds on a 28-core machine and we only need 165 seconds to achieve the same accuracy using a $500 GPU, or 300 seconds with a $230 GPU; comparing with histogram based method used in LightGBM, our GPU algorithm is also 7 to 8 times faster, while achieving a similar level of accuracy.* Compared with existing GPU acceleration implementations for decision tree building, our scalability is much better.The exact-split based GPU implementation in XGBoost fails due to insufficient memory on 4 out of 6 datasets we used, while our learning system can handle datasets over 25 times larger than on a single GPU, and can be trivially extended to multi-GPUs.§ RELATED WORKDecision tree has become one of the most successful nonlinear learning algorithms in many machine learning and data mining tasks. Many algorithms are proposed based on decision trees and tree ensemble methods, such as random forest <cit.>, gradient boosting decision trees (GBDT) <cit.>,and regularized greedy forest <cit.>. These algorithms have shown superb performance in regression, classification, and ranking <cit.> tasks. Among these tree ensemble methods, GBDT has gained lots of attention recently due to its superb performance and its flexibility of incorporating different loss functions, such as square loss, logistic loss and ranking loss.Although the idea of GBDT is simple, it is actually non-trivial to have an implementation that performs well in practice, leading to a need of developing efficient and easy-to-use software packages of GBDT. XGBoost <cit.> is the most widely used package for training GBDT, and has shown lots of success in many data mining challenges. In terms of implementation, XGBoost uses several tricks: it uses the sort-and-scan algorithm discussed in Section <ref> to find the exact best split on each leaf, designs regularization terms to prevent over-fitting, and optimizes the code to handle different types of data and improve cache locality. Recently, another GBDT package,LightGBMproposes to use histogram-building approach tospeed up the leaf split procedure when training decision trees. Although the split of leaves is approximate, it is much more efficient than the exact-split method[XGBoost recently also added a histogram based learner]. We will discuss histogram based tree splitting in detail in Section <ref>. As the size of data grows dramatically, there has been an increasing need for parallelizing decision tree training. The crucial part of decision tree training is to determine the best split of each leaf which turns out to be the main parallelizable component. One category of parallel decision tree algorithms is to partition the training data across machines, examples include PLANET <cit.> and Parallel Voting Decision Tree (PV-Tree) <cit.>. They select top-k features within each machine, and then communicate to select the best split based on the feature histogram. Another group of approaches partition data by feature, and YGGDRASIL<cit.> is a representative of this category.The main idea of YGGDRASIL is to divide features into different machines, compute a local optimal split, and then master will decide the best split among them. XGBoost and LightGBM also have implemented distributed decision tree training implementation for both cases: partition over features and partition over data samples. Theprevious works mainly focus on using CPU and multiple machines to parallelize decision tree training, however, as an important parallel computing resource, GPU is rarely exploited for this problem. Among these packages, only XGBoost utilizes GPU to accelerate decision tree training, but the speedup is not that significant, e.g., training on a top-tier Titan X GPU is only 20% faster than a 24-core CPU[http://dmlc.ml/2016/12/14/GPU-accelerated-xgboost.html].There are also some other early attempts on building decision trees using GPUs, for instances, CUDATree<cit.>. All these GPU implementations use a similar strategy to find the best split, which mimics the exact-split method on CPU. For the first a few tree levels, specialized multi-scan and multi-reduce operations are used to find splits among all leaves, and then it switches to radix sort and prefix-scan on re-partitioned data on each leaf when the tree goes deeper. It requires a lot of irregular memory access and its computation pattern does not fit into GPU's parallelization model well, so they can hardly compete with optimized multicore implementations on modern server CPUs. § PROBLEM FORMULATION AND BACKGROUND We first describe the standard procedure for training a decision tree, and then introduce a popular tree ensemble algorithm, Gradient Boosted Decision Tree (GBDT) that constructs a bunch of decision trees in a boosting fashion. After that, we will discuss the histogram based method to approximately construct decision trees, and how to efficiently use this method on GPU. §.§ Decision TreeIn this section we use regression tree to introduce the algorithm, while a similar approach can be used for classification trees.Given training data X={_i}_i=1^N and their target Y={y_i}_i=1^N, where _i∈^d and _i can be either continuous or categorical features. We use X∈^N× d to denote the data matrix, and _j to denote the j-th column of X, which contains all the data points' value for j-th feature. A decision tree learns a model f such that f(_i) ≈ y_i with respect to some user defined loss functions.With square loss, the objective function for training a decision tree can be written asmin_f∈∑_i=1^N (f(_i) - y_i)^2. For illustration and simplicity, we use square loss throughout the paper and omit the discussion of the regularization term. We refer the readers to <cit.> for more technical details on tree boosting. The standard procedure to build a regression tree starts from the root node (containing all the training samples), and grows the tree by keeping splitting existing leaf nodes in the tree until some stopping conditions are met. Let V_s denote the set of examples that pass through the leaf s and define a split as t = [feature id,threshold], consisting of the feature variable to split and at what threshold it has to be split. Based on the split, V_s is partitioned into two disjoint sets: a set V_r associated with the right node and a set V_l associated with the left node. For each split pair we can compute the prediction values (h_r and h_l) associated with the right and left nodes based on the loss function restricted to the corresponding sets of examples:min_h_l, h_r∑_i∈ V_r(h_r-y_i)^2 + ∑_i∈ V_l(h_l-y_i)^2,and the optimal assignment for h_l and h_r is h_l=(∑_i∈ V_ly_i)/N_l, h_r=(∑_i∈ V_ry_i)/N_r,where N_l and N_r are the number of data examples landed in left and right child respectively. After plugging (<ref>) into the loss function of (<ref>), the objective value for a given split t becomesL(t) = ∑_iy_i^2- (∑_i∈ V_ry_i)^2/N_r -(∑_i∈ V_ly_i)^2/N_l.To find the best split, we need to test on all possible split pairs including all the feature id and their feature values and choose the one that achieves the lowest objective value in (<ref>). To reduce redundant computation, we first sort the j-th feature values for all examples on this leaf. Then, we scan through the sorted feature values one by one to enumerate all possible split points, and at each step we move one examplefrom right child to the left. Assume knowing ∑_iy_i (constant), then by maintaining the prefix sum ∑_i∈ V_l y_i when moving examples from right to left,the new loss (<ref>) can be computed in constant time at each step. After going over all examples on this leaf we find the exact feature value that minimizes (<ref>).In summary, there are two major computation steps in this exact-split finding method: (1) sort feature values; (2) update prefix sum (∑_i∈ V_ry_i). All the previous attempts for GPU-based decision tree training rely on parallelizing these two steps. §.§ Gradient Boosted Decision Tree Gradient Boosted Decision Trees (GBDT)is a tree ensemble algorithm, thatbuilds one regression tree at a time by fitting the residual of the trees that preceded it. Mathematically, given a twice-differentiable loss function ℓ(y,X), GBDT minimizes the loss (for simplicity, we omit the regularization term here): L = ∑_i=1^N ℓ(y_i,F(_i)),with the function estimation F() represented in an additive form:F() = ∑_m=1^T f_m(),where each f_m() is a regression tree and T is the number of trees. GBDT learns these regression trees in an incremental way: at m-stage, fixing the previous m-1 trees when learning the m-th trees. More specifically, to construct the m-th tree, GDBT minimizes the following loss:L_m = ∑_i=1^N ℓ(y_i,F_m-1(_i)+f_m(_i)),where F_m-1() = ∑_k=1^m-1 f_k(). A popular way to solve the optimization problem in (<ref>) is by Taylor expansion of the loss functionL_m≈L̅_m= ∑_i=1^N [ ℓ(y_i,F_m-1(_i))+g_if_m(_i)+h_i/2f_m^2(_i) ], withg_i = ∂ℓ(y_i,F(_i))/∂ F(_i)|_F(_i)=F_m-1(_i) h_i = ∂^2 ℓ(y_i,F(_i))/∂^2 F(_i)|_F(_i)=F_m-1(_i)It is easy to see that minimizing L̅_m is equivalent to minimizing the following function: min_f∈∑_i=1^N h_i/2 ( f_m(_i) + g_i/h_i)^2. Interestingly, this optimization problem is equivalent to training a regression tree, as shown in (<ref>):Therefore, we can follow the similar procedure as discussed in <ref> to build a regression tree. For each partition, the new objective function can then be computed bysomething like ∑_i∈Left g_i)^2/∑_i∈Left h_i +∑_i∈Right g_i)^2/∑_i∈Right h_iwe need to compute ∑_i∈Left g_i and ∑_i∈Left h_i for each potential cut value. §.§ Approximate Split Finding Using Feature HistogramsAs discussed in <ref>, finding the exact best split for a feature requires going through all feature values and evaluating objective function values for each of them. For large datasets, it is unnecessary and repetitious to check every possible position to find the exact split location; instead, an approximately best split often works quite well.One way to find the approximate best split is to test only k split positions, and this can be done efficiently using feature histograms. We first convert continuous feature values into k discrete bins, and then construct a histogram with k bins for each feature. To find the split, we canevaluate (<ref>) only at these k points. Because building histograms is a rather straight-forward process, it is easy to implement efficiently on hardware. LightGBM and the “hist” tree builder in XGBoost use this approach to speed up decision tree training.As we have shown in (<ref>), the objective function value for each decision tree in GBDT can be evaluated as long as we have the values of g_i and h_i.Therefore, the histogram-based algorithm usually build two histograms for g_i and h_i, and then use them to find the best split.Algorithm <ref> shows how to build a feature histogram. Note that we assume the feature values have been converted into k integer bin values when data is loaded, thus _j ∈{1, ⋯, k}^N. It is clear that this algorithm is memory-bound, as building the histograms requires non-sequential scattering access to large arrays. We utilize the high bandwidth memory and large computation power on GPUs to accelerate this operation. § PROPOSED ALGORITHM A Brief Review of Programming Challenges on GPUs. Although a GPU can easily handle a million threads, there are many constraints on how these threads can interact and work efficiently. First, a GPU executes a bundle of threads (called a “warp” or a “wavefront”) in lock-step; in other words, threads within a bundle must be executing exactly the same sequence of instructions. Branching within a bundle will cause all threads in the bundle execute both directions of the branch, with certain threads masked off if the branching condition does not hold. Moreover, unlike threads on a CPU, GPU threads cannot synchronize and communicate with each other in an arbitrary manner. Only a small group of threads (called a workgroup, usually in size of a few hundreds), can synchronize and efficiently exchange data with each other. Each workgroup has access to a dedicated high-bandwidth and low latency local memory, which can be used as a scratchpad or to exchange data between threads within a workgroup. Local memory usually has a very limited size (for example, ≤ 64 KBytes per workgroup), and using too much local memory can affect performance adversely. Although GPU's main memory (global memory) can have 10 times more bandwidth than CPU's main memory, loading data from global memory can be quite expensive with a very long latency. One remarkable feature of GPUs is that they can make context switches between thread bundles at very little cost, so when a bundle of threads are stalled due to global memory access, other available threads can be dispatched. As long as we have enough threads to schedule, the long memory latency can be hidden; thus it is important to occupy the GPU with a sufficiently large number of threads. Building Histograms on GPU: problem of having too many threads.Building image histograms using GPUs is a classical task in general-propose GPU computing <cit.>. Given an image as an array, we maintain a counter for each bin, and increment the corresponding counter if a image pixel value falls into that bin. The single thread implementation of this algorithm is trivial; however, problems arise when there are a large number of threads computing one histogram. One way to build histogram in parallel is that each thread builds its private histogram using part of the data, preferably in its local memory, and in the end all threads reduce their private histograms into a single final histogram.When the number of threads is large, the reduction step will incur a large overhead; also, the limited size of local memory prevents us from building too many private histograms. Thus, we want the number of private histograms to be much smaller than the total number of threads. However, if two or more threads update the bin counters of the same histogram, their updates may conflict with each other, i.e., two or more of them may want to increment the same counter. In this case, we have to guarantee that when a conflict occurs, threads resolve it by updating the counter sequentially, one by one. To do this efficiently without explicit locking, hardware atomic operations are necessary. Fortunately, most recent GPUs (AMD GPUs past 2012 and NVIDIA GPUs past 2014) support hardware atomic operations in local memory <cit.>, but it is still important to reduce conflicts for best performance.Constructing Feature Histograms on GPU. When we build feature histograms for decision tree learning, our input is a set of features with corresponding statistics (gradient and hessian) rather than an image. There are some important differences between building image histograms and feature histograms. First, unlike constructing the histogram for an image, building feature histograms involves building a large number of histograms at the same time, one for each feature. Second, besides incrementing an integer counter for each bin, we need to accumulate gradient and hessian statistics, two floating point numbers for each bin. Third, since we only need to access samples on the current leaf, the memory accessing pattern for loading each sample's feature is non-sequential. Data Structure and Memory Allocation.Since we need non-sequential scatter access to the feature array and global memory access is expensive, it is very inefficient to just read one byte of feature data. Thus, we bundle every 4 binned features (one byte each) into a 4-feature tuple (4-byte) and store feature tuples in GPU memory. Each GPU thread will work on 4 features of one sample at once. Since GPU is built for single precision (4-byte) arithmetic, 4-byte elements usually yields best efficiency. This strategy also requires that each workgroup maintains 4 set of histograms in local memory, and each set of histogram consists of 3 statistics: gradient, hessian and a counter. Each value takes 4 bytes (assuming single precision is used), so the total local memory requirement is 4 × 3 × 4 × k bytes. When k=256, we need 12 KB local memory per workgroup. This allows 5 workgroups per compute unit of GPU , which is an acceptable occupancy.Reduce Atomic Update Conflicts. For simplicity, here we focus on discussing how to build 8 histograms (gradient histogram and hessian histogram for 4 features) at the same time. In our GPU algorithm, each thread processes one sample of the 4-feature tuple, and a total of m (m is the size of workgroup) samples are being processed at once. Remember that GPUs execute a bundle of threads in lock-step. It is easy to write the program as every thread updates feature 0's gradient histogram all together, then updates feature 1's gradient histogram all together, etc, until all 4 features' gradient and hessian histograms are updated. The update operation needs to be atomic. However, when all m (usually 256) threads update a single histogram with k bins simultaneously, it is very likely that some threads have to write to the same bin because they encounter the same feature value, and the atomic operation becomes a bottleneck since hardware must resolve this conflict by serializing the access. To reduce the chance of conflicting updates, we exploit a special structure that occurs in our feature histogram problem but not in traditional image histogram problem—we construct multiple histograms simultaneously instead of just one. We want m threads to update all 8 distinct histograms in each step, as shown in Algorithm <ref>. To understand this algorithm, we can consider a special case where m=8. In line <ref>, when l=0, thread 0, 1, 2, 3 update the gradient histogram of feature 0, 1, 2, 3 using data sample i=0, 1, 2, 3's feature value, while thread 4, 5, 6, 7 update the hessian histogram of feature 0, 1, 2, 3 using data sample i=0, 1, 2, 3's feature value. In this case, m threads are updating 8k histogram bins at each step, greatly reduce the chance that two threads write to the same bin. For real implementation, line <ref> and <ref> require some tricks because we must avoid thestatement.Parallel By Features and Data. For simplicity, in Algorithm <ref> we only show the case where a 4-feature tuple is entirely processed by this workgroup, and this will require ⌈ d/4 ⌉ workgroups to process d features. In our implementation, we split the work to GPU workgroups both by features and by samples. If more than one workgroup is processing the same 4 features, a final reduction program (also runs on GPU) is required to merge their private histograms into the final histogram.Use of Small Bin Size. A major benefit of using GPU is that we can use a less than 256 bin size to further speedup training, potentially without losing accuracy. On CPU it is not very beneficial to reduce the bin size below 256, as at least one byte of storage is needed for each feature value. However, in our GPU algorithm, using a smaller bin size, for example, 64, allows us to either add more private histograms per workgroup to reduce conflict writes in atomic operations, or reduce local memory usage so that more workgroups can be scheduled to the GPU, which helps to hide the expensive memory access latency. Further more, using a smaller bin size can reduce the size of histograms and data transfer overhead between CPU and GPU. We observe significant performance gain by using a bin size of 64, without losing training accuracy, as we will show in section <ref>.§ EXPERIMENTAL RESULTSWe compare the following algorithms for decision tree learning in GBDT in this section:Histogram:We will compare our proposed histogram-based algorithm on GPU[Our GPU algorithm was first released at <https://github.com/huanzhang12/lightgbm-gpu> on Feb 28, 2017, and has been merged into LightGBM repository on April 9, 2017, in commit ] with other methods. LightGBM is a representative CPU implementation of this algorithm and we use it as the reference[XGBoost recently implemented Histogram based tree construction using a similar algorithm as LightGBM]. Exact: the traditional way to learn a decision tree as described in section <ref>, which enumerates all possible leaf split points. We use the “exact” tree learner in XGBoost for the implementation on CPU, and the “grow_gpu” learner <cit.> in XGBoost on GPU as the reference implementation of Exact [].Sketching: proposed in <cit.>, which also uses histogram for approximately finding the split, however features are re-binned after each split using sketching. We use the “approx” tree learner in XGBoost for this algorithm. No GPU implementation is available for this algorithm due to its complexity.Datasets. We use the following six datasets in our experiments: ,,,, , and, as shown in Table <ref>. The six datasets represent quite different data characteristics. The first three are very large and dense datasets collected for classification tasks; and are for learning to rank tasks with a mixture of dense and sparse features. The dataset has categorical features. For comparisons involving XGBoost, we did not include the dataset, as it does not support categorical features directly andhas to convert the dataset to one-hot encoding, which makes the comparison unfair.Parameters. We follow a publicly available benchmark instruction[https://github.com/Microsoft/LightGBM/wiki/Experiments] for setting training parameters, so that our results are comparable to public results. For Histogram in LightGBM, we set the total number of leaves to 255, and each leaf has at least one example. Bin size k is set to 255[LightGBM uses one bin as sentinel, thus a byte can only represent 255 bins. Similarly, only 63 bins are used when using a 6-bit bin value representation.] and 63. For Exact and Sketching in XGBoost, we set the maximum tree depth to be 8. The GPU implementation of Exact algorithm in XGBoost only works on and datasets; other datasets do not fit into the 8 GB GPU memory. For all experiments, we use learning rate η = 0.1 and run 500 boosting iterations, except for we set η = 0.015.Hardware. In all our experiments, we use two representative, main-stream GPUs from the latest production line of AMD and NVIDIA: Radeon RX 480 and GTX 1080. The two GPUs are installed to a dual-socket 28-core Xeon E5-2683 v3 server with 192 GB memory, and we use the same machine to collect results for the CPU algorithms. For all CPU results, we run 28 threads. We list the characteristics of these hardware in Table <ref>. Note that the GPUs we used are not the best ones in the market, and our results can be further improved by using a more expensive GPU. We hope that even a budget GPU can show significant speedup in training, making GPU a cost-effective solution.Memory Usage Comparison. As shown in Table <ref>, our histogram-based method uses at most 1 GB GPU memory for all datasets, thanks to the fact that each feature value after binning only takes 1 byte. A GPU with 16 GB memory can deal with datasets at least 16 times larger than , or over 25 times larger than . For even larger datasets, we can trivially extend our algorithm to multi-GPU cases where each GPU holds a disjoint set of features. As a comparison, we also include the GPU memory usage of Exact. Unfortunately, it can easily run out of memory because GPU memory is usually much smaller than CPU. Considering most GPUs have 8 GB to 16 GB memory, Exact algorithm on GPU requires too much memory for training most large scale datasets. Since GPU is particularly useful for training large datasets, the usefulness of Exact on GPU is very limited.Training Performance Metrics Comparison. Since we use reduced precision and less number of bins for training on GPU, it is interesting to see whether training on GPU can obtain the similar level of performance metrics (AUC, NDCG) with training on CPU. In Table <ref>, we can see that training with a bin size of 64 does not affect training performance metrics on both CPU and GPU. Also, our Histogram based method on GPU can get very similar AUC and NDCG with the one on CPU despite using single precision. This table, on the other hand, justifies the use of a smaller bin size. XGBoost has a different tree building procedure from LightGBM. More specifically, XGBoost grows leaves for one level and thengoes to the next level, rather than splits leaves one by one. Thus, it is impossible to find a configuration for XGBoost that exactly matches LightGBM. We use the parameters suggested by the LightGBM wiki for making our results comparable and reproducible, however we admit that their parameter settings is more favorable for LightGBM, as XGBoost with a tree of depth 8 can have a maximum of 255 leaves; LightGBM has the chance to build a deeper tree since we don't constrain the depth as long as the total number of leaves is less than 255. However, since our main focus is to speed up the training process, tuning XGBoost's parameter to get better training metrics is not our interest.Training Speed.As shown in Figure <ref>, on dataset and , our speedup is most significant: Using the GTX 1080 GPU and 63 bins, we are7-8 times faster than Histogram algorithm on CPU, and up to 25 times faster than Exact on CPU. On , and , we also have about 2-3 times speedup. Even using a low-cost RX 480 GPU (less than half of the price of GTX 1080), we can still gain significant amount of speed up, as it is only 30% to 50% slower than GTX 1080. We should reemphasize that this comparison is made between a powerful 28-core server, and a budget or main-stream (not the best) GPU.Also, Exact on GPU cannot even beat 28 CPU cores on and , and for all other datasets it runs out of memory.Thus, we believe that the Exact decision tree construction algorithm using parallel multi-scan and radix sort on GPU does not scale well. Our histogram based approach can utilize the computation power of GPU much better.We encourage the reader to read the appendix for more design details and experimental results.§ CONCLUSIONS We develop a new parallel algorithm for accelerating the decision tree building process on GPUs. Unlike the traditional approaches based on parallel sorting and prefix-scan, we develop an efficient GPU-based algorithm for constructing feature histograms and approximately find the best split. With the help of our highly efficient GPU kernel, we are able to accelerate the training of large GBDTs on a main-stream GPU up to 10 times faster than LightGBM or 30 times faster than XGBoost on a dual-socket 28-core Xeon server, while achieving a similar prediction accuracy. unsrt § ADDITIONAL DESIGN CONSIDERATIONS§.§ Further Reduce Atomic Update Conflicts Using Banked Histogram Counters Along with the techniques we described in section 4 to reduce atomic update conflicts and increase performance, we also used banked histogram counters which is a traditional technique to speedup image histogram building on GPU. Instead of building just one histogram per feature that is shared by all threads, we build B banks of histograms for each feature, and each thread in a workgroup only updates one of the counter banks. Finally, counter values of all banks will be summed up to one value. In this way, the conflict rate on updates is reduced by a factor of B. However this option is severally limited to feature histogram building because of the limited local memory space. With bin size 256, feature histogram takes 12 KBytes local memory per workgroup and we do not use any additional bank. With bin size 64, feature histogram takes only 3 KBytes and thus we can afford 4 banks, and still use 12 KBytes per workgroup. This greatly improves performance for some datasets, as we have shown in section 5. §.§ Sparse Features When GPU is constructing histograms, we also want the CPU to do some useful work at the same time to maximize resource utilization. Currently, our strategy is to process sparse features on CPU and process dense features on GPU. Processing sparse features requires more bookkeeping during leaf split and histogram construction, thus it is beneficial to treat a feature as sparse only when it is sparse enough. By default, if a feature has more than 80% zeros, LightGBM will treat it as a sparse feature and use special data structure to process it. We change this compile time threshold constant to a new configuration variable(denoted as t, with a default value of 0.8). LightGBM will process a feature as sparse only when there are more than t × N zeros. By selecting an appropriate threshold, we can balance the load on both computing resources and maximize performance.In our experiments, we found that for and , it is best to make features completely dense to process them on GPU (t=1), because our GPU histogram construction algorithm has a large speedup factor. For datasets , , and , all features are already dense and processed on GPU. §.§ Bin Redistribution In some cases, only a few distinct values appear in a binned feature (for example, a feature only contains numbers drawn from {1, 2, 3}), thus the effective bin size will be set to 3 instead of specified 64 or 256. Although this is not a problem for CPU, on GPU this causes performance degradation as only a few bins are being written and the conflict rate during atomic operations will be high. Because threads within a bundle (“warp”) execute in locksteps, this can slow down an entire bundle of threads. Our benchmark shows that in the extreme case, where there exists one feature with only one bin value such that every thread contents to update the same bin, the workgroup (assuming a bin size of 256) processing that 4-feature tuple can be 4 times slower comparing with random feature values.We solve this problem by redistributing the bin values. Before copying features to GPU, if we find that one feature only lies in k^' < k/2 bins, where k is the desired maximum bin size for the dataset, we will redistribute the feature bin value with at most k^' numbers using the strategy shown in Algorithm <ref>.After we transfer the constructed histogram from GPU back to CPU, we will simply accumulate the values from bin i × m to (i + 1) × m - 1 to get values of the i-th bin of the original histogram. §.§ Dealing With Repeated Feature Values In some datasets, we found that some binned features range from 1 to k but most of them are just a single value. It occurs quite often when a features has some sparsity and thus there are many repeated zeros. This can slow down the histogram construction as there are a lot update conflicts for bin 0. We add a fast path in our implementation for this case: we maintain the most recently used bin of the histogram counters in GPU registers, and when the same bin value occurs again we don't need to update the histogram in local memory. Our benchmark shows that it incurs very little overhead, but noticeably decreases the training time of and .§.§ Data Movement Before we launch the GPU program to construct feature histograms, we need to prepare for its input. All feature values are copied as 4-feature tuples only once before training starts, and stay unchanged during the entire training process. After one leaf is built, CPU updates the list of sample indices and the corresponding gradient and hessian statistics for samples on that leaf. Thus three arrays (indices, gradients, hessians) need to be copied to GPU after each split. We make the copy asynchronous when possible, to hide the data transfer time.Instead of transferring all histograms to CPU memory after all of them are built in GPU memory,we start transferring the histogram for each feature back to CPU memory immediately after it has been built. This also helps hiding data transfer time from histogram computation, especially when the number of features is large. §.§ Numerical IssuesLimited precision training has been successfully used in training of deep neural networks <cit.>. Theoretical analysis for limited precision training is also available for some algorithms like SGD <cit.>. In histogram based decision tree training, the main numerical issue comes from the accumulation of gradient and hessian statistics of each bin. The current CPU implementation of lightGBM uses 32-bit floating point to store statistics, but the accumulation is done by 64-bit floating point arithmetic.However, unlike CPUs, most GPUs are relatively weak to compute in double precision. For example, the NVIDIA Titan X Pascal has over 10 TFLOPS peak processing power for single precision operations, but this number becomes only 0.3 TFLOPS for double precision floating point operations. Moreover, double precision histograms also increase the memory pressure of local memory. Thus, it is necessary to avoid double precision computation on GPUs in order to have good performance. In our implementation, we use 32-bit single precision numbers by default.We also provide a configuration parameter to switch our algorithm into double precision mode for the cases when users do have a GPU with good double precision processing power (like NVIDIA Tesla series), or want better arithmetic precision for reproducibility.§ IMPLEMENTATION DETAILS§.§ GPU Algorithm ImplementationOur GPU algorithm for building feature histograms is implemented in OpenCL 1.2 and can target a large range of GPU devices from different vendors. We guarantee our implementation quality by using inline assembly when possible and check the compiler generated GPU assembly code manually, to ensure that our code runs with best efficiency. §.§ Integrating Our Algorithm Into LightGBM GPU code is well known for being tricky, unfriendly to ordinary programmers and hard to maintain. Instead of re-implementing the entire tree building logic of LightGBM on GPU, we only implement the procedure of building feature histograms, which is the most time-consuming operation in LightGBM. Thus, our GPU algorithm has only weak interactions with other parts of LightGBM, and the new learning system immediately gains all other good capabilities of LightGBM, without re-implementing these features on GPU. Thanks to our implementation's modularity, our accelerated histogram building algorithm also works for distributed GBDT building methods in LightGBM, thus we make very large scale distributed GPU training possible.We replace the ConstructHistogram function in LightGBM with our GPU implementation, and add necessary code for GPU initialization and data movement to LightGBM. Our implementation is publicly available[<https://github.com/huanzhang12/lightgbm-gpu>] since Feb 28, 2017, and has been officially merged into LightGBM in commiton April 9, 2017.§.§ Atomic Operations on GPUMost modern GPU architectures (NVIDIA Maxwell or later, AMD Graphic Core Next 1.0 or later) support atomic operations in local memory space. However, these atomic instructions only work on integer data types.Although NVIDIA provides an pseudo (PTX) instructionto perform atomic floating point addition in local memory, it does not translate to a single hardware instruction. Rather, the compiler will generate a small loop consisting of a floating point addition and an atomic compare-and-swap operation in local memory. On AMD GPUs, no such pseudo instruction is available, so we directly implement atomic floating point addition using a loop of local memory compare-and-swap instructions. §.§ Disabling Power Saving FeaturesMany GPUs have aggressive DVFS (dynamic voltage and frequency scaling) behaviour which downclocks the GPU when it is not under full load. Since our algorithm does not work on GPU with full load (CPU is used to finalize the tree with GPU constructed histograms), GPU drivers are likely to automatically select a lower power state with reduced frequency. To overcome this problem on AMD GPUs, we manually set performance mode using control knobs located at . For the NVIDIA GPU we used, we cannot find a reliable way to put it into the(highest) performance mode. Thus, our code runs inmode with reduced GPU memory clock. Even though, we are still able to show significant speedup. §.§ Turbo Boost and Hyper-threadingFor better reproducibility, we disable Turbo Boost on CPU and run the CPU in performance mode (fixed maximum frequency). We also do not use hyper-threading, as we found that LightGBM becomes slower with hyper-threading in our case (with 28 CPU cores) due to additional threading overhead.§ ADDITIONAL EXPERIMENTAL RESULTS§.§ Performance Characterization of Histogram-based Algorithm on CPUWe make an instruction-level profiling of the Histogram algorithm in LightGBM on CPU, and identified the major bottlenecks over several different datasets. We found that over 85% of time is spent on the four functions in Table <ref>. Function BeforeFindBestSplit() mainly spends its time on generating three arrays: the indices of training samples on this leaf, and the corresponding hessian and gradient values for them. Its time complexity is O(N_r), where N_r is the number of samples on the current leaf. Function ConstructHistogram() implements Algorithm 1, which goes over one feature data to construct its feature histogram in O(N_r) time, and there are d calls to this function (one for each feature), so the total complexity is O(N_r d). Function FindBestThreshold() goes over one feature histogram to find the best split point with complexity O(k), and there are d calls to this function so the overall complexity is O(kd).Function Split() is to split the samples on a node to its left and right children given the split value and the feature index computed by the previous function. Its complexityis O(N_r).As shown in Table <ref>, on 3 different datasets with different d, n and we fix k=255, the majority of time in LightGBM is spent on constructing histograms for all features in function ConstructHistogram(), which is expected, as this process needs to go over all feature values on one leaf (O(N_r d) elements) to build d histograms, each with k bins. Since k (usually 255 or smaller) is much smaller than N_r, the time spent on finding the best split inside histograms is not significant.[caption=, the most time consuming function in LightGBM, is our target for acceleration,captionpos=b,label=lst:lightgbm-histogram,basicstyle=0.7,language=C] void ConstructHistogram(char* feature,char* example_indices, int num_data, float* leaf_gradients, float* leaf_hessians, struct HistogramBins* histogram)for (int i = 0; i < num_data; i++)char bin = feature[example_indices[i]]; histogram[bin].sum_gradients += leaf_gradients[i]; histogram[bin].sum_hessians += leaf_hessians[i]; histogram[bin].cnt++;[language=C] void Split(int threshold, int* data_indices, int num_data, int* less_indices, int* greater_indices)int greater_count = 0, greater_count = 0; for (int i = 0; i < num_data; ++i)int idx = data_indices[i]; if (this_feature[idx] > threshold)greater_indices[greater_count++] = idx;elseless_indices[less_count++] = idx; Our goal is thus to improve the performance of function ConstructHistogram() on GPU. It is clear that this function is memory-bound, as building the histograms requires non-sequential scattering access to large arrays in memory, in which case the cache system does not work quite well. Thus, our key to success is to utilize the high bandwidth memory and large computation power on GPUs to accelerate this operation. §.§ Synthetic benchmark on GPU Feature Histogram ConstructionWe first test the speedup for building feature histograms on synthetic data. To construct synthetic data, we generate n=8,000,000 samples and each sample is d=500 dimensional. Each feature value is a byte holding a random bin number ranging from 1 to k, where k ∈{64, 256} is the total number of bins.During the decision tree building process, each leaf node usually only holds a small portion of data samples. Considering a tree of depth D, one leaf will on average has n/2^D training samples, and these training samples can spread far away in memory. Building histograms for a leaf node requires scanning data with a large, random stride in memory, which can be very slow because it is impossible to fit the whole dataset in cache. We emulate this behavior in our benchmark by generating a permutation of n numbers, truncating it into size of n/2^D, sorting the truncated array, and then using it as indices to access data samples. In Figure <ref>, we compare the performance of our GPU algorithm with CPU for building feature histograms with k=64 and 256. The CPU implementation is directly extracted from LightGBM's source code. The metric we used for comparison is the effective bandwidth; that is, how much feature data can be processed during one unit time. We observe that the largest bandwidth occurs when D=0. In this case, all the data are used to build histograms (e.g., histogram for the root of a tree), and memory access is sequential. When D increases, processing bandwidth is reduced dramatically on all devices because of the large-stride memory access. GPU is much faster than CPU over all Ds especially when k=64. However, when D becomes too large, there are not many samples left, so the overhead of invoking a GPU function becomes significant, which limits the available speedup.On dataset , we only show about 50% - 80% improvements. The reason is two-fold: first, the relatively smaller data size makes the communication overhead between GPU and CPU significant; second, according to our synthetic benchmarks (in appendix), GPU prefers large workload to gain good speedup.§.§ Load Balancing Between CPU and GPU For datasets with a mixture of dense and sparse features, when we are building the feature histograms for dense features on GPU, CPU is responsible for building feature histograms for sparse features at the same time. We denote the configuration parameteras t. By varying the threshold t, we can vary the workload allocated to GPU and CPU. Table <ref> shows the performance comparison with different t ranging from 0.8 (LightGBM default) to 1.0 (all features are treated as dense features). We can see that for and , it is worthwhile making features completely dense to process them on GPU, because our GPU histogram construction algorithm has a large speedup factor. For datasets , , and , all features are already dense and processed on GPU.§.§ Convergence Behaviour of Our GPU Histogram Algorithm Since our GPU algorithm uses a smaller bin size and single precision for training, we want to make sure that there is no instability during training. In Figure <ref>, we show the AUC or NDCG@10 with respect to the number of boosting iterations for and . As we can see, despite of the different bin sizes, Histogram on CPU (with double precision math) and GPU (with single precision math) both converge to the same metric value. Other datasets also exhibit a similar behaviour, so we omit their figures.§.§ Use 4-bit Bins (k=16) It is possible to just use 4 bits to store a binned feature value (i.e., two feature values packed into one byte), by using a bin size of 16 (practically 15 bins in LightGBM as one bin is used as a sentinel). The benefits of using a 4-bit bin is three-fold: First, this reduces memory usage for storing feature values by half and also reduces required memory bandwidth; second, this allows 8 features to be packed into one 4-byte tuple, increasing the available workload per workgroup; third, a smaller bin size also reduces memory pressure in local memory, and allows multiple banks of histogram counters to be built to further reduce atomic update conflicts.We conduct experiments for the same six datasets with the same settings as in the Experimental Results section, and compare training accuracy and speed in Table <ref> and Figure <ref>. We can observe that although using 4-bit bins significantly decreases training time for some datasets (like , and ), we cannot achieve the same accuracy as using larger bin sizes with the same number of boosting iterations. However, sometimes the difference is very small, like in , , and ; thus, using a bin size of 15 may help us produce a reasonably good model within a very short time. | http://arxiv.org/abs/1706.08359v1 | {
"authors": [
"Huan Zhang",
"Si Si",
"Cho-Jui Hsieh"
],
"categories": [
"stat.ML",
"cs.DC",
"cs.LG"
],
"primary_category": "stat.ML",
"published": "20170626132729",
"title": "GPU-acceleration for Large-scale Tree Boosting"
} |
http://arxiv.org/abs/1706.09030v2 | {
"authors": [
"Joshua Aftergood",
"So Takei"
],
"categories": [
"cond-mat.mes-hall",
"cond-mat.str-el"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170627200209",
"title": "Noise in tunneling spin current across coupled quantum spin chains"
} |
|
http://arxiv.org/abs/1706.08953v2 | {
"authors": [
"Laura Fanfarillo",
"Lara Benfatto",
"Belen Valenzuela"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20170627173137",
"title": "Orbital mismatch boosting nematic instability in iron-based superconductors"
} |
|
Perpendicular magnetic anisotropy in insulating ferrimagnetic gadolinium iron garnet thin films S. T. B. Goennenwein December 30, 2023 =============================================================================================== There is a strong demand for precise means for the comparison of logics in terms of expressiveness both from theoretical and from application areas. The aim of this paper is to propose a sufficiently general and reasonable formal criterion for expressiveness, so as to apply not only to model-theoretic logics, but also to Tarskian and proof-theoretic logics. For model-theoretic logics there is a standard framework of relative expressiveness, based on the capacity of characterizing structures, and a straightforward formal criterion issuing from it. The problem is that it only allows the comparison of those logics defined within the same class of models. The urge for a broader framework of expressiveness is not new. Nevertheless, the enterprise is complex and a reasonable model-theoretic formal criterion is still wanting. Recently there appeared two criteria in this wider framework, one from Garca-Matos & Vnnen and other from L. Kuijer. We argue that they are not adequate. Their limitations are analysed and we propose to move to an even broader framework lacking model-theoretic notions, which we call “translational expressiveness”. There is already a criterion in this later framework by Mossakowski et al., however it turned out to be too lax. We propose some adequacy criteria for expressiveness and a formal criterion of translational expressiveness complying with them is given. § INTRODUCTIONIt is very common for those who work with logic to make comparisons such as “the logic Ł' is more expressive than Ł”, “Ł' is stronger than Ł”, “Ł is included in Ł'”, “Ł can be reduced to Ł'”, etc. Such assertions are often made on imprecise grounds and, though possibly being non-ambiguous and non-problematic, the lack of clarity around the usage of these concepts can generate terminological confusion across the literature (e.g. <cit.>) and harden the comparison of formal results.In the literature, the notion of logic inclusion or sub-logic (these terms will be used interchangeably here) is pretty much linked with language and axiomatic extensions, which on their turn are linked with “strength”, that is, the capacity of proving theorems or having valid formulas.Now the concept of sub-logic is sometimes associated with strength and sometimes associated with expressiveness, and sometimes with both(e.g. in <cit.>), which is known to be the case of paradoxes <cit.>. Three kinds of systems are relevant here: model-theoretic logics, Tarskian and proof-theoretic logics, they will now be briefly defined. A logic Ł is called model-theoretic if it is defined semantically and presented as a sequence (,,), whereis a set of formulas,is a class of models andis a satisfaction relation on ×. A logic Ł is Tarskian if it is defined as (,⊢), where ⊢ is a consequence relation on(possibly multi-consequence). Finally, Ł is a proof-theoretic logic if it is defined as(, ℛ), where ℛ is a set of inference rules.[Some additional criteria are usually imposed for asystem to qualify as one of these three kinds, but they are immaterial here.] In model-theoretic logics there is a straightforward approach to expressiveness that is also reasonably taken as a definition of logic inclusion: a logic Ł_2 is at least as expressive/includes Ł_1 if every class of structures characterizable in Ł_1 is also characterizable in Ł_2 (see e.g. <cit.> and <cit.>). This naturally only holds for logics defined within the same class of structures.If one wants also to compare logics defined within different classes of structures, then it does not seem adequate to use the concept of sub-logic, as we shall see below. It is better to use the concept of expressiveness. There is no straightforward approach to expressiveness for Tarskian and proof-theoretic logics (TPL, for short). As for sub-logic, in TPL it is also linked with language and axiomatic extensions. However, we can often see “sub-logic” relations taken in a wider sense, i.e. when, for two given logics Ł and Ł', it happens that Ł' is not a language/axiomatic extension of Ł, but there is a certain mapping of Ł-formulas into Ł'-formulas respecting the consequence relation. These cases are normally interpreted as saying that Ł is included/embeddable/reconstructible/interpretable/can be simulated in Ł'. We propose to call these as expressiveness relations whenever they can be seen as modeling the following intuition0.1 (E) 0.8 For every Ł-sentence ϕ, there is an Ł'-sentence ψ with the same meaning. This same intuitive explanation of expressiveness holds for model-theoretic logics, and is used as a basis for formal criteria therein (e.g. <cit.>). Thus we can have a reasonably homogeneous concept for comparing logics: that of expressiveness. We shall reserve the term “sub-logic" just when there are axiomatic or language extensions, and we shall not use the term “strength” because it is ambiguous betweenexpressive and deductive strength.A precise definition for the notion of relative expressiveness for model-theoretic logics was given already in the 1970s (e.g. in <cit.> and <cit.>). As we said, this definition is based on the capacity of characterizing structures and underlies each of the so-called Lindstrm-type theorems,[That is, theorems of the form “If a logical system Ł' is at least as expressive as Ł and have properties P_1,...,P_n, then Ł' is as expressive as Ł”; see e.g. <cit.>, <cit.> and<cit.>.] which form the basis of abstract model theory.*Single-class expressiveness Considering model-theoretic logics defined within the same class of structures, the above intuition can be captured easily since there is a common ground where sentences can be compared. This common ground is easily achieved by defining the meaning of a sentence ϕ in a logic Ł =(, , _Ł) as {∈ | _Łϕ} (Mod_Ł(ϕ), for short). Thus we call this framework single-class expressiveness.Since every sentence in Ł_1 is mapped to a sentence in Ł_2 having the same meaning, this framework of expressiveness can be seen as consisting of certain formula-mappings between model-theoretic logics. A formal definition for it is then straightforward. Let τ be a signature and let Ł_1=(_1,,_Ł_1) and Ł_2=(_2,,_Ł_2) be model-theoretic logics. Ł_2 is at least as expressive as Ł_1 (Ł_1 _ECŁ_2) if and only if (iff, for short) for every τ-sentence ϕ∈_1 there is a τ-sentence ψ∈_2 such that Mod_Ł_1(ϕ)=Mod_Ł_2(ψ). Notice that here the class of modelsis the same for both Ł_1 and Ł_2, and ϕ,ψ share the same non-logical symbols. The above definition can be paraphrased in terms of elementary classes:[For some signature τ, a class 𝒦 of τ-structures is elementary in a logic Ł iff there is an Ł-sentence ϕ such that 𝒦={|_Łϕ}. A class 𝒦 of τ-structures is a projective class of Ł if for some τ' ⊇τ there is an Ł-elementary τ'-class 𝒦' such that 𝒦={' τ |' ∈𝒦'}, where ' τ is the τ-reduct of '.]Ł_ECŁ' iff every elementary class of Ł is an elementary class of Ł'.Despite being the basis for many important results, _EC is very limited. It is not only restricted to model-theoretic logics, but it requires the classes of structures being compared to share the same signature. As a consequence, it only allows the comparison of logics defined within the same class of structures. The urge for a broader definition is not new.[See <cit.>, <cit.>, <cit.> and <cit.>.] A straightforward means of extension already appears in <cit.> and is examined in <cit.>. Using the notion of projective class, one can loosen the above definition allowing that Ł' is at least as expressive as Ł iff every elementary class of Ł is a projective class in Ł' (Ł_PCŁ') (, p. 232).Even among those expressiveness results using _EC, we can notice some flexibility in its application.One such example appears in <cit.>, where the definition of _EC above is given, but afterwards (p. 307) it is informally relaxed in order to allow changes of signature, thus the proper definition being used appears to be the one based on projective classes (_PC). The problem is that elsewhere we get different results depending on whether we use _EC or _PC, as Shapiro showed <cit.>: Ł(Q_0) _ECŁ(A) and Ł(A) _ECŁ(Q_0), but Ł(Q_0) _PCŁ(A) and Ł(A) _PCŁ(Q_0).[The logic Ł(Q_0) is the first-order logic extended with the quantifier “there exists infinitely many", and Ł(A) is the first-order logic with the “ancestral” operation A, i.e. Axy(Rxy) says that x is an ancestor of y in the relation R.]Remaining within model-theoretic logics, a wider framework —let us call it multi-class— would comprise besides formula-mappings also structure-mappings, thus allowing structures of one logic to be mapped to structures of the other. This would enable the comparison of logics defined within different classes of structures. Recently there appeared two formal definitions of multi-class expressiveness, to wit <cit.> and <cit.>. In the sequence we will present them and argue that they are not adequate. There have been also early claims outside abstract model-theory relating logics in the sense of (E) above, but no explicit definitions of the main concepts involved were given. Gdel used his result on the interpretation of classical into intuitionistic logic to infer that, contrary to the appearances, it is classical logic that is contained in intuitionistic logic <cit.>. Since then, there followed many results of interpretations, embeddings, reconstructions,simulations, etc. among Tarskian and proof-theoretic logics. Such results have often been used to justify some statement of inclusion or relative expressiveness between the logics at issue.[E.g. <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.> and the recent <cit.>.]We proposed to call those with the underlying intuition (E) as expressiveness results. Naturally, this notion of expressiveness is no longer directly linked with the capacity of characterizing structures as in model-theoretic logics, rather it resides in the capacity of a logic to “encode” another. Let the framework of expressiveness based on such capacitybe named “translational expressiveness”.[The term is borrowed from <cit.>. Curiously, the same kind of problem appeared in computer science: there was a multitude of programming languages and process calculi and many informal claims relating the expressive power of such, through the existence of certain encodings of one into another. This situation fomented a series of works aiming at a standardization of such “expressibility results” (e.g. <cit.>, <cit.> and <cit.>). Though aimed at different objects, it is still possible to learn from this enterprise and propose the first steps of a standardization of a definition of relative expressiveness.]As opposed to the case of model-theoretic logics, until recently there was no attempt to give a precise definition of relative expressiveness in this framework. To the best of our knowledge, Mossakowski et al. <cit.> were the first to give an explicit formal definition of translational expressiveness for logics, that is, an expressiveness relationbased on the existence of certain kinds of formula-mappings.We will expose their definition and show that it is still not adequate. Then, some adequacy criteria for expressiveness are proposed and a formal criterion for translational expressiveness is given. §.§ Structure of the paper This paper presents the following panorama on relative expressiveness between logics: (*)Relative expressiveness between logics (intuitive concept as given by (E)) (a) Adequacy criteria for expressiveness → Approaches to (*) hopefully satisfying (a) * single-class * formal proposals: _EC, _PC * multi-class * formal proposals: _gv, expressiveness_g * translational * formal proposals: Mossakowski et al.'s and expressiveness_gg. In 2 the framework of multi-class expressiveness will be presented and two formal criteria will be analysed, one from <cit.> (_gv) and other from <cit.> (expressiveness_g). We argue that, using the intuitive explanation of expressiveness given above, there are counterexamples to both. In the sequence, we investigate what is wrong with them and propose that moving to an even wider framework, encompassing a greater range of logics and lacking structure-mappings, might be promising. In 3 we present Mossakowski et al.'s formal criterion for translational expressiveness and show that, due to a result of <cit.>, it is still not adequate. Then, some basic adequacy criteria for expressiveness will be proposed. In the sequence we analyse some formal conditions related to translations already appearing in the literature and investigate whether they satisfy the adequacy criteria. Finally, a formal sufficient condition for translational expressiveness (expressiveness_gg) is proposed. We will argue that expressiveness_gg satisfies the criteria and is materially adequate. § MULTI-CLASS EXPRESSIVENESS§.§ M. Garca-Matos and J. Vnnen on sub-logic Garca-Matos and Vnnen gave a multi-class definition of sub-logic. Their definition is similar to one given in <cit.> but is laxer.[Garca-Matos and Vnnen's approach is a non-signature indexed version of the “map of logics” in <cit.>. In Meseguer's paper, it is not allowed for sub-logic mappings that sentences in the source logic be mapped to theories in the target logic, and the formula-mappings must be injective.]Seemingly, they treat the term “sub-logic” as synonymous with “expressiveness” (exchanging the order of terms, naturally), since they present the Lindstrm theorems as being about sub-logic, whereas they are presented by one of the authors elsewhere as being about expressiveness (e.g. <cit.>). We shall argue that the relation defined must be seen as an expressiveness relation, and it will be shown that as an expressiveness relation, it has important downsides. Let us consider their definition of sub-logic <cit.>: A logic Ł=(, , ) is a sub-logic of Ł'= (', ', ') (in symbols Ł_gvŁ') if there are a sentence θ∈' and functions f: ' ⟶, : ⟶' such that: (a) For every ∈ exists a ' ∈' such that f(')= and ' ' θ(b) For every ϕ∈ and for every ' ∈', if ' ' θ, then (' ' (ϕ) iff f(') ϕ)Thus, if the class of structures ' of a logic Ł' is richer than the class of structuresof a logic Ł, one could still allow a comparison between Ł and Ł', by restricting ' to the translatable structures, i.e. those ' whichsatisfy some condition θ and then use a function f to translate this reduced class of Ł'-structures into Ł-structures.§.§.§ A problem with _gv Let Ł=(, ℳ, ) be a trivial propositional logic in some given signature, and let (𝔐, v) be the set of is truth tables together with a valuation. Let Ł'=(', ℳ', ') be any logic that has at least one valid sentence δ and let the formula θ of the definition above be such δ. Define the following mappings * f: ℳ'⟶ℳ. For every ' ∈ℳ', f(')= (𝔐,v). * : ⟶'. For every ϕ∈, (ϕ) = δ.Then it is easily seen that both items (a) and (b) above are satisfied. Thus, according to this definition of sub-logic, every logic containing at least one valid formula has a trivial sub-logic. If we think on the usual meaning given to “sub-logic”, this not plausible at all, since the logic (', ℳ', ') could be non-trivial and might even lack a trivializing particle, so how come it could have a trivial sub-logic?[This counter-example was based on another one given in <cit.>, which was given as an argument for strengthening the notion of translation used.] It is not enough to require that the mappingbe injective. Using an idea of <cit.>, take for target logic any Ł^*=(^*, ℳ^*, ^*) that has a denumerable number of valid formulas δ_1, δ_2, ... and define the mapping from the formulas of the trivial logic ={ϕ_1, ϕ_2, ...} to Ł^*-formulas as (ϕ_i)= δ_i. Still we have that Ł^* has a trivial sub-logic, once more, Ł^* may be any logic with a denumerable number of validities, also lacking a trivializing particle.Naturally, the usual senses of logic inclusion, that is, through language or axiomatic extensions do not apply here. The only way to make sense of this is to interpret the above cases as saying that a trivial logic can be simulated in any logic containing at least one validity. This capacity of simulating a logic is an expressive capacity, therefore the definition above is better seen as a definition of expressiveness. Yet, as an expressiveness relation, it is noteworthy that no restriction on the translation functions f andare imposed, so one may wonder whether the definition over-generates.We are not in position to settle definitively this question. However we will give a plausibility argument to the effect that we should impose stricter conditions on model- and formula-mappings, since there is a natural and reasonable extension of the above definition that indeed over-generates. Though not, strictly speaking, a counter-example, the case to be presented below shall give evidence that there is an intrinsic problem with the above proposal for multi-class expressiveness. As we said, the sentence θ on the above definition of _gv is intended to cut Ł'-structures that are meaningless from the point of view of Ł. Apparently, it would do no harm to the idea behind _gv to allow θ to be a recursive set of sentences, as it is normally done in works dealing with translations of logics and conversion of structures (e.g. <cit.>). This would be useful if the logics at issue have no conjunction, so that θ could be a finite set of sentences; or if the low expressive power of the logics Ł and Ł' makes that the Ł'-structures to be reduced into Ł-structures be only characterizable through an infinite but recursive set of Ł'-sentences. This happens in the case of many-sorted logic (ℳ𝒮ℒ) and . If θ is not allowed to be an infinite set of sentences, then ℳ𝒮ℒ would not be a sub-logic, in the above sense, of , which is implausible. Though the conversion of -structures into -structures is mentioned <cit.>, the case of a given -signature τ containing infinitely-many unary symbols S_1,S_2,... is not considered. To convertτ-structures into ℳ𝒮ℒ-structures then one needs to make sure that unary predicates S_1,S_2,.. to be converted to many-sorted domains are non-empty. This would only be accomplished by setting θ={∃ x S_1(x), ∃ x S_2(x), ...} <cit.>.However, if one allows such modification another implausible situation occurs. Consider the classical propositional logic () and a propositional logic , defined by Bziau <cit.>.shares all the definitions of the classical propositional connectives, except for negation, where it has only one “half” of its clause: for a -model M and formula ϕ, if M(ϕ)= T, then M(ϕ)=F; the converse direction does not hold.Bziau shows that there is a translation frominto . Below we will give Mossakowski et al.'s presentation of it, which includes also a model translation <cit.>. Given an n-ary connective #, a translationis literal for # if (#(ϕ_1,...,ϕ_n))= #((ϕ_1),...,(ϕ_n)); for an atomic formula p,is literal when (p)=p. Define the mapping (,f): ⟶ as follows: * : ^⟶^ * (ϕ)= (ϕ) → ((ϕ)), * literal for ,,→ and atomic formulas; * and f: ^⟶^ * f(𝔐^,v) = (𝔐^,v), where 𝔐 comprises the truth-tables for each connective and v a valuation on the propositional variables. Notice that f takes a -model, keeps the valuation v and replaces the truth-tables for the correspondingones.Then we have that f(𝔐^,v) _ϕ if and only if (𝔐^,v) _(ϕ). The model mapping f is surjective, so that it obeys (a) above.Now Mossakowski et al. (, p. 100) define a mapping also fromtousing an auxiliary set of formulas Δ constructed out of -formulas.Define the mapping (',f', Δ): ⟶ as follows: * ':^⟶^ * For every ϕ∈^, '(ϕ)=p_ϕ, where p_ϕ is a propositional variable. Define Δ as the following set of formulas, for ϕ, ψ∈^:2 * '(ϕψ) ↔'(ϕ) '(ψ) * '(ϕψ) ↔'(ϕ) '(ψ) * '(ϕ→ψ) ↔'(ϕ) →'(ψ) * '(ϕ) →'(ϕ). The purpose of Δ is to encode the semantics ofinto the propositional variables {p_1,p_2,...}, since every -formula is translated into one of such p_i, in a -model satisfying Δ the valuation of the propositional variables p_i is forced to respect the semantics of . For example, in , if (𝔐^,v) _ r, then it holds that (𝔐^,v) _ r, but the converse direction does not hold.This is simulated in the -models satisfying Δ by the fourth clause above: if (𝔐^,v) _ p_r, then (𝔐^,v) _ p_r which implies that (𝔐^,v) _ p_r. But, as in , it does not hold that if (𝔐^,v) _ p_r, then (𝔐^,v) _ p_r. Now define the model-translation f': ^⟶^: * Let(𝔐,v) be a -model satisfying Δ. Then f'(𝔐,v) is defined as follows: * For every -formula ϕ, f'(𝔐,v) _ϕ iff (𝔐,v) _'(ϕ). f' is also surjective (so it obeys (a) in the criterion for sub-logic above). Then we have that f'(𝔐^,v) _ϕ iff (𝔐^,v) _Δ and (𝔐^,v) _'(ϕ). Therefore, by the above results and according to the extended definition of sub-logic, we would have thatandare one sub-logic of another, which is not plausible.is not a sub-logic ofin the sense of language/axiomatic extension. Neither they are expressively equivalent, using (E) above, since the “half-negation” present inis not available in . The problem is that the translation fromtouses a trick to sneak in the semantics ofinto Δ. Restricting the -models that satisfy Δ, one simulates the behaviour of -formulas in the propositional variables p_i and sustain such behaviour through the model-translation.The modified version of _gv, allowing θ to be a recursive set of sentences looks at least as “natural” as the original one. Even considering the original definition<ref> we can see that there is something wrong with it, in not requiring any kind of preservation of the structure of formulas e.g. by forcingto be inductively defined through the formation of formulas. Then one may conjecture that, among more expressive logics, there be translations (,f) wheremaps entire formulas ϕ to propositional variables p_ϕ and, with a sentence θ restricting the target structures, f is able to mimic the semantic behavior of ϕ. Then it is very doubtful that the obtained p_ϕ would have the same meaning as ϕ. Thus, we think we have good reasons to consider that Garca-Matos and Vnnen's definition of sub-logic is not adequate. It would certainly be better to use a stronger notion of translation, paying attention to the structure of formulas. Only then the meaning of the target-formulas could be said to match the meaning of the source-formulas. Below we will see that a development along this line appeared in the literature.Nevertheless, there is still a structure-attentive translation that “cheats” similarly as the one above,mimicking the semantics of one logic into the other. §.§ L. Kuijer on multi-class expressivenessIn his doctorate thesis <cit.> Kuijer studies the expressiveness of various logics of knowledge and action, these logics are taken in the model-theoretic sense.He notices that there are some results relatinglogics similarly as in single-class expressiveness.[The referred results are: <cit.>, <cit.>, <cit.>, <cit.> and <cit.>.] These works were selected as prototypical for a criterion in the wider framework of multi-class expressiveness.The purpose is to investigate features shared by all the results and construct a criterion, to be called “expressiveness_g”, based on these features.Similarly with the work of Garca-Matos and Vnnen exposed above, these prototypes involve translations of sentences and translations of structures. So a translation from Ł_1 to Ł_2 is a pair (,f), with : _1 →_2 and f: _1 →_2 or f: _2 →_1, such that (,f) satisfies some given conditions. A first plausible condition is that (,f) must preserve and respect truth:A translation (,f): Ł_1 →Ł_2with: _1 →_2 and f: _1 →_2 is truth preserving if, for every ϕ∈_1 and ∈_1 _Ł_1ϕ if and only iff() _Ł_2(ϕ).Then a tentative definition of expressiveness_g could be Ł_2 is at least expressive_g as Ł_1 iff there is a (,f): Ł_1 →Ł_2 that is truth preserving.The problem is that the requirement of truth preservation is very weak, indeed there are several trivial truth-preserving translations among almost every logic. Kuijer gives the following example <cit.>.§.§.§ A trivial translation Let Ł_1 = (_1, _1, _Ł_1) be any logic on possible world semantics such that _1 is countable and let Ł_2=(_2, _2, _Ł_2) be a logic where _2 is a countable set of propositional variables but with no connectives and where _2 is a class of models with possible worlds. Thus, every ' ∈_2 is a set of possible worlds with a valuation.Define a truth-preserving translation (_t, f_t) from Ł_1 to Ł_2 in the following way: map every ϕ∈_1 to a propositional variable p_ϕ∈_2, f_t maps a model ∈_1 to a model ' ∈_2 taking the set of possible worlds ofand removing every other structure, and with the following valuation v(p_ϕ)= { w ∈| (, w) _Ł_1ϕ}. Then clearly, by definition, (_t,f_t): Ł_1 ⟶Ł_2 is a truth preserving translation.§.§.§ Defining expressiveness_g Since Ł_1 in the above example is an arbitrary logic on possible world models, if truth preservation were the only condition for multi-class expressiveness, Ł_2 would be at least as expressive as Ł_1, which is absurd, given that Ł_2 has scarce expressive means. Nevertheless, truth-preservation is clearly a necessary condition. Thus, one must find other features P_1,...,P_n a translation must satisfy in order to serve as a formal elucidation of the notion of multi-class expressiveness. Another immediate criterion that comes to mind in order to avoid the trivial translations is to require the preservation of validities and entailment relations. However, some of the chosen prototypical translations do not preserve validity and some do not preserve entailment. Since the idea was to capture the essential features shared by all prototypical translations inexpressiveness_g, none of these can be imposed as a necessary condition. Kuijer then goes through a number of tentative criteria, e.g. preservation of atomic formulas, of sub-formulas, etc., and shows that they are either too lax or too restrictive. Among the lax criteria, that is, the ones that are satisfied by some trivial translation, is one that Kuijer considers nonetheless important, the criterion of being model based:A translation (,f) is model based if there are two functions f_1,f_2 such that, for all (𝔐,w) ∈_1, we have that f(𝔐,w)= (f_1(𝔐),f_2(𝔐,w)).A model based translation would force f to preserve some structure of 𝔐 andprevent that the pointed models (𝔐,w) and (𝔐,w') be translated to completely unrelated models. Finally, the condition that apparently divides the good from bad translations and gives a reasonable notion of multi-class expressiveness is the criterion of being finitely generated.For the sake of simplicity, some aspects of the definition below are not completely formalized.[For the complete formal definition, the reader may consult <cit.>.]Letbe a set of formulas generated by a set 𝒫 of propositional variables and a set 𝒞 of connectives. Let X={x_1,x_2,...} be a set of variables with 𝒫∩ X = ∅, and let ^X be the set of formulas generated by 𝒫∪{x_1,x_2,...} with the connectives 𝒞. Then we have (, p. 115): Let Ł_1 and Ł_2 be such that _i is generated by a set 𝒫_i of propositional variables and a finite set𝒞_i of connectives, for i ∈{1,2}. Let ϕ^X ∈^X_1, then a translation (,f): Ł_1 →Ł_2 is finitely generated ifcan be inductively defined by a finite number of clauses of the form (ϕ^X)= ψ^X for (x_1,...,x_n) ∈Ψ where ψ^X is an ^X_2-sentence constructed out of x_1,...,x_n and possibly containing (x_i), for x_i ∈^X_1; and where Ψ is the range ofthe x_i, e.g. if a given x_i is to be replaced by a formula or only by an atomic formula.The set X contains the special propositional variables to be used in the translation clauses, for which one can substitute formulas. An example of such a translation clause is: (x_1 → x_2) = ((x_1) (x_2)) for (x_1,x_2) ∈_1 ×_1; and (x_1)=x_1 for x_1 ∈𝒫. The idea is that (, p. 110) it is the fact of being inductively defined and thus respecting (some) of the structure of the formulas that sets the finitely generated translations apart from the trivial translations. Thus Kuijer concludes that the truth-preservingtranslations giving rise to an expressiveness relation could be characterized as the ones being finitely generated and model-based. Therefore, the final criterion given for multi-class expressiveness is (, p. 111) Let Ł_1 and Ł_2 be such that _iis generated by a set 𝒫_i of propositional variables and a finite set𝒞_i of connectives for i ∈{1,2}.Then Ł_2 is at least as expressive_g asŁ_1 iff there is a translation (,f) from Ł_1 to Ł_2 that is model based, finitely generated and truth preserving.§.§.§ A problem with expressiveness_g Kuijer had no pretensions that his multi-class definition were to be the generalization of expressiveness as given by the single-class framework. The aim was to find only a “reasonable generalization” (, p. 83). While keeping this in mind, we would like to argue that his proposal is still not good enough as a criterion for multi-class expressiveness. This is because one can find a pair of logics Ł, Ł' such that Ł' is intuitively more expressive than Ł, althoughŁ is at least as expressive_g as Ł'.The logics at issue are Epstein's relatedness logic () <cit.> and classical propositional logic (). The logicbesides the truth-functional connectives, has a relevant implication “→", which is the reason it is intuitively more expressive than , which lack such a connective.The referred translation would imply thatis at least as expressive_g as .Despite the circumscribed character of Kuijer's criterion, we think that a reasonable generalization of single-class expressiveness should be able to deal with a reasonable amount of logics, not only with a handful of them. Particularly when the logics at issue are in the literature, and have not been constructed in an ad-hoc fashion just to give a counter-example. Finally, there is nothing specific about the logics appearing in the counter-example, so it is quite possible that there are also modal counter-examples.Epstein presentswith the connectives ,, →. The first two are defined as usual and the underlying idea for interpreting the relevant implication symbol “→" is as follows. It holds that p → q whenever p materially implies q and both are subject-matter related to each other through a relation ℛdefined on all propositional variables. Specifically, for propositional variables p_i,p_j and -sentences ϕ and ψ, ℛ(ϕ,ψ) holds if and only if for some p_i occurring in ϕ, and p_j occurring in ψ, it holds that ℛ(p_i,p_j). Thus, the truth table for “→" is the one for material implication with an additional column for ℛ, so that if ℛ(ϕ,ψ) holds and (ϕψ) is true, then ϕ→ψ is true; else, if ℛ(ϕ,ψ) does not hold, then ϕ→ψ is false. Let τ= {p_0,p_1,..., , →, } be a signature for . An -model (𝔐,ℛ,v) is formed by the truth-tables for , , →, a symmetric and reflexive relation ℛ on τ-formulas and a valuation v.For propositional variables d_i,j, let τ^+ = {p_1,p_2,...}∪{d_i,j| i,j ∈ℕ}∪{,, ⊃}. Letbe defined on τ^+ (note we use ⊃ here to emphasize that it is a material implication).[The use of new propositional variables is for the sake of simplicity, as we could arrange the p_1,p_2,... inso as to assign some of the p_is the role of such d_i,j.] We will see below that there is a truth-preserving, model-based and finitely generated translation (^E, f^E): ⟶. The mapping ^E is defined as follows:[The mapping presented was adapted from (, p. 299). It was given a simpler form which makes the proof of the theorem below straightforward.We refer to Epstein's mapping as ^E^*, which is identical with ^E except for →, where^E^*(ϕ→ψ)=(^E^*(ϕ) ⊃^E^*(ψ))[ (p_iin ϕ,p_jin ψ⋁ d_i,j)(p_nin ϕ,p_nin ψ⋁ (d_n,nd_n,n))]. Notice that our mapping ^E below is only truth-preserving while Epstein's ^E^* is also validity-preserving, as e.g. ^E(p → p) = (p ⊃ p)d_p,p and ^E^*(p → p)= (p ⊃ p)[d_p,p (d_p,p d_p,p)].] * ^E(ϕ→ψ) = (^E(ϕ) ⊃^E(ψ))d_ϕ,ψ * literal for , and atomic formulas.[Kuijer requires alsothat no propositional variable occurs outside the scope of a translation function, so for atomic formulasone should useadditional functions s: 𝒫⟶𝒫. Thus we can take the identity function as such s.] Here the basic idea for the translation of ϕ→ψ comes from the definition of “→”: ϕ materially implies ψ and both formulas are related through ℛ. As the translation is defined inductively through the formation of formulas by a finite number of clauses, it is finitely generated.Now, from an -model (𝔐, ℛ, v), one easily defines a transformation f^E from -models to -models. Let f^E(𝔐, ℛ, v)= (𝔐^*, v^*), where, for 𝔐^* take all the truth-tablesin 𝔐, excluding the one for →. Define v^* as follows (adapted from <cit.>): * v^*(p_i)= v(p_i); * v^*(d_ϕ,ψ) = T iff ℛ(ϕ,ψ) holds.Clearly f^E is model-based. Bothandsatisfy a semantic deduction theorem (, p. 299). To prove that (^E,f^E) is truth-preserving, one has to prove only that, for an arbitrary -model (𝔐, ℛ,v), it holds that (𝔐, ℛ,v) _ϕ if and only if f^E(𝔐, ℛ,v) _^E(ϕ). is at least as expressive_g as .The main question now is: does (^E,f^E): ⟶ show thatis at least as expressive as ? We do not think it is reasonable to say so, since the extra expressiveness brought about by the implication connective inis only by a trick mimicked in . Independently of the model-translation f^E to give the intended truth values for the “relevance-mimicking" variables d_ϕ,ψ, it is not possible to have a relevant conditional in , by say, adjoining to a conditional ϕ⊃ψ such variables d_ϕ,ψ. To do so, would require too much for the intended meaning of such variables. Surely this would not augment the expressive power of the propositional logic, as it concerns only an interpretation of propositional variables, and intuitively, specific interpretations of propositional variables do not influence the expressiveness of a logic.Anyway, the model-mappings are not essential for these translations using indexed variables,they only facilitate their definition. An early example was given by Richard Statman in <cit.> where a translation ofinto its implicational fragment ^{→} is presented. There, the conjunctions pq are mapped to implications containing x_p q, among formulas of the sort x_p → (x_q → x_pq), x_pq→ x_p, etc. Here the situation is entirely different since the proof-theoretic behaviour of individual conjunctions are encoded in specific variables using implicational axioms. Coming back to Kuijer's criterion, we argued above that it is not enough to give an intuitively adequate account of expressiveness. If the model mapping were not from the source logic to the target logic but vice-versa, then there would not be such truth preserving mappings fromto , as there would be no way to construct the relatedness predicate ℛ out of a -model. Kuijer discarded such a definition of the model mappings f since itimplies that any truth-preserving translation is also validity preserving,[Suppose that for logics Ł=(, , _Ł) and Ł'=(','_Ł') that (,f): Ł⟶Ł' is truth-preserving, with :⟶' and f: ' ⟶. Suppose ϕ is Ł-valid, then for any model ' ∈', f(') _Łϕ, thus, by truth-preservation, ' _Ł'(ϕ), but ' is any Ł'-model, thus, (ϕ) is Ł'-valid.]and some of his paradigmatic examples of multi-class expressiveness are not validity preserving. Let us analyse a possiblestrengthening on the formula translation. We will not give a detailed analysis of features of translations since it suffices to notice that Epstein's translation preserves completely the structure of the formulas, except for →. For this case, additional propositional variables d_ϕ,ψ must be introduced to bear the intended meaning of ℛ (variables whose interpretation inis sustained by the model translation.) If one required thatbe compositional, that is,every n-ary connective C(ϕ_1, ..., ϕ_n) of the source logic is translated by a schema C^((ϕ_1)/ξ_1,...,(ϕ_n)/ξ_n) of the target logic, then the above translation would not pass the test. This is because p_1 → p_2 is translated through the schema (ξ_1 ξ_2)d_p_1,p_2, and p_3 → p_4 by the schema (ξ_1 ξ_2)d_p_3,p_4. If the translation were compositional, dealing with the same connective, the same translation schema would be used. The problem of adopting this criterion is that it implies that the connectives be translated one at a time, and again some of the paradigmatic translations selected by Kuijer takes into consideration sequences of connectives, so they would not satisfy it.Therefore, to prevent translations such as those above from passing the test for multi-class, one would have to use a criterion forthat is stronger than being finitely generated, but weaker than being compositional. Nevertheless, the enterprise of placing restrictions on the formula translationsalone seems not to be promising, as the model-translations play a major role in the counter-examples presented above. On the other hand, placing also restrictions on model-translations and making them fit with the restrictions on formula-translations is a very complex enterprise, and there may be better alternatives.Given this situation, we would like to suggest a change of perspective as regards relative expressiveness between logics. Below, some comments will be made regarding the nature of the notion of expressiveness and its relation with the concept of logical system it applies to. §.§ Single-class expressiveness vs multi-class expressiveness vs translational expressivenessNow we would like to make some remarks on the study of the relation of expressiveness between logics. As we commented before, in the single-class framework it is very simple to define relative expressiveness, since there is a common ground, the structures, where one can compare whether the sentences have the same meaning. Now consider the multi-class framework, if Ł=(,,) and Ł'=(',',') are defined on different classes of structures, how would we know whether an Ł-sentence ϕ and an Ł'-sentence ψ have the same meaning? After all, in this case it trivially holds that Mod_Ł(ϕ)≠ Mod_Ł'(ψ). As we saw, for this task new tools are needed: a model-mapping f: ℳ⟶ℳ' or f': ℳ'⟶ℳ;[The translation f presupposes a mapping σ of signatures: for each Ł[τ]-structure, there would correspond a Ł'[σ(τ)]-structure, respectively for f'.] and a formula-mapping : ⟶' or ': ' ⟶. Now, for an Ł-formula ϕ and Ł'-formula ψ, we would have some possibilities for guessing when ϕ and ψ have the same meaning:2 * Mod_Ł(ϕ) = Mod_Ł('(ψ)), * Mod_Ł'(ψ)= Mod_Ł'((ϕ)), * f[Mod_Ł(ϕ)]= Mod_Ł'(ψ),[Let f[Mod_Ł(ϕ)]= {f() |∈ Mod_Ł(ϕ)}.] * f'[Mod_Ł'(ψ)] = Mod_Ł(ϕ). Thus, now the weight goes on the notion of translation (,f). As we saw in the examples presented above, for (,ϕ) in Ł, and (',ψ) in Ł', the task of establishing the congruence between the pairs (, ϕ) and (', ψ)by means of translations is very difficult. Basing it on satisfaction is far away from being sufficient, since we can easily devise translation functions such thatsatisfies ϕ iff ' satisfies ψ.On the other hand, imposing conditions on (,f) is a complex enterprise, because either it under-generates or, by a little breach, it over-generates. Moreover, the need to have model-mappings besides formula-mappings may open up a back door to undesirable translations, to see it, consider again the examples offered against Garca-Matos & Vnnen's and Kuijer's approaches. All of them use some “trick” in the formula-translation function and sustain it through the model-translation. Then it is of little help to place structural restrictions on formula-translations, as did Kuijer. He also tried placing restrictions on model-translations, but it did not help either. Therefore, it might be more promising to move to a wider framework of relative expressiveness, dispensing with the semantic notions altogether.In this framework, to be called “translational expressiveness”, we would then concentrate the investigations on the conditions on formula translations. The aim is to find the set of conditions that better preserve/respect the theoremhood/consequence relation and the structure of formulas of each logic. This way a reasonable formal criterion of expressiveness for Tarskian and proof-theoretic logics (TPL, for short) would be obtained, and a bigger range of logics would be comparable. Finally, these advantages would arguably come at no cost, since this wider enterprise would not be more difficult than multi-class expressiveness. The big difference between the approaches of expressiveness is not in the division between expressiveness for model-theoretic logics and for TPL, but in the division, in model-theoretic logics, of expressiveness within the same and within different classes of structures. Naturally the most direct concepts of expressiveness are linked with the capacity of characterizing structures, but this only applies when comparing the same class of structures. If one allows translations between structures, such capacity is no longer at issue.Once we depart from the safe harbour of a single class of structures for comparing logics, then all bets are off. Multi-class expressiveness does not guarantee a firmer grasp of the intuitive concept of expressiveness anymore than translational expressiveness. Since the move to a wider framework might not only free us from problems inherent to multi-class expressiveness, but also allow a bigger range of comparison of logics, then the prospects for the enterprise are better. As we said in the introduction, people have been using informally some concepts of translational expressiveness between logics. However, as opposed to what happens with model-theoretic logics, to the best of our knowledge, in the literature there is only one explicit and formal criterion in this framework, that of <cit.>. In the next section, their proposal will be analysed and we will show that it is not adequate. We shall then propose some adequacy criteria for expressiveness and a formal criterion in the framework of translational expressiveness will be given. We then argue that the criterion satisfies the adequacy criteria.§ TRANSLATIONAL EXPRESSIVENESS: OBTAINING A STILL WIDER NOTION OF EXPRESSIVENESS In this section we will deal with logics in the Tarskian and proof-theoretic sense. We also mention logics taken as a closed set of theorems/validities, to be called simply “formula logics". Let Ł_1 and Ł_2 belogics, Γ∪{ϕ} be a set of Ł_1-formulas anda translation mapping Ł_1-formulas into Ł_2-formulas in such a way that for each Ł_1-formula ϕ: ⊢_Ł_1ϕ if and only if ⊢_Ł_2(ϕ). In this case Ł_1 is translatable into Ł_2 with respect to theoremhood. If it is the case that Γ⊢_Ł_1ϕ if and only if (Γ) ⊢_Ł_2(ϕ) thenŁ_1 is translatable into Ł_2 with respect to derivability<cit.>. The later translations are known as conservative translations <cit.>. A conservative translation is a translation with respect to derivability. Whenever we want to refer indistinctly to translations with respect to theoremhood or conservative translations, the term back-and-forth will be employed. A translation is back-and-forth if it is either a theoremhood preserving or a conservative translation.§.§ Mossakowski et. al.'s approachAs far as we know, Mossakowski et al. <cit.> proposed the first explicit criterion for the concept of sub-logic and expressiveness in the framework of translational expressiveness:Ł_1 is a sub-logic of Ł_2 if and only if there is aninjective conservative translation from Ł_1 to Ł_2;Ł_1 is at most as expressive as Ł_2 iff there is a conservative translation α: Ł_1 ⟶Ł_2.The authors do not explain why sub-logic requires injective conservative mappings while expressiveness does not. Anyway, we will see that these criteria for sub-logic and expressiveness via conservative mappings do not work. The conception that conservative translations could give rise to a notion of expressiveness and also a notion of logic inclusion has been supported more than once. For example, in<cit.> it is said that the existence of a conservative translation (maybe injective or bijective) would give rise to some kind of logic inclusion between Tarskian logics.[The author says ():If we assume (...) a Tarskian perspective, then a logic system is nothing more than a set of formulas together with a [consequence] relation (...) Thus, the preservation of that relation by a conservative translation [from Ł_1 to Ł_2] would reveal that, as structures,Ł_2 “contains” Ł_1 (Probably we should add the requirement that f is an injective or even a bijective mapping.)] Also for Kuijer, conservative translations give an adequate concept of expressiveness for Tarskian logics <cit.>.[The author says ():There is a conservative translation from Ł_1 to Ł_2 if and only if everything that can be said in Ł_1 can also be said in Ł_2. ]Unfortunately, conservative translations will not make a reasonable concept neither of sub-logic nor of expressiveness. Due to a result of Jeřbek <cit.>, explaining expressiveness and sub-logic through conservative translations would makeinclude and be at least as expressive as many familiar logical systems, e.g. first-order logic. He proved the following result (, p. 668), where for a logic Ł, a translation is most general whenever it is equivalent to a substitution instance of every other translation of Ł to . For every finitary deductive system Ł= (, ⊢) over a countable set of formulas , there exists a conservative most general translation : Ł→. If ⊢ is decidable, then f is computable.The defined mapping is injective.[For the sake of brevity, we omit the definition of the translation and simply point out that it is a non-general-recursive translation (to be defined below).] Let a logic be called “reasonable” if it is a countable finitary Tarskian logic. Jeřbek managed to generalize even more his results so that almost any reasonable logic can be conservatively translated into the usual logics dealt with in the literature.[Among others, classical, intuitionistic, minimal and intermediate logics, modal logics (classical or intuitionistic), substructural logics, first-order (or higher-order) extensions of the former logics.]Now one would hardly accept that everycountable finitary logic has the same expressiveness or is one sub-logic of the other.The author criticizes the notion of conservative translation for not requiring the preservation of neither the structure of the formulas nor the properties of the source logic <cit.>. Thus, it must be strengthened in order to serve for an expressiveness measure.This could be done in a simpler way by requiring injective, surjective or bijective mappings. As Jeřbek's mapping is injective, only requiring injectiveness will not do. As a matter of fact, it seems that already requiring injectiveness one is overshooting the mark. Since in this way ^{, } would not be as expressive as ^{, , }. Any mapping g:^{, , }⟶^{, } would have to map both ^{, , }-sentences ϕψ and (ϕψ) to the same ^{, }-sentence (g(ϕ)g(ψ)), so it would not be injective. Other kinds of strengthening hinted byJeřbek's () are:* force the mappings to preserve more structure of the source logic sentences in the target logic;* force the mappings to preserve more properties of the source logic.The adequacy criteria for expressiveness to be given below will require to some extent (1) and (2). §.§ Adequacy criteria for expressivenessAs we saw above, Mossakowski et al. <cit.> gave a proposal for a wide notion of expressiveness: by means of the existence of conservative translations. Due to Jeřbek's results on the ubiquity on this kind of translation, their definition is not adequate. Maybe we should step back and think about some adequacy criteria every approach to expressiveness ought to accomplish. The intuitive explanation for expressiveness (E) given in the beginning elucidates relative expressiveness in terms of a certain congruence of meanings. It appears already in a more direct form in Wjcicki's Theory of Logical Calculi <cit.>, and we place it as the first adequacy criterion [Adequacy Criterion 1] Ł_2 is at least as expressive as Ł_1 only if everything that can be said in terms of the connectives of Ł_1 can also be said in terms of the connectives of Ł_2. Here, for “being said in terms of the connectives” there can be stricter interpretations (as proposed by Wjcicki, Humberstone, Epstein) and wider interpretations (as proposed by Mossakowski et al. and us), to be developed below. There are some meta-properties of logics that are intuitively known to limit or increase expressiveness. Thus, the presence/absence of such properties can be used to test whether there can be or not an expressiveness relation between the given logics. A first one coming to mind is that nothing can be expressed in a trivial logic, so it cannot be more expressive than any logic. Another one has to do with the relation between expressiveness and computational complexity.This relation has even been stated as the “Golden Rule of Logic” by van Benthem in <cit.>, where he says “gains inexpressive power arelostinhigher complexity”. Nevertheless, the “Golden Rule” is not quite useful here, since we know that in general neither a low expressiveness means low complexity,[For example, there are propositional logics whose complexityis ineach arbitrary degree of unsolvability (e.g. see <cit.>).] nor a high complexity means high expressiveness.[There can be equally expressive logics that, though both decidable, have very different computational complexities (e.g. see <cit.>).] Nevertheless the complexity levels of decidability/undecidability can be useful for expressiveness comparisons: if a logic is decidable, then it cannot describe Turing machines, Post's normal systems, or semi-Thue systems. Therefore, a decidable logic Ł cannot be more expressive than an undecidable logic Ł', otherwise, Ł would not be decidable! The third meta-property that could be useful when evaluating expressiveness relations (except, naturally, when dealing with formula-logics) is the deduction theorem. Though involved in many formulation issues, as we shall see, a logic has a deduction theorem when it has the capacity to express in the object language its deductibility relation. Thus, other things being equal, a logic having this capability is intuitively more expressive than another one lacking it. Therefore, it is desirable that an expressiveness relation carries with it the deduction theorem, so that (a) below apparently should hold 0.1 (a)0.9 if Ł_2 is more expressive than Ł_1, and Ł_1 has a deductiontheorem, then so does Ł_2.We have some issues here. Being formulation sensitive, it is complicated to define in which circumstances the existence of a deduction theorem for a logic implies its existence in another logic, whenever there is an expressiveness relation between them.For example, a less expressive logic might have the standard deduction theorem,[To be defined below.] while the more expressive logic has only a general version of it, or perhaps lacks it completely.This happens with Mendelson's ,[A Hilbert-style first-order calculus with the generalization rule “from ϕ infer ∀ x ϕ”. For more, see <cit.>.] the propositional fragment of it still satisfies the standard deduction theorem, though it fails for quantified formulas. So, it does not seem reasonable to say that this formulation ofis not more expressive than , because it does not satisfy the standard deduction theorem, since the fragment ofas expressive assatisfies it.[The same considerations apply to Ł_TK described in <cit.> and <cit.>.] Cases like these constrain us to limit the role of the deduction theorem in expressiveness relations, admitting wider formulations of it.Thus we are forced to adapt (a) accordingly so as to be able to take into account such phenomena. Finally, we have the meta-property related adequacy criterion.[Adequacy Criterion 2] It cannot hold that Ł_2 be more expressive than Ł_1 when * Ł_1 is non trivial and Ł_2 is trivial; * Ł_1 is undecidable and Ł_2 is decidable; * Ł_1 satisfies the standard deduction theorem and the language fragment of Ł_2 purportedly as expressive as Ł_1 does not satisfy (not even) the generaldeduction theorem; The last criterion reflects the intuition that expressiveness is a transitive relation and there are logics that are more expressive than others.[Adequacy Criterion 3] (Taken from <cit.>) The expressiveness relation should be a non-trivial pre-order, that is, it should be a transitive and reflexive relation, and there must be some pair of logics Ł_1 and Ł_2 such that Ł_2 is not at least as expressive as Ł_1.We now analyse with greater detail the criteria 1 and 2. §.§.§ Criterion 1- on “whatever can be said in terms of the connectives”We can understand this criterion as saying “every connective of Ł_1 is definable in Ł_2”. But the usual notion of definability is either treated within the same logic, or between different logics within the same class of structures.As we intend to deal with translations between logics, the usual notion of definability is too rigid. We must give a broader reading of the criterion 1 in order to understand it as imposing an intuitive restriction on translations between logics. Thus the idea is to impose restrictions P_1,P_2,... on translations so that: Ł_1 ⟶Ł_2satisfies P_1,P_2,...only if, intuitively, everything that can be said in terms of the connectives of Ł_1 can also be said in terms of the connectives of Ł_2; let us say in shorter terms that this happens only if the connectives of Ł_1 are generally preserved in Ł_2.In the sequence some candidates for such P_1,P_2,... are listed, the back-and-forth condition was given before. A translation : Ł_1 ⟶Ł_2 is compositional whenever for every n-ary connective # of Ł_1 there is an Ł_2-formula ψ^# such that (#(ϕ_1,...,ϕ_n))= ψ^#((ϕ_1),...,(ϕ_n)).A grammatical translationis a back-and-forth compositional translationsuch that, for a sentence ϕ, (ϕ) may contain no other formulas other than the ones appearing in (p), where p appears in ϕ (thus, no parameters are allowed). A definitional translationis a grammatical translation for which (p)=p for every atomic p. We have four proposals for filling the above list of restrictions. All of them require basically two conditions, taking as P_1 the back-and-forth condition.In decreasing order of strictness, there is divergence in taking P_2 as a * definitional translation (Wjcicki and Humberstone), * grammatical translation (Epstein and apparently Koslow), * general-recursive translation (to be defined below), * surjective conservative translation (Mossakowski et al.). Humberstone <cit.>, recalling Wjciki's definitional translations and intuitions about expressiveness, guessed that if there is a definitional translation between Ł_1 and Ł_2, then all connectives in Ł_1 are preserved in Ł_2.[However, it seems that in <cit.> he allows that connectives are preserved in a weaker way, through compositional translations.] For us, the existence of a definitional translation from Ł_1 to Ł_2 is the strongest guarantee that the connectives of Ł_1 are generally preserved in Ł_2. Nevertheless, it is too strict a requirement, and there are weaker forms of translations that can also do the job. For Epstein <cit.>, a grammatical translation is a homomorphism between languages and thus it yields a translation of the connectives. The justification is that such translationsare only possible when for each connective in the source logic, there corresponds a specific structure in the target logic that behaves similarly. Thus, through a grammatical translation, the connectives of the source logic are generally preserved in the target logic. Koslow <cit.> also allows that a connective from one logic Ł_1 “persists” in Ł_2 if there is a homomorphism from Ł_1 to Ł_2. According to Mossakowski et al. <cit.>, grammatical translations are too demanding for the task, as many useful and important translations are non-grammatical (e.g. the standard modal translation). For them, instead of seeking to preserve the structure of the formulas, it would be better to preserve the proof-theoretic behaviour of the connectives and to treat the connectives only as regards this behaviour (, p. 100). In this paper, some proof-theoretic conditions on the connectives are listed, e.g. for conjuntction the condition is Γ⊢ϕψ iff Γ⊢ϕ and Γ⊢ψ. This formulation may lead one to think thathere shall be a logical constant, and not possibly a formula γ(ϕ,ψ) (think of ^{¬,}, where γ(ϕ,ψ)= (ϕψ)); naturally in the first case, the whole proposal would make no sense. In table <ref>we reformulate the conditions to reflect their proposal more clearly, where δ^# is an arbitrary formula that stands for the connective #. A proof-theoretic connective is present in a logic if it is possible to define the corresponding operations on sentences satisfying the conditions given in table <ref>. We shall now investigate this idea in detail and argue that, as it is, the preservation of connectives would require mappings stricter than conservative translations otherwise the notion of the “presence” of a connective must be relaxed.Drawbacks on the preservation of proof-theoretic connectives A translation :Ł_1 ⟶Ł_2 transports a given Ł_1-connective # if its presence in Ł_1 implies its presence in Ł_2, the converse implication is called reflection <cit.>.It is claimed () that if a mapping : Ł_1 ⟶Ł_2 is conservative and surjective, then all proof theoretic connectives of Ł_1 are transported to Ł_2 and all proof-theoretic connectives present in Ł_2 are reflected in Ł_1. However, this claim must be taken with a grain of salt, let us see why. Let Ł_1 be a logic having a proof-theoretic conjunction according with the table <ref> above and suppose there is a surjective conservative mapping : Ł_1 ⟶Ł_2. For Ł_2-formulas δ_1,δ_2, let Γ∪{ϕ,ψ} be a set of Ł_1-formulas with (ϕ)=δ_1 and (ψ)=δ_2. Then it holds that ((Γ) ⊢_Ł_2(ϕ) and (Γ) ⊢_Ł_2(ψ))iffΓ⊢_Ł_1δ^(ϕ,ψ)iff(Γ) ⊢_Ł_2(δ^(ϕ,ψ)).Thus, Ł_2 would have proof-theoretic conjunction. The grain of salt is that, once no structural restriction is imposed upon , it is not necessary that(δ^(ϕ,ψ)) be constructed out of (ϕ) and (ψ). In this case, it seems at least unnatural to say that(δ^(ϕ,ψ))is an operation on the sentences (ϕ) and (ψ). Therefore,we must relax what it means for a connective to be present in a logic. One has to say that e.g. the proof-theoretic conjunction is present in a logic Ł_2 if, for all formulas δ_1,δ_2 and set of formulas Δ, there is a formula γ such that (Δ⊢_Ł_2δ_1 and Δ⊢_Ł_2δ_2) iff Δ⊢_Ł_2γ.A similar reformulation should be given for the other connectives. In this case, though, whenever it holds that Δ⊢_Ł_2δ_1 and Δ⊢_Ł_2δ_2, then any Ł_2-theorem in the place of γ serves to satisfy this condition for conjunction. For example, take a Tarskian logic Ł defined on the signature {p,q,r, ⊤}, where p,q,r are propositional variables and ⊤ the constant for logical truth. Then Ł has proof-theoretic conjunction since p,q ⊢ pand p,q ⊢ q holds iff p,q ⊢⊤. This is probably unproblematic and a consequence of the meaning of ⊤.Nevertheless, for some casesthis approach to the presence of connectives has some downsides. For example, restrict Ł to the signature {p,⊤}. Then Ł has the proof-theoretic conditional, since it holds thatp ⊢ p iff ⊢⊤, ⊤⊢ p iff ⊢ p, p ⊢⊤ iff ⊢⊤and ⊤⊢⊤ iff ⊢⊤. But if the signature were incremented by another variable q, then the resulting system would no longer have a proof-theoretic conditional, since for no δ it would hold that p ⊢ q iff ⊢δ. This volatility of the presence of proof-theoretic connectives is unreasonable. Recapitulating, the idea of this approach is that one shall define the mappings so as to preserve the proof-theoretic connectives, instead of requiring the mappings themselves to preserve the structure of the formulas.But if the mappings do not respect the structure of the formulas, what shall be called the presence of a connective, must also be relaxed.Besides the inconvenients mentioned above, this proposal would be too restrictive in some cases. For example, Statman's translation <cit.> of into its implicational fragment showshow can one “express” (in some sense of the term) conjunctions using only implicational formulas; recent works have generalized this result so that any logic having a certain natural deduction formulation and having the sub-formula principle is translatable into the implicational fragment of minimal logic <cit.>.[The idea of these translations is the following: for a given -formula ϕ, take all sub-formulas δ_1,δ_2 and associate to it implicational axioms of the sort x_δ_1 δ_2→ x_δ_1,x_δ_1 δ_2→ x_δ_2 and x_δ_1→ (x_δ_2→ x_δ_1 δ_2), where x_δ_1, x_δ_2 and x_δ_1 δ_2 are fresh variables.]Nevertheless, not even in the weaker sense given above the conjunctions are “present” in ^{→}. Anyway, it must be borne in mind that to give a good and general definition of when a connective or operator is generally preserved is a difficult and spinous topic. Below we give another proposal, which is at the same time weaker (the translation mentioned above would enter) and stronger (requires structure-attentive mappings). Let us now consider the structure-attentive translations and think on the minimum conditions on the preservation of the structure of formulas that would allow for a reasonable and general notion of preservation of connectives. General-recursive translations: allowing context-sensitivity in a general preservation of connectives The criterion of compositionality given above a priori seems a reasonable condition for the preservation of connectives through translations. Notice that in the criterion the functionthat translates #(ϕ_1,...,ϕ_n) is the same that translates the sub-formulas ϕ_i. From this comes the compositionality: a translationof a formula is obtained through the same translationof its sub-formulas. Thinking about the issue of translating a connective, it is also reasonable that the translation be sensitive to the context where the connective is inserted.This is the case in the translation (_+): Grz ⟶ S4 in <cit.>, where_+(□ p)= □ p, but _+(□ p) = □ (□(p →□ p) → p). Therefore, _+ distinguishes between translating □-formula and □-formula, and this is done through the help of an auxiliary translation (see complete definition in section <ref>).Thus, there are translations between some logics where the mappings must be context-sensitive, so as to convey the proper meaning of some source connectives in the target logic. There are also those cases where the connectives can be dealt context-independently but auxiliary translations are needed anyway. The standard translation of modal logic to , besides some parameters, needs n auxiliary translations for each formula of modal degree n e.g.as ^x(p)=Px but ^x(□ϕ)= ∀ y (Rxy →^y(ϕ)). For the sake of simplicity, we will restrict our notion of context-sensitivity to whether or not the connective to be translated is in the scope of an unary operator. When the translation of a n-ary connective # is sensitive as to whether it is on the scope of an unary ∘, a simple solution is to treat ∘# as a composite n-ary connective to be translated. With the aim of capturing these cases, let us considera sufficiently general kind of translation. French in <cit.> presents a concept of recursively interdependent translation that includes non-compositional translations that are still defined recursively through the formation of formulas. A generalization of his concept will be employed here, since the original has an unmotivated restriction allowing only unary auxiliary mappings. The generalization allows auxiliary mappings of any arity and also has a simpler notation. Let Ł_1=(_1, ⊢_Ł_1) and Ł_2=(_2,⊢_Ł_2) be logics, Let '_1,...,'_w be auxiliary mappings of any arity defined inductively on _1-formulas.A translation : _1 ⟶_2 from Ł_1 to Ł_2 is general-recursive if, for every n-ary connective # and formulas ϕ_1,...,ϕ_n ∈_1, there is an Ł_2-formula #^(p_1,...,p_m) containing only the shown propositional variables p_1,...,p_m, such that (#(ϕ_1,...,ϕ_n)) = #^('_1(ϕ_i,...,ϕ_j)/p_1,...,'_w(ϕ_h,...,ϕ_l)/p_m)where {ϕ_i,...,ϕ_j}∪{ϕ_h,...,ϕ_l}⊆{ϕ_1,...,ϕ_n}. Notice that the clauses must be given for each single connective in the source logic. If there is a need to translate a composite connective, an additional clause for it should be given. Therefore, the general-recursive translations are still structure-preserving and must be defined inductively through the formation of formulas. Later in section <ref>we argue that, together with some other conditions, general-recursive translations preserve, in a general but reasonable sense of the term, the connectives of the source in the target logic. *Another issue with translated connectives One might insist whether the behaviour of the defined connective in the target logic would indeed be equivalent with the behavior of the original connective. Corcoran argues that this is often not the case. In <cit.> he defines a notion of “deductive strength” which is based on the capacity of a logic to introduce and eliminate a connective occurring as a principal sign in a formula. Considering this notion, it can be that a given connective # of a logic Ł_1 be definable in a logic Ł_2 through a translation, nevertheless, the “deductive strength” of Ł_2 as regards # is lower than the corresponding one in Ł_1. For example, consider the two classical propositional logics ^{,→} and ^{,}, formulated as natural deduction systems. Translating the conditional from ^{,→} to ^{, } one obtains the following rule of inference: from the pattern of reasoning from ϕ to ψ, infer (ϕψ). Contrary to the rule for → in Ł_1 (the usual natural deduction rule), according to Corcoran this rule of Ł_2 is not rigorous, since it depends on the rules of the other connectives. Corcoran's considerations are very interesting, but we think that despite the fact that the defined connectives can lose “deductive strength” (in his terms) it is reasonable to say that they maintain expressive strength. The loss of “deductive strength” can influence other issues such as modularity, normalization, etc. but it does not affect directly expressiveness. Having revised the literature linked with the adequacy criterion 1 and stated our proposal, now the same will be done with respect to the adequacy criterion 2. §.§.§ Criterion 2- On the preservation of (some) meta-propertiesThere are two important issues here:(i) what is being understood as a meta-property(ii) what does it mean for a translation to preserve a meta-property of one logic into another It would seem desirable to have a general formal framework so that one could give precise answers to (i) and (ii). Nevertheless, the adequacy criterion 2 asks for preservation of specific meta-properties, and not of every meta-property of a certain kind. Thus, there is no need to place them in a fixed framework. Moreover the first two (non-triviality and decidability) have simple and exact formulations, so it is straightforward to stablish whether they are preserved by a translation. The only meta-property whose statement and definition of preservation need elucidation is the deduction theorem.As the framework(s) of (hyper) contextual translations[See <cit.>, <cit.> and <cit.> for a detailed presentation of both.]offers exact answers to (i) and (ii) above, we will investigate whether they are adequate for our purposes.In both, a logic is taken as an assertion calculus containing a set of formulasand a set of rules of inference between sequents. The difference between the frameworks is the kind of sequent allowed.The language of the assertion calculus includes schematic variables for sentences ξ_1,ξ_2,... and set of sentences X_1,X_2,..., so the calculus has also substitution and instantiation rules for dealing with those. In these frameworks, P is a meta-property of a logic Ł defined in the above terms whenever P can be formulated as an inference between sequents (or hyper-sequents), that is, if P can be formulated as a derived rule of Ł.For example, the deduction theorem can be formulated this way: from Γ, ϕ⊢ψ, infer Γ⊢ϕ→ψ. Now for the disjunctive property, one needs the richer framework of the hyper-sequents: from Γ⊢ϕψ, infer Γ⊢ϕ or infer Γ⊢ψ. A (hyper) contextual translation : Ł_1 ⟶Ł_2 is a mapping that is transparent to the schematic variables such thatif P is a meta-property of Ł_1 then, (P) is a meta-property of Ł_2. The transparency to schematic variables implies that (hyper) contextual translations by definition preserve structural properties such as left weakening: from X ⊢ξ, infer X,X' ⊢ξ.Consider now the finiteness property, i.e. if Γ⊢ϕ, then for a finite Δ⊆Γ, Δ⊢ϕ. It cannot be formulated in neither of the cited frameworks, indeed the same holds for the majority of other relevant meta-properties of logics: decidability, interpolation, cut-elimination, etc. Despite the limitation on expressible meta-properties, the framework of (hyper) contextual translations has also a clear answer to item (ii) above: a translationpreserves a meta-property P of the source logicif (P) is a derived rule of the target logic.In the sequence we use the example of the deduction theorem to argue that there can be some problems even with this strict notion of meta-property preservation. A limitation of the formulation of meta-property in the framework of (hyper) contextual translationsEven in the strict framework of (hyper) contextual translations, meta-properties are formulation-sensitive, so that one formulation of a meta-property P may hold for a logic Ł while other formulation P' fails for Ł. A paradigmatic example is the deduction theorem. In most formulations, e.g. for classical propositional logic () and intuitionistic propositional logic (), it is read as: If Γ, ϕ⊢ψ, then Γ⊢ϕ→ψ. Nevertheless, only a generalized version holds for Lukasiewicz Ł^3: If Γ, ϕ⊢ψ, then Γ⊢ϕ→ (ϕ→ψ) <cit.>. A similar issue occurs in systems containing proof-rules besides inference-rules,[A proof rule is of the form: from ⊢ϕ, infer ⊢ψ. An inference rule is of the form: from ϕ, infer ψ. The necessitation rule and generalization rules are sometimes defined as proof-rules: from ⊢ϕ, infer ⊢□ϕ; from ⊢ψ(x) infer ⊢∀ x ψ(x). Notice that both ϕ and ψ(x) must be theorems in their respective systems, otherwise one gets implausible inferences: from p it follows □ p, and from P(x) it follows that ∀ x P(x).] for example, Mendelson'sand modal logic with the necessitation rule. In both cases, only a modified version of the deduction theorem holds. For modal logic K, among other possibilities, the following deduction theorem holds <cit.>: if Γ, ϕ⊢_K ψ, and each of the propositional variables appearing in hypothesis Γ∪{ϕ} is in the scope of a modal operator, then Γ⊢_K ϕ→ψ.[Other formulation is given in <cit.>: if Γ, ϕ⊢_K ψ, and the rule of necessitation is applied m ≥ 0 times to formulas that depend on ϕ, then Γ⊢_K (□^0 ϕ ... □^m ϕ) →ψ, where □^0 ϕ = ϕ, □^1 ϕ= □ϕ, etc.]For Mendelson's system, the formulation of the deduction theorem is also clumsy <cit.>: “Assume that, in some deduction showing that Γ, ϕ⊢ψ, no application of [the generalization rule] to a wff that depends upon ϕ has as its quantified variable a free variable of ϕ. Then, Γ⊢ϕ→ψ.”So what is a deduction theorem? According to Zeman, the general statement of it might be <cit.>If there is a proof from the hypotheses ϕ_1,...,ϕ_n for the formula ψ, then there is a proof from the hypotheses ϕ_1,...,ϕ_n-1 for the formula ϕ_n ⊃ψ. For Zeman, the problem of the formulation of the deduction theorem for each system lies in the proper understanding in the system of what it is meant by a “proof from hypotheses”. Thus, the different results cited above for Ł^3, modal logic and Mendelson'sare different ways —seemingly equivalent modulo the specificities of each system— of capturing the idea of DT above. The situation is explained by Hakli and Negri <cit.> as follows. For some logics, either one modifies their rules in order for them to deal adequately with assumptions, and get the “standard formulation” of the deduction theorem, or leave the rules from the logic intact and obtain a “non-standard” form of the deduction theorem.[Although the moral of the story holds for first-order logic, as shown above, they only mentioned modal logics. Nevertheless, we do not know of any such rectification for the formulation of the deduction theorem in Lukasiewicz Ł^3.]Now let us come back to the issue of preservation of meta-properties by translations. Considerand modal logic S4,presented in the framework of (hyper) contextual translations, e.g. both equipped with a common set of propositional variables p_1,p_2,... and schematic variablesξ_1,ξ_2,..., X_1,X_2,..., for formulas and sets of formulas, respectively, etc.Consider Gdel's translation ^g:⟶ S4 (defined to be literal to schematic variables):2 * ^g(p_i) = □ p_i* ^g(X_i) = X_i* ^g(ξ_i) = ξ_i* ^g(ϕ) = □^g(ϕ)* ^g(ϕ→ψ)= □(^g(ϕ) →^g(ψ))* literal for ,,. Then the deduction theorem foris defined as the following meta-property 0.2 (P)0.7 if X, ξ_1 ⊢ξ_2, then X ⊢ξ_1 →ξ_2.Carnielli et al. (<cit.>) notice that ^g above is not a contextual translation since S4 does not satisfy 0.2^g(P)0.7 if X, ξ_1 ⊢ξ_2, then X ⊢□(ξ_1 →ξ_2). To see why, instantiate X to p_1 → p_2 and ξ_i to p_i, for i ∈{1,2}. In S4 it holds that p_1 → p_2, p_1 ⊢ p_2, but it does not hold that p_1 → p_2 ⊢□(p_1 → p_2).One can see clearly that this is caused by the transparency given in ^g to the schematic variables of P. If P were formulated in terms of non-schematic formulas, e.g. 0.2(P')0.7 if Γ, p_1 ⊢ p_2, then Γ⊢ p_1 → p_2,then its translation ^g(P') into S4 would be0.2^g(P')0.7 if ^g[Γ],□ p_1 ⊢□ p_2, then ^g[Γ] ⊢□(□ p_1 →□ p_2), which is satisfied in S4. Although P is a correct formulation of DT (see above) for , ^g(P) is not the correct formulation of DT for S4. Therefore, the claim that contextual and hyper-contextual translations preserve the meta-properties of logics (expressible in the framework) is not entirely justified. The opacity given for the schematic variables may give a“false negative” as regards the presence of some meta-property in the target logic, this would prevent the definition of such translations. Therefore this framework is not adequate for our purposes. General statement and preservation of the deduction theorem Recall our discussion on the deduction theorem. We saw that there are many formulations of it, and it depends on how the notion of proof from assumptions is treated in each logic. Now to talk about the preservation of deduction theorem through the translations, we have to give a sufficiently general formulation of it, but such that it still carries the spirit of Zeman's definition. Let us give it a more direct formulation:A logic Ł_1 has the standard deduction theorem whenever it holds that ϕ_1,...,ϕ_n ⊢_Ł_1ψ if and only if ϕ_1,...,ϕ_n-1⊢_Ł_1ϕ_n →ψ. The general formulation has to be lax enough so as to enable one to say that, for example, the translation ^l:⟶Ł^3 preserves the deduction theorem, since it holds that “if Γ, ϕ⊢_Ł^3ψ, then Γ⊢_Ł^3ϕ→ (ϕ→ψ)”; analogously for the translation ofinto S4. The general version of the deduction theorem we propose is the following: A logic Ł_1 has the general deduction theorem whenever ϕ_1,...,ϕ_n ⊢_Ł_1ψ iff ϕ_1,...,ϕ_n-1⊢_Ł_1α^→(ϕ_n,ψ), where α^→ is an Ł_1-formula, with one or more occurrences of ϕ_n and ψ. In abstract algebraic logic this formulation is known as the uniterm global deduction-detachment theorem <cit.>. A translation :Ł_1 ⟶Ł_2 is said to preserve the general deduction theorem whenever Ł_1 has the standard deduction theorem and (Ł_1) has the general deduction theorem.The case where Ł_1 satisfies only the general deduction theorem is more complex, as it will be seen below.§.§ expressiveness_gg: a sufficient condition for expressivenessIn the adequacy criteria we proposed for expressiveness, there appears two informal necessary conditions: preserving the connectives and behaving in the appropriate way as regards the selected meta-properties. The other condition of being a non-trivial pre-order is already given precisely. The first two conditions are open to interpretation, so we proposed a precise formulation of the minimal requirements such interpretations would have to satisfy. This amounted on requiring the translation to preserve the general deduction theorem, and to be back-and-forth general-recursive. The one-way mappings between logics are a very weak in the sense they are almost omnipresent, so that requiring back-and-forth mappings as a formal necessary condition forexpressiveness is rather uncontroversial. Nevertheless, requiring structure-attentive translations in order to preserve the connectives has been questioned by Mossakowski et al. as we have seen above. They proposed other way to preserve the connectives without requiring such translations. We think this approach has some downsides and proposed a different one, based on general-recursive translations. Being a general-recursive mapping is a relatively weak condition on translations. If some translation does not comply with it is because at least some connective of the source logic is only translated “globally", i.e. a formula containing it is translated as a whole, and the translation ignores its eventual sub-formulas, e.g. Glivenko's double negation translation ofinto<cit.>.Thus, it is reasonable to require general-recursiveness as a formal necessary condition for expressiveness, along with theback-and-forth condition. One very important issue to be dealt with in a future work is already pointed out by Mossakowski et al. (<cit.>): this minimal notion of structure preservation is up to now only defined for propositional logics. It is to be investigated how it should deal with quantifiers. Notice, however, that this limitation does not weaken the necessary character of general-recursive translations, as an eventual wider approach should include it.Before we present a sufficient formal criterion for our concept, whose content surely is no surprise by now, some adjustments must be made concerning the preservation of the general deduction theorem. The general statement of the deduction theorem still involves compositionality: the formula α^→(ϕ_n,ψ) at issue should have ϕ_n and ψ as sub-formulas. In order to assure this, we have to define a slightly stricter notion of general-recursive translation, which we call general-recursive^C: A translationis general-recursive^C iffis general-recursive and it is compositional for the conditional symbol, that is, for a formula ϕ→ψ in the source logic and a template-formula C^(p_1,...,p_n) in the target logic, (ϕ→ψ)= C^((ϕ),...,(ψ)). Thus the translated formula may contain as sub-formulas one or more occurrences of (ϕ) and (ψ). The restriction on general-recursive translations is intended to assure that for the logics satisfying the standard deduction theorem, at least the translation clause for the conditional is compositional. This will rule out clauses such as (ϕ→ψ)= C^(_i_1(ϕ),...,_i_n(ϕ),...,_j_1(ψ),...,_j_n(ψ)), for _ij different from . Otherwise, it could happen that the resulting translation C^ of the conditional ϕ→ψ does not contain (ϕ) and (ψ) as sub-formulas. Then it would not be reasonable to say that such formula C^ expresses the deductibility relation between (ϕ) and (ψ).Now we present a sufficient criterion for expressiveness A logic Ł_2 is at least as expressive_gg as Ł_1 if and only ifthere is a back-and-forth general-recursive (for short, B&F-GR) translationfrom Ł_1 to Ł_2, such thatdoes not require model-mappings. If Ł_1 satisfy the standard deduction theorem, thenmust be general-recursive^C. Below it will be shown that expressiveness_gg satisfies the adequacy criteria given above. §.§.§ Adequacy criterion 1 As we mentioned before (section <ref>) there is some consensus in the literature that preservation of connectives requires at least compositional back-and-forth translations. The various non-compositional translations show that we could have a wider notion of preservation of connectives. We now argue that through general-recursive translations and some other conditions, it is still guaranteed that whatever can be said in terms of the source connectives can be said in terms of the target connectives. If a translation : Ł_1 ⟶Ł_2is back-and-forth and general-recursive (B&F-GR), anddoes not require model translations to convey the meaning of some connective in Ł_1, then the connectives of Ł_1 are preserved (in a general sense) in Ł_2. The back-and-forth condition shall mean either a theoremhood- or derivability-preserving translation, depending on whether one is considering formula logics or Tarskian logics, respectively. A back-and-forth translation assures a certain similarity between the global deductive behaviour of the source andtarget formulas. Though, as Jeřbek's result shows, it is not enough for any reasonable notion of connective preservation and there must be some extent of structure preservation.In this sense, the advantage of compositional translations is that they are particularly regular, so that each connective in the sourceis associated to a fixed schema in the target logic, and the translation clauses are clearer. But this can be also a limitation on the means of translation, comparable to restricting translations in ordinary language to word-to-word mappings. There are many cases of logics where the translation of certain operators must consider their context, so that they have to be translated in block. We gave before some examples, for another one consider Balbani and Herzig's <cit.> translation ^bh of modal provability logic G into K4. For ^bh the result of translating □ϕ also depends on whether it occurs in the scope of a negation sign:^bh(□ p)= □ p, but (□ p) = □ (□ p → p). These cases cannot be captured in compositional translations and can only be dealt with in more complex non-compositional ones. The following clause is proposed as a refinement of adequacy criterion 1, capturing more precisely when a connective or group of connectives is generally preserved by a translation from Ł_1 to Ł_2.0.1 (α)0.9 for each n-ary (composite) connective ⊗ in Ł_1 andŁ_1-formulas ϕ_1,...,ϕ_n, there must be Ł_2-formulas δ^⊗(p_1,...,p_m) (possibly m ≠ n) and ψ_1,...,ψ_m such that⊗(ϕ_1, ..., ϕ_n) has asimilar deductive behaviour with δ^⊗(ψ_1/p_1,...,ψ_m/p_m). It is easy to see that back-and-forth general-recursive (B&F-GR) translations satisfy clause (α). In general-recursive translations, every connective of the source logic must be given inductive translation clauses, and translations for composite connectives may be given either as extra clauses, or by means of auxiliary translations. Thus, for a n-ary (composite) connective ⊗ in Ł_1, the formula ⊗(ϕ_1,...,ϕ_n) must be mapped by a GR-translation to a formula δ^⊗(ψ_1,...,ψ_m), where each ψ_i is obtained from the translation of some ϕ_k. Now if the translation is back-and forth, then ⊗(ϕ_1,...,ϕ_n) will have a similar deductive behaviour with δ^⊗(ψ_1,...,ψ_m), since Γ⊢_Ł_1⊗(ϕ_1,...,ϕ_n) iff (Γ) ⊢_Ł_2δ^⊗(ψ_1,...,ψ_m). Nevertheless, the satisfaction of (α) is still not enough guarantee for general preservation of connectives, as Epstein's translation ^E:⟶ we saw above (section <ref>) is B&F-GR.We asserted then that the relatedness implication “→” is not expressible in . The issue is that ^E uses the backdoor of the model-mapping to make the translated formula(p ⊃ q)d_p,q true whenever the source formula p → q is true. However, if uninterpreted, the translated formula does not have the same meaning as the original formula.Thus, the meaning of the relatedness implication is not really expressed in terms of the connectives of . Thus, the last part of the proposition above is intended to force the source connectives to be defined entirely in terms of the target connectives and not smuggled by the model-translations.[This might be seen as forcing the connectives in a certain logic to be given first an adequate set of axioms/rules of inference in order to be translatable. This would agree with Zucker maxim that the meanings of the connectives must not be imposed from the outside <cit.>. Nevertheless, this restriction does not prohibit non-axiomatizable model-theoretic logics to be translated into each other. For example, the identity mapping from Ł(Q_0) to Ł(Q_0,Q_1) is a perfectly reasonable translation and would comply with the criterion above.] Therefore, if : Ł_1 ⟶Ł_2 is B&F-GR, then the clause (α) is satisfied. If besides the translations of the connectives are not aided by model-mappings, then it is reasonable to say that everything expressible in terms of theconnectives of Ł_1 are expressible in terms of the connectives of Ł_2. In the beginning of the section we cited proposals for preservation of connectives via translations by Wjcicki, Epstein and Mossakowski et al. The proposal above is much weaker than the first two. As regards the Mossakowski et al.'s, this approach is both weaker and stronger:stronger since it requires some preservation of structure; and weaker since it does not require the preservation of the proof-theoretic connectives.§.§.§ Adequacy criterion 2Deduction-theoremIn the adequacy criteria we asked that if a logic Ł_1 has the standard deduction theorem, and Ł_2 is at least as expressive as Ł_1, then the language fragment of Ł_2 as expressive as Ł_1 has the general deduction theorem. We formulated above the standard deduction theoremand a general version of it. As it will be seen below, in order to guarantee the preservation ofthe general deduction theorem, the source logic must have the standard deduction theorem.We fist remark that conservative general-recursive^C translations preserve the general deduction theorem:Let Ł_1 with conditional symbol “→" satisfy the standard deduction theorem. If : Ł_1 ⟶Ł_2 is a conservative general-recursive^C translation, then (Ł_1) has the general deduction theorem. Let the hypotheses of the proposition be satisfied. Then (ϕ_1),...,(ϕ_n)⊢_Ł_2(ψ) iff ϕ_1,...,ϕ_n ⊢_Ł_1ψ iff ϕ_1,...,ϕ_n-1⊢_Ł_1ϕ_n →ψ, (by the standard deduction theorem) iff (ϕ_1),...,(ϕ_n-1) ⊢_Ł_2(ϕ_n →ψ).By the definition of general-recursive^C translations, (ϕ_n →ψ) is a formula containing one or more occurrences of (ϕ_n) and (ψ). Thus, the image of Ł_1 underhave a general deduction theorem.To preserve the general deduction theorem the source logic must have the stronger one. If Ł_1 only satisfies the general deduction theorem and : Ł_1 ⟶Ł_2 is B&F-GR, we cannot guarantee that (Ł_1) satisfies the general deduction theorem.If Ł_1 satisfies the general deduction theorem, then ϕ_1,...,ϕ_n ⊢_Ł_1ψ iff it holds that ϕ_1,...,ϕ_n-1⊢_Ł_1δ^→(ϕ_n, ψ), for some Ł_1-formula δ^→.Then we have that (ϕ_1),...,(ϕ_n) ⊢_Ł_2(ψ) iff (ϕ_1),...,(ϕ_n-1) ⊢_Ł_2(δ^→). As δ^→ contains ϕ_n and ψ as sub-formulas andis general-recursive, (δ^→) will contain _j1(ϕ_n),...,_jn(ψ) as sub-formulas. If _j1,...,_jn are equal to , then the compositionalityof the deduction theorem is saved and (Ł_1) has also a general deduction theorem. Else, if _j1,...,_jn are distinct from , then the general-deduction theorem is not preserved in (Ł_1).Therefore the present approach is limited in that the source logics to be analysed in terms of expressiveness have to be put in “proper form” so that they satisfy the standard deduction theorem or an even wider version of it must be defined, dropping the compositionality requirement. Non-trivialityA logic Ł is non-trivial if for some Ł-formulas ϕ and ψ it holds that ϕ⊬_Łψ. By the adequacy criteria, a trivial logic cannot be more expressive than any logic. Thus, we have to make sure that a translation intended to induce expressiveness must reflect triviality or, alternatively, preserve non-triviality.That is, if :Ł_1 ⟶Ł_2, and Ł_2 is trivial, then Ł_1 is trivial.All back-and-forth translations reflect triviality. Undecidability We commented before that the presence of decidability could be used as an indicator of the adequacy of our definition. This is because decidability is a limitation of expressiveness of a logic. Thus the condition requires that if Ł_1 is undecidable and Ł_2 is decidable, then Ł_2 is not more expressive than Ł_1. The result below shows this condition is normally satisfied.If Ł_1 is undecidable, then there is no computable back-and-forth translation :Ł_1 ⟶Ł_2, where Ł_2 is decidable. If a logic Ł_1 is undecidable, Ł_2 is decidable and : Ł_1 ⟶Ł_2 is B&F, then it follows thatwould not be computable. In this case,apparently would not be general-recursive. §.§.§ Adequacy criterion 3Non-trivial pre-orderThe non-triviality part of the pre-order is already fulfilled by a proposition above: there are two logics Ł_1,Ł_2, such thatŁ_1 is non-trivial, Ł_2 is trivial and there is no back-and-forth translation from Ł_1 to Ł_2. We have to prove that back-and-forth general-recursive (recall the abbreviation B&F-GR) translations are transitive, that they are reflexive is clear.If : Ł_1 ⟶Ł_2 is surjective back-and-forth and ': Ł_2 →Ł_3 is back-and-forth, then ' ∘: Ł_1 ⟶Ł_3 is a back-and-forth translation <cit.>. Now if :Ł_1 ⟶Ł_2 and ':Ł_2 ⟶Ł_3 are both B&F-GR, beingadditionally surjective, then ' ∘ is also B&F-GR. To see it, let C_1,...,C_n and C'_1,...,C'_n be the translation clauses forand ', respectively. By surjectivity, each Ł_2-formula is reached by some Ł_1-formula through the applications of C_1,...,C_n. Thus to obtain a general-recursive mapping, is just to combine the application of the two set of clauses. Take a Ł_1-formula ϕ and obtain through C_1,...C_n an Ł_2-formula (ϕ). Now apply C'_1,...,C'_n to (ϕ) to obtain an Ł_3-formula '((ϕ)). This translation is B&F, and is general-recursive^C, since it is obtained through the clauses C_1,...,C_n,C'_1,...,C'_n. Nevertheless, the surjectiveness requirement may be difficult to comply with, if one is comparing increasingly expressive logics (through language extension), e.g. propositional logic, modal logic, first-order logic. We should find a way to guarantee that whenever : Ł_1 ⟶Ł_2 and ': Ł_2 ⟶Ł_3 are B&F-GR, then there is a B&F-GR translation ^* (not necessarily ' ∘) from Ł_1 to Ł_3.Let us suppose :Ł_1 ⟶Ł_2 is a non-surjective B&F-GR and that ': Ł_2 ⟶Ł_3 is B&F-GR. There is naturally a weakened version ^w of ' ∘, the weaker part being in the way back of Ł_3-formulas to Ł_1-formulas. That is, for Ł_1-formulas ϕ_1,ψ_1, it holds that if ϕ_1 ⊢_Ł_1ψ_1, then '((ϕ_1)) ⊢_Ł_3'((ψ_1)). But the converse direction only holds partially.In order that the converse direction hold, instead of normally taking Ł_3-formulas ϕ_3,ψ_3 in the range of '(Ł_2), one has to take Ł_3-formulas in the intersected range of ' ∘. For such Ł_3-formulas ϕ_3,ψ_3, with ' ∘(ϕ_1) = ϕ_3 and ' ∘(ψ_1)= ψ_3 for some Ł_1-formulas ϕ_1,ψ_1, it holds that if ϕ_3 ⊢_Ł_3ψ_3, then ϕ_1 ⊢_Ł_1ψ_1.But this is exactly what we wanted. The backward direction should hold only for those Ł_3 formulas that are linked with Ł_1-formulas through Ł_2-formulas. To translate Ł_1-formulas into Ł_3 formulas ^w takes only the '-clauses C'_1,...C'_n involved in translating (Ł_1) formulas.Thus, this weakened version of ' ∘ will suffice for us to conclude that whenever there are B&F-GR translations :Ł_1 ⟶Ł_2 and ':Ł_2 ⟶Ł_3, then there is a B&F-GR translation ^w:Ł_1 ⟶Ł_3.There remains the question whether this ^w will preserve the general deduction theorem. This will happen wheneveris compositional for “→” and the translation clause of ' for the formula (ϕ→ψ) is compositional. Then '((ϕ→ψ)) is an Ł_3-formula containing '((ϕ)) and '((ψ)) as sub-formulas, which implies that ^w preserves the general deduction theorem. Therefore we have that Back-and-forth general-recursive translations form a non-trivial pre-order on logics. From the above propositions we can conclude that Every back-and-forth general-recursive translation preserves the connectives and the selected meta-properties (undecidability, non-triviality and general deduction theorem) and form a non-trivial pre-order on logics. This implies that Every back-and-forth general-recursive translation not aided by model-mappings agrees with adequacy criteria 1,2 and 3.Many well known translations intuitively giving rise to an expressiveness relation satisfy expressiveness_gg. In the sequence, we briefly present them. §.§ Corroborating expressiveness_gg: the structure preserving translationsFor the sake of supporting our notion of translational expressiveness, there follows some translations obeying the criterion that are reasonably taken as inducing an expressiveness relation. * (Wjcicki) frominto Ł^3: * ^l(p_i)=p_i^l(ϕ)= ^l(ϕ) →^l(ϕ) * ^l(ϕ→ψ) = ^l(ϕ) → (^l(ϕ) →^l(ψ)) * (Gentzen) from classical first-order logic 𝒞ℒ into intuitionistic first-order logic (ℐℒ), and also fromto Minimal first-order logic () <cit.>:[ is the intuitionistic logic without the rule of ex falso quodlibet. An interesting result due to Luiz Carlos Pereira and Herman Haeusler<cit.> is that this translation mapsto any intermediate logic betweenand .] * ^c(Pt_1...t_n) = Pt_1...t_n ^c(ϕψ) = (^c(ϕ) ^c(ψ)) * ^c(∃ x ϕ) = ∀ x ^c(ϕ) literal for , , → and ∀; * (Gdel) fromtoextended with the modal system S4: * ^s(R_it_1...t_n) = □ R_it_1...t_n ^s(∀ x ϕ) = □∀ x(^s(ϕ)) * ^s(ϕ→ψ) = □(^s(ϕ) →^s(ψ)) literal for , , and ∃; * (Prawitz and Malmns) fromto(for #∈{, , →}): * ^m1(R_it_1,...,t_n)= R_it_1,...,t_n ^m1(∀ x ϕ) = ∀ x(^m1(ϕ) ) * ^m1(ϕ#ψ)= (^m1(ϕ) #^m1(ψ)) ^m1() =; * (Demri and Gor) from Grz to S4: * _+(□ϕ)= □ ( □ [_+(ϕ) →□_-(ϕ)] →_+(ϕ)) _-(□ϕ)=□_-(ϕ) * _+(ϕ) = _-(ϕ) _-(ϕ) = _+(ϕ) * _+(ϕ→ψ)= _-(ϕ) →_+(ψ)_-(ϕ→ψ) = _+(ϕ) →_-(ψ) * _+ and _- are literal forand atomic formulas; * (Van Benthem) Standard translation from modal logic to : * ^x(p_i)= P_ix ^x(◊ϕ)= ∃ y (Rxy ^y(ϕ)) * ^x(□ϕ) = ∀ y (Rxy →^y(ϕ)) literal for ,,, →,. § CONCLUSIONS The commonly used precise notions of expressiveness are defined within a framework which is based on the capacity of characterizing structures, thus they apply only to model-theoretic logics. As this framework of expressiveness is defined only with respect to logics sharing the same class of structures, it was called in this text “single-class expressiveness”. This framework can be seen as consisting of certain formula-mappings between model-theoretic logics. We saw two formal criteria for expressiveness, due to Garca-Matos and Vnnen and Kuijer, constructed in a wider framework which we called “multi-class expressiveness”. This wider framework encompasses besides formula-mappings, also model-mappings.We argued that both criteria are inadequate for multi-class expressiveness. Then it was defended that moving to an even broader framework might be more promising, this is because the possibility of using model-mappings, as it happens with the counter-examples presented, opens a backdoor for “undesirable” translations. In the broader framework, which we called “translational expressiveness”, a criterion for expressiveness would lack semantic notions and be based exclusively in terms of the existence of certain formula-mappings preserving the consequence relations of the logics at issue. A proposal in this direction due to Mossakowski et al. was analysed and criticized, since it also over-generates. Studying the reasons for the over-generation, we proposed some adequacy criteria for relative expressiveness and a formal criterion of translational expressiveness satisfying them. The criterion is still limited in some aspects, as the notion of a structure-preserving translation is up to now only precisely defined with respect to propositional logics. The definition of structure-preserving translation is only intuitively extrapolated to quantifiers, that is, one would normally recognize a structure-preserving translation clause for a quantifier, though there is still no formal definition of it. Therefore, a truly broad formal criterion for expressiveness encompassing preservation of quantifiers is sill wanting, and we leave it for a future work.[As regards this limitation, it is curious to see that in Lindstrm's characterization of first-order logic as the most expressive logic satisfying countable compactness and the Lwenheim-Skolem theorem <cit.>,the notion of logic used does not have anything like a quantifier. Perhaps this is due to the wide interpretation of “quantifier” as a class of structures. Something like quantification is only required to prove a characterization with respect to the upward Tarski-Lwenheim-Skolem theorem <cit.>. Only after sometime a proper “quantifier property” was required of an abstract logic extending first-order logic <cit.>.]alpha | http://arxiv.org/abs/1706.08481v1 | {
"authors": [
"Diego Pinheiro Fernandes"
],
"categories": [
"math.LO",
"cs.LO",
"03Axx"
],
"primary_category": "math.LO",
"published": "20170626170621",
"title": "Translations: generalizing relative expressiveness between logics"
} |
Skating on Slippery Ice J.M.J. van Leeuwen ... =======================Instituut-Lorentz, Universiteit Leiden,Niels Bohrweg 2, 2333 CA Leiden, The Netherlands. The friction of a stationary moving skate on smooth ice is investigated, in particular in relation to the formation of a thin layer of water betweenskate and ice. It is found that the combination of ploughing and meltinggives a friction force that is rather insensitive for parameterssuch as velocity and temperature. The weak dependence originates from the pressure adjustment inside the water layer. For instance, higher velocities, giving rise to higher friction, also lead to larger pressures,which, in turn, decrease the contact zone and so lower the friction. By treating ice as a Bingham solid the theory combines and completes two existing but conflicting theories on the formation of the water layer.keywords: solid friction, fluid mechanics, lubrication.§ INTRODUCTION Ice seems to be the only substance on which one can conveniently skate, which prompts the question: “what sort of special properties does ice haveas compared to other solids?” Moreover one can glide on ice over a wide range of velocities, types of skates and temperatures. Ice is in many respects a peculiar solid and there is much folklore about the mystery of skating.Ice is one of the few substances where the solid is less dense than the liquid, which has a profound impact in nature.Skating is a minor beneficiaryof this property, as canals freeze on top, so one does not have to wait tillthe canal is solidly frozen.Another interesting property of water is that the melting line in thepressure-temperature plane has an unusual slope: with increasing pressurethe freezing temperature lowers, while usually pressure favours the solid phase. It is illustrated in the famous high-school experiment where a steel cable with weights on both sides, melts itself through a block of ice at temperatures below zero, such that the block refreezes on top of the steelcable! This property has featured for quite a while as explanation forskating: due to the pressure exerted by the skater on the ice, a water layerforms and the skates glide on this water layer. It has been demonstratedseveral times that this explanation is not feasible <cit.>. Although the lowering of the melting point under pressure does notexplain the skating phenomenon, its influence can not be dismissedat low temperatures, as we will show. The slipperiness of ice has also been attributed to the special structure of the free ice surface. The existence of a water layer on the surface,even without skating, was already suggested by Faraday <cit.>.Computations and measurements indicate that this layer is only a few molecules thick, such that one cannot speak of this water layer as a hydrodynamic system, see e.g. <cit.>. For slow velocities and low temperatures the structure of the surface plays an important role on the friction properties <cit.>.In this paper we study the formation and influence of the water layerunderneath the skate for usual conditions, i.e for sliding velocities ofmeters per second and temperatures of a few degrees below the melting point of ice. Gliding is only a part of the physics of skating.Also important is the ability to push oneself forward, which is possible due to the shape of the skate and to the fact that ice is easily deformable. The main argument for the formation of a water layer, is that friction generates heat and that heat melts ice. How much of the heat melts ice and which part leaks away, is an important issue, which we address in this paper. We will treat the water layer as a hydrodynamic system, which implies that its thickness has to be at least of the order of 10 nm. If such a layer of water is formed, the hydrodynamic properties of the layer determine the friction, which then becomes independent of the surface properties.The physics of the water layer between skate and ice is not simple, with a rich history, see e.g. <cit.>. In spite of the fact that the problem is century old, the water layer hasnever been directly observed.A potential method for observation is basedon the difference in dielectric properties of ice and water at highfrequencies <cit.>. Indirect evidence for the water layer may result from measurements of the friction of a skate on ice.If friction is mediated through a water layer, then its characteristics can be checked. This paper deals with a calculation of the friction.It is well known that a skater on virgin ice leaves a trail. Is this trail due to melting or to plastic deformation (ploughing)? The deformation is plastic if the exerted pressure exceeds the hardness of ice. The trail is an indication that the deformation of ice is plastic. Indeed, the weight of a skater of, say 72 kg, cannot be supported by an elastic deformation of ice. Moreover skates have sharp edges which will make kinks in the surface of ice (even in horizontal position) and near a kink thepressure will always exceed thehardness of ice. Therefore we focus on plastic deformation of the ice and justify this a posteriori by the high pressures occurring in the water layer for skating speeds. At the moment there are two quantitative but competing theories for theformation of a water layer and the furrow of the trail. The one by Lozowski and Szilder <cit.>, assumes that most of the dent in the ice is the result of ploughing. The other theory, by Le Berre and Pomeau <cit.> assumes that the dentis due to melting only. We will show which fraction of the trail isdue to melting and which is due to ploughing.The two regimes, melting and ploughing merge continuously. Although ourdescription is a unification of both theories, the results are substantiallydifferent from both theories. In this paper we discuss the issue in the simplest possible setting: a speed skater moving in upright position over the ice with a velocity V on perfectly smooth ice and skates.The skater stands with his mass M on one skate. For skating near the melting point of ice, heat flows into theskates and into the ice are less important and we discuss their influencelater on.Our main concern is the thickness of the water layer underneath the skate; the water films that form at the sides of the skate,play a minor role.The only measurements of the friction of skates under realisticconditions, that we are aware of, have been performed by de Koning et al.<cit.>. Their skater had a velocity of speed of V=8 m/s and a weight of 72 kg. Together with the standard parameters of skates: curvature R=22 m and width w=1.1 mm, we call these specifications theskating conditions. Unless otherwise stated, our calculations are carriedout for temperature T=0 ^0C. We take the skating conditions asreference point and vary the parameters individually with respect to this point.The various aspects of the theory are presented in Sections in the followingorder: <ref>describes the used coordinate systems and the geometry of the skate.<ref> provides the necessary information on the material constants of water and ice.<ref> gives the force balance for a static skater.<ref> derives the heat balance, determiningthe thickness of the water layer.<ref> solves the equations for the thickness of the water layer in the regime where only melting plays a role.<ref> summarises the necessary formulas for hydrodynamicsand pressure of the water layer.<ref> yields the shape of the water layer in theploughing regime.<ref> treats the cross-over from the ploughing to themelting regime.<ref> calculates the pressure in both regimes.<ref> relates the weight of the skater to the pressure in the water layer. Also the slowing down force of the ice is computed,which is the sum of the friction in the water layer and the ploughing force.<ref> contains the velocity dependence of the friction.<ref> discusses the influence of the ice temperature on the friction.<ref> closes the paper with a discussion of theapproximations and a comparison with the existing theories.In addition a number of separate issues are treated in Appendices.§ GEOMETRY OF THE SKATES For the description of the phenomena we need two coordinate systems: the ice fixed system and that of the moving skater. If x,y,z are the coordinates in the ice system, then the coordinatesx',y',z' of the same point in the skate system arex'= x - V t,y'=y,z'=z, where V is the velocity of the skate. The x coordinate points in the forward direction of the skate. The origin of the skate coordinates is in the middle of the skate at the level of the ice. The lowest point ofthe skate, the depth of the trail, is a distance d below the originalice level. The y direction is horizontally and perpendicular to the skate blade and the z direction points downward into the ice. At time t=0 the two coordinate systems coincide. See Fig. <ref> for a cross-section in the longitudinal direction.d(x') is the locus of the bottom of the skate. With R the curvature radius of the skate it is given by the equation[R-d +d(x')]^2 + x'^2 = R^2,ord(x') = [R^2 -x'^2]^1/2 + d -R. In the ice system we have correspondinglyd(x,t)=d(x')=[R^2 -(x-Vt)^2]^1/2 + d -R. So for a fixed point x in the ice system, the downward velocity of the skatev_ sk (x) is at t=0v_ sk (x) = (∂ d(x,t)/∂ t)_t=0= V x/[R^2 -x^2]^1/2≃ Vx/R.The last approximation uses that x is a few centimeters and R about20 meters. v_ sk is also the velocity with which the top of the water layer, in contact with the skate, comes down.Later on we need alsov_ ice (x), being the velocity at the bottomof the layer with which the ice recedes due to the pressure. The thickness of the water layer at a point x is denoted by h(x,y). So in the downward direction we have the skate between 0<z<d(x), water betweend(x)<z<d(x)+h(x,y) and ice below z>d(x)+h(x,y). The water at the sides of the skate is of minor influence, since the depth d(x)measures in μm, while the width of the skate is around 1 mm. In order to focus on the essentials we restrict the discussion to the treatment of the layer underneath the skate. In Fig. <ref> we give a sketch of the transverse cross-section in the y,z plane. As indicated in this figure, the water layer may vary in the transverse y direction. In thecoming sections we approximate h(x,y) by a function h(x) of x alone. In Appendix <ref>, we show that this is a good approximation forcalculating the friction.§ MATERIAL CONSTANTS OF WATER AND ICE In the Table <ref>we have listed the relevant material constantsof water and ice. Apart from these well known constants, there are two more material properties relevant for skating: the hardness of ice p_H and the deformation rate γ. The Brinell hardness number is measured by pushing with a force F,an “undeformable” spherical ball into the material.After lifting the force, the material shows a dent, with surface S.The ratio F/S is independent of F and equal to the hardness p_H.This means that the material reacts upondeforming forces with a fixed counter pressure p_H, such that the contactsurface S times p_H balances the applied force F.[The Brinell hardness takes as contact surface the spherical surface of the dent, whichis slightly larger than the top circle of the dent. In contrast to theBrinell hardness, we measure the contact area in the horizontal directionand not along the skate, since the horizontal surface matters for the force balance Eq. (<ref>).]For the hardness dependence of ice on the temperature Pourier et al.<cit.> give the relationp_H = (14.7 - 0.6T)*10^6Pa, with T the temperature in centigrades. An earlier measurement gave quite different values <cit.>. The value depends on the method of measurement<cit.>. We take the viewpoint that the hardness is defined by the response to a quasi-static deformation of the ice.Mostly the hardness comes into our analysis as a multiplicative constant. Although the measurements of Pourier et al. were not carried out quasi-statically, we stick to the value given inEq. (<ref>) for the hardness in our calculations, when explicitly needed.However, skating is a dynamic event. For instance a forward skatingvelocity of 10 m/s implies, a downward velocity of about 1 cm/sat the tip of contact. In order that the ice recedes at such a large rate,one needs pressures far exceeding the hardness. Such large pressures require a relation between the applied pressure and the velocity withwhich the ice recedes. With p(x,y,d(x)+h(x)) the pressure in the water layer in contact with the ice,we will use for the downward velocity of ice the relationv_ ice (x,y) = γ [p(x,y,d(x)+h(x))-p_H],where γ is a material constant with the dimension [m/(Pa s)]. Eq. (<ref>) takes the receding velocity proportional to the pressure excess. This is similar to treating ice as a Bingham solid <cit.>, where one puts, for plastic flow, the shear rate proportional to pressure excess. The deformed region of the ice is of the order of the width w. So dividingv_ ice by w gives the order of the occurring shear rates. In this way we deduce, from the measured shear rates <cit.>, a valueγ p_H ≃ 1 mm/s. This is not more than an order of magnitude estimate, since glaciers and laboratory experiments induce plasticflows on a time scale much lower than in speed skating. § STATIC DEFORMATIONElastic deformations of ice are controlled by the elastic coefficient (Youngs modulus).By calculating the elastic deformation field due to a skate which bears aweight M, one estimates that for M below 10kg, the skate makes anelastic deformation. The estimate is hampered by role of the edges of the skate. If they are not rounded off a bit, they produce a kink in the deformation field, which leads to unlimited pressures in the ice. The estimate shows, however, that for practical skater masses the deformation is plastic. Static inelastic deformations are determined by the hardness p_H. At rest, the skater exerts a pressure on the ice equal to the hardness p_H.The contact area times the pressure balances the weight of the skater.The contact area is the width w of the skate times the contact length 2l_0. So one has the force balanceM g = 2 p_H w l_0, which gives the value of l_0. The static depth d_0 of the dent inthe middle of the skate is related to l_0 by geometry R^2 = (l_0)^2 + (R-d_0)^2,ord_0 ≈l_0^2/2 R.The two equations (<ref>) and (<ref>) determine the static valuesof l_0 and d_0. We find for a weight of 72 kg the valuesl_0 = 2.2cm and d_0 = 11 μm. We note that this estimate assumes that the pressure distribution in the ice underneath the skate is uniformlyequal to p_H. If one calculates, for small weights, the pressuredistribution for elastic deformations, one finds that the pressure is largest at the edges of the skate and in the middle where the deformation is deepest. Thus at the point where the elastic deformation turns gradually into a plastic deformation the above estimate does not apply. It only applies for a fully developed plastic deformation. The calculation of contact length l and the depth d for a moving skateris a major part of the problem. The relation between l and d is the same as Eq. (<ref>) between l_0 and d_0, since it is geometric.We will see that for a fast moving skater the contact length l issubstantially shorter than the 2l_0 needed at rest. While for static contact the total length, forward and backward, 2 l_0 counts, for the dynamic contact only the forward section 0 ≤ x ≤ l is relevant.What happens in the backward section -l ≤ x < 0 does not contribute to the heat balance nor to the friction, since the contact between ice and skate is broken.§ THE HEAT BALANCEThe heat generated by friction in the water layer leads to melting ofice. The first point for establishing the heat balance is to compute the melting velocity v_ m (x). The trough made by the skate has a width w and a depth d(x) + h(x). So the trough grows downwards at a ratev_ tr = (∂ [d(x,t) + h(x,t )]/∂ t)_t=0. Since the trough grows by melting with a velocity v_ m and ploughing, which has a downward velocity v_ ice, we have the equalityv_ m (x)+v_ ice (x)= v_ tr. Working out the right hand side of Eq. (<ref>) gives the expressionfor the melting velocityv_ m (x) = v_ sk (x) - v_ ice (x) - V ∂ h/∂ x,with v_ sk given by Eq. (<ref>) and v_ ice by Eq. (<ref>).The main source of heat is the friction in the water layer dueto the gradient in v_x. The gradient of the transverse flow v_y contributes an order of magnitude less to the heat generation. So the frictional heat generated in a time dt and a volume h(x) w dx equalsd H (x) = ηV^2/h^2(x) h(x) w dx dt.The heat gives rise to melting of a volume d V (x), but it is a delicate question which fraction of the heat is effective. There are two competitors for melting. Inside the water layer a fraction ζ_ w will flow towardsthe ice and the remainder will flow towards the skate. In Appendix <ref> it is shown that the fraction ζ_ w≥ 1/2,but usually equal to 1/2, when the difference between skate and ice temperature is small. The second competitor is the heat flow inside the ice, which is a subtle point, playing a role at low-temperature skating. We discuss this effect in Section <ref>. We stick here to the fraction 1/2 and get for the molten volumed V (x) = d H (x)/2ρ L_H,with ρ L_H the latent heat per volume.Equating this molten volume with the increase in water due to v_ m (x) leads to the balance equationv_ m (x) wdx dt = d V (x) = k V/h(x) w dx dt, where k is the important parameter introduced byLe Berre and Pomeau <cit.> k = η V/2 ρ L_H.k is a (microscopic) small length. We find for skating conditionsk = 2.1*10^-11m.[Actually k is about a factor10^3 smaller than the value 1.8*10^-8m given by the authors of<cit.>, since they erroneously take for the water density ρ=1, while ρ=10^3 in SI-units.].We now turn this equation into a differential equation for h(x) bysubstituting Eq. (<ref>) into Eq. (<ref>). Bringing the difference v_ sk-v_ ice to the right hand side yields-V∂ h/∂ x= kV/h(x) -[v_ sk(x)-v_ ice(x)]. This equation becomes useful if we have an expression for the receding velocity v_ ice (x). For the ice to recede, the water layermust have a pressure p exceeding the hardness p_H of ice. The pressure in the water layer will be lower than p_H near the midpoint x=0,where the layer is close to the open air. We will show that near the tip x=l the pressure will exceed p_H. We call the fraction with p>p_H the ploughing regime and the fraction with p<p_H the melting regime.In the melting phase we have v_ ice (x)=0 and with v_ sk (x) from Eq. (<ref>), we get the layer equation-d h(x)/dx = k/h(x) - x/R,which is the equation derived by Le Berre and Pomeau <cit.>.In the ploughing phase we need the expression Eq. (<ref>)for the receding speed v_ ice.The pressure in the water layer has to depend on y, since it drives out the water sideways. This causes the receding velocity to depend on y and that in turn makes the layer thickness h also dependent on y. In order to stick to the approximation where h depends only on x, we replace Eq. (<ref>) by its average over yv_ ice (x) = γ∫^w/2_-w/2dy/w [p(x,y,d(x)+h(x)) - p_H].In Appendix <ref> it is outlined how the y dependence in v_ ice can be accounted for. Eq. (<ref>) is derived without information about the hydrodynamics of the water layer, other than that the gradient in v_x is the main source of friction. In the ploughing regime, where v_ ice (x) ≠ 0 we have to resolve the pressure dependence from the flow pattern.§ THE MELTING REGIMEIn order to analyse the layer equation (<ref>), we introducetwo length scales as a combination of the microscopiclength k and the macroscopic length R. The longitudinal length s_l and the depth length s_d are defined ass_l = (k R^2)^1/3,ands_d = (k^2 R)^1/3. For the skating velocity V=8m/s, we have as the scale for the contact length s_l = 2.16 mm and as scale for the thickness s_d = 0.21 μm. Both are rather small.[The water layer thickness s_hwould multiply with afactor 100 for ρ=1 and the length s_l with a factor 10. These values are comparable with the values found by Le Berre and Pomeau <cit.>.]If we use s_l as a scale for the longitudinal coordinate x ands_d for the thickness hx = s_lx̅ andh = s_d h̅,Eq. (<ref>) becomes dimensionless -d h̅ (x̅)/ d x̅=1/h̅ (x̅)-x̅. The advantage of this scaled equation is that no external parameters occur in the equation. The skating velocity V and radius of curvature Rcome in via the scales s_l and s_d through the parameter k. Eq. (<ref>) is easy to integrate numerically, starting from a guess for the contact length l̅. At x̅=l̅ the thickness h̅ vanishesand thus the first term on the right hand side of Eq. (<ref>) dominates and the solution behaves ash̅ (x̅) ≃√(2 (l̅-x̅)), x̅→l̅. In Fig. <ref> we have given the curves for a few values of l̅. The curves distinguish themselves only near the tip x̅=l̅.Integrating the equation from below starting at x̅=0, there is a value h̅_0 ≃ 1.284 such that the curves with h̅(0)>h̅_0 curve upwards asymptotically and the curves with h̅(0)<h̅_0bend downwards hitting the axis.The seperatrix starting at h̅(0)=h̅_0 behaves asymptotically ash̅(x̅) ≃ 1/x̅.The value of the contact length follows from thebalance between the pressure in the water layer and the weight M of the skater, for which we need the pressure in the water layer.§ THE HYDRODYNAMICS OF THE WATER LAYERThe pressure is determined by the hydrodynamic equations of the water layer.The pressure distribution has been derived both in <cit.> and <cit.>. Here we give the expressions which are important for the next section. In Appendix <ref> we sketchhow the pressure follows from the assumption that the transverse flow has aPoisseuille form v_y (x,y,z) = a(x) y [z-d(x)][h(x)-z+d(x)].The amplitude a(x) determines, through the fluid equations, the pressure behaviour. At the top and bottom of the layer we havep(x,y,d(x)) = p(x,y,d(x)+h(x))= η a(x) ( w^2/4 - y^2 ).The pressure is maximal in the middle of the skate blade and drops offtowards the edges. The y dependence of the pressure is essential for pushing out the water towards the edges of the skate. (It causes also an y dependence in the layer thickness h, see Appendix <ref>.) The incompressibility of water implies the connection of a(x) with the downward velocities of the top and bottom of the water layer v_ sk (x)-v_ ice (x) = a(x) h^3 (x)/6Eq. (<ref>) holds both in the melting and the ploughing phase. In the melting regime, where v_ ice=0, it implies a simple relation between a(x) and h(x)V x/R = a(x) h^3 (x)/6. Using Eq. (<ref>) for a(x) and Eq. (<ref>), gives for theaverage pressure in themelting phase the expression1/w∫^w/2_-w/2 dy p(x,y,d(x)+h(x)) =η w^2 V/R x/h^3 (x). This presents a problem for the weight balance, if the melting phase would apply all the way to the tip, where h(x) behaves as given by Eq. (<ref>). Thatleads to a diverging pressure, which is non-integrable. So some regularisation near the tip is necessary, see <cit.>. In our treatment this problem does not occur, since the the ploughing regime takes overas soon as the pressure exceeds the hardness p_H.§ THE PLOUGHING REGIMEAs follows from the analysis of the previous section, part of the deformation of ice is due to the force on the ice.With Eq. (<ref>) and Eq. (<ref>) we find v_ ice (x) = γ [η a(x) w^2 /6 -p_H]. Using this expression in Eq. (<ref>) we obtain the following relation between a(x) and h(x)a(x) = 6 V x/R +γ p_H/h^3(x)+γη w^2. The heat balance equation (<ref>) can be cast, with Eq. (<ref>),into the form-d h(x)/dx = k/h(x) -a(x) h^3 (x)/6V.Then using a(x) from Eq. (<ref>), turns it into an explicitlayer equation for h(x)-d h(x)/dx =k/h(x) -x/R +γ p_H/V/h^3(x)+γη w^2 h^3(x).We note that putting γ =0, which is equivalent to putting v_ ice=0, reduces indeed the equation to Eq. (<ref>) of the melting regime. On the other hand, the limit γ→∞ reduces the equation to -d h(x)/dx = k/h(x) -p_H /η w^2 V h^3 (x), which is the backbone of the equation derived by Lozowski and Szilder<cit.>. A very large γ implies that the pressure at the bottom of the layer stays equal to the hardness p_H and that is an implicit assumption in <cit.>. Eq. (<ref>) can be solved analytically, see Appendix <ref>.In order to get a better insight in Eq. (<ref>), we make the equationdimensionless by introducing the same scaling as in Eq. (<ref>), yielding the layer equation-d h̅/d x̅ = 1/h̅ (x̅)-x̅ + c_1/c_2+ h̅^3 (x̅)h̅^3(x̅), with the dimensionless constantsc_1= γ p_H/V(R/k)^1/3,c_2=γη w^2/k^2 R. The magnitude of these constants depends on the value of γ, on which we have little experimental evidence. With the value γ p_H = 10^-3 m/s, we get for skating conditionsc_1=1.27,c_2=15.0 c_3 = c_1/c_2 =0.085. Note that the ratio c_3 is independent of γ.§ THE CROSS-OVER FROM PLOUGHING TO MELTINGWe must integrate Eq. (<ref>) starting from a value l̅ tilla point where the velocity v_ ice(x) tends to become negative.Thus with Eq. (<ref>) we have to obey the condition η a(x) w^2 > 6 p_H. With the expression (<ref>) for a(x) this translates to η w^2 V x/R > p_H h^3 (x),orx̅ > c_3 h̅^3 (x̅). At the top x̅=l̅ we haveh̅ (l̅)=0. So there theinequality is certainly fulfilled. At the midpoint x̅=0, sothere the inequality is certainly violated. Somewhere in between, at the cross-over point l̅_c, the ploughing regime merges smoothlyinto the melting regime. In dimensionless units, l̅_c is thesolution of the equationl̅_c = c_3 h̅^3 (l̅_c).At the cross-over point the layer thickness h̅_c=h̅ (l̅_c) is the same in both regimes. The derivative is also continuous at the cross-over point. We find in the ploughing regime- (d h̅/d x̅)_l̅_c =1/h̅ (l̅_̅c̅) - c_3 h̅^3 (l̅_c) + c_1/c_2 + h̅^3 (l̅_c)h̅^3 (l̅_c) =1/h̅ (l̅_̅c̅) - c_3 h̅^3 (l̅_c) =1/h̅ (l̅_̅c̅) - l̅_c, which equals the value in the melting regime.§ SCALING THE PRESSURE IN THE WATER LAYERThe pressure at the top of the water layer is given by Eq. (<ref>) and withEq. (<ref>) for the amplitude a(x) we get in the ploughing regimep(x)= η w^2 V x/R +γ p_H/γη w^2+h^3(x). It is interesting to compare this value with the hardness p_H of ice andto express this ratio in dimensionless unitsp̅ (x̅) = p(x)/p_H = η w^2/p_H/V x/R+γ p_H/V/γη w^2+h^3(x) = x̅ +c_1/c_1+ c_3 h̅^3 (x̅). This expression holds in the ploughing regime. In the melting regime wehavep̅ (x̅) = 1/c_3 x̅/h̅^3 (x̅). Note that, with Eq. (<ref>),both expressions (<ref>) and(<ref>) yield p̅ (l̅_c) = 1.p̅ (x̅) is larger than 1 in the ploughing phase and smallerthan 1 in the melting phase. The maximum pressure occurs at the tip,x̅=l̅, where h̅=0, with the valuep̅_t = p̅ (l̅) = 1 + l̅/c_1. As l̅ will turn out to be around 6, this is a substantial ratio.§ THE MACROSCOPIC FORCESThe skate feels a normal and tangential force. The normal force F_N=Mg is the weight of the skater. The tangential friction force hastwo ingredients: the friction force F_ fr, due to the water layer and theploughing force F_ pl, which pushes down the ice. All three forces are related to integrals over the contact zone. The weight M of the skater is balanced by the pressure at the top in the water layerF_N = w ∫^l_0 dx p(x). The friction force is given by the gradient of the flow in the water layerF_ fr = η w ∫^l_0 dx V/h(x) .The ploughing force results from the force that the pressure in thewater layer exerts on theice in the forward direction. It is given byF_ pl = w ∫^l_0 dxx/Rp(x).The ratio x/R gives the component of the force in the forward direction.Applying the scaling Eq. (<ref>) on x and h(x) and scaling thepressure with the hardness p_H, we get the expressions{[ F_N =a_N ∫^l̅_0 d x̅ p̅( x̅),; F_ pl = a_ pl∫^l̅_0 d x̅ x̅ p̅ ( x̅),; F_ fr =a_ fr∫^l̅_0 d x̅ 1/h̅(x̅). ]. The integrals are dimensionless and the constants have the dimension of a forcea_N = p_H w s_l,a_ pl = p_H w s_d, a_ fr= ηV ws_l/s_d. Note that the ratio a_ pl/a_N involves the ratio of the scaless_d/s_l, which is a reflection of the fact that the normal force actsover the longitudinal length l and the ploughing over the depth d.In order to compare friction with ploughing, we use the number λ introduced in Eq. (<ref>), leading to η V = 2 kρ L_H = 2 kλp_H. This gives for the relation between a_ fr anda_ pla_ fr= 2 kλp_H ws_l/s_d=2 p_Hwλ s_d= 2 λa_ pl.An interesting feature of pressures p(x), exceeding the hardness p_H in the ploughing regime, is that they shrink the contact length l and the penetration depth d, since d goes with the square of l.So the skater “rises” due to his velocity. We find in the limitV → 0 an indentation depth d ≃ 44 μm and for V=8m/s a value d=4.5 μm.[See Appendix <ref> for the relationbetween the static d_0 and d in the slow limit.]For slow velocitiesthe F_ frvanishes and F_ pl has alimit ≃ 0.7 N for a skater of 72 kg. For the V=8m/s we find F_ fr =0.84 N and F_ pl = 0.29 N.So the large pressure build-up near the tip, reduces the ploughing force, from dominant at V → 0, to a fraction of the total friction force.§ VELOCITY DEPENDENCE OF THE FRICTION The integration of the layer equation is straightforward once we know the contact length l. The value of l determines the weightof the skater. Since theweight is given, we must find the contact length by trial and error.In Fig. <ref> we have drawn the shape of the water layer for a few values of the deformation rate γ and a skater weight of 72 kg.The curves end at x=l and one observes that the contact length is rather sensitively dependent on the value of γ. This is notsurprising since γ has a direct influence on the pressure in the water layer and the pressure determines the weight. The smallup-swing of the thickness in the middle of the skate (x=0) is amanifestation of the melting phase. On the other hand the overal thickness of the layer does not depend strongly on the value of γ.The next result is the friction as function of the velocity.In Fig. <ref> we have drawn how the ploughing and water friction combine to the total strength of the friction. While both components vary substantially with the velocity, the combination is remarkably constant over a wide range of velocities. One observes that the low V limit (exhibiting a square root dependence on V), covers only a very small region of velocities. In the Fig. <ref> we have also plotted the influence on thecontributions, if one takes the y dependence of the thickness into account. The effects onfriction and ploughing are small and opposite, such thatthe change of the total friction is not visible in the Fig. <ref>.In order to see how much the value of γ influences the friction, we have drawn in Fig. <ref> the total friction as function of the velocity for some values of γ. The influence of γ is noticeable, but not dramatic. A factor 16 difference in γ p_H, betweenγ p_H=4 mm/s and γ =0.25 mm/s, gives a factor 2 in the friction for large velocities. But there is a substantial difference with respect to thetheory of Lozowski and Szilder <cit.>, using γ=∞. Usually the friction is expressed in terms of the friction coefficientσ, being the ratioof the tangential and the normal force. In the present case it readsσ = F_ fr + F_ pl/F_N .However, for skating the friction coefficient is not independent of thenormal force. In a standard friction experiment the contact surface isproportional to the normal force and the friction force is proportional tothe contact area, such that in the friction coefficient the contact areadrops out. This proportionality does not hold for skating.The order of magnitude of the friction coefficients is 0.002 for skating conditions. We estimate the contact area for skating conditions as s_ll̅ w ≃ 14.3 mm^2.§ TEMPERATURE DEPENDENCE OF THE FRICTIONSo far we have considered temperatures close to the melting point of ice, where temperature gradients and associated heat flows are small. At lower temperatures they start to play a role.In order to melt ice, one first has to heat it to the melting temperatureT_m. If the difference between the melting temperature and the surface temperature T_s is positive, i.e. when the surface temperature is lower than the melting temperature, one has to increase the latent heat L_H with the amount needed to heat the ice.Since the latent heat is 80 times the heatnecessary to raise the temperature of ice by one degree,this is usually a small correction. Another small correction comes from the heat flux which may exist in the ice layer. In a skating rink the ice is cooled from below and there is a heat flow downwards. Natural ice freezes by cooling the top layer and correspondingly the heat flow is upwards. But the temperature gradients are small with respect to the temperature gradients in the water layer, so the effect on the amount of ice that melts is small and we leave it out.However, as pointed out in <cit.>, there is another heat flow,which can have an important effect on the frictionat low ice temperatures. If the surface temperature is low, one has to heatthe surface, before it melts. This causes a temperature gradient in the ice and an associated heat leak into the ice. The melting occurs under pressure and one has to raise the temperature, not to zero centigrade, but to the melting temperature T_m at that pressure.Since the pressure p in the water layer is large, T_m can be substantial below zero degree centigrade.The lowering of the melting temperature is approximately given by T_m = - 0.1 p*10^-6.The maximum value of T_m occurs at p=2*10^8 Pa, producing a T_m of -20 degrees centigrade. Since the pressure varies strongly with the position x of the contact, T_m varies also with x. In the middle of the skate, where the contact ends, the pressure vanishes and the melting temperature T_m(0)=0. At the tip the pressure is maximaland T_m(l) reaches its lowest point. We have to distinguish two cases: T_m(l)<T_s and T_m(l)>T_s. In the former case, there is a point x_0 where T_m(x_0)=T_s. For x>x_0 the melting temperature is then below the surface temperature and no heat is needed to raise the ice to T_m. In the latter case the ice is heated all along the contact line and at the tip asudden jump in the surface temperature occurs. In Appendix <ref> we have given the derivation of the temperature gradient in the ice at the surface. It reads(∂ T(x,z)/∂ z)_z=0 =(V/πα_ ice)^1/2( -∫_x^x_0 dx' 1/√(x'-x)∂ T_m(x') /∂ x'+ T_m(l)-T_s/√(l-x)) The understanding is that the last term is absent for T_m(l)<T_s and in the other case the integral extends to x_0=l.In <cit.> only this last term is taken into account, together with settingT_m(l)=0.The gradient gives a downward heat flow at the surface z=0J_ ice (x) = -κ_ ice(∂ T(x,z)/∂ z)_z=0. This gradient takes away a fraction of the heat supplied by J_ w-ice ζ(x)= 1 - J_ ice(x)/J_ w-ice, with J_ w-ice given by Eq. (<ref>).In the layer equation we have to replace the first term of the right hand side byk/h(x)→k ζ(x)/h(x).In Fig. <ref> we have drawn the friction as function of thesurface temperature for skating conditions. Note that the friction hardly changes in the region 0>T_s>-5^0C, after which the friction starts toincrease. The influence of γ is similar for all temperatures. § DISCUSSION We have investigated the thickness of the water layer underneath the skate as a result of melting of ice by the frictional heat.In skating two processes take place: a plasticdeformation of the ice (ploughing) and the generation of a water layer by melting. For low velocities ploughing dominatesand for high velocities friction in the water layer dominates.In the skating range of velocities,the total friction is rather independent on the velocity.The friction in the water layer increases with the velocity, which is compensated by an almost equal decrease in the ploughing force. A high skating velocity causes a high pressure in thewater layer, lifting the skater. Consequently the skate penetrates lessdeep into the ice and the ploughing force decreases. The theory assumes that the thickness of the water layer is large enough to treat ithydrodynamic system. For low velocities and low temperatures this assumption breaks down (see Appendix <ref>). There are two important material constants of ice,determining the friction: the hardness p_H and the deformation rate γ. Unfortunately no accurate data exist on these constants, which hampers a quantitative calculation of the friction. In particular the value of γ is poorly known, while it has a substantial influence on the magnitude of the friction. The relevance of γ becomes clear from an estimate of the speed at which ice has to be pushed down at the tip of contact. For a forward velocity of 10 m/s, the downward speed of the ice is about 1 cm/s. Such high deformation rates require large pressures,several times the hardness.The most important theoretical parameter is k, defined in Eq. (<ref>), which is microscopically small for reasonable values of the velocity of theskate. In combination with the macroscopic curvature R of the skate,two length scales follow: the longitudinal scale s_land the depthscale s_d defined in Eq. (<ref>). s_l is a measure for the contactlength and s_d gives the magnitude of the thickness of thewater layer. Our analysis combines elements of the theory of Le Berre and Pomeau <cit.>, which only accounts for the effects of melting and the theory of Lozowski andSzilder, which equals the pressure in the water layerto the hardness of ice. The new element is that we propose that the ice recedes with a velocityproportional to the excess pressure with respect to the hardness. In thetheory of Le Berre and Pomeau the ice does not recede (which is equivalent with γ=0), in spite of the fact that, in their approach, the pressure grows unlimitedly near the tip.In the theory of Lozowski and Szilder the ice adapts instantaneously (which is equivalent with γ=∞), keeping the pressure equal to the hardness.We have mainly considered skating near the melting point. At low temperatures a number of new elements come into play, which we have indicated in Section<ref>. A quantitative discussion of these effects is delicate, since they depend not only on the conditions of the ice, but also on the value of the constant γ in the Bingham Eq. (<ref>). Since the hardness p_H and the deformation rate γ, are not very well known as function of the temperature, a precise measurementof these properties would be very welcome. We have left out a number of refinements in order to focus on the essential features of skating. Refinements that can be treated in the presented context are: * We have omitted the influence of the melting of the ice at the sides of the skate. A simple treatment adds to the width w on both sides the amount d(x). Since the indentation depth d of the skate is very small compared to the width w of the blade (we find a ratio 1/500) it gives a small correction.* We have assumed that only the gradient of the forward velocitycontributes to the friction and the corresponding heat generation. It is easy to take into account the contributions of the gradient in the transverse velocity. The relative importance of the longitudinal and transverse heat generation is of the order 1/λ, see Eq. (<ref>).This means that the transverse velocity gradient contributes only a fewpercent to the generated heat.* Most of our calculations are based on the assumption that thethickness h depends only on the longitudinal coordinate x. In Appendix <ref> we have made a start of taking the transverse y dependence into account. A fully consistent treatment, including the hydrodynamic equations, is computationally quite involved and as far as the friction is concerned not very encouraging, as the effect is quite smallin lowest order (see Fig. <ref>). The reason is that the variation ofh(x,y) with y is modest except at the edges of the skate.There are several influences outside our scope,such as the humidity of the air and theaddition of suitable chemicals to the surface layer, which are importantfor speed skating records, but not essential for the phenomenon of skating. Apart from a more accurate measurement of p_H and γ, it would be interesting if the deformation of the ice, due to the skate, could be observed. Presumably the 10% difference in density between ice and water, which we ignored, plays an important role for the form of the deformation.De Koning et al. <cit.> report a friction force of 3.8 N for the straight strokes and 4.9 N for the curves. The difference is due to the factthat in the curves the skate is at an angle with the ice. In the straights there are also parts, at the begin and end of the stroke, where the skate makes an angle with the ice. So for the upright part, for which we perform the calculation, one estimates a friction force around 2 N. This compares well with the values we see in Fig. <ref>. A fit might be seen asa measurement of γ and tends to the value γ p_H = 2 mm/s.In Appendix <ref> we discuss the slow velocity limit V → 0,which is hardly relevant for skating, but may be useful for measurementsin the laboratory, involving low V.Acknowledgments. The author is indebted to Tjerk Oosterkamp fordrawing his attention to the problem and for careful reading of the manuscript and to Tom van de Reep for explaining the details of the measurements in Leiden. He also acknowledges discussions with the experimental group of Daniel Bonn in Amsterdam, in particular the discussions with Bart Weber on ploughing. Numerous conversations about the properties of ice with Henk Blöte are highly appreciated. § VELOCITY AND PRESSURE IN THE WATER LAYERIn this Appendix we discuss the hydrodynamics in the waterlayer underneath the skate. We take advantage of the fact that we have three different length scales: in the x direction the scale is incentimeters, in the y direction in millimeters and in the z directionin microns. So the gradients in the z direction are much larger than in the other directions and we may use the lubrication approximation ofthe Navier-Stokes equations for an incompressible fluid∇ p = η Δ v,and∇· v = 0.The velocity in the x-direction is forced by the motion of the skatev_x = V (1 - z-d(x)/h(x)). At the top of the layer z=d(x), the velocity of water equals that of theskate and at the bottom, z=d(x)+h(x), it vanishes at the solid ice surface.The velocity in the y direction has a Poisseuille formv_y (x,y,z) = a(x) y [z-d(x)][h(x)-z+d(x)], This velocity component vanishes at the skate blade z=d(x) as well as at the bottom of the layer at z=d(x)+h(x). The linear dependence on y is a consequence of the incompressibility of water. To see this, consider avolume between x and x+δ x, y and y+δ y and z=d(x) andz=d(x)+h(x). At the top it goes down withthe velocity v_ sk (x) and at the bottom it may go down with a velocityv_ ice (x).The total decrease of the volume due to vertical motion of the top and bottom boundary equalsΔ V_v = [v_ sk (x)-v_ ice (x)] δ xδ y δ t.In the horizontal direction we have an inflow at y and an outflow aty+δ y resulting in the net displaced volume∫^h(x)+d(x)_d(x) dz [v_y (x,y+δ y,z) -v_y (x,y,z)]δ x δ t =a(x) h^3(x)δ yδ x δ t /6. As water is incompressible we have the balancev_ sk (x)-v_ ice (x) = a(x) h^3 (x)/6. The linear dependence of v_y on y makes the right hand sidein Eq. (<ref>) independent of y.The third component of the velocity is given byv_z =v_ ice(x)+ a(x) ([z - d(x)]^3/3 -[z-d(x)]^2 h(x)/2 + h^3 (x)/6). Note that we have chosen the constants such that at the top v_z(x,y,d(x))=v_ sk(x) and at the bottomv_z(x,y,d(x)+h(x))=v_ ice(x).The pressure distribution compatible with this flow field is fixed up to a constant. Here we take the boundary condition p(x,w/2,d(x)) = 0, using that at the corners of the furrow the pressure is (nearly)zero. This gives the pressure the formp(x,y,z) = ηa(x) ( w^2/4 - y^2 - [z - d(x)][d(x) +h(x)-z] ). At the top and the bottom the pressure equalsp(x,y,d(x)) = p(x,y,d(x)+h(x))= η a(x) ( w^2/4 - y^2 ). which is maximal in the middle of the skate blade.It is easy to verify that the flow field and the pressure fulfil theNavier-Stokes equations (<ref>), provided that we consider for the differentiation only the explicity and z dependence and ignore the x dependencies of a(x) and h(x) for the calculation of the gradients.§ THE Y DEPENDENCE OF THE WATER LAYERWe see from Eq. (<ref>) that the pressure depends explicitlyon y. This implies, through the expression (<ref>) forv_ ice, that also v_ ice is dependent on x and y. That in turn forces the function a and h to depend also on x and y. Taking the y dependence fully into account, also for the detailed solutionof the hydrodynamic equations in the layer of varying thickness, is quiteinvolved. Here we give a first step, which focusses on the explicit y dependence that enters into the equations. With Eq. (<ref>) and (<ref>) one hasv_ ice (x,y) = γ[η a(x,y)(w^2/4-y^2) - p_H]. If v_ ice depends on x and y, we also must change the layerEq. (<ref>) into-∂ h(x,y)/∂ x = k/h(x,y) -1/V [v_ sk (x) - v_ ice (x,y)].Finally the connection between a and v_ ice, as given byEq. (<ref>), changes intov_ sk (x) - v_ ice (x,y) = ∫^h(x,y)+d(x)_d(x) dz∂ v_y/∂ y = 1/6 ∂/∂ y a(x,y) y h^3 (x,y).The last step uses that v_y vanishes at the boundaries. The three equations (<ref>)-(<ref>) determine the behaviour of the three quantities a(x,y), h(x,y) and v_ ice (x,y). We first eliminate v_ ice by inserting Eq. (<ref>) into Eqs. (<ref>) and(<ref>), which leads to the set{[ -∂ h(x,y)/∂ x =k/h(x,y) - (x/R + γ p_H/V) +γη/V a(x,y) [w^2/4-y^2],; x/R + γ p_H/V = 1/6V ∂/∂ ya(x,y) y h^3 (x,y)+γη/V a(x,y) [w^2/4-y^2]. ]. The equations simplify in the center y=0 where we may use∂/∂ ya(x,y) y h^3 (x,y) ≃ a(x,y) h^3 (x,y).In the anticipation that the variation of a and h with y is modest, we use the approximation (<ref>) for the whole width. Then a(x,y) can be expressed in terms of h(x,y) asa(x,y)=6(V x/R +γ p_H)/h^3(x,y)+γη [3w^2/2-6y^2] For calculational purpose we give the scaled version of the equations,using for y and a the scalingy=wy̅,a(x,y) = V/k^2 R(k/R)^1/3a̅(x̅,y̅).For a̅ expression (<ref>) becomes a̅(x̅,y̅) = 6(x̅+c_1)/h̅^3(x̅,y̅)+ c_2(3/2 -6 y̅^2).The constants c_1 and c_2 are defined in Eq. (<ref>). With this value of a inserted into the first Eq. (<ref>), we get for h̅ the equation -∂h̅(x̅,y̅)/∂x̅=1/h̅(x̅,y̅) - (x̅+c_1) h̅^3(x̅,y̅)/h̅^3(x̅,y̅)+c_2(3/2 -6 y̅^2). This equation has to be solved, starting from a value x̅=l̅, where the thickness behaves as indicated in Eq. (<ref>). The transition to the melting regime occurs atx̅ (3/2- 6 y̅^2) = c_3 h̅^3(x̅,y̅). From there on the equation reads in the melting regime as before-∂h̅(x̅,y̅)/∂x̅=1/h̅(x̅,y̅) - x̅. In contrast to the equation where the average pressure was employed, the transition from the ploughing to the melting regime is y dependent. It occurs immediately at the edges y̅=± 1/2 and lastly in the middle y̅=0. In Fig. <ref> we have plotted the solution of Eqs. (<ref>) and (<ref>) for skating conditions.Only at the edges there is a substantial y̅dependence. It is a consequence of the boundary condition that the pressure should vanish at the edges of the skate. In fact the pressure is alwayshigher than the atmospheric pressure,but that is a small value as compared to the hardness of ice, which is the scale for the pressure in the water layer. The approximation Eq. (<ref>) can be improved by computed the derivatives of a̅ and h̅ from the solution of Eqs. (<ref>) and (<ref>) and adding that as a correction to Eq. (<ref>).In view of the small influence on the friction by the first approximation outlined in this section,(see Fig. <ref>), such a further refinement is not worth while. § THE SLOW VELOCITY LIMITIn this Section we discuss the limit of the velocity V → 0. When the velocity V of the skate becomes small, the scaling used in the previous sections is not adequate because the scales s_l and s_dvanish in the limit of V → 0. The constants c_1 and c_2, on the other hand start to diverge asc_1 ∼ V^-4/3,and c_2 ∼ V^-2.Using these limits in Eq. (<ref>) for p̅ we see that p̅ approaches 1, implying that for low velocities the pressure in the water layer hardly rises above the hardness p_H. Consequently the ice will recede also slowly. But if the pressure equals p_H, Eq. (<ref>) of Lozowski and Szilder <cit.> becomes valid. Fortunately Eq. (<ref>) can be solved exactly.Using that the water layer vanishes at the top x=l of the skate yields the expression for the layerh (x) = A [tanh ((l-x)/l_a]^1/2. The asymptotic thickness A of the layer is given byA =(η^2 w^2/p_H ρ L_H)^1/4V^1/2 =(k^2 λ)^1/4and the length l_a of the onset of the asymptotic value readsl_a = w/2(ρ L_H/p_H)^1/2= w/2λ^1/2.The ratio λ= ρ L_H/p_H=22.72, is a number, yielding l_a=2.62mm.In principle, we still have to match this solution with the solution in the melting regime. However, for V → 0 the melting regime shrinks to zero and the solution Eq. (<ref>) applies to the whole region.In the low velocity limit the expressions simplify, since the pressure in the layer approaches p_H. Going back to the first three expression (<ref>)-(<ref>), we have the equation for lF_N = Mg = p_H w l,or l= F_N/p_H w and l becomes equal to the static contact length 2 l_0. The indentation depth d approaches therefore 4 d_0, with d_0 the static value. The ploughing force reads in the limit V → 0F_ pl = p_H w d = p_H w l^2/2 R = F^2_N/2 p_H w R, using lfrom Eq. (<ref>). This is an interesting relation. At zero velocity there is no water layer and the ploughing force is the only friction. It shows that Amonton's law does not hold, since the friction is not proportional to the normal force.Note that the relation contains only the hardness p_H and that it is therefore a relation to measure the hardness. The integral for the friction due to the water layer becomes elementaryF_ fr = η w V/A∫^l_0dx/[tanh ((l-x)/λ)]^1/2= p_H w k^1/2λ^3/4 [l + l_a (0.5 log(2)+ 0.25 π)]. As k is proportional to V the friction force vanishes as V^1/2.§ HEAT TRANSFER IN THE WATER LAYERThe heat flow J in the water layer is related to the temperature T by theequationJ = - κ_ w∇ T In the stationary state the divergence of J equals the heat source density,which is given by Eq. (<ref>) κ_ w∇^2 T = - ηV^2/h(x)^2. The solution of this equation has to be supplemented by the boundaryconditions at the skate side T=T_ sk and the ice side T=T_ ice.The main variation is parabolic in the downward z direction. In terms of the coordinate z' with respect to the center of the layerz'=z-d(x)-h(x)/2,we get the solutionT(z') = a + b z' - c z'^2. The constant c follows from Eq. (<ref>) asc = η/2 κ_ w V^2/h(x)^2. The boundary conditions give the values of the constants a and b.{[T_ sk= a - b h(x)/2 - c [h(x)/2]^2,; T_ ice= a + b h(x)/2 - c [h(x)/2]^2. ]. With Δ T =T_ ice-T_ sk we find for b b = Δ T/h(x).The unimportant parameter a follows by using this value in one of theEqs. (<ref>).With the temperature profile given we can determine the heat flowstowards the skate and the ice. At the skate side we have a flow out of the water layerJ_ w-sk = k_ w [b +ch(x)]=k_w/h(x) [Δ T + Δ T_V],where Δ T_V is a temperature difference depending only on the velocity V and given byΔ T_V = η/2 κ_wV^2 =1.47 * 10^-3V^2.(With V the numerical value in m/s and Δ T_Vin centigrade.) Likewise we have for theflow towards the ice the valueJ_ w-ice = -κ_ w [b -ch(x)]= -κ_w/h(x)[Δ T - Δ T_V]. The fraction ζ_ w of thetotal heat produced in the layer towards the ice, is given byζ_ w = J_ w-ice/J_ w-sk= 1/2(1 - Δ T/Δ T_v). The temperature at the ice side equals the melting temperature at the pressure in the water layer. At the skate side the temperature may be higher than this meltingtemperature, but cannot be lower. So Δ T < 0 and the fraction will always be higher than or equal to 1/2. If- Δ T > Δ T_v, all heat flows towards the ice.We note that, due to the layer thickness h(x) in the denominator of Eq. (<ref>), the temperature gradient at the water-ice interface is huge.§ HEAT FLOWS IN THE ICE The temperature distribution in the ice is governed by the heat equation∂ T/∂ t = α_ iceΔ T +(∂ T/∂ t)_ forced. First we have to find the expression for the temperature forcing. Takea point x in the ice at time t=0. This point has experienced for earlier times t a temperature raise T_m(x-Vt)-T_s at the surface, which we locate for convenience at z=0. For the gradient inthe z direction this means a δ(z) dependence. So we find for the temperature forcing(∂^2 T(x,z,t)/∂ z ∂ t)_ forced = ∂ [T_m(x-Vt) -T_s]/∂ tδ (z) = -V ∂ T_m(x-Vt) /∂ xδ (z). This holds for times in the past up to t_0t_0=-(l-x)/V, with l the contact length. Let us first discuss the case where the pressure at the tip has a melting temperature T_m(l) below the surfacetemperature. Then we have to solve the following equation in the time intervalt_0 < t <0 ∂^2T(x,z,t)/∂ z ∂ t = α_ iceΔ∂ T(x,z,t)/∂ z -2 V ∂ T_m(x-Vt) /∂ xδ (z). We have inserted a factor 2 in the source term as it is easier to solve the equation in the complete space -∞ < z < ∞ and to use the symmetry between the upper and lower half z-plane. The differentiations in the Laplacian Δ may be restricted to those in the z direction, since the variation in the z direction is much larger than in the xdirection. The solution follows by Fourier transform in the z directionR_k (x,t) = ∫^∞_-∞ dz ∂ T(x,z,t)/∂ ze^i k z . The equation for R_k (t) reads∂ R_k (x,t)/∂ t = - α_ ice k^2 R_k (x,t) - 2 V ∂ T_m(x-Vt) /∂ x, with the solutionR_k (x,t) = - 2 V ∫^0_t_0 dt'∂ T_m(x-Vt') /∂ xe^α_ ice k^2 t'. The inverse Fourier transformation yields for the gradient at z=0(∂ T(x,z,t)/∂ z)_z=0 =- V ∫^0_t_0 dt' 1/√(-πα_ ice t')∂ T_m(x-Vt') /∂ x.Then changing the integration variable t' to x'=x-Vt' gives(∂ T(x,z,t)/∂ z)_z=0=-( V/πα_ ice)^1/2∫_x^l dx' 1/√(x'-x)∂ T_m(x') /∂ x'.This expression holds for the case T_m(l)<T_s.In the other case, when T_m(l)>T_s, the ice temperature is suddenly raised at the tip by the amount T_m(l)-T_s and one has in addition to theintegral the contribution from this jump (leading to a δ functionin the integral)δ(∂ T(x,z,t)/∂ z)_z=0 =( V/πα_ ice)^1/2T_m(l)-T_s/√(l-x). The combination of the integral and the jump are given in Eq. (<ref>). In <cit.> only this jump is taken into account with T_m(l)=0.Here we give for completeness the change in the layer equation as due to this jump, in order to show how the layer equation of Lozowski and Szilder <cit.> results. The fraction of heatavailable for melting is then reduced by the factorζ = 1- 2 κ_ ice [T_m(l)-T_s] h(x)/η V^3/2√(πα_i (l-x)). We find ζ(x̅) by scaling h(x) and l-xζ(x̅) = 1- q h̅ (x̅)/[l̅ -x̅]^1/2, with the constantq = √(2)κ_ ice/√(ηπα_ iceρ L_h)T_m(l)-T_s/V= 1.825 T_m(l)-T_s/V. The correction due to ζ changes the scaled equation (<ref>) to-d h̅/d x̅ = 1/h̅ (x̅)- q/√(l̅-x̅)-x̅ + c_1/c_2+ h̅^3 (x̅)h̅^3(x̅), Eq. (<ref>) is thescaled version of a similar equation for the layer given in <cit.>.It is interesting that ζ in Eq. (<ref>) approaches at thetip a finite value ζ(l̅),since h̅(l̅-x̅) vanishes in thesame way as the square root h̅(l̅-x̅) ≃ a √(l̅ -x̅). The amplitude a satisfies the equationa/2 = 1/a - q, ora = √(2 + q^2)- q. For q → 0 the amplitude a=√(2) (as before inEq. (<ref>)) and for q large, the amplitude vanishes as 1/q. So for low temperatures and slow velocities the value of a rapidly decreases, rendering the thickness of thewater layer too thin to treat the layer as a hydrodynamic system.999 bowden F. P. Bowden and T. P. Hughes, Proc. R. Soc. A 2172(1939) 280-298.F. P. Bowden,Proc. R. Soc. A 271 (1953), 462-478.schenau J. J. de Koning, G. de Groot and G. J. van Ingen Schenau, J. Biomechanics 25 (1992) 565-571.faraday M. Faraday, Experimental Researches in Chemistryand Physics, Taylor and Francies, London (1859) p. 372.rosenberg B. Rosenberg, Physics Today (2005) Dec. 50-54.amsterdam B. Weber, Y. Nagata, S. Ketzetzi, F. Tang, W. J. Smit, H. J. Bakker, E.H.G. Backus, M. Bonn and D. Bonn. “Molecular insight into theslipperiness of ice.” Under review. B. Weber, PhD. thesis (2017) Univ. Amsterdam.persson B. N. J. Persson, J. Chem. Phys. 143 (2015) 224701.dx.doi.org/10.1063/1.4936299leiden T. H. A. van der Reep, Masters Thesis, (2014) Univ. Leiden.lozowski E. P. Lozowski and K. Szilder, Int. Journ. of Offshore and Polar Engineering 23 (2013) 04.pomeau M. Le Berre and Y. Pomeau, Int. Journ. of Non-linear Mech. 75 (2015) 77-86.pourier L. Pourier, R. I. Thompson, E. P. Lozowski, S. Maw and D. J. Stefanyshyn, 21st Int. Offshore and Polar Eng. Conf. (2011)Maui, ISOPE, 3 1071.new A. Penny, E. P. Lozowski, T. Forest, C. Fong, C. Maw,P. Montgomery and N. Sinha in Physics and Chemistry of Ice (2007) 495, W. F. Kuhn, editor, Roy. Soc. Chem.nye J. F. Nye Proc. Roy. Soc. A219 4 (1953) 477-489.barnes P. Barnes, D. Tabor and J. C. F. Walter, Proc. R. Soc. Londen A 324 (1971) 127-155.karna T. Karna, Annals of Glacialogy 19 (1994) 114-120. | http://arxiv.org/abs/1706.08278v3 | {
"authors": [
"J. M. J. van Leeuwen"
],
"categories": [
"cond-mat.other",
"cond-mat.soft",
"physics.flu-dyn"
],
"primary_category": "cond-mat.other",
"published": "20170626084324",
"title": "Skating on slippery ice"
} |
Reiner–Stanton–White <cit.> defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action and a polynomial. A key example arises from the length generating function for minimal length coset representatives of a parabolic quotient of a finite Coxeter group. In type A, this result can be phrased in terms of the natural cyclic action on words of fixed content.There is a natural notion of refinement for many CSP's. We formulate and prove a refinement, with respect to the major index statistic, of this CSP on words of fixed content by also fixing the cyclic descent type. The argument presented is completely different from Reiner–Stanton–White's representation-theoretic approach. It is combinatorial and largely, though not entirely, bijective in a sense we make precise with a “universal” sieving statistic on words, .A building block of our argument involves cyclic sieving for shifted subset sums, which also appeared in Reiner–Stanton–White. We give an alternate, largely bijective proof of a refinement of this result by extending some ideas of Wagon–Wilf <cit.>.Optically induced transparency in bosonic cascade lasers A. V. Kavokin December 30, 2023 ======================================================== § INTRODUCTIONSince Reiner, Stanton, and White introduced the cyclic sieving phenomenon (CSP) in 2004 <cit.>, it has become an important companion to any cyclic action on a finite set. Some remarkable examples of the CSP involve the action of a Springer regular element on Coxeter groups <cit.>, the action of Schutzenberger's promotion on Young tableaux of fixed rectangular shape <cit.>, and the creation of new CSPs from old using multisets and plethysms with homogeneous symmetric functions <cit.>. See <cit.> for Sagan's thorough introduction to the cyclic sieving phenomenon. More recent work on the CSP includes <cit.>. Here we are concerned with cyclic sieving phenomena involving cyclic descents on words. Cyclic descents were used implicitly by Klyachko <cit.> and independently introduced by Cellini <cit.>. Since then, cyclic descents have been used by Lam and Postnikov in studying alcoved polytopes <cit.> and by Petersen in studying P-partitions <cit.>. They also appear prominently in an ongoing line of research on cyclic descent extensions for standard tableaux by Adin, Elizalde, Reiner, and Roichman <cit.>.An earlier “extended abstract” for the present work appeared in <cit.>. We assume some familiarity with the CSP, though we recall certain key statements.Suppose C_n is a cyclic group of order n generated by σ_n, W is a finite set on which C_n acts, and f(q) ∈[q]. We say the triple (W, C_n, f(q)) exhibits the cyclic sieving phenomenon (CSP) if for all k ∈,# W^_n^k#{ w ∈ W: _n^kw = w } = f(_n^k),where _n is any fixed primitive n-th root of unity.Representation theoretically, evaluations of f at n-th roots of unity yield the characters of the C_n-action on W.In many instances of cyclic sieving, and all of those considered here, f(q) is the generating function for some statistic on W. Given a statistic W →, letW^(q) ∑_w ∈ W q^ w∈[q].We say two statistics , 'W → are equidistributed on W if W^(q) = W^'(q), and we say they are equidistributed modulo n on W if W^(q) ≡ W^'(q) ( q^n - 1).Our main result is a refinement of a CSP triple first observed by Reiner–Stanton–White, which we now summarize; see Section <ref> for missing definitions. Consider words in the alphabet {1, 2, …}. Given a word w = w_1⋯ w_n of length n, let (w) denote the content of w and write_α{words w : (w) = α}for the set of words with content α. Write (w) for the major index of w. The cyclic group C_n acts on words of length n by rotation.The following expresses an interesting result of Reiner, Stanton, and White in our notation.<cit.>.Let α n. The triple(_α, C_n, _α^(q))exhibits the CSP. Reiner, Stanton, and White deduced Theorem <ref> from the following more general result about Coxeter systems.<cit.>.Let (W,S) be a finite Coxeter system and J ⊆ S. Let W_J be the corresponding parabolic subgroup, W^J the set of minimal length representatives for left cosets XW/W_J, and X^ℓ(q) ∑_ w ∈ W^J q^ℓ(w). Let C be a cyclic subgroup of W generated by a Springer regular element. Then (X, C, X^ℓ(q)) exhibits the cyclic sieving phenomenon. Theorem <ref> follows from Theorem <ref> when W = S_n by identifying W/W_J with words of fixed content , whereis the composition recording the lengths of consecutive subsequences of J, and C is generated by an n-cycle. One must also use the classical result of MacMahon thatis equidistributed with the inversion statistic on words, from which it follows that _^(q) = X^ℓ(q) <cit.>. A refinement of a CSP triple (W, C_n, W^(q)) is a CSP triple(V, C_n, V^(q))where V ⊂ W has the restricted C_n-action. If (V, C_n, V^(q)) refines (W, C_n, W^(q)), then so does (U, C_n, U^(q)) where UW - V. Thus, a CSP refinement partitions W into smaller CSPs with the same statistic. If W is an orbit, its only refinements are W and ∅. In Section <ref>, we define a statistic on words, flex, which is universal in the sense that it refines to all C_n-orbits. Such universal statistics are essentially equivalent to the choice of a total ordering for each orbitof W.We partition words of fixed content into fixed cyclic descent type (). One computes (w) by building up w by adding all 1's, 2's, …, and counting the number of cyclic descents introduced at each step. For precise details, see Definition <ref> and Example <ref>. We write the set of words with fixed content α n and cyclic descent type δ as_α,δ{ w ∈_n : (w) = α, (w) = δ}. Our main result is the following.Let α n andbe any composition. The triple(_α,δ, C_n, _α,δ^(q))refines the CSP triple (_α, C_n, _α^(q)). It is not at all clear how to modify Reiner–Stanton–White's representation-theoretic approach to Theorem <ref> to give Theorem <ref>, since _α, δ is not closed under the S_n-action. Finding a representation-theoretic interpretation of Theorem <ref> would be quite interesting.In the course of proving Theorem <ref>, we derive an explicit product formula for _α,δ^(q) mod (q^n - 1) involving q-binomial coefficients, Theorem <ref>. The formula results in a q-identity similar to the Vandermonde convolution identity; see Corollary <ref>. The argument involves constructing _α, δ algorithmically by recursively building a certain tree.The two-letter case of Theorem <ref> can be rephrased as follows. Fix n ∈_≥ 1 and k, b ∈_≥ 0. Let _k,b denote the set of subsets Δ of /n of size k where #{i ∈Δ : i+1 ∉Δ} = b. Define the statistic _k,b→ by identifying /n with {1, …, n} and setting (Δ) ∑_i ∈Δ : i+1 ∉Δ i, which sums the maximum of the cyclic blocks of Δ.The triple(_k,b, C_n, _k,b^(q))exhibits the CSP.When n = 5, k = 3, b = 2,_k,b = {{ 1, 2, 4 }, { 2, 3, 5 }, { 3, 4, 1}, { 4, 5, 2 }, { 5, 1, 3 }},which havestatistic 6, 8, 5, 7, 4, respectively, so _k,b^(q) = q^4 + q^5 + q^6 + q^7 + q^8. We then have _k,b^(_5) = 0, _k,b^(1) = 5, in agreement with (<ref>).Theorem 8.3 in <cit.> and hence Theorem <ref> builds on a representation-theoretic result due to Springer <cit.>. Our argument is highly combinatorial, but it is not entirely bijective. Finding an explicit bijection would be quite interesting. See Section <ref> for more details.A key building block of our proof of Theorem <ref> involves cyclic sieving on multisubsets and subsets, which was also first stated in <cit.>. We describe refinements of these results as well, Theorem <ref> and Theorem <ref>, restricting to certain gcd requirements in the subset case. We present a completely different inductive proof of our subset refinement in the spirit of our proof of Theorem <ref>. Both our proof of Theorem <ref> and Theorem <ref> use an extension lemma, Lemma <ref>, which allows us to extend CSPs from smaller cyclic groups to larger ones.The rest of the paper is organized as follows. In Section <ref>, we recall combinatorial background. In Section <ref>, we introduce the concept of modular periodicity and prove our extension lemma, Lemma <ref>. In Section <ref>, we define cyclic descent type. In Section <ref>, we decompose words with fixed content and cyclic descent type and prove a product formula for _,^(q) modulo q^n - 1, Theorem <ref>. Section <ref> uses the results of Section <ref> to prove our main result, Theorem <ref>. Section <ref> refines cyclic sieving on multisubsets and subsets with respect to shifted sum statistics. In Section <ref>, we introduce the flex statistic and use it to reinterpret Theorem <ref>.§ COMBINATORIAL BACKGROUNDIn this section, we briefly recall or introduce combinatorial notions on words and fix our notation. We use the alphabet of positive integers {1, 2, …} throughout unless otherwise noted. We also write #S or |S| for the cardinality of a set S. For W →, recall the notationW^(q) ∑_w ∈ W q^(w).A word w of length n is a sequence w = w_1 w_2 ⋯ w_n of letters w_i ∈. Let |w| denote the length of a word w. Let _n denote the set of all words of length n. The descent set of w is (w) {1 ≤ i < n : w_i > w_i+1}, and the number of descents is (w) #(w). The major index of w is (w) ∑_i ∈(w) i. The cyclic descent set of w is (w) {1 ≤ i ≤ n : w_i > w_i+1}, where now the subscripts are taken mod n, and we write (w) #(w) for the number of cyclic descents. Any position 1 ≤ i ≤ n that is not a cyclic descent is a cyclic weak ascent. The inversion number of w is (w) #{(i, j) : 1 ≤ i < j ≤ nand w_i > w_j}. We use lower dots between letters to indicate cyclic descents and upper dots to indicate cyclic weak ascents throughout the paper as in the following example. If w = 155.3.155.3. = 15531553, then |w| = 8, (w) = { 3, 4, 7 }, (w) = 3, (w) = {3, 4, 7, 8}, (w) = 4, (w) = 14, and (w) = 9. A composition or weak composition of n is a sequence = (_1, …, _m) of non-negative integers summing to n, typically denoted α n. A composition is strong if α_i > 0 for all i. The content of a word w, denoted (w), is the sequencewhose j-th part is the number of j's in w. For w ∈_n, (w) is a weak composition of n. We write_α {w ∈_n : (w) = α}.The cyclic group C_n ⟨_n ⟩ of order n acts on _n by rotation as_n · w_1 ⋯ w_n-1 w_nw_n w_1 ⋯ w_n-1.Typically we consider σ_n to be the long cycle (1 2 ⋯ n) ∈ S_n.The set of all words inis a monoid under concatenation. A word is primitive if it is not a power of a smaller word. Any non-empty word w may be written uniquely as w = v^f for f ≥ 1 with v primitive. We call |v| the period of w, written (w), and f the frequency of w, written (w). An orbit of _n under rotation is a necklace, usually denoted [w]. We have (w) = #[w] and (w) (w) = |w|. Content, primitivity, period, frequency, andare all constant on necklaces.The necklace of w = 15531553 = (1553)^2 is[w] {15531553, 55315531, 53155315, 31553155 }⊂_(2, 0, 2, 0, 4)⊂_8,which has period 4, frequency 2, and4. Reiner–Stanton–White gave equivalent conditions for a triple (W, C_n, f(q)) to exhibit the CSP. In place of (<ref>) in Definition <ref>, we may instead requiref(q) ≡∑_orbits ⊂ Wq^n - 1/q^n/|| - 1 ( q^n - 1),where the sum is over all orbitsunder the action of C_n on W. Note that for d | n,q^n - 1/q^d - 1 = ∑_i=0^n/d - 1 q^di≢0 ( q^n - 1).This means every C_n-action on a finite set W gives rise to a CSP (W, C_n, f(q)), where f(q) is the right hand side of (<ref>). We refer the interested reader to <cit.> for the proof of the equivalence of (<ref>) and (<ref>).If (V, C_n, f(q)) exhibits the CSP, then so do both of the triples (V, C_g, f(q)) and (V, C_n, f(q^-1)) when g | n by (<ref>). In the latter case we have relaxed the constraint f(q) ∈[q] to f(q) ∈[q, q^-1], which does no harm since (<ref>) involves evaluations at roots of unity. Further, if (V, C_n, f(q)) and (W, C_n, h(q)) exhibit the CSP, then (V ∐ W,C_n, f(q) + g(q)) and (V × W, C_n, f(q)h(q)) exhibit the CSP, where C_n acts on V × W by τ (v,w)(τ v, τ w) <cit.>. For a set S, writeSk {all k-element subsets of S},Sk {all k-element multisubsets of S}.Let α = (α_1, …, α_m)n. We use the following standard q-analogues:[n]_q 1 + q + ⋯ + q^n-1 = q^n - 1/q - 1, [n]_q! [n]_q [n-1]_q ⋯ [1]_q,nα_q[n]_q!/[α_1]_q! ⋯ [α_m]_q!∈[q],nk_qn+k-1k_q n+k-1k, n-1_q. We write [a, b] {i ∈ : a ≤ i ≤ b}. Observe that the cyclic group C_n = _n of order n acts on [0, n - 1] by _n(i)i + 1 ( n). This induces actions of C_n on [0,n - 1]k and [0,n - 1]k by acting on values in each subset or multisubset. For example, _4 ·{0,0,0,2,2,3} = {0,1,1,1,3,3}. These actions, in slightly more generality, appear in one of the original, foundational CSP results as follows.<cit.>.In the notation above, the triples[0,n - 1]k, C_n, nk_q and[0, n - 1]k, C_n, nk_qexhibit the CSP. We will also have use of the following principal specializations (see <cit.> or <cit.>):[0,n - 1]k^(q)= e_k(1, q, q^2, …, q^n - 1)= q^k2nk_q, [0,n - 1]k^(q)= h_k(1, q, q^2, …, q^n - 1)=n k_q.Here thestatistic denotes the sum of the elements of a subset or submultiset of .Recall that the length function ℓ on S_n coincides with the inversion statistic defined above on words of content (1, 1, …, 1). More generally, minimal length coset representatives of parabolic quotients S_n/S_J also have length given by the inversion statistic on the corresponding words _α. The following classical result is due to MacMahon.<cit.>.For each α n,andare equidistributed on _α with_α^(q) = nα_q = _α^(q).§ MODULAR PERIODICITY AND AN EXTENSION LEMMAWe now introduce the concept of modular periodicity and use it to give an extension lemma, Lemma <ref>, which allows us to extend CSP's from certain subgroups to larger groups. We will verify the hypotheses of Lemma <ref> in the subsequent sections to deduce Theorem <ref>.We say a statistic W → has period a modulo b on W if for all i ∈,#{ w ∈ W : (w) ≡_b i }= #{ w ∈ W : (w) ≡_b i + a }.Similarly, we say a Laurent polynomial f(q) ∈[q, q^-1] has period a modulo b ifq^a f(q) ≡ f(q)( q^b - 1),or equivalently if (q^b - 1) | (q^a - 1)f(q). For example, 1 + 5q + q^2 + 5q^3 + q^4 + 5q^5 has period 2 modulo 6. Note thathas period a modulo b on W if and only if W^(q) has period a modulo b. The following basic properties of modular periodicity will be useful throughout the paper.Let f(q) ∈[q,q^-1] and a, b, c ∈. * If f(q) has period a modulo c and period b modulo c, then f(q) has period u a + v b modulo c for any u, v ∈. In particular, f(q) has period (a,b) modulo c.* If f(q) has period a modulo b and period b modulo c, then f(q) has period a modulo c.* If f(q) has period a modulo c and b | c, then f(q) has period a modulo b.* If f(q) has period a modulo b, then so does f(q) h(q) for any Laurent polynomial h(q).* If f(q) has period a modulo b and a | b, thenf(q) ≡a/bq^b - 1q^a - 1 f(q) ( q^b - 1).(i), (iii), (iv), and (v) are straightforward. For (ii), suppose(q^b - 1) | (q^a - 1)f(q),(q^c - 1) | (q^b - 1)f(q).Write q^c - 1 = ∏_k=1^c (q - _c^k). If q - _c^k does not divide f(q), then it must divide q^b - 1 and hence q^a - 1. It follows that(q^c - 1) | (q^a - 1)f(q). Suppose C_n = ⟨σ_n⟩ acts on W. Let g | n and C_g ⟨^n/g_n ⟩⊂ C_n. If * (W, C_g, f(q)) exhibits the CSP,* f(q) has period g modulo n, and* for all C_n-orbits ⊂ W, we have n/||| g,then (W, C_n, f(q)) exhibits the CSP.LetF(q) ∑_C_n-orbits ⊂ Wq^n - 1/q^n/|| - 1.By (<ref>), (W, C_n, F(q)) exhibits the CSP, so (W, C_g, F(q)) also exhibits the CSP by Remark <ref>. Thus, by (<ref>) and condition (i),f(q) = F(q) + p(q)(q^g - 1)for some p(q) ∈[q]. Each summand of F(q) has period g modulo n since(q^n - 1) | (q^g - 1)q^n - 1/q^n/|| - 1,by condition (iii). Putting this together with condition (ii), f(q) and F(q) have period g modulo n. Using Lemma <ref>(v) twice along with (<ref>) now givesf(q)≡g/nq^n - 1/q^g - 1 f(q) = g/nq^n - 1/q^g - 1 (F(q) + p(q) (q^g - 1) ) ≡g/nq^n - 1/q^g - 1 F(q) ≡ F(q) (q^n - 1).§ CYCLIC DESCENT TYPEIn this section, we introduce the cyclic descent type of a word. We also verify hypothesis (iii) of Lemma <ref> for _α, δ for a particular g; see Lemma <ref>.Let w^(i) denote the subsequence of w with all letters larger than i removed. We have a “filtration”∅≼ w^(1)≼ w^(2)≼…≼ w^(m - 1)≼ w^(m) = w,where u ≼ v means that u is a subsequence of v. We think of this filtration as building up w by recursively adding all of the copies of the next largest letter “where they fit.” The cyclic descent type of a word w, denoted (w), is the sequence which tracks the number of new cyclic descents at each stage of the filtration. Precisely, we have the following.The cyclic descent type (CDT) of a word w is the weak composition of (w) given by(w)((w^(1)),(w^(2)) - (w^(1)), …, (w^(m)) - (w^(m - 1)) ).Note thatis constant on necklaces since rotating w rotates each w^(i) andis constant under rotations. Furthermore, (w^(1)) = 0 always, so (w) always begins with 0.Suppose w = 143124114223, sow^(1) = 1111 (w^(1)) = 0,w^(2) = 112.1122.(w^(2)) = 2,w^(3) = 13.12.11223. (w^(3)) = 3,w^(4) = 14.3.124.114.223.(w^(4)) = 5.Hence, (143124114223) = (0, 2-0, 3-2, 5-3) = (0, 2, 1, 2). Recall from (<ref>) that_α, δ {w ∈_n : (w) = α,(w) = δ}.We could define _α, δ more “symmetrically” by replacingwith “cyclic weak ascent type,” which would be the point-wise difference ofand . However, content is ubiquitous in the literature, so we use it. Despite (<ref>),andare not equidistributed even modulo n on _, in general, so (_,, C_n, _,^(q)) does not generally exhibit the CSP. For example, _(2, 2), (0, 2) = {1212, 2121}, which has_(2, 2), (0, 2)^(q) =q^2 + q^4, _(2, 2), (0, 2)^(q) = q^1 + q^3,which are not even congruent modulo q^4 - 1.If α = (α_1, …, α_m), δ = (δ_1, …, δ_m), N _, is a necklace, and ggcd(_1, …, _m, _1, …_m), then n|N|| g. Suppose N is the necklace of w, meaning (w) = n|N|, so we can write w = u^n|N|. Hence, using pointwise multiplication,(w) = n|N|(u), (w) = n|N|(u).In particular, n|N| divides _1, …, _m, _1, …_m, so n|N|| g.§ RUNS AND FALLSIn this section, we give a method to algorithmically construct _α, δ and use it to prove a product formula for _α, δ^(q) modulo q^n - 1, Theorem <ref>. We conclude the section by using this formula to verify hypothesis (ii) of Lemma <ref> for _α, δ; see Proposition <ref>. §.§ A Tree Decomposition for _α, δ We now describe a way to create words with a fixed content and CDT in terms of insertions into runs and falls. This procedure is organized into a tree, Definition <ref>, whose edges are labeled with sets and multisets. Lemma <ref> describes changes in the major index upon traversing an edge of this tree. Write w = w_1 ⋯ w_n ∈_n. A fall in w is a maximal set of distinct consecutive indices i, i+1, …, j-1, j such that w_i > w_i+1 > ⋯ > w_j, where we take indices modulo n. A run in a non-constant word w is a maximal set of distinct consecutive indices i, i+1, …, j such that w_i ≤ w_i+1≤⋯≤ w_j, where we take indices modulo n. The constant word w = ℓ^n by convention has no runs and n falls. Note that each letter in w is part of a unique fall and a unique run, except when w = ℓ^n is constant. It is easy to see that w has n - (w) falls and (w) runs, since they are separated by cyclic weak ascents and cyclic descents, respectively. Note that this holds if w is constant since then w by convention has no runs. Index falls from 0 from left to right starting at the fall containing the first letter of w, and do the same with runs. We writeF(w)[0, |w| - (w) - 1] and R(w)[0,(w) - 1]for the indices of the falls and runs of w, respectively.Let w = 26534611 = 26534611 = 26.5.346.11 ∈_8, where upper dots indicate cyclic weak ascents and lower dots indicate cyclic descents. Since (w) = 3, we have F(w) = [0, 4] and R(w) = [0, 2]. The 5 falls of w are 2, 653, 4, 61, 1, with respective indices 0, 1, 2, 3, 4. The 3 runs of w are 1126, 5, 346, with respective indices 0, 1, 2.Let w be a word. Fix a letter ℓ and pick a subset F of the falls F(w). Assume ℓ does not appear in any of the falls in F. We insert ℓ into falls F by successively inserting ℓ into each fall w_i > w_i+1 > ⋯ > w_j in F so that w_i ⋯ℓ⋯ w_j is still decreasing.Similarly, we may fix a letter ℓ and pick a multisubset R of R(w) (this time ℓ may already appear in a run in R). We insert ℓ into runs R by successively inserting ℓ into each run w_i ≤ w_i+1≤⋯≤ w_j in R so that w_i ⋯ℓ⋯ w_j is still weakly increasing. When inserting ℓ into a run already containing ℓ, the resulting word is independent of precisely which of the possible positions is used. This is the reason we insert into runs and falls instead of positions.Note that there is a slight ambiguity in our description of insertion into falls and runs, since it may be possible to insert either at the beginning or at the end of w while still satisfying the relevant inequalities. Given the choice, we always insert at the beginning of w. Let w = 26534611 . Insert 7 into falls of w with indices 0 and 3 to successively obtain 726534611 and then w'7265347611 . Note that w'= 7.26.5.347.6.11 has two more runs (or cyclic descents) than w. Now insert 7 into the runs of w' with multiset of indices { 0,2,3,3 } to successively obtain 77.26.5.347.6.11, 77.26.57.347.6.11, 77.26.57.3477.6.11, and w” 77.26.57.34777.6.11. Let_n = { w ∈_n : wends in a 1},_, = { w ∈_,: wends in a 1}.We restrict to _n and _α, δ since the major index generating function is easier to find and extends to _α, δ^(q) (mod q^n - 1).Fix w ∈_n, a letter ℓ not in w, andF ⊂ F(w) = [0, |w| - (w) - 1] andR mult.⊂ [0, (w) + |F| - 1]where mult.⊂ denotes a multisubset. Let w' be obtained by inserting ℓ into falls F of w. Note that [0, (w) + |F|-1] = R(w') indexes the runs of w'. Now let w” be obtained by inserting ℓ into runs R of w'. We say w” is obtained by inserting the triple (ℓ, F, R) into w. Observe that (w”) = (w') = (w) + |F| and w”∈W_n+|F|+|R|. We next describe the effect of inserting a single letter on . We restrict to _n so we preserve a cyclic weak ascent at the end and never add a letter to the end. The fact that the increments in major index from inserting a new letter into all possible positions form a permutation was first observed by Gupta <cit.>. Lemma <ref> tells us exactly the increment in major index based on which run or fall the newly inserted letter fits into. Suppose w' ∈_n + 1 is obtained by adding a letter ℓ to w ∈_n in any position. Then w' is obtained by inserting ℓ into some run or fall of w, and(w') - (w) = (w) - rif ℓ is inserted into run r of w (w) + 1 + f if ℓ is inserted into fall f of w. If (w') = (w), then ℓ is inserted into some run of w, and otherwise (w') = (w) + 1 and ℓ is inserted into some fall of w. Inserting ℓ into run r of w will increment the position of (w) - r descents by 1 each, so(w') - (w) = (w) - r.Let (w)1+2+⋯ + (|w| - 1) - (w), which is the sum of i ∈ [|w| - 1] where w_i ≤ w_i+1. Inserting ℓ into fall f of w will increment the position of (|w|-1) - (w) - f weak ascents by 1 each, so(w') - (w) = (|w|-1) - (w) - f,from which it follows that(w') - (w) = (w) + 1 + f. Suppose w” is obtained by inserting the triple (ℓ, F, R) into w ∈_n. Then(w”) - (w) = |F|+12 + ((w))(|F| + |R|)+ |F||R| + ∑_f ∈ F f - ∑_r ∈ R r. Let w' be obtained by inserting ℓ into falls F of w. It suffices to show(w') - (w) = |F|+12 + ((w))|F| + ∑_f ∈ F fand(w”) - (w') = ((w'))|R| - ∑_r ∈ R rsince (w') = (w”) = (w) + |F|. Both (<ref>) and (<ref>) follow from iterating Lemma <ref> and recallingis incremented by 1 each time we insert into a fall.For the rest of this section, fix a strong composition α = (_1, …, _m) of n ≥ 1 and δ = (_1, …, _m)k with _1 = 0. We emphasize that α and δ have the same number, m, of parts. For ℓ = 1, …, m, letn_ℓ α_1 + ⋯ + α_ℓ, k_ℓ δ_1 + ⋯ + δ_ℓ.For w ∈_,, we have the defining conditions |w^(ℓ)| = n_ℓ and (w^(ℓ)) = k_ℓ. Furthermore, let_ℓ[0, n_ℓ - 1 - k_ℓ - 1 - 1]_ℓ, _ℓ[0, k_ℓ - 1]_ℓ - _ℓandg (α_1, α_2, …, α_m, δ_1, δ_2, …,δ_m).If w ∈_α, δ, then the set _ℓ consists of all subsets of the falls F(w^(ℓ-1)) which, when ℓ is inserted into those falls of w^(ℓ-1), result in a word w' with k_ℓ cyclic descents. The multiset _ℓ similarly consists of all choices of runs R(w') which, when ℓ is inserted into those runs, result in a word with length n_ℓ. We restrict to strong compositions α for notational simplicity, though the results in this section may easily be generalized to arbitrary weak compositions by “flattening” weak compositions to strong ones by removing zeros. Construct a rooted, vertex-labeled and edge-labeled tree T_α,δ recursively as follows. Begin with a tree T^(1) containing only a root labeled by the word 1^_1. For ℓ=2, …, m, to obtain T^(ℓ), do the following. For each leaf w of T^(ℓ - 1) and for each triple (ℓ, F, R) with F ∈_ℓandR ∈_ℓ,add an edge labeled by (F, R) to T^(ℓ - 1) from w to w” where w” is obtained by inserting (ℓ, F, R) into w. Define T_,T^(m). Let α = (3,1,1) and δ = (0,1,0). Figure <ref> is the tree T_α,δ.Let α = (4, 2, 3) and δ = (0, 2, 1). Figure <ref> is the subgraph of T_α,δ consisting of paths from the root to leaves that are rotations of 112113323. For this full T_α, δ, the root has 42 = 6 children since 1111 has 4 falls. Each child of the root itself has 4132 = 24 children. Hence, T_α,δ has 144 leaves. Notice that the cyclic rotations of 311211332 appearing as leaves in Figure <ref> are precisely those ending in 1. It will shortly become apparent that in this example, #_α, δ = 9/4· 144 = 324.The vertices of T_α, δ which are ℓ < m edges away from the root are precisely the elements of {w^(ℓ + 1) : w ∈_α, δ}, each occurring once. In particular, the leaves of T_α, δ are precisely the elements of _α, δ, each occurring once. By definition of _ℓ and _ℓ, any leaf of T^(ℓ)has content (_1, …, _ℓ), cyclic descent type(_1, …, _ℓ), and ends in a 1, so is in{w^(ℓ) : w ∈_α, δ}. Conversely, given anyw ∈_α, δ, the word w^(ℓ) is obtainedby inserting a unique triple (ℓ, F, R) into w^(ℓ-1) by repeated applicationsof Lemma <ref>. By Lemma <ref>, the tree T_α, δ encodes a bijectionΦ_α, δ∼∏_ℓ=2^m _ℓ×_ℓgiven by reading the edge labels from the root to w. We suppress the dependence of Φ on α and δ from the notation since they can be computed from the input w. For any w ∈_α, δ,#[w] = n/α_1·# [w] ∩_α, δ.Consequently,#_α, δ = n/_1·#_α, δ. Each w ∈_α, δ has (w) = n/(w) distinct cyclic rotations, of which α_1/(w) end in 1. Using Notation <ref>, we have#_α, δ= n/α_1∏_ℓ=2^mn_ℓ-1 - k_ℓ-1δ_ℓk_ℓα_ℓ - δ_ℓ.In particular, _α, δ≠∅ if and only if0≤δ_ℓ≤α_ℓ for all 1 ≤ℓ≤ m, and δ_1 + ⋯ + δ_ℓ+1 ≤α_1 + ⋯ + α_ℓ for all 1 ≤ℓ < m. The product in (<ref>) is #∏_ℓ=2^m _ℓ×_ℓ, which is #_α, δ by the bijection Φ. Now (<ref>) follows from (<ref>), and (<ref>) follows from (<ref>).§.§ Major Index Generating Functions We next use the bijection Φ and Lemma <ref> to give a product formula for _α, δ^(q), Theorem <ref>. We then use modular periodicity to obtain an analogous expression for _,^(q) modulo q^n - 1, Theorem <ref>. Using Notation <ref>, we have _α, δ^(q)= ∏_ℓ=2^m q^k_ℓ_ℓn_ℓ - 1 - k_ℓ - 1δ_ℓ_qk_ℓα_ℓ - δ_ℓ_q^-1 = q^η(,)∏_ℓ=2^mn_ℓ - 1 - k_ℓ - 1δ_ℓ_qk_ℓα_ℓ - δ_ℓ_qwhereη(,)n - α_1 + k2+ ∑_ℓ=2^m δ_ℓ2. Recall (A) denotes the sum of the elements of a set or multiset A. Combining Φ with Lemma <ref> shows that_α, δ^(q)= ∏_ℓ=2^m∑_F ∈_ℓR ∈_ℓq^ϵ(ℓ, F, R),where ϵ(ℓ, F, R) δ_ℓ+12 + k_ℓ - 1α_ℓ + δ_ℓ (α_ℓ - δ_ℓ) + (F) - (R). Noting thatδ_ℓ+12 + k_ℓ-1α_ℓ+ δ_ℓ(α_ℓ - δ_ℓ)= k_ℓα_ℓ - δ_ℓ2,simplifying (<ref>) gives_α, δ^(q)= ∏_ℓ=2^m q^k_ℓα_ℓ - δ_ℓ2_ℓ^(q)_ℓ^(q^-1).Equation (<ref>) now follows from (<ref>), (<ref>), and the definition of _ℓ and _ℓ. As for (<ref>), consider the reversal bijection r _ℓ→_ℓ induced byx ↦ k_ℓ - 1 - xon [0, k_ℓ-1]. This bijection satisfies (r(A)) = (k_ℓ - 1)(_ℓ - _ℓ) - (A), so_ℓ^(q^-1) = q^-(k_ℓ - 1)(_ℓ - _ℓ)_ℓ^(q).Plugging (<ref>) into (<ref>) and noting that∑_ℓ=2^m (k_ℓα_ℓ - δ_ℓ2- (k_ℓ - 1)(α_ℓ - δ_ℓ))= ∑_ℓ=2^m (α_ℓ -δ_ℓ/2 - δ_ℓ^2/2 + k_ℓδ_ℓ) = n-α_1 - k/2 + ∑_ℓ=2^m (-δ_ℓ^2/2 + ∑_j=2^ℓδ_j δ_ℓ) = n-α_1 - k/2 + 1/2∑_ℓ=2^m ∑_j=2^m δ_j δ_ℓ= n-α_1 - k/2 + k^2/2gives_α, δ^(q)= q^n - _1 + k2∏_ℓ=2^m_ℓ^(q) _ℓ^(q). Using (<ref>) and (<ref>) now yields (<ref>). Let α n, δ k. The statistichas period k modulo n on _α, δ. Moreover,is constant modulo d (n, k) on necklaces in _α, δ, andW^_α, δ(q) ≡n/α_1_α, δ^(q)(mod q^d-1).Since cyclically rotating w ∈_α, δ increments each cyclic descent by 1 modulo n, we have(σ_n · w) ≡_n (w) + k.In particular,has period k modulo n on necklaces in _,. Furthermore,is constant on necklaces in _, modulo d. By (<ref>), each necklace [w] ∈_, has the same fraction, _1n, of its elements in _,, so (<ref>) follows. Using Notation <ref>, let d (n, k). Then, modulo q^n - 1,_,^(q)≡d_1q^n - 1q^d - 1∏_ℓ=2^m q^k_ℓ_ℓn_ℓ - 1 - k_ℓ - 1_ℓ_qk_ℓ_ℓ - _ℓ_q^-1≡d_1q^n - 1q^d - 1q^k2 + ∑_ℓ = 2^m _ℓ2 - _1 ∏_ℓ = 2^m n_ℓ - 1 - k_ℓ - 1_ℓ_qk_ℓ_ℓ - _ℓ_q. By Lemma <ref>,has period k modulo n on _α, δ. Hence by Lemma <ref>(i),has period d modulo n on _α, δ. Using Lemma <ref>(v) and (<ref>) gives_α, δ^(q)≡d/nq^n - 1q^d - 1_α, δ^(q) ≡d/nq^n - 1q^d - 1n/α_1_α, δ^(q) + p(q)(q^d - 1) ≡d_1q^n - 1q^d - 1_α, δ^ ( q^n - 1),where p(q) ∈[q]. Theorem <ref> now follows from Theorem <ref>. Using Notation <ref>, let d (n, k). Then, modulo q^n - 1,_α^(q) = nα_q≡∑_δd_1q^n - 1q^d - 1∏_ℓ=2^m q^k_ℓ_ℓn_ℓ - 1 - k_ℓ - 1_ℓ_q k_ℓ_ℓ - _ℓ_q^-1where the sum is over weak compositions δ of k satisfying (<ref>). In particular,#_α = nα= ∑_δn_1∏_ℓ=2^mn_ℓ - 1 - k_ℓ - 1_ℓk_ℓ_ℓ - _ℓ.Note that the two-letter case of (<ref>) is a special case of the classical Vandermonde convolution identity <cit.>. §.§ Verifying Hypothesis (ii) of Lemma <ref> for Words of Fixed Content and CDTUsing Notation <ref>, _,^(q) has period g modulo n.Let d = (n,k). By Theorem <ref>,_,^(q) ≡d_1q^n - 1q^d - 1∏_ℓ=2^m q^k_ℓ_ℓn_ℓ - 1 - k_ℓ - 1_ℓ_q k_ℓ_ℓ - _ℓ_q^-1modulo q^n - 1. The action of rotation on elements of _ℓ = [0, n_ℓ-1 - k_ℓ-1]δ_ℓ increases their sum by _ℓ modulo n_ℓ-1 - k_ℓ-1. Thus by (<ref>), n_ℓ - 1 - k_ℓ - 1_ℓ_q has period _ℓ modulo n_ℓ - 1 - k_ℓ - 1. Similarly by (<ref>), k_ℓ_ℓ - _ℓ_q^-1 has period _ℓ - _ℓ modulo k_ℓ. For ℓ=2, …, m, by Lemma <ref>(iv) we then have_,^(q) has period _ℓ modulo n_ℓ - 1 - k_ℓ - 1, and _,^(q) has period _ℓ - _ℓ modulo k_ℓ. We show _,^(q) has period α_ℓ and δ_ℓ modulo n by downward induction on ℓ, for m ≥ℓ≥ 2. Note that the base case ℓ = m is accounted for by our argument as well.Suppose _,^(q) has period α_j and δ_j modulo n for all j > ℓ. By Lemma <ref>, _α, δ^(q) has period k modulo n. By Lemma <ref>(i), _,^(q) thus has periodk_ℓ = k - (δ_m + ⋯ + δ_ℓ+1)modulo n. Since _,^(q) has period _ℓ - _ℓ modulo k_ℓ, _,^(q) has period α_ℓ-δ_ℓ modulo n by Lemma <ref>(ii).As noted, _,^(q) has period δ_ℓ modulo n_ℓ-1 - k_ℓ-1. By Lemma <ref>(i), _,^(q) also has periodn_ℓ-1 - k_ℓ-1= n - (α_m + ⋯ + α_ℓ+1)- k + (δ_m + ⋯ + δ_ℓ+1)- (α_ℓ - δ_ℓ)modulo n. Hence, as _,^(q) has period _ℓ modulo n_ℓ - 1 - k_ℓ - 1, _,^(q) has period δ_ℓ modulo n by Lemma <ref>(ii). By another application of Lemma <ref>(i), _,^(q) has period α_ℓ modulo n as well, completing the induction.Indeed, _,^(q) has period δ_1 = 0 modulo n trivially, and _,^(q) has period α_1 = n - (α_m + ⋯ + α_2) modulo n by Lemma <ref>(i). Putting everything together, _,^(q) has periods _1, …, _m, _1, …, _m modulo n, so by one more application of Lemma <ref>(i), _,^(q) has period g modulo n. § REFINING THE CSP TO FIXED CONTENT AND CYCLIC DESCENT TYPEIn this section, we verify the final hypothesis (i) of Lemma <ref> for _α, δ and deduce Theorem <ref>. Throughout this section we continue to follow Notation <ref>. We recall in particular that_ℓ[0, n_ℓ - 1 - k_ℓ - 1 - 1]_ℓ, _ℓ[0, k_ℓ - 1]_ℓ - _ℓandg (α_1, …, α_m, δ_1, …, δ_m).§.§ A Fixed Point LemmaTo prove our main result, Theorem <ref>, one approach would be to find a C_n-equivariant isomorphism between a known CSP triple and (_α, δ, C_n, _α, δ^(q)). Such a triple is hinted at by (<ref>) and the bijection Φ using products of CSP's coming from Theorem <ref>, though the approach encounters immediate difficulties. For instance, _α, δ is not generally closed under the C_n-action. In this section, we instead give a fixed point lemma, Lemma <ref>, which is intuitively a weakened version of the equivariant isomorphism approach.We define C_g-actions on _ℓ, _ℓ, and _α, δ as follows. Since g | n_ℓ-1 - k_ℓ-1, g | k_ℓ, and g | n, C_g acts on each of _ℓ, _ℓ, and _α, δ by restricting the actions of C_n_ℓ-1-k_ℓ-1, C_k_ℓ, and C_n to their unique subgroups of size g. For instance, the action of C_g on _α, δ is generated by rotation by n/g.We additionally define C_g-actions on _ℓ×_ℓ and ∏_ℓ=2^m _ℓ×_ℓ by letting C_g act diagonally. We emphasize that despite having C_g-actions on _α, δ and ∏_ℓ=2^m _ℓ×_ℓ, the bijection Φ_α, δ∼∏_ℓ=2^m _ℓ×_ℓ is not in general equivariant since _α, δ is not closed under the C_g action on _,. Given a multisubset of some set [0, a], we may encode it as a multiplicity word w_0 w_1 … w_a where w_i is the multiplicity of i. In particular, we may consider the bijection Φ_α, δ∼∏_ℓ=2^m _ℓ×_ℓ as mapping words to sequences of pairs of certain words.Consider the leaf w = 211332311 in Figure <ref> from Example <ref>. Reading edge labels gives Φ(w) = (({0, 2}, ∅), ({2}, {1,2})). Recalling that _2 consists of subsets of [0, 4-1], _2 consists of multisubsets of ∅, _3 consists of subsets of [0, 4-1], and _3 consists of multisubsets of [0, 3-1], the corresponding sequence of words is ((1010, ϵ), (0010, 011)), where ϵ denotes the empty word. Table <ref> summarizes several similar translations. Suppose w = u^k for some word u. IfΦ(u) = ((x_2, y_2), …, (x_m, y_m))encoded as multiplicity words as in Definition <ref>, thenΦ(w) = ((x_2^k, y_2^k), …, (x_m^k, y_m^k)). The insertion triples needed to build w are the sequences of k shifted copies of the insertion triples needed to build u. An element τ∈ C_g fixes w ∈_α, δ if and only if τ fixes Φ(w).For τ∈ C_n, let o(τ) denote the order of τ. It is easy to see that τ∈ C_n fixes w ∈_n if and only if there is some word u such that w = u^o(τ).Suppose τ∈ C_g fixes w, so that w = u^o(τ). By Lemma <ref>,Φ(w) = ((x_2^o(τ), y_2^o(τ)), …,(x_m^o(τ), y_m^o(τ))).Each of the words x_i^o(τ) and y_i^o(τ) is fixed by τ, so Φ(w) is fixed by τ. The reverse implication follows analogously using the fact that Φ is a bijection.§.§ Verifying Hypothesis (i) of Lemma <ref> for Words of Fixed Content and CDTUsing Notation <ref>, (_, , C_g, _, ^(q)) exhibits the CSP.We use the notation and actions in Definition <ref>. Recall that_ℓ[0, n_ℓ-1 - k_ℓ-1 - 1]δ_ℓ,_ℓ[0, k_ℓ-1]α_ℓ - δ_ℓ.From Theorem <ref>, for each 2 ≤ℓ≤ m,_ℓ, C_g, n_ℓ - k_ℓ_ℓ_q and_ℓ, C_g, k_ℓ_ℓ - _ℓ_q^-1exhibit the CSP. Taking products,∏_ℓ = 2^m _ℓ×_ℓ, C_g, ∏_ℓ = 2^mn_ℓ - k_ℓ_ℓ_q k_ℓ_ℓ - _ℓ_q^-1exhibits the CSP. Comparing this to Theorem <ref>, we have^_α, δ≡∏_ℓ = 2^mn_ℓ - k_ℓ_ℓ_q k_ℓ_ℓ - _ℓ_q^-1modulo q^g - 1, as ∑_ℓ=2^m k_ℓα_ℓ≡_g 0 because g |_ℓ for all ℓ. Thus,∏_ℓ = 2^m _ℓ×_ℓ, C_g,_, ^(q)exhibits the CSP.By Lemma <ref>, for any w ∈_α, δ,#[w] = n/_1·#([w] ∩_α, δ).Since [w] is an orbit under C_n, an element τ∈ C_n fixes w if and only if τ fixes [w] pointwise. Thus, for any τ∈ C_n,#_α, δ^τ = n/α_1·#_α, δ^τ.Combining (<ref>) and Lemma <ref> now shows that for any τ∈ C_g,#_α, δ^τ = n/α_1·#(∏_ℓ=2^m _ℓ×_ℓ)^τ. Hence, by (<ref>), the CSP in (<ref>),and (<ref>), _α, δ, C_g,n/α_1_α, δ^(q) exhibits the CSP. By (<ref>), n/α_1_α, δ(q) ≡_α, δ^(q) modulo q^d - 1, hence also modulo q^g - 1 since g | d, completing the proof. We have now finished the verification of the conditions in Lemma <ref> for _α, δ. Condition (i) is Theorem <ref>, Condition (ii) is Proposition <ref>, and Condition (iii) is Lemma <ref>. This completes the proof of Theorem <ref>.§ REFINEMENTS OF BINOMIAL CSP'SA key step in the proof of Theorem <ref> was Theorem <ref> due to Reiner–Stanton–White, which says that the triples([0,n-1]k, C_n, nk_q) and([0,n-1]k, C_n, nk_q)exhibit the CSP. Indeed, <cit.> contains two proofs, one via representation theory <cit.> and another by direct calculation <cit.>. In this section, we give two refinements of related CSP's involving an action of C_d on sets of subsets (Theorem <ref>) and multisubsets (Theorem <ref>) for all d | n, using shifted sum statistics. Our proof of the subset refinement, Theorem <ref>, does not use Theorem <ref>, so it can be used as an alternative proof of the subset case of Theorem <ref>. Our method is inspired by the rotation of subintervals used by Wagon and Wilf in <cit.>. §.§ Cyclic Actions and NotationWe define two different cyclic actions of the cyclic group of order d on [0, n - 1] and induce these actions to [0,n - 1]k and [0,n - 1]k. We also fix notation for the rest of the section.Fix n ∈_≥ 1, k ∈_≥ 0, and d | n. Let= [0,n - 1]k,= [0,n - 1]kFor all j ∈ [1, nd ], letI_d^j[(j - 1)d, jd - 1],which we call a d-interval. For any composition = (_1, …, _n/d)k with n/d parts, let_ { A ∈ : #(A ∩ I_d^j) = _jfor allj },_ { A ∈ : #(A ∩ I_d^j) = _jfor allj },where the intersection in (<ref>) preserves the multiplicity of A. We also fix cyclic groups C_d, C_d' of order d whose actions are described below.Let C_d act on [0, n-1] by simultaneous rotation of d-intervals, which is generated by the permutation_d(0 1…(d - 1)) …((n - d) (n - d + 1)…(n - 1))in cycle notation. On the other hand, C_n has a unique subgroup C_d' of order d which also acts on [0, n-1] and is generated by the permutation_n^n/d =0nd … n - nd…nd - 12nd - 1… (n - 1) .Induce these actions of C_d and C_d' up toandbyg { a_1, …, a_k }{ ga_1, …, ga_k }.Notice that the action of C_d restricts to _ and _ for any = (_1, …, _n/d)k.Let (G, X) be a pair where G is a group acting on a set X. A morphism of group actions (G, X) → (G', X') is a pair (ϕ, ψ) where ϕ G → G' is a group homomorphism and ψ X → X' is a map of sets which satisfyψ(g · x) = ϕ(g) ·ψ(x) for allg ∈ G, x ∈ X. The actions of C_d and C_d' on [0, n-1] are isomorphic since σ_d and σ_n^n/d have the same cycle type. This isomorphism explicitly arises from ϕσ_d ↦σ_n^n/d with ψ 0 ↦ 0, 1 ↦n/d, etc. Thus the actions of C_d and C_d' onandare isomorphic as well. Recall thestatistic sums the elements of a set or multiset. We also use the following shifted sum statistic. For A ∈, let'(A) ∑_a ∈ A a - ∑_i=0^k-1 i = (A) - k2.Recall from (<ref>) and (<ref>) that^'(q) = nk_q, ^(q) = nk_q.Using (<ref>), we may restate Theorem <ref> as saying that(, C_n, ^'(q)) and(,C_n,^(q))exhibit the CSP. Moreover, under the restricted action of C_d'C_n onand ,(, C_d', ^'(q)) and(, C_d', ^(q))exhibit the CSP by Remark <ref>. By Remark <ref>,(, C_d, ^'(q)) and (, C_d, ^(q))also exhibit the CSP.Let n = 8, k = 4, and d = 4. Abbreviating {0, 4, 5, 6} as 0456, etc., gives_(1,3) = {0456,0457,0467,0567, 1456,1457,1467,1567, 2456,2457,2467,2567, 3456,3457,3467,3567 }.Here, C_4 acts on [0, 8-1] by the permutation (0123)(4567), and C_4' acts by (0246)(1357). _(1, 3) contains _(1, 3) in addition to, for instance, 0444.§.§ A Multisubset RefinementWe next prove a refinement of the CSP triple (, C_d, ^(q)) in (<ref>) by fixing sizes of intersections with the d-intervals.Recall Notation <ref>, and fix a composition = (_1, …, _n/d)k. Then, (_, C_d, _^(q)) refines the CSP triple (, C_d, ^(q)).Separating the d-intervals into different multisubsets gives_≅[0,d - 1]_1⋯[0,d - 1]_n/d,which preserves the natural C_d-action andstatistic modulo d. Since[0,d - 1]_j, C_d, [0,d - 1]_j^(q)exhibits the CSP for all j, the result follows from Remark <ref>. The following analogous result holds for subsets.Recall Notation <ref>, and additionally fix a composition = (_1, …, _n/d)k. Then (_, C_d, _^^∗(q)) exhibits the CSP, where^∗(A) (A) - ∑_ j = 1^n/d_j2. Separating the d-intervals into different subsets gives_≅[0,d - 1]_1⋯[0,d - 1]_n/d,which preserves the C_d-action andstatistic modulo d. Since[0,d - 1]_j, C_d, [0,d - 1]_j^ - _j2(q)exhibits the CSP for all j, (_, C_d, _^^∗(q)) exhibits the CSP by Remark <ref>. Since we must shift thestatistic by different amounts depending on , Proposition <ref> is not a CSP refinement, in contrast to Theorem <ref>.§.§ A Subset RefinementWe next prove an honest refinement of the CSP triple (, C_d, ^'(q)) in (<ref>). To do so, we restrict to certain subsets of S for each divisibility chain ending in n. Our proof again inductively extends CSP's up from cyclic subgroups of C_d using Lemma <ref>. In this subsection we first define our restricted subsets and give some examples. We then present a series of lemmata verifying the conditions of Lemma <ref> before proving our refinement, Theorem <ref>. Suppose e | d | n. Let_d, e{ A ∈ :(d, #(A ∩ I_d^1), #(A ∩ I_d^2),…, #(A ∩ I_d^n/d) ) = e }.We have _n, (n, k) = and _n, e = ∅ for all other e. By conditioning on the sizes of the intersections with d-intervals, _d,e decomposes as the disjoint union_d,e = ∐_,ranging over all = (_1, …, _n/d)k satisfying(d, _1, …, _n/d) = e. If n = 4, k = 2, then abbreviating {0, 2} as 02, etc., gives_1,1 = {01,02,03,12,13,23} = ,_2,1 = {02,03,12,13}, _2,2 = {01,23},_4,1 = ∅, _4,2 = {01,02,03,12,13,23} = , _4,4 = ∅.Consequently, _4,2∩_2,1 = {02,03,12,13} and _4,2∩_2,2 = {01,23}. Suppose D is a totally ordered chain in the divisibility lattice ending with (n, k) | n, i.e. D = d_p | d_p - 1|⋯| d_0 | n where d_0 (n, k). Write_D _n,d_0∩_d_0, d_1∩⋯∩_d_p - 1,d_p⊂.We may now state our subset refinement. The proof is postponed to the end of this subsection.Using Notation <ref>, let D be a totally ordered chain in the divisibility lattice ending with (n, k) | n and starting with e | d. Then, (_D, C_d, _D^'(q)) refines the CSP triple (, C_d, ^'(q)). If n = 4, k = 2, and D = 1 | 2 | 4, then G = _4,2∩_2,1 has C_2 orbits { 02,13 } and {03,12}. Moreover,G^'(q) = q^1 + 2q^2 + q^3 ≡ 2(q^0 + q^1) ( q^2 - 1),so (G, C_2, G^'(q)) exhibits the CSP by (<ref>). In fact, the subset case of Theorem <ref> is the special case D = (n, k) | n of Theorem <ref>, so the proof below of Theorem <ref> yields an alternative proof of the subset case of Theorem <ref>. (, C_n, ^'(q)) exhibits the CSP. Let D be a totally ordered chain in the divisibility lattice ending with (n, k) | n and beginning with e | d. Suppose C_e' is the unique subgroup of C_d of order e. * _D = ∐_α, where the disjoint union is over a subset of the sequences α satisfying α = (α_1, …, α_n/d)k and (d, α_1, …, α_n/d) = e.* _D is closed under the C_d and C_e-actions on .* The C_e' and C_e-actions on _D are isomorphic.* For any C_d-orbitof _D, we have d/||| e.* The ' statistic has period e modulo d on _D. For (i), by (<ref>) we have _d, e = ∐_α whereα = (α_1, …, α_n/d)k and(d, α_1, …, α_n/d) = e.Write D = e | d | d_1 |…| d_r = n. For all j = 1, …, r, since d | d_j, each d_j-interval is a union of d-intervals. Thus, for A ∈_, whether A ∈_d_j, d_j + 1 for any j depends only on , so _d_j, d_j + 1∩_ = or _d_j, d_j + 1∩_ = _. Now (i) follows from _D = _d,e∩_d, d_1∩…∩_d_r - 1, d_r.For (ii), by (i) it suffices to show that each _α is closed under the C_d and C_e-actions. Since σ_d rotates d-intervals, it preserves the size of each d-interval, so σ_d indeed maps _α to itself. The same argument applies with σ_e in place of σ_d.For (iii), by (i), it suffices to show the C_e and C_e'-actions on _α are isomorphic. Recalling (<ref>), we have_≅[0,d - 1]_1⋯[0,d - 1]_n/d.By Remark <ref>, the actions of C_e and C_e' on [0,d - 1]_j are isomorphic for each j, so their actions on _ are isomorphic as well.For (iv), pick A ∈ with A ∈_α for α as in (i). Let A_jA ∩ I_d^j, which has α_j elements. Viewing A_j as a multiplicity word w_j as in Definition <ref>, we see that A_j has d-_j zeros and _j ones. For all j, w_j is some word repeated d|| times. Using the two-letter case of Lemma <ref>, we have d|||(w_j) |_j. Thus d|||(d, _1, …, _n/d) = e.For (v), it suffices to show that ' has period e modulo d on _α for α as in (i). By the gcd condition, there exist c_1, …, c_n/d∈ such thatc_1 _1 + … + c_n/d_n/d≡ e ( d).For some particular A ∈_, consider cyclically rotating the elements of A ∩ I_d^j forward by c_j in I_d^j for all j. The result is a bijection ϕ_→_ that satisfies '(ϕ(A)) ≡'(A) + e ( d), from which (v) follows.Let n=12, k=8, and D = 1 | 2 | 4 | 12. Then_D = _12, 4∩_4, 2∩_2, 1= _4, 2∩_2, 1.We have _2, 1 = ∐_α where α = (α_1, …, α_6)8 and (2, α_1, …, α_6) = 1. Similarly _4, 2 = ∐_β where β = (β_1, β_2, β_3)8 and (4, β_1, β_2, β_3) = 2. In fact,∅⊊_D ⊊_2, 1since, for instance, _α⊂_D when α = (4, 0, 1, 1, 1, 1) while _α⊂_2, 1 - _D when α = (2, 0, 2, 1, 2, 1). Let d | n. The C_d action on _d, d is trivial and(_d, d, C_d, _d, d^'(q))exhibits the CSP.All subsets in _d, d have each d-interval either full or empty, so C_d fixes every A ∈_d,d. By (<ref>), (_d, d, C_d, _d, d^'(q)) thus exhibits the CSP if and only if _d, d^'(q) ≡#_d, d mod (q^d - 1). If _d, d = ∅ the result is trivial, so take _d, d≠∅. For any A ∈_d, d, since each d-interval is full or empty, we have d | k and'(A) ≡kdd2 - k2≡k(d - k)2≡ 0 ( d).We may now prove Theorem <ref>.[Proof of Theorem <ref>] We induct on d. If d=1, then the relevant triple ( _D, C_1, _D^'(q)) exhibits the CSP trivially. For the induction step, we first claim that ( _D, C_e, _D^'(q)) exhibits the CSP. If e=d, then _D _d,d, so by Lemma <ref> the C_e action is trivial. It is easy to see that CSP's with trivial actions refine to arbitrary subsets, so (_D, C_e, _D^'(q)) exhibits the CSP in this case. If e < d, by conditioning on the sizes of the intersections of the e-intervals, we can write_D = ∐_f | e_f | Dwhere f | D denotes the chain with f prepended to D. Hence (_f | D, C_e, _f | D^'(q)) exhibits the CSP by induction for each f | e, since f | D begins with f | e. Thus (_D, C_e, _D^'(q)) exhibits the CSP by (<ref>), proving the claim.In order to realize the (_D, C_d, _D^'(q)) CSP triple from the CSP triple (_D, C_e, _D^'(q)), we verify the conditions of Lemma <ref>. From Lemma <ref>(ii), the restriction of the C_d-action on _D to the subgroup C_e' ⊂ C_d of size e is isomorphic to the C_e-action on _D, giving Condition (i). Condition (ii) is Lemma <ref>(v), and Condition (iii) is Lemma <ref>(iv). Thus (_D, C_d, _D^'(q)) exhibits the CSP by Lemma <ref>. § THE FLEX STATISTICWe conclude by formalizing the notion of universal sieving statistics and giving an example, flex, in the context of words. We end with an open problem. Given a set W with a C_n-action, we say W →_≥ 0 is a universal CSP statistic for (W, C_n) if (, C_n, ^(q)) exhibits the CSP for all C_n-orbitsof W.Let (w)denote the index at which w appears when lexicographically ordering the necklace [w], starting from 0. Let flex be the product(w) (w) (w).For example, listing N = [15531553] in lexicographic order givesN = {15531553, 31553155, 53155315, 55315531 },so, noting (N) = 2, we have(15531553) = 0,(15531553) = 0,(31553155) = 1,(31553155) = 2,(53155315) = 2,(31553155) = 4, (55315531) = 3,(55315531) = 6. The functionis a universal CSP statistic for (_n, C_n).Let N be any necklace of length n words. Since (N) = n|N|, and {(w) : w ∈ N } = {0, 1, …, |N| -1}, we haveN^(q) = ∑_j = 0^|N| - 1 q^j n|N| = q^n - 1q^n/|N| - 1,so (N, C_n, N^(q)) exhibits the CSP by (<ref>). Given a universal sieving statisticon some set W,takes on precisely the values {0, n/d, …, n-n/d} modulo n on any orbit of size d. The converse holds as well. In this sense, up to shifting values by n, universal sieving statistics are equivalent to total orderings on each orbitof W.Standing in contrast to Lemma <ref>, (N, C_n, N^(q)) does not exhibit the CSP when N = [123123], sois not a universal CSP statistic on (_n, C_n). However,trivially refines to the orbit N={1^n} for any n. Since refinement is not generally closed under intersections, it is not clear if there is any useful sense in whichon words can be “maximally refined.”It follows from Lemma <ref> and (<ref>) that Theorem <ref> is equivalent to the following.The statisticsandare equidistributed modulo n on _,. Indeed, we were originally led to Theorem <ref> through an exploration of the irreducible multiplicities of the so-called higher Lie modules (see e.g. <cit.>), which uncovered the fact thatandare equidistributed modulo n on _. Data exploration led us to conjecture this equidistribution refined to fixed cyclic descent type as in Theorem <ref>. These connections will be explained in a future publication. They also naturally suggest the problem of finding explicit bijections proving Theorem <ref>, which we leave as an open problem. For α n and δ any weak composition, find a bijection_, →_,satisfying((w)) ≡(w)( n).§ ACKNOWLEDGMENTS We were partially supported by the National Science Foundation grant DMS-1101017. We sincerely thank our advisor, Sara Billey, for her many helpful suggestions, including connections to cyclic sieving, and for her very careful reading of numerous versions of the manuscript. We also thank Vic Reiner for fruitful early discussions related to Lie multiplicities, Yuval Roichman and his collaborators for generously sharing their preprints, and the anonymous referees for their thoughtful comments. alpha | http://arxiv.org/abs/1706.08631v2 | {
"authors": [
"Connor Ahlbach",
"Joshua Swanson"
],
"categories": [
"math.CO"
],
"primary_category": "math.CO",
"published": "20170627003707",
"title": "Refined Cyclic Sieving on Words for the Major Index Statistic"
} |
ANU]J.P.A. Pittcor1 [email protected]]C. Zoppou [email protected] ANU]S.G. Roberts [email protected][cor1]Corresponding author [ANU]Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, AustraliaWe use numerical methods to study the behaviour of the Serre equations in the presence of steep gradients because there are no known analytical solutions for these problems. In keeping with the literature we study a class of initial condition problems that are a smooth approximation to the initial conditions of the dam-break problem. This class of initial condition problems allow us to observe the behaviour of the Serre equations with varying steepness of the initial conditions. The numerical solutions of the Serre equations are justified by demonstrating that as the resolution increases they converge to a solution with little error in conservation of mass, momentum and energy independent of the numerical method. We observe four different structures of the converged numerical solutions depending on the steepness of the initial conditions. Two of these structures were observed in the literature, with the other two not being commonly found in the literature. The numerical solutions are then used to assess how well the analytical solution of the shallow water wave equations captures the mean behaviour of the solution of the Serre equations for the dam-break problem. Lastly the numerical solutions are used to evaluate the usefulness of asymptotic results in the literature to approximate the depth and location of the front of an undular bore. Serre equationssteep gradients dam breakBehaviour of the Serre Equations in the Presence of Steep Gradients Revisited [ December 30, 2023 =============================================================================§ INTRODUCTIONThe behaviour of flows containing steep gradients are important to a range of problems in shallow water such as the propagation of a bore, the dam-break problem and shoaling waves on a beach.The Serre equations are used as a compromise between the non-dispersive shallow water wave equations and the incompressible inviscid Euler equations for modelling dispersive waves of the free surface in the presence of steep gradients, which are present for the Euler equations <cit.> but not for the shallow water wave equations. The Serre equations like the shallow water wave equations produce methods <cit.> that are computationally easier and quicker to solve than the best methods for the Euler equations. The Serre equations are considered the most appropriate approximation to the Euler equations for modelling dispersive waves up to the shore line <cit.>. Therefore, understanding the behaviour of the Serre equations in the presence of steep gradients offers some insight into the behaviour of steep gradients for fluids more generally.There are no known analytical solutions to problems containing steep gradients for the Serre equations. To infer the structure of solutions to problems containing steep gradients we have to resort to investigating numerical solutions of the Serre equations for these problems. There are few examples in the literature which depict the behaviour of numerical solutions to the Serre equations in the presence of steep gradients <cit.>. These papers all present problems with discontinuous initial conditions <cit.> or a smooth approximation to them when the numerical method requires some smoothness of the solutions <cit.>. Among these papers there are differences in the structures of the numerical solutions, with some demonstrating undulations in depth and velocity throughout the bore <cit.> and others showing a constant depth and velocity state in the middle of the bore <cit.>.The mean behaviour of numerical solutions to the dam-break problem for the Serre equations is consistent across the literature <cit.> and was demonstrated to be well approximated by the analytical solution to the dam-break problem by the shallow water wave equations <cit.>. Expressions for the leading wave amplitude and speed of an undular bore for the Serre equations were derived and verified for a range of undular bores by <cit.>. These expressions were also shown to be valid for all the different structures found in the literature <cit.>. The first aim of this paper is to investigate and explain why different behaviour has been published in the literature for numerical solutions of the Serre equations for problems containing steep gradients. We find that the undulations of a bore can be damped to a constant depth and velocity state by the numerical diffusion introduced by the method, as is the case for <cit.>. Oscillation damping can also occur due to the particular smoothing of the initial conditions, as is the case for <cit.>, <cit.> and <cit.>. We do find that over long time periods the Serre equations damp these oscillations as they propagate, but this natural decay is dominated by other factors in the literature.The second aim of this paper is to assess the utility of the shallow water wave equations and the results of <cit.> as guides for the evolution of an undular bore. We find that for a range of dam-break problems the analytical solution of the shallow water wave equations is a good approximation for the mean depth and velocity of the Serre equations, extending the findings of <cit.> and <cit.> to a larger range of dam-break problems. It was also found that the results of <cit.> are a good approximation to our numerical solutions.The first aim of this paper is achieved by demonstrating that our numerical solutions are good approximations to the true solutions of the Serre equations. This is accomplished by demonstrating that as the resolution of a particular method is increased, the numerical solutions converge to a numerical solution with little error in the conservation of mass, momentum and energy. The numerical solution is also consistent across the five different numerical methods. Three of the methods are the first, second and third-order methods presented by <cit.>. The first-order method is equivalent to the method of <cit.>. The fourth method is a recreation of the second-order method used by <cit.>. Lastly, the fifth method is a second-order finite difference approximation to the Serre equations. The second aim is accomplished by comparing our verified numerical solutions to the analytical solutions of the shallow water wave equations and the Whitham modulation results presented by <cit.>.The paper is organised as follows, in Section <ref> the Serre equations and the quantities they conserve are presented. In Section <ref> the smoothed dam-break problem is defined, the measures of the relative difference between numerical solutions and the relative error in the conserved quantities are presented. The analytical solution of the shallow water wave equations and the expressions for the amplitude and speed of the leading wave of an undular bore are presented. In Section <ref> the numerical methods and their important properties are presented. In Section <ref> the four different structures in the solutions of smoothed dam-break problem for the Serre equations are determined using verified numerical solutions. The verified numerical solutions are also used to evaluate how well the analytical solution of the shallow water wave equations captures the mean behaviour of the solution of the Serre equations for the dam-break problem. The Whitham modulations results are also compared to the verified numerical solutions to test their veracity. § SERRE EQUATIONS The Serre equations can be derived by integrating the full inviscid incompressible Euler equations over the water depth <cit.>. They can also be derived as an asymptotic expansion of the Euler equations <cit.>. Assuming a constant horizontal bed, the one-dimensional Serre equations are <cit.>∂ h∂ t + ∂ (uh)∂ x = 0and∂ (uh)∂ t + ∂∂ x ( u^2h + gh^22 )_Shallow Water Wave Equations + ∂∂ x (h^33 [ ∂ u ∂ x∂ u∂ x - u∂^2 u∂ x^2- ∂^2 u∂ x ∂ t ])_Dispersion Terms = 0._Serre EquationsWhere u(x,t) is the horizontal velocity over the depth of water h(x,t), g is the acceleration due to gravity, x is the horizontal spatial variable and t is time. The Serre equations are conservation laws for `mass' (<ref>), `momentum' (<ref>) and the Hamiltonian <cit.> ℋ(x,t) = 1/2(hu^2 + h^3/3(∂ u/∂ x)^2 + gh^2) which is the total energy.The total amount of a quantity q in a system in the spatial interval [a,b] at a particular time t, is measured by 𝒞_q(t) = ∫_a^b q(x,t)dx . Conservation of a quantity q implies that 𝒞_q(0) = 𝒞_q(t) for all t provided the interval is fixed and the system is closed. Our numerical methods should demonstrate conservation for the quantities h, uh and ℋ.§ SMOOTHED DAM BREAK PROBLEMIn this section we define a class of initial condition problems, called the smoothed dam-break problem that we use throughout our numerical investigation. This class of initial conditions are used in the literature <cit.> to smoothly approximate the discontinuous initial conditions of the dam-break problem, as some numerical methods require smoothness of the solutions.The smoothed dam-break problem has the following initial conditionsh(x,0) = h_0 + h_1 - h_0/2(1 + tanh(x_0 - x/α))m, andu(x,0) = 0.0m/s. This represents a smooth transition centred around x_0 between a water depth of h_0 on the right which is smaller than the water depth of h_1 on the left. Here α measures the distance over which approximately 46% of that smooth transition between the two heights occurs. Decreasing α increases the steepness of the initial conditions as can be seen in Figure <ref> where h_0=1m and h_1=1.8m. These are the same h_0 and h_1 values as those of the smoothed dam-break problem of <cit.> and the dam-break problem of <cit.>.There are no known analytical solutions of the Serre equations for the dam-break problem or an arbitrary smoothed dam-break problem. Therefore, to demonstrate that our numerical solutions converge we use the relative difference between numerical solutions. To demonstrate that our numerical solutions also have small errors in the conserved quantities we use the relative error of their conservation. Both of these measures are defined in this section.§.§ Assessing validity of Numerical SolutionsTo demonstrate that our numerical solutions converge to a solution with little error in the conserved quantities as the spatial resolution is increased we use two measures; the relative difference between numerical solutions of different resolutions and the error in the conservation of a quantity. The relative difference between numerical solutions measures their convergence, while the error in conservation measures how well the numerical solutions conserve the quantities h, uh and ℋ.We introduce the following notation for the spatial grids defined by x_i and the temporal grids defined by t^n upon which the numerical solutions are calculated. These grids are uniform so that Δ x = x_i - x_i-1 for all i and Δ t = t^n - t^n-1 for all n. We use subscripts and superscripts to denote where a quantity q is evaluated in the following way q_i^n = q(x_i,t^n). Finally, the ith cell is the interval [x_i -Δ x/2,x_i +Δ x/2] centred around x_i. §.§.§ Convergence of Numerical ResultsTo measure the convergence of the numerical solutions we ensured all grids had common locations to compare them by dividing Δ x by 2 to create finer grids. Therefore, the finest grid with the smallest Δ x contains all the locations x_i in any coarser grid. To measure the relative difference between quantities on these grids we compare them only on the coarser grid points x_i. For some quantity q we have our numerical approximation to it on the finest grid q^* and on the coarser grid q', with the relative difference between the two being L_1^q = ∑_i| q'(x_i)- q^*(x_i)|∑_i| q^*(x_i)|.§.§.§ Conserved QuantitiesTo calculate the error in conservation of a quantity, we must first calculate the total amount of the conserved quantities for the initial conditions. For the smoothed dam-break problem the initial conditions (<ref>) were integrated to obtain expressions for the total mass C_h(0), the total momentum C_uh(0) and the total Hamiltonian C_ℋ(0). Provided x_0 is the midpoint of the spatial domain [a,b ] the total amounts for the conserved quantities are 𝒞_h(0) = h_1 + h_0/2(b- a), 𝒞_uh(0) = 0 and 𝒞_ℋ(0) = g/4(h_0^2 - h_1^2 + α(h_1 - h_0)^2tanh(a - b/2 α)). To calculate how well we approximate the total amount of a quantity q in our numerical solution we fit a quartic interpolant of the primitive variables h and u over a cell utilising neighbouring cells and then apply Gaussian quadrature with 3 points. The amount of q in each cell is summed across all cells to get the total amount of q in our numerical solution at time t, which we call 𝒞^*_q(t). The error in conservation of a quantity q for a numerical solution is C_1^q = | 𝒞_q(0) - 𝒞^*_q(t) | /|𝒞_q(0)|. Note that for uh the denominator is 0 and that there is a flux of momentum due to the unequal heights at both ends of the domain. To resolve this issue for uh the error in the conservation of uh is measured by C_1^uh = | 𝒞_uh(0) - 𝒞^*_uh(t) - gt/2(h(b)^2 - h(a)^2)|.§.§ Background for derived and observed comparisonsIt was demonstrated by <cit.> and <cit.> that the analytical solution of the shallow water wave equations for the dam-break problem captures the mean behaviour of the numerical solutions of the Serre equations to the dam-break problem <cit.> and the smoothed dam-break problem <cit.>. <cit.> derived an expression for the long term amplitude of the leading wave of an undular bore A^+ for the Serre equations. Since the front of an undular bore decomposes into solitons, the speed of the leading wave S^+ can be calculated from its amplitude.To be self contained we present the analytical solution of the shallow water wave equations to the dam-break problem and the expressions derived by <cit.>.§.§.§ Shallow Water Wave Equation Analytical SolutionFor the dam-break problem the shallow water wave equations, which are the Serre equations with dispersive terms neglected, can be solved analytically. An example of the analytical solution of the shallow water wave equations for the dam-break problem is presented in Figure <ref>. Region I is the undisturbed water upstream of the dam-break at constant height (h_1) and velocity (0m/s). Region II is the rarefaction fan connecting regions I and III. Regions III and IV represent the shock with constant height (h_2) and constant velocity (u_2), these regions are separated by x_u_2 = x_0 + u_2t. Region V is the undisturbed water downstream at constant height (h_0) and velocity (0m/s) separated from Region IV by a shock which travels at velocity S_2. Expressions for the unknown quantities h_2, u_2 and S_2 in terms of h_0 and h_1 were given by <cit.> ash_2 = h_0/2(√(1 + 8 (2h_2/h_2 - h_0√(h_1) - √(h_2)/√(h_0))^2) - 1), u_2 = 2(√(gh_1) - √(gh_2))and S_2 = h_2 u_2/h_2 - h_0. Applying (<ref>) to our dam-break heights of interest; h_0 =1m and h_1 = 1.8m results in h_2 = 1.36898m , u_2 = 1.074975 m/s and S_2 = 3.98835 m/s which are shown in Figure <ref> for t=30s. The location of the front of the bore for the shallow water wave equations at time t is thus x_S_2(t) = x_0 + S_2 t so that x_S_2(30s) = 619.6505m.§.§.§ Whitham Modulation for Undular Bores of the Serre EquationsUtilizing Whitham modulation theory for a one-phase periodic travelling wave an asymptotic analytical expression for the amplitude A^+ and speed S^+ of the leading wave was derived by <cit.>. An example of an undular bore is shown in Figure <ref>. The derived expressions for A^+ and S^+ are Δ/(A^+ + 1)^1/4 - (3/4 -√(A^+ + 1))^21/10(2/1 + √(A^+ + 1))^2/5 = 0 and S^+ = √(g (A^+ + 1)) where Δ = h_b / h_0, and h_b is the height of the bore. The height of the bore created by the dam-break problem in (<ref>) used by <cit.> wash_b = 1/4(√(h_1/h_0) + 1)^2. For our dam-break heights of interest h_0 = 1m and h_1 = 1.8m we obtain h_b = 1.37082m, Δ = 1.37082, A^+ = 1.73998m and S^+ = 4.13148m/s. The location of the leading wave of an undular bore at time t is then x_S^+(t) = x_0 + S^+ t so that x_S^+(30s) = 623.9444m.§ NUMERICAL METHODS Five numerical schemes were used to investigate the behaviour of the Serre equations in the presence of steep gradients, the first (𝒱_1), second (𝒱_2) and third-order (𝒱_3) finite difference finite volume methods of<cit.>, the second-order finite difference method of <cit.> (ℰ) and a second-order finite difference method (𝒟) that can be found in the Appendix.The 𝒱_i methods are stable under a Courant-Friederichs-Lewy (CFL) conditionpresented by <cit.>. The 𝒱_i methods have demonstrated the appropriate order of convergence for smooth problems <cit.>. Furthermore, 𝒱_2 and 𝒱_3 have been validated against experimental data containing steep gradients <cit.>. The two methods 𝒟 and ℰ were found to be stable under the same CFL condition.Generally, we found that 𝒱_1 is the worst performing method due to its numerical diffusion <cit.>. Of the high-order methods ℰ is the worst performing, introducing dispersive errors. § NUMERICAL RESULTSWe investigate the behaviour of the Serre equations in the presence of steep gradients by numerically solving the smoothed dam-break problem while varying the steepness of the initial conditions. As Δ x → 0 our numerical solutions should represent a good approximation of the true solution of the Serre equations. If our numerical solutions to a smoothed dam-break problem converge to the same numerical solution with little error in conservation of mass, momentum and energy as Δ x → 0 for each method, then this numerical solution is considered an accurate approximate solution to that smoothed dam-break problem for the Serre equations.This process validates our numerical solutions for the smoothed dam-break problem, and thus validates our numerical methods to approximate the solution of the Serre equations in the presence of steep gradients, if it exists. With a validated model we can compare the numerical solution to the analytical solution of the shallow water wave equations for the dam-break problem and the results of <cit.>.Throughout most of this section we are interested in the numerical solution at t=30s to the smoothed dam-break problem with h_0 = 1m, h_1 = 1.8m and x_0 = 500m while allowing for different α values. All numerical methods used Δ t = 0.01 Δ x which is smaller than required by the CFL condition, ensuring stability of our schemes. The method 𝒱_2 requires an input parameter to its slope limiter and this was chosen to be θ = 1.2 <cit.>. The spatial domain was [0m,1000m] with the following Dirichlet boundary conditions, u = 0m/s at both boundaries, h =1.8m on the left and h =1m on the right. §.§ Observed Structures of the Numerical Solutions We observe that there are four different structures for the converged to numerical solution depending on the chosen α. They are the `non-oscillatory' structure 𝒮_1, the `flat' structure 𝒮_2, the `node' structure 𝒮_3 and the `growth' structure 𝒮_4. An example of each of these structures is shown in Figure <ref> which were obtained using 𝒱_3 with Δ x = 10/2^11m.The four structures are identified by the dominant features of the numerical solutions in regions III and IV. They also correspond to different structures in the numerical solutions that have been presented in the literature. From Figure <ref> it can be seen that as α is decreased, steepening the initial conditions, the numerical solutions demonstrate an increase in the size and number of oscillations particularly around x_u_2. We observe that the difference between 𝒮_2, 𝒮_3 and 𝒮_4 is the amplitude of the oscillations in regions III and IV.For the non-oscillatory and flat structures there is excellent agreement between all higher-order numerical methods at our highest resolution Δ x = 10/2^11m. An illustration of this agreement is given in Figure <ref> for 𝒮_2 which is the most difficult to resolve of the two structures. However, the first-order method 𝒱_1 suppresses oscillations present in the numerical solutions of other methods due to its diffusive errors <cit.>. To resolve these oscillations with 𝒱_1 much lower of values of Δ x are required.§.§.§ Non-oscillatory StructureThe 𝒮_1 “non-oscillatory” structure is the result of a large α, which causes the front of this flow to not be steep enough to generate undulations over short time periods. As the system evolves the front will steepen due to non-linearity and undulations will develop.The structure 𝒮_1 is not present in the literature as no authors chose large enough α because, such a large α poorly approximates the dam-break problem. An example of this structure can be seen in Figure <ref> for α = 40m using 𝒱_3 with various Δ x values. Because this is not a very steep problem all numerical results are visually identical for all Δ x < 10 / 2^4m.From Table <ref> it can be seen that not only have these solutions converged visually but the L_1 measures demonstrate that we have reached convergence to round-off error by Δ x = 10 / 2^8m after which the relative difference between numerical solutions plateau. Table <ref> also demonstrates that the error in conservation of the numerical solutions are at round-off error for h and ℋ. The conservation of uh is poor because the smoothed dam-break has such a large α that h(0m) ≠ 1.8m and h(1000m) ≠ 1m, causing unequal fluxes in momentum at the boundaries. As stated above when Δ x = 10/2^11m the numerical solutions from all methods are identical for this smoothed dam-break problem. The convergence of the numerical solutions as Δ x → 0 to a numerical solution with small error in conservation, independent of the method demonstrates that we have accurately solved the smoothed dam-break problem with α = 40m. Therefore, the 𝒮_1 structure should be observed in the solutions of the Serre equations for the smoothed dam-break problem for sufficiently large α.§.§.§ Flat StructureThe most common structure observed in the literature <cit.> is the “flat structure” 𝒮_2. It is observed when the initial conditions are steep enough such that the bore that develops has undulations. This structure consists of oscillations in regions III and IV which are separated by a constant height state around x_u_2. An example of the 𝒮_2 structure can be seen in the numerical solutions presented in Figure <ref> where α = 2m.As Δ x decreases the numerical solutions converge so that by Δ x = 10 / 2^8m the solutions for higher Δ x are visually identical. Table <ref> demonstrates that although we have convergence visually, the L_1 measures are still decreasing and are larger than round-off error. Likewise the C_1 measures are still decreasing and have only reached round-off error for h. This indicates that to attain full convergence of the numerical solutions of this smoothed dam-break problem down to round-off error using 𝒱_3 would require an even smaller Δ x. The relative difference between numerical solutions is small and the numerical solutions exhibit good conservation. Therefore, our highest resolution numerical solution is a good approximation to any numerical solutions with lower Δ x values. Figure <ref> demonstrates that at Δ x = 10 / 2^11m the numerical solutions of all higher order methods are the same. These results demonstrate that our highest resolution numerical solution is an accurate approximate solution of the Serre equations for the smoothed dam-break problem with α = 2m. This implies that the 𝒮_2 structure should be observed in solutions of the Serre equations for smooth dam-break problems with similar α values.These numerical solutions compare well with those of <cit.> who use the same α but different h_0 and h_1 values and observe the 𝒮_2 structure. We found that we observed this structure for all numerical method's numerical solutions to the smoothed dam-break problem with α values as low as 1m and Δ x = 10/2^11m. The numerical solutions of <cit.> use α=1m but different heights and observe the structure 𝒮_2. Therefore <cit.> and <cit.> observe the 𝒮_2 structure in their numerical results due to their choice of α for the smoothed dam-break problem. The first-order method 𝒱_1 is diffusive <cit.> and damps oscillations that are present in the numerical solutions of higher-order methods as in Figure <ref>. We find that for any smoothed dam-break problem with α≤ 4m and the dam-break problem only the 𝒮_2 structure is observed for the numerical solutions of 𝒱_1 at t=30s with Δ x = 10/2 ^11m. This is evident in Figure <ref> with the numerical solutions of 𝒱_1 using our finest grid where Δ x = 10/2^11m on our steepest initial conditions where α = 0.001m. Therefore, <cit.> using the diffusive 𝒱_1 with their chosen Δ x and Δ t, which are larger than our Δ x and Δ t could only observe the 𝒮_2 structure. §.§.§ Node StructureThe “node” structure, 𝒮_3 was observed by <cit.>. The 𝒮_3 structure has oscillations throughout regions III and IV that decay to a node at x_u_2 as can be seen in Figure <ref> where α = 0.4m.Figure <ref> demonstrates that our numerical solutions have not converged, however this is only in the area around x_u_2. Due to the large difference in numerical solutions around x_u_2 the L_1 measure over the area around x_u_2 would not be insightful. However, by omitting this region we can gain some knowledge about how well our solutions agree away from x_u_2. This was performed for the relevant L_1 measures in Table <ref> by omitting the interval [520m, 540m]. These modified L_1 measures demonstrate that while our numerical results have visually converged outside this interval, they have not converged down to round-off error. Table <ref> demonstrates that the C_1 measures are still decreasing and have only attained round-off error for h. Therefore, to resolve the desired convergence of the numerical solutions to one with small error in conservation using 𝒱_3 would require even smaller Δ x values.There is good agreement across different numerical methods for Δ x = 10/2^11m as can be seen in Figure <ref>. In particular all the higher-order methods exhibit the same structure and only disagree in a very small region around x_u_2. We observe that the numerical solution of the worst higher-order method ℰ has not converged well to the numerical solutions of the other higher-order methods. We have only obtained a good approximation to the desired numerical solution as Δ x → 0 away from x_u_2. However, our highest resolution numerical solutions from various higher-order methods are very similar. This suggests that again although we do not have full convergence, our highest resolution numerical solution is a good approximation to the desired numerical solution over the whole domain. Therefore, our highest resolution numerical solutions are an accurate representation of the solutions of the Serre equations for this smoothed dam-break problem. Therefore, the 𝒮_3 structure should be observed in the solutions of the Serre equations for the smoothed dam-break problem with α = 0.4m.These numerical solutions support the findings of <cit.> who also use some smoothing <cit.> but do not report what smoothing was performed. Using their method ℰ and similar Δ x to <cit.> we observe the 𝒮_4 “growth” structure in the numerical solution for α values smaller than 0.1m, indicating that the smoothing performed by <cit.> limited their observed behaviour to just the 𝒮_3 structure. §.§.§ Growth StructureThe 𝒮_4 “growth” structure, which has hitherto not been commonly published in the literature features a growth in the oscillation amplitude around x_u_2. An example of the growth structure can be seen for 𝒱_3's numerical solutions in Figure <ref> to the smoothed dam-break problem with α = 0.1m. This structure was observed in the numerical solutions of 𝒱_3 for Δ x = 10/2^11m at t=30s for α values as low as 0.001m and even for the dam-break problem.Figure <ref> shows that this structure can only be observed for Δ x = 10 / 2^10m, with poor convergence of the numerical results around x_u_2. Again our L_1 measures in Table <ref> omit the interval [520m,540m] in the numerical solutions. This demonstrates that although we have visual convergence away from x_u_2 our numerical solutions have not converged to round-off error as Δ x → 0. The C_1 measures in Table <ref> are still decreasing and have only attained round-off error for h, although for uh and ℋ the errors in conservation are small. These measures continue the trend in Table <ref> where smaller α's and thus steeper initial conditions lead to larger L_1 and C_1 measures because steeper problems are more difficult to solve accurately.Figure <ref> demonstrates that our numerical solutions for Δ x = 10 /2^11m with the best methods 𝒟, 𝒱_3 and 𝒱_2 disagree for only a few oscillations around x_u_2. Since both 𝒟 and ℰ are second-order finite difference methods their errors are dispersive. These dispersive errors cause the numerical solutions to overestimate the oscillation amplitude of the true solution, particularly around x_u_2. Because the dispersive errors of ℰ are larger than 𝒟 more oscillations are observed for the numerical solutions produced by ℰ. The 𝒱_3 method was shown to be diffusive by <cit.> and therefore its numerical solutions underestimate the oscillation amplitude in the true solution. Therefore, the true solution of the Serre equations should be between the dispersive method 𝒟 and the diffusive method 𝒱_3, and thus will possess the 𝒮_4 structure.The numerical solutions of 𝒟 and 𝒱_3 acting as upper and lower bounds respectively for the oscillation amplitude as Δ x is reduced is demonstrated in Figure <ref> using the maximum of h in the interval [520m, 540m]. From this figure it is clear that the amplitudes of the numerical solutions of 𝒟 converge downto the limit as the resolution is increased while the numerical solution amplitudes of 𝒱_3 converge up to it. This shows that we have effectively bounded the true solution of the Serre equations. Unfortunately, 𝒱_3 could not be run in reasonable computational times with lower Δ x, but the numerical solutions of 𝒟 show that doing so is unnecessary.These results indicate that the solutions of the Serre equations to the smoothed dam-break problem with sufficiently small α values should exhibit a growth structure at t=30s, even though we have not precisely resolved all the oscillations in our numerical solutions. It was found that decreasing α did increase the amplitude of the oscillations around x_u_2. For 𝒱_3 with Δ x= 10/2^11m and α = 0.001m the oscillations in h were bounded by the interval [1.28m,1.46m]. Of particular note is that the number of oscillations are the same in Figures <ref> and <ref> for the best methods even though they have different structures.By changing the interval and desired time for the numerical solution, Δ x could be lowered further so that by t=3s our numerical solutions have fully converged for α values as low as 0.001m. This allows us to show that the height of the oscillations around x_u_2 for the solution of the Serre equation to the smoothed dam-break problem are bounded at t=3s as α→ 0. Figure <ref> demonstrates this for the numerical solutions of 𝒱_3 with Δ x = 10/2^13m. §.§ Shallow water wave equation comparisonThe analytical solutions of the shallow water wave equations have been used as a guide for the mean behaviour of the solution of the Serre equations for the dam-break problem in the literature <cit.>.To assess the applicability of this the mean bore depth and mean fluid velocity in the interval [x_u_2-50m,x_u_2+50m] were calculated from our numerical solution to the smoothed dam-break problem with various height ratios. These means were compared to their approximations from the analytical solution of the dam-break problem for the shallow water wave equations h_2 and u_2. The results of this can be seen in Figure <ref> for numerical solutions of 𝒱_3 with Δ x = 10/2^9m to the smoothed dam-break problem at t=100s with α = 0.1m where h_0 is fixed and h_1 is varied.We use a final time of t=100s as it allows the internal structure of the bore to develop more fully giving a more reliable mean estimate, as a consequence we resort to a coarser grid to keep the run-times reasonable. We find that decreasing Δ x does not significantly alter the mean of h and u. We also find that increasing α also does not significantly alter the mean of h and u. Therefore, the mean behaviour of the true solution of the Serre equations to the dam-break problem is captured by these numerical solutions, if it exists.It can be seen that h_2 and u_2 are good approximations to the mean behaviour of the fluid inside the bore for a range of different aspect ratios. Although, as h_1/h_0 increases this approximation becomes worse, so that h_2 becomes an underestimate and consequently u_2 is an overestimate.We find that for h_1/h_0 = 1.8 the mean values of h and u inside the bore for the Serre equations are not equal to h_2 and u_2. This can be seen in Figure <ref> for the numerical solutions of 𝒱_3 with Δ x = 10/2^9m to the smoothed dam-break problem with α = 0.1m at t=300s. It can be seen that h_2 is an underestimate of h and u_2 is an overestimate of u although the difference between these values and the mean behaviour of the Serre equations is small and only noticeable over long time periods.The location of the leading wave of the Serre equations slowly diverges from the location of the front of a bore in the shallow water wave equations over long periods of time. This divergence causes the small difference evident in 𝒱_3's numerical solution to the smoothed dam-break problem with α =0.1m at t=300s using Δ x = 10/2^9m, which is shown in Figure <ref>.We note that the 𝒮_4 structure present in the numerical solutions using this method and parameters at t=30s in Figure <ref> has decayed away by t=300s in Figure <ref>. This is a trend throughout our numerical solutions where oscillation amplitude decreases over time around x_u_2, changing the structure of the solution. This can be seen by obtaining full convergence of the numerical solutions to the smoothed dam-break problem at t=3s. The converged to numerical solutions for 𝒱_3 are shown in Figure <ref>. From this figure it can be seen that the oscillation amplitudes for the numerical solutions for the smoothed dam-break problems with α = 0.4m and α = 0.1m are much larger at t=3s than they are at t=30s in Figure <ref>. Since we have demonstrated that our numerical solutions are good approximations to the true solution of the Serre equations at t=30s and t=3s, decreasing oscillation amplitude around x_u_2 over time is probably a property of the Serre equations. This implies that bounding the oscillation amplitudes at time t=3s as was done above, bounds the oscillation amplitudes at all later times. §.§.§ Contact discontinuity<cit.> noted the presence of a `degenerate contact discontinuity' which is the node in the 𝒮_3 structure and travels at the mean fluid velocity in the bore.We observe that as our numerical solutions evolve over time, oscillations appear to be released from the contact discontinuity and travel away from it in both directions, leading to decay of amplitudes around the contact discontinuity. Therefore, the contact discontinuity is an important feature and its behaviour determines the structure of the oscillations in the middle of the undular bore. The different speeds of the oscillations are determined by the phase velocity, which for the Serre equations linearised around the mean height h̅ and mean velocity u̅ in regions III and IV of the solution to the dam-break problem is υ_p = u̅±√(gh̅)√(3/h̅^2 k^2 + 3) with wave number k. It can be seen that as k →∞ then υ_p →u̅ and as k → 0 then υ_p →u̅±√(gh̅). Since the contact discontinuity travels at the mean velocity inside the bore, it corresponds to very high wave number oscillations. The oscillations on the left travel slower than the contact discontinuity and are therefore lower wave number oscillations associated with the phase velocity u̅ - √(gh̅)√(3/ (h̅^2 k^2 + 3)). The oscillations on the right travel travel quicker than the contact discontinuity and are therefore lower wave number oscillations associated with the phase velocity u̅ + √(gh̅)√(3/ (h̅^2 k^2 + 3)).These different phase velocities have two different behaviours for h and u. When the phase velocity is u̅ + √(gh̅)√(3/ (h̅^2 k^2 + 3)) we have oscillations where h and u are in-phase, while when the phase velocity is u̅ - √(gh̅)√(3/ (h̅^2 k^2 + 3)) we have oscillations where h and u are out-of-phase. This can be seen in Figure <ref> for the numerical solutions of 𝒱_3 with Δ x = 10/2^9m for the smoothed dam-break problem with α = 0.1m at t=30s.§.§ Whitham Modulation Comparsion<cit.> demonstrated that their Whitham modulation results approximated the numerical solutions of the smoothed dam-break problem well for a range of aspect ratios. We observed that the Whitham modulation results are an underestimate compared to our numerical solutions.This can be seen in Figure <ref> as the relative difference between A^+ from <cit.> and the leading wave amplitude of our numerical solution A does not converge to 0 over time. Since we find that the numerical solutions for the smoothed dam-break problem with α = 0.1m have converged for the front of the undular bore by Δ x = 10/2^8m as in Figure <ref>, our numerical solutions for A are considered reliable. We also note that unlike the oscillations around x_u_2 the leading wave amplitude increases over time.The Whitham modulation results for the location of the leading wave x_S^+ is a better approximation than that given by the shallow water wave equations x_S_2, as can be seen in Figure <ref>. § CONCLUSIONSUtilising two finite difference methods of second-order and three finite difference finite volume methods of various orders to solve the nonlinear weakly dispersive Serre equations an investigation into the smoothed dam-break problem with varying steepness was performed. Four different structures of the numerical solutions were observed and demonstrated to be valid, the general trend of these structures is that an increase in steepness increases the size and number of oscillations in the solution. This study explains the different structures exhibited by the numerical results in the literature for the smoothed dam-break problem for the Serre equations and uncovers a new result. These results demonstrate that other methods in the literature could replicate our results if their simulations are extended. Furthermore, these results suggest that this new result and its associated structure is to be expected for the solution of the Serre equation to the dam-break problem at least for short enough time spans, if it exists.We find that the analytical solution of the shallow water wave equations for the dam-break problem provides a reasonable approximation to the mean height and velocity inside the bore formed by the smoothed dam-break problem for the Serre equations. Finally, we observe that the Whitham modulations results for the leading wave of an undular bore provide a more accurate approximation to the location and depth of the front of an undular bore than the shallow water wave equations.§ REFERENCESelsarticle-num-names§The methods ℰ and 𝒟 use the centred second-order finite difference approximation to the momentum equation (<ref>), denoted as 𝒟_u. For the mass equation (<ref>) ℰ uses the two step Lax-Wendroff method, denoted as ℰ_h while 𝒟 uses a centred second-order finite difference approximation, denoted as 𝒟_h.§.§ 𝒟_u for the Momentum EquationFirst (<ref>) is expanded to get h∂ u∂ t - h^2∂^2 u/∂ x ∂ t - h^3/3∂^3 u/∂ x^2 ∂ t= -X where X contains only spatial derivatives and is X = uh∂ u/∂ x + gh∂ h/∂ x + h^2∂ u/∂ x∂ u/∂ x + h^3/3∂ u/∂ x∂^2 u/∂ x^2 - h^2u∂^2 u/∂ x^2- h^3/3u∂^3 u/∂ x^3 . All derivatives are approximated by second-order centred finite difference approximations on a uniform grid in space and time, which after rearranging into an update formula becomes h^n_iu^n+1_i - (h^n_i)^2 (u^n+1_i+1 -u^n+1_i-1/2 Δ x) - (h^n_i)^3/3(u^n+1_i+1 - 2u^n+1_i + u^n+1_i-1/Δ x^2) = - Y^n_i where Y_i^n = 2Δ tX_i^n - h_i^nu_i^n-1 + (h_i^n)^2(u^n-1_i+1 -u^n-1_i-1/2 Δ x) + (h_i^n)^3/3(u^n-1_i+1 - 2u^n-1_i + u^n-1_i-1/Δ x^2) and X_i^n = u_i^nh_i^nu^n_i+1 -u^n_i-1/2 Δ x + gh^n_ih^n_i+1 -h^n_i-1/2 Δ x + (h^n_i)^2(u^n_i+1 -u^n_i-1/2 Δ x)^2+ (h^n_i)^3/3u^n_i+1 -u^n_i-1/2 Δ xu^n_i+1 - 2u^n_i + u^n_i-1/Δ x^2 - (h^n_i)^2u_i^nu^n_i+1 - 2u^n_i + u^n_i-1/Δ x^2 - (h^n_i)^3/3u^n_i u^n_i+2 - 2u^n_i+1 + 2u^n_i-1 - u^n_i-2/2Δ x^3.Equation (<ref>) can be rearranged into an explicit update scheme 𝒟_u for u given its current and previous values, so that [[ u^n+1_0; ⋮; u^n+1_m ]] = A^-1[[ -Y^n_0;⋮; -Y^n_m ]] =: 𝒟_u(u^n,h^n, u^n-1, Δ x, Δ t ) where A is a tri-diagonal matrix. §.§ Numerical Methods for the Mass Equation The two step Lax-Wendroff update ℰ_h for h is h^n + 1/2_i+ 1/2 = 1/2(h^n_i+1 + h^n_i) - Δ t/2Δ x(u^n_i+1h^n_i+1 - h^n_iu^n_i), h^n + 1/2_i- 1/2 = 1/2(h^n_i + h^n_i-1) - Δ t/2Δ x(u^n_ih^n_i - h^n_i-1u^n_i-1) and h^n+1_i = h^n_i - Δ t/Δ x(u^n + 1/2_i+ 1/2h^n + 1/2_i+ 1/2 - u^n + 1/2_i- 1/2h^n + 1/2_i- 1/2). The quantities u^n + 1/2_i ± 1/2 are calculated using u^n+1 obtained by applying 𝒟_u (<ref>) to u^n then linearly interpolating in space and time to give u^n + 1/2_i+ 1/2 = u^n+1_i+1 + u^n_i+1 + u^n+1_i + u^n_i/4 and u^n + 1/2_i- 1/2 = u^n+1_i + u^n_i + u^n+1_i-1+ u^n_i-1/4. Thus we have the following update scheme ℰ_h for (<ref>) h^n+1 = ℰ_h(u^n,h^n,u^n+1, Δ x, Δ t ). The second order centered finite difference approximation to the conservation of mass equation (<ref>) is h^n+1_i = h^n-1_i - Δ t (u^n_ih^n_i+1 - h^n_i-1/Δ x + h^n_iu^n_i+1 - u^n_i-1/Δ x). Thus we have an update scheme 𝒟_h for all i h^n+1 = 𝒟_h(u^n,h^n,h^n-1 ,Δ x, Δ t ). §.§ Complete MethodThe method ℰ is the combination of (<ref>) for (<ref>) and (<ref>) for (<ref>) in the following way . [ u^n+1 =𝒟_u(u^n,h^n, u^n-1, Δ x, Δ t ); h^n+1=ℰ_h(u^n,h^n,u^n+1, Δ x, Δ t ) ]}ℰ(u^n,h^n, u^n-1,h^n-1, Δ x, Δ t ).The method 𝒟 is the combination of (<ref>) for (<ref>) and (<ref>) for (<ref>) in the following way . [ h^n+1=𝒟_h(u^n,h^n,h^n-1Δ x, Δ t ); u^n+1 =𝒟_u(u^n,h^n, u^n-1, Δ x, Δ t ) ]}𝒟(u^n,h^n, u^n-1,h^n-1, Δ x, Δ t ). | http://arxiv.org/abs/1706.08637v1 | {
"authors": [
"Jordan Pitt",
"Christopher Zoppou",
"Stephen Roberts"
],
"categories": [
"math.NA",
"physics.flu-dyn"
],
"primary_category": "math.NA",
"published": "20170627012957",
"title": "Behaviour of the Serre Equations in the Presence of Steep Gradients Revisited"
} |
We present a generalization of Bloch's theoremto finite-range lattice systems of independent fermions, in which translation symmetry isbroken solely due to arbitrary boundary conditions, by providing exact,analytic expressions for all energy eigenvalues and eigenstates. Starting with a re-ordering of the fermionic basis that transforms the single-particleHamiltonian into a corner-modified banded block-Toeplitz matrix, a key step is aHamiltonian-dependent bipartition of the lattice, which splits the eigenvalue probleminto a system of bulk and boundary equations. The eigensystem inherits most of itssolutions from an auxiliary, infinite translation-invariant Hamiltonian that allows for non-unitary representations of translation – hence complex valuesof crystal momenta with specific localization properties.A reformulation of the boundary equation in termsof a boundary matrix ensures compatibility with the boundary conditions, and determines the allowed energy eigenstates in the form of generalized Bloch states. We show how the boundarymatrix quantitatively captures the interplay between bulk and boundary properties, leading to theconstruction of efficient indicators of bulk-boundary correspondence.Remarkable consequences of our generalized Bloch theorem are the engineering of Hamiltonians that host perfectly localized, robust zero-energy edge modes, and the predicted emergence,for instance in Kitaev's Majorana chain, of localized excitations whose amplitudes decay inspace exponentially with a power-law prefactor. We further show how thetheoremmay be used to construct numerical and algebraic diagonalization algorithms for the class ofHamiltonians under consideration, and use the proposed bulk-boundary indicator tocharacterize the topological response of a multi-band time-reversal invariant s-wave topologicalsuperconductor under twisted boundary conditions,showing how a fractional Josephson effect can occur without entailing a fermionic parity switch.Finally, we establish connections to the transfer matrix method anddemonstrate, using the paradigmatic Kitaev's chain example, that a defective (non-diagonalizable)transfer matrix signals the presence of solutions with a power-law prefactor.A generalization of Bloch's theorem for arbitrary boundary conditions:Theory Lorenza Viola December 30, 2023 ==============================================================================§ INTRODUCTION Modern electronic transport theory in crystalline solids relies on twofundamental tenets. On the one hand, because of the Pauli exclusion principle, electrons satisfy Fermi-Dirac statistics; on the other,Bloch's theorem allowslabeling of the one-electron wave-functions in terms of their crystal momenta. The setof allowed momenta, defining the so-called Brillouin zone,is determinedby symmetry and the fact that Born-von-Karman(periodic) boundary conditions (BCs) are enforced on the system<cit.>. It isthe organization of electrons within the Brillouin zone that is key to defining its conduction properties. While the assumption of a perfect crystal with a unit cell that is periodically repeated emphasizesthe (discrete) symmetry of translation, the torus topological constraint imposed bythe Born-von-Karman condition further eliminates the potential emergence of edge or boundaryelectronic states in a real, finite crystal. Although much of the transport properties are determinedby bulk electrons, technologically relevant processes on the surface of solids are known tolead to intriguing phenomena, such as surface superconductivity <cit.> orKondo screening of magnetic impurities resulting in exotic surface spin textures <cit.>.Early theoretical investigations by Tamm and Shockley <cit.> initiated thesystematic study of surface state physics, that witnessed a landmark achievement with thediscovery of the quantum Hall effect <cit.>, and that today finds its most striking applicationsin topological insulating and superconducting materials <cit.>.The organization of bulk electronscomes with a twist. The quantum electronicstates labeled by crystal momenta organize in ways subject to classificationaccording to integer values of topological invariants defined over the entire Brillouin zone <cit.>.The first Chern number, determined in terms of the Berry connection, is one of thosetopological invariants, defining a topologically non-trivial electronic phase wheneverits value differs from zero <cit.>. For instance, the transverseconductivity of a quantum Hall fluid is proportional to such a Chern number.Perhaps surprisingly, there appears to be a connection betweena non-vanishing value of the topological invariant, a bulk property, and the emergenceof “robust” boundary states, an attribute of the surface. This principle is known as the bulk-boundarycorrespondence <cit.>. At first, this relation seems odd, since surface propertiesaretotally independent from those of the bulk; for example, one can deposit impurities,generate strain and reconstruction, or add externally applied electric fields only on the surface.Nonetheless, it seems reasonable to assume that as long as the symmetry protecting the surfacestates is not broken by external means, a bulk-boundary correspondence will still hold,although the quantum surface state will, in general, get transformed<cit.>. In other words,although the mere existence of a boundary mode may be robust, onlyclassical information may be protected in general <cit.>.It is apparent that Bloch's theorem and its consequences pertain to the realm of bulk physics.A crystal without boundaries is required to establish it. But, can one generalize Bloch'stheorem for independent electrons to arbitrary BCs, so that bulk and surface statescan be handled on an equal footing, and physical insight about the interplay betweenbulk and boundary may be gained?In light of our previous discussion, it is clear that to accomplishsuch a task one needs to give up on some concepts, such as the notion of a Brillouin zone.If possible, such a generalization would allow us to formulate a bulk-boundary correspondenceprinciple that makes use of both bulk and boundary information. It is tempting to arguethat the relative importance of BCs diminishes as the size of the crystal grows. Notwithstanding, for example, recent work shows that BCs impact the quasi-conserved local charges of one-dimensional systems, with important consequences for bulk quench dynamics <cit.>. More generally, the statisticalmechanics of topologically nontrivial systems begs some answers directly relevant tothe questions above <cit.>. In this paper, we generalize Bloch's theorem to systems of independent electronssubject to arbitrary BCs. Intuitively speaking, one may expect such a result on thebasis that translation symmetry is only mildly broken by BCs – namely, clean (disorder-free)systems are translationally-invariant away from the boundary. Our generalized Blochtheorem makes this idea precise, by providing an exact (often in fully closed-form) description of the eigenstates of the system's Hamiltonian in terms of generalized eigenstates ofnon-unitary representations of translation symmetry in infinite space, that is,with boundaries at infinity and no torus topology <cit.>. As aresult, both exponentially decaying edge modes and more exotic modes with power-lawprefactors can emerge, provided the BCs allow them. Our generalized Bloch theorem leverages the bulk-boundary separation of the Schrödinger equation we introduced in Ref. [abc] and the full solution of the bulk equation rigorously established in Ref. [JPA]. It extends the diagonalization procedure described in Ref. [abc], and recently used in Ref. [KatsuraTwisted], to a more general class ofHamiltonians and BCs, which in particular allows for different modifications to beimposed on different boundaries. A unifying theme behind these results is an effective analytic continuation to the complexplane of the standard Bloch's Hamiltonian off the Brillouin zone. This analytic continuation is remarkably useful because the original problem reduces toa matrixpolynomial function <cit.>.Interestingly, a recent study made use of similar polynomial structures for the purpose of topological classification <cit.>.The outline of this paper is as follows.In Sec. <ref> we discuss a re-arrangement of the fermionic basis that allows us to reduce the diagonalization of the original many-electron finite-range quadratic Hamiltonian in second quantization, subject to specified BCs, to the one of a single-particle Bogoliubov-deGennes Hamiltonian that has the structure of a corner-modified block-Toeplitz matrix, as introduced in Ref. [JPA].Section <ref>developsa structural characterization of the energy eigenstates for the many-electronsystems under consideration, culminating into our generalization ofBloch's theorem. Like the usual Bloch's theorem, such a generalization is firstand foremost a practical tool for calculations, granting direct access to exactenergy eigenvalues and eigenstates.In Sec. <ref>, we provide two new procedures – one numericaland another algebraic – for carrying out the exact diagonalization of the single-particle Hamiltonian,based on the generalized Bloch theorem. The algebraic procedure, which may provide closed-formsolutions to the problem, is explicitly illustrated through a number of examples inSec. <ref>. While, in order to illustrate our methodology,we focus largely on one-dimensional systems here,we anticipate that additional applications to higher-dimensional problemswill be addressed in a companion paper <cit.>. Remarkably, while mid-gap modes with power-law prefactors have been predicted for systems with long-range couplings,we show analytically that they can alsoprominently manifest in short-range tight-binding models of topological insulators andsuperconductors <cit.>.Crucially, our generalized Bloch theorem also allows derivation of a boundary indicator for the bulk-boundary correspondence, which contains information from both the bulk and the BCs and, as remarked in Ref. [abc], is computationally more efficient than other indicators also applicable in the absence of translational symmetry <cit.>.This is the subject of Sec. <ref>. In the same section, we expand on the analysisof the two-band time-reversal invariant s-wave topological superconductingwire we introduced previously <cit.>, by employingour newly defined indicator of bulk-boundary correspondence – constructed byusing the generalized Bloch theorem, as opposed to the simplified Ansatz we presented in Ref. [abc]. Specifically, this indicator is employed in the analysis of theJosephson response of the s-wavesuperconductor in a bridge configuration, sharply diagnosing the occurrenceof a fractional 4π-periodic Josephson effect. Remarkably, we find thatthis is possible without a conventional fermionic parity switch, whichwe explain based on a suitable transformation into two decoupled systems, each undergoing a parity switch. Section<ref> establishes some important connections between our generalized Bloch theorem and the widely employed transfer matrix approach <cit.>. Interestingly, from the standpointof computing energy levels, our bulk-boundary separation is in many ways complementary tothe transfer matrix method.While the latter canhandle bulk disorder (at a computational cost), it does not, a priori, lend itself to investigating thespace of arbitrary BCs in a transparent way.On the contrary, our generalized Bloch theorem can handle arbitrary BCsefficiently, as long as the bulk respects translational invariance –with arbitrary (finite-range) disorder on the boundary being permitted. Looking afresh at the transfer matrix approach from thegeneralized Bloch theorem's perspective yields a remarkable result: the generalized eigenvectorsof the transfer matrix, whose role has been appreciated only recently <cit.>, describe energy eigenstates with power-law corrections to an otherwise exponential behavior.Our generalized Bloch theorem further suggests a way to extend the transfer matrix approachto a disordered bulk and arbitrary BCs. A discussion of the main implications of our work, along with outstanding research questions,concludes in Sec. <ref>, whereas additional technical material is included in separate appendixes.§ FROM INDEPENDENT FERMIONS TO TOEPLITZ MATRICESWe begin by describing the class of model Hamiltoniansinvestigated in this and the companion paper <cit.>.The upshot of this section will be a non-conventional re-ordering of thephysical subsystems' labels that allows recasting the single-particle(Bogoliubov-de Gennes (BdG)) Hamiltonians in Toeplitz form,essential forthe exact diagonalization procedure we will describe. Consider a D-dimensional, translation-invariant infinite system of independent fermions.Such a system is described in full generality by a quadratic, not necessarilyparticle-number-conserving, Hamiltonian in Fock space.In a lattice approximation,the vector position of a given fermion in the regular crystal lattice can be writtenas the sum of a Bravais lattice vector and a basis vector <cit.>. We will include these basis vectors as part of the internallabels, and denote Bravais lattice vectors asȷ≡∑_μ=1^D j_μ_μ,with _1,…,_D primitive vectors and eachj_μ∈ℤ. An orthonormalbasis of the Hilbert space of single-particlestates is thus labeled by Bravais lattice vectors ȷ, and a finite number of internal labels m=1,…,d_ int.We denote by c^ _ȷ m (c^†_ȷ m) the fermionic annihilation (creation) operatorcorresponding to lattice vector ȷ and internal state m. The Hamiltonian of a translation-invariantsystem can then be written asH = ∑_∑_ȷ[Φ̂^†_ȷK_Φ̂^ _ȷ+ +1/2(Φ̂^†_ȷΔ_Φ̂_ȷ+^† +h.c.)],with Φ̂^†_ȷ≡[c^†_ȷ 1⋯ c^†_ȷ d_ int ],$̊ a Bravais lattice vector, and thed_int×d_inthopping and pairingmatricesK_,Δ_satisfying K_-=K_^†,Δ_-=-Δ_^ T, where the superscriptTdenotes the transpose operation. Forarrays, such asΦ̂^†_ȷandΦ̂_ȷ^ , we stick to the convention that those appearing on the left (right) of a matrix are row (column) arrays.Since the infinite system is translation-invariant in allDdirections, it is customary to introduce the volume containing the electrons by imposingBorn-von Karman (periodic) BCs over a macroscopic volumecommensurate with the primitive cell of the underlying Bravais lattice. If the allowedȷ's in the macroscopic volume correspond toj_μ=1,…,N_μ,then, Φ̂_^†≡∑_ȷe^i·̨ȷ/√(M)Φ̂_ȷ^†defines the Fourier-transformed array of creation operators ofreal Bloch wavevector (or crystal momentum),≡̨∑_μ=1^D k_μ/N_μ _̱μ, withk_μintegers such that$̨ lies inside the Brillouin zone. Thetotal number of primitive cellsis given by M=N_1 N_2 … N_D, and _̱μ defines the reciprocal lattice vectorssatisfying _μ·_̱ν = 2πδ_μν, with δ_μν representing Kronecker's delta<cit.>. Finally, by letting * denote complex conjugation,one can express the Hamiltonian of Eq. (<ref>) in momentum space asH = 1/2∑_[̨Φ̂_^† K_Φ̂^ _ + Φ̂_-^† K_-^*Φ̂^ _-+ Φ̂_^†Δ_Φ̂_-^† + Φ̂_Δ_-^*Φ̂_-],which has a block structure in terms of the matricesK_≡∑_e̊^i·̨K_,Δ_≡∑_e̊^i·̨Δ_. Now let us turn our attention to systems that are periodic along (D-1) directions andterminated by two parallel hyperplanes perpendicular to the direction _1.We then write the allowed values of ȷ as ȷ = j _1+ȷ_⊥, j=1,…,N=N_1, ȷ_⊥ = ∑_μ=2^Dj_μ_μ .In this scenario, each Bloch wavevector $̨ is no longera good quantum number. However, we can still block-diagonalize the Hamiltonian in the partial basis Φ̂__̨⊥^† = √(N)∑_ȷ_⊥e^i_̨⊥·ȷ_⊥/√(M)Φ̂_ȷ_⊥^†,_̨⊥ =∑_μ=2^Dk_μ/N_μ_̱μ ,whereΦ̂_ȷ_⊥^†is defined to be the arrayΦ̂_ȷ_⊥^†≡[Φ̂__1+ȷ_⊥^† Φ̂_2_1+ȷ_⊥^† Φ̂_N_1+ȷ_⊥^† ]. A system with sudden termination at hyperplanes correspondig toj=1andj=Nis modeled by open (or hardwall) BCs, in which case the Hamiltonian can be expressedasH_N ≡∑__̨⊥H_N,_̨⊥,H_N,_̨⊥ = 1/2(Φ̂__̨⊥^† K__̨⊥Φ̂^ __̨⊥ + Φ̂_-_̨⊥^† K_-_̨⊥^*Φ̂^ _-_̨⊥+ Φ̂__̨⊥^†Δ__̨⊥Φ̂_-_̨⊥^† + Φ̂__̨⊥Δ_-_̨⊥^*Φ̂_-_̨⊥),in terms ofNd_int ×Nd_intmatrices[K__̨⊥]_jj' = K_j'-j,_̨⊥≡∑__̊⊥ e^i·̨K_, =̊ (j'-j)_1+_̊⊥ ,and analogously defined matricesΔ__̨⊥. We will henceforthassume that therangeRof hopping and pairing along the_1direction is finite. This means that K_r,_̨⊥ = Δ_r,_̨⊥=0, ∀ _̨⊥if|r|>R. In this paper, we are interested in BCs more general than open BCs. They are modeled by a Hermitian many-body operatorWon Fock space whichsatisfies the following restrictions (see also Appendix <ref>):* W has no effect beyond the “boundary slab”, containing basis vectorsȷ = b_1 +ȷ_⊥, b=1,…,R, N-R+1,…,N;* W is periodic along the D-1 directions _2,…,_D, and has a decomposition analogous to that of H_N.Because of the latter restriction,W ≡∑__̨⊥ W__̨⊥with[W__̨⊥]_b b' = 1/2(Φ̂_b,_̨⊥^† W^(K)__̨⊥Φ̂^ _b',_̨⊥ + Φ̂_b,-_̨⊥^† (W^(K)_-_̨⊥)^*Φ̂^ _b',-_̨⊥+ Φ̂_b,_̨⊥^† W^(Δ)__̨⊥Φ̂_b',-_̨⊥^† + Φ̂_b,_̨⊥ (W^(Δ)_-_̨⊥)^*Φ̂_b',-_̨⊥),whereb,b' ∈{1,…,R, N-R+1,…,N},W__̨⊥^(K)is Hermitian andW__̨⊥^(Δ)isantisymmetric for each_̨⊥. Then, the model Hamiltonian, with arbitrary BCs, becomesH=H_N+W=∑__̨⊥H__̨⊥, H__̨⊥ = H_N,_̨⊥ + W__̨⊥.From now on, we will focus on diagonalizing one such blockH__̨⊥, for a fixed value of_̨⊥. We will investigate the interplay between _̨⊥ and our diagonalization algorithm, (and, more generally, disordered BCs), in Ref. [PRB2]. The next step consists of deriving the BdG Hamiltonianfor this block. The conventional way <cit.> is to use the (Nambu) basisΨ̂__̨⊥^†≡[ Φ̂__̨⊥^†Φ̂_-_̨⊥ ],withΦ̂^†__̨⊥defined in Eq. (<ref>), so thatH__̨⊥can be expressed in the form,H__̨⊥ =1/2Ψ̂__̨⊥^†H__̨⊥Ψ̂__̨⊥+ 1/2tr(K__̨⊥+W__̨⊥^(K))in terms of a Hermitian matrixH__̨⊥(note thatthe matrixW__̨⊥^(K)has entries[W__̨⊥^(K)]_jj'=0if any ofj,j'take values from the set{R+1,…,N-R}). This relation leads us to a BdG HamiltonianH__̨⊥ ≡H_N,_̨⊥+W__̨⊥withH_N,_̨⊥ = [ K__̨⊥ Δ__̨⊥; -Δ_-_̨⊥^* -K_-_̨⊥^* ], W__̨⊥ = [ W__̨⊥^(K) W__̨⊥^(Δ); -W_-_̨⊥^(Δ)^* -W_-_̨⊥^(K)^* ].The diagonalization of the BdG HamiltonianH__̨⊥ implies that of H__̨⊥, as detailed forexample in Ref. [blaizot].The2×2block-structure ofH__̨⊥emphasizes the intrinsiccharge-conjugation symmetryunder the anti-unitary operator𝒞 ≡(1_Nd_int τ_x) 𝒞_cc,i.e., 𝒞H__̨⊥𝒞^-1 = -H_-_̨⊥, whereτ_xis the Pauliσ_x-matrix in the Nambu basis,and𝒞_ccdenotes complex conjugation. Such a block-structure, however, doesnot explicitly highlight the role oftranslation invariance. For this reason, we reorder the (Nambu) basis according to <cit.> Ψ̂__̨⊥^†≡[ Ψ̂_1,_̨⊥^†⋯ Ψ̂_N,_̨⊥^† ],Ψ̂_j,_̨⊥^†≡[Φ̂_j,_̨⊥^† Φ̂^ _j,-_̨⊥ ] ,so thatthe BdG Hamiltonian transforms to H__̨⊥↦ H__̨⊥≡ H_N,_̨⊥+W__̨⊥,in terms of abanded block-Toeplitz matrixH_N,_̨⊥= H_N,with entries[H_N]_jj'=h_j'-jalong the diagonals, and a block matrixW__̨⊥= W, whereh_r = [ K_r,_̨⊥ Δ_r,_̨⊥; -Δ_r,-_̨⊥^* -K_r,-_̨⊥^* ],[W]_bb' = [ W^(K)_bb',_̨⊥ W^(Δ)_bb',_̨⊥; -(W_bb',-_̨⊥^(Δ))^* -(W_bb',-_̨⊥^(K))^* ].Explicitly, in array form, we have:H_N= [ h_0 … h_R 0 ⋯ 0; ⋮ ⋱ ⋱ ⋱ ⋮; h_R^† ⋱ ⋱ 0; ⋱; ⋱; 0 ⋱ ⋱ h_R; ⋮ ⋱ ⋱ ⋱ ⋮; 0 ⋯ 0 h_R^† ⋯ h_0 ],W = [ w^(l)_11… w^(l)_1R 0w_11… w_1R;⋮⋱⋮ ⋮ ⋮⋱⋮; w^(l)_R1… w^(l)_RR ⋮w_R1… w_RR;;;0⋯⋯ 0 ⋯⋯0;;; w^†_11… w^†_1R ⋮w^(r)_11… w^(r)_1R;⋮⋱⋮ ⋮ ⋮⋱⋮; w^†_R1… w^†_RR 0w^(r)_R1… w^(r)_RR ],where we have used the notationw^(l)_bb'≡W_bb' , w^(r)_bb'≡ W_N-b+1,N-b'+1,w_bb'≡ W_b,N-b'+1 .Here, the superscript(l)[or(r)] indicates the entries that allowhoppings only near the left [or right] boundary, whereas the oneswithout superscript allow hoppings from the left to the right boundary slabs. The matrixH=H_N+Wis acorner-modified banded block-Toeplitz matrix as defined in Ref. [JPA], and is amenable to the exact solution approach described therein <cit.>.This transformed BdG Hamiltonian allows us to write the second-quantized HamiltonianH__̨⊥in the formH = 1/2∑_j=1^NΨ̂^†_j h_0Ψ̂^ _j + 1/2∑_r=1^R(∑_j=1^N-rΨ̂^†_j h_rΨ̂^ _j+r +h.c.)+1/2∑_b,b'Ψ̂^†_b W_bb'Ψ̂^ _b' + 1/2tr(K+W^(K)),where we have dropped the label_̨⊥everywhere. In particular, for one-dimensional systems (D=1), we recover (up to a constant) theclass of Hamiltonians considered in Ref. [abc], provided thatWis expressibleas W = 1/2∑_r=1^R∑_b=N-R+1^N(Ψ̂^†_b g_rΨ̂^ _b+r-N + h.c.),for some2d_int×2d_intmatricesg_r.Notice that for particle number-conserving systems (Δ=0=W^(Δ)),the single-particle Hamiltonian is justH = K+W^(K), which is already a corner-modified, banded block-Toeplitz matrix. In such cases, the re-ordering of the basis is not required, and one may directly apply the diagonalization procedure described in the following sections toH, with internal blocks of dimensiond_int. In order to have a uniform notation, we shall used ≡{[d_ int Δ=0 (number-conserving); 2d_ int Δ≠ 0(number-non-conserving) ]. . § ALGEBRAIC CHARACTERIZATION OF ENERGY EIGENSTATES A main goal of this work is to diagonalize the single-particle HamiltonianH=H_N+W, which is a corner-modified, banded block-Toeplitz matrix.In this section, we investigate the structure of its energy eigenstates,which will culminate in a generalization of Bloch's theorem to systems described bysuch model Hamiltonians. Our analysis will illustrate, in particular, that for non-generic parameter values,Hamiltonians may display a finite number of exceptional (singular) energies corresponding todispersionless,flat bands. The latter represent a macroscopic number of energy eigenstates that are localized in the bulk and, thus, are completely insensitiveto BCs. It is remarkable that the analytic continuation of the Bloch Hamiltonian canstill encompass this situation. We will show how to use it to construct the localized flat band energyeigenstatesdirectly in real space.§.§ An impurity problem as a motivating example Consider the simple tight-binding HamiltonianH_N = -t∑_j=1^N-1(c^†_j c_j+1+ c^†_j+1 c_j),defined on an open chain ofN(even) lattice sites with nearest-neighbor hopping strengtht, and lattice constanta=1. The corresponding single-particle Hamiltonian isH_N = -t∑_j=1^N-1(|j⟩⟨ j+1|+|j+1⟩⟨ j|),and breaks translation-invariance due to the presence of the boundary, so that the crystal momentum is not a good quantum number. In fact, for anyk ∈(0,2π], the state|k⟩= 1/√(N)∑_j=1^Ne^ikj|j⟩(labeled byk) obeys H_N|k⟩ = -2tcos k |k⟩+t/√(N)(|1⟩ + e^ik(N+1)|N⟩),with a similar relation holding for-kH_N|-k⟩ = -2tcos k |-k⟩ + t/√(N)(|1⟩ + e^-ik(N+1)|N⟩).The first term on the right-hand side of Eqs. (<ref>)-(<ref>) indicates that|k⟩and|-k⟩“almost” (for largeN) satisfy the eigenvalue relation with energy-2tcosk,while the two terms in the brackets show that the eigenvalue relation is violatednear the two edges of the chain. Underperiodic BCs,-2tcoskis the actual energy eigenvalue of the eigenstate|k⟩(and|-k⟩),andkis the crystal momentum, given byk=2πq/N, q=1,…,N ∈(0,2π]<cit.>.Because of the identical first term-2tcoskin Eqs. (<ref>)and (<ref>), the states|k⟩and|-k⟩can belinearly combined in order to cancel off the similar-looking boundarycontributions. Forα,β∈C, the eigenvalue relation H_N(α|k⟩ + β|-k⟩)=-2tcos k (α|k⟩ + β|-k⟩),is recovered provided that the constraintt/√(N)(α +β )|1⟩ +t/√(N)(α e^ik(N+1) + β e^-ik(N+1))|N⟩=0is satisfied. For this to hold, the coefficients of both|1⟩and|N⟩must vanish, which leads to the kernel equationt [11;e^ik(N+1) e^-ik(N+1) ][ α; β ]≡ B[ α; β ]=0.The determinant of the above “boundary matrix”Bmust vanish, which happensif the conditione^i 2k(N+1)=1is satisfied, that is, when k=πq/(N+1),q=1,…,N. For each of these values of k,α=-β=1/√(2)provides therequired kernel vector of the boundary matrix, with the resultingNeigenvectors|ϵ_k⟩≡|k⟩ - |-k⟩/√(2) =i √(2/N)∑_j=1^Nsin (k j)|j⟩,ofenergyϵ_k = -2tcosk. Notice that the allowed values ofkdiffer from thecase of periodic BCs <cit.>. Encouraged by these results, let us change the Hamiltonian by adding an on-site potential at the edges, W= w(|1⟩⟨ 1|+|N⟩⟨ N|), w ∈ℝ,so that thetotal single-particle Hamiltonian becomesH=H_N+W. The boundarymatrixBchanges toB ≡[ t+w e^ikt+w e^-ik; te^ik(N+1)+w e^ikN te^-ik(N+1)+w e^-ikN ].While it is harder to predict analytically the values ofkfor which it has a non-trivial kernel,it is interesting to examine the limit w≫ t.Then, we can approximate the relevant kernel condition as B [ α; β ]≈ w [ e^ike^-ik;e^ikN e^-ikN ][ α; β ]=0,showing nontrivial solutions if e^i 2k(N-1)=1.There are now(N-2)k-values yielding stationary eigenstates as before.The two missing eigenstates are localized at the edges, and can be taken to be|1⟩ and |N⟩, to leading order in t/w ≪ 1. These localized statesare reminiscent of Tamm-Shockley modes <cit.>.In hindsight, it is natural to ask whether this approach to diagonalization may be improved and extended to more general Hamiltonians. The answer is Yes, and this paper provides the appropriate tools. §.§ The bulk-boundary system of equations The above motivating example suggests that it may be possible toisolate the extent to which boundary effects prevent bulkeigenstates from becoming eigenstates of the actual Hamiltonian.Consider Eqs. (<ref>) and (<ref>) in particular. We may condense them into a singlerelative eigenvalueequation, P_BH_N|± k⟩= (- 2tcos k) P_B|± k⟩, in terms of the projectorP_B ≡∑_j=2^N-1|j⟩⟨ j|. The extension of this observation to the general class ofHamiltonians H=H_N+W requires only knowledge of the range R in Eq. (<ref>).The block-structure ofH_Ndefines a subsystem decomposition of thesingle-particle state space <cit.>,ℋ≅C^N⊗C^d≡ℋ_L ⊗ℋ_I,whereℋ_Landℋ_Iare lattice and internal state spacesof dimensionsNandd, respectively.Let{|j⟩, j=1,…,N}and{|m⟩, m=1,…,d}be their respective orthonormal bases.Definebulk and boundary projectors, P_B ≡∑_j=R+1^N-R|j⟩⟨ j|⊗1_d,P_∂≡1-P_B,with1 ≡1_N⊗1_dthe identity matrix onℋ, and 1_N, 1_dthe identity matrices onℋ_Landℋ_I,respectively (see Fig. <ref>).The defining property of the bulk projector is that it annihilates anyboundary contributionW, that is,P_BW=0. Because P_B+P_∂=1, thebulk-boundary system of equations,{[P_B H_N|ϵ⟩=ϵ P_B|ϵ⟩,; (P_∂ H_N+W)|ϵ⟩=ϵ P_∂|ϵ⟩ , ].may be seen to be completely equivalent to the standard eigenvalue equation, H|ϵ⟩=ϵ|ϵ⟩<cit.>.This bulk-boundary separation of the eigensystem problem is advantageousbecause the bulk equation is, in a well-defined sense,translation-invariant.Let us define aleft-shift operator T ≡∑_j=1^N-1|j⟩⟨ j+1| on the lattice spaceH_L(see Appendix <ref>). Then, one may verify that H_N= 1_N⊗ h_0+∑_r=1^R (T^r ⊗ h_r+T^†^r ⊗ h_r^†) .By extendingTinfinitely on both directions, we obtain a translation-invariantauxiliary Hamiltonian,H≡1⊗ h_0+∑_r=1^R (T^r⊗ h_r+T^-r⊗ h_r^†),where T≡∑_j∈ℤ|j⟩⟨ j+1| now denotes the generator of discrete translations on the (infinite-dimensional)vector space spanned by{|j⟩}_j∈Z, and1the corresponding identity operator. The subtle difference between HamiltoniansH_NandHis that whileTis not invertible,Tis,and in factT^-1=T^†. This difference is decisive in solving the corresponding eigenvalue problems. On the one hand, the eigenvalue equation H|Ψ_ϵ⟩=ϵ|Ψ_ϵ⟩ is equivalent to the infinite system of linear equationsh_0|ψ_j⟩ + ∑_r=1^R(h_r|ψ_j+r⟩+h^†_r|ψ_j-r⟩) =ϵ|ψ_j⟩,j∈Z,where|Ψ_ϵ⟩≡∑_j∈ℤ|j⟩⊗ |ψ_j⟩.On the other, the bulk equation P_BH_N|ϵ⟩=ϵ P_B|ϵ⟩, with|ϵ⟩≡∑_j=1^N|j⟩⊗|ψ_j⟩is equivalent to Eq. (<ref>) but restricted to thefinite domainR < j≤N-R. Hence, the bulk equation is underdetermined (there are 2R morevector variables than constraints).In particular, if |Ψ_ϵ⟩is an eigenstate of the infinite Hamiltonian as above, then |ϵ⟩≡∑_j=1^N|j⟩⟨ j|Ψ_ϵ⟩=P_1,N|Ψ_ϵ⟩is a solution of the bulk equation. It is in this sense of sharedsolutions with H that the bulk equation is, as anticipated, translation-invariant. §.§ Exact solution of the bulk equation Let us revisit the energy eigenvalue equation, Eq. (<ref>).If the goal were to diagonalize the infinite-system Hamiltonian H, then one should focus on finding energy eigenvectorsassociated to normalized states in Hilbert space.However, our model systems are offinite extent, and we are only interested in using H as an auxiliary operator for finding the translation-invariant solutions of the bulk equation. Hence, we will allow H to act onarbitrary vector sequencesof the form Ψ= ∑_j∈Z|j⟩|ψ_j⟩,possibly “well outside" the Hilbert state space, and so we will dropDirac's ket notation. From the standpoint ofsolving the bulk equation,every sequence that satisfies HΨ=ϵΨ is acceptable, so one must find them all.In the space of all sequences, the translation symmetry T remainsinvertible but is no longer unitary, because the notion of adjoint operatoris not defined.This is important,because it means that translations need not have their eigenvalues onthe unit circle, or be diagonalizable. Nonetheless, [T,H]=0, and so both features have interesting physical consequencesfor finite systems. We will refer to the space of solutions of the bulk equation as the bulk solution space and denote it byℳ_1,N(ϵ) ≡ KerP_B(H_N-ϵ1_d),for anyfixed energyϵ. Let ℳ_-∞,∞(ϵ) ≡ Ker (H-ϵ 1) denote the space of eigenvectors of H of energyϵ within the space of all sequences. In terms of these spaces, our arguments in Sec. <ref> establish the relation P_1,Nℳ_-∞,∞⊆ℳ_1,N,where we dropped the argumentϵ. Translation invariance is equivalent to the propertiesTℳ_-∞,∞⊆ℳ_-∞,∞ and T^-1ℳ_-∞,∞⊆ℳ_-∞,∞<cit.>. If the matrixh_Ris invertible, Eq. (<ref>) becomesP_1,Nℳ_-∞,∞=ℳ_1,N<cit.>.SinceTcommuteswithT^-1, the generator of translations to the right, these two symmetries share eigenvectors of the formΦ_z,1|u⟩≡∑_j∈Zz^j|j⟩|u⟩, with z an arbitrary non-zero complex number and|u⟩any internal state: there are d linearly independent eigenvectors oftranslations for each z≠ 0.As a simple but important consequence of the identitiesTΦ_z,1|u⟩ = zΦ_z,1|u⟩,T^-1Φ_z,1|u⟩ = z^-1Φ_z,1|u⟩, one finds that HΦ_z,1|u⟩ =Φ_z,1 H(z)|u⟩,where the linear operatorH(z)=h_0+∑_r=1^R(z^rh_r+z^-rh_r^†) ,acts on the internal spaceH_Ionly. This H(z)is precisely thereduced bulk Hamiltonianh_B(z)of Ref. [abc], obtained here by way of a slightly different argument. Since H_k=H(z=e^ik)is the usual Bloch Hamiltonian of a one-dimensional system withBorn-von-Karman BCs,H(z)is the analyticcontinuation of H_k off the Brillouin zone.One can similarly continue the energy dispersion relation off theBrillouin zone, by relating ϵ to z via(H(z)-ϵ1_d)=0.In practice, it is advantageous to use the polynomialP(ϵ,z) ≡ z^dR (H(z)-ϵ1_d).We will say that ϵ isregular if P(ϵ,z) is not the zero polynomial, andsingular otherwise. That is,P(ϵ,z)=0 identically for allzif ϵ is singular.Such a (slight) abuse of language<cit.> is permitted since we areinterested in varyingϵfor a fixed Hamiltonian. For any given Hamiltonian of finite rangeR, thereare at most a finite number of singular energies. Physically, singularenergies correspond to flat bands, as one can see by restriction to theBrillouin zone. We can now state a first useful result, whose formalproof follows from the general arguments in Ref. [JPA]: Theorem 1.If ϵ is regular, the number of independent solutions of the bulk equationis ℳ_1,N (ϵ) =2Rd, for any system size N>2R.This result ties well with thephysical meaning of the number2Rd = dim(Range P_∂) as countingthe total number of degrees of freedom on the boundary, which is equalto the dimension of the boundary subspace.The condition N>2R implies that the system is big enough to contain at least one site in the bulk.§.§.§ Extended-support bulk solutions at regular energies The solutions of the bulk equation that are inherited from Hhave non-vanishing support on the full lattice spaceH_L, and are labeled by the eigenvalues of,possibly together with a second “quantum number” that appears becauseis not unitary on the space of all sequences.For anyz 0, if|u⟩satisfies the eigenvalueequationH(z)|u⟩ = ϵ|u⟩,then Eq. (<ref>) implies thatΦ_z,1|u⟩is an eigenvector ofHwith eigenvalueϵ.In order to be more systematic, let{z_ℓ}_ℓ=1^ndenote thendistinct non-zeroroots of Eq. (<ref>), and{s_ℓ}_ℓ=1^ntheir respective multiplicities. For generic values ofϵ,H(z_ℓ)has exactlys_ℓeigenvectors{|u_ℓs⟩}_s=1^s_ℓinH_I, satisfying H(z_ℓ)|u_ℓ s⟩ = ϵ|u_ℓ s⟩, s=1,…,s_ℓ.SinceHΦ_z_ℓ,1|u_ℓ s⟩=ϵΦ_z_ℓ,1|u_ℓ s⟩, the statesP_1,NΦ_z_ℓ,1|u_ℓ s⟩ = ∑_j=1^Nz_ℓ^j|j⟩|u_ℓ s⟩≡ |z_ℓ,1⟩ |u_ℓ s⟩are solutions of the bulk equation. Intuitively, thesestates are “eigenstates of the Hamiltonian up to BCs." For a few isolated values ofϵ,H(z_ℓ)can have less thans_ℓeigenvectors. However, the number of eigenvectors ofHis stills_ℓ<cit.>, as weillustrate here by example. Suppose for concreteness thatH-ϵ1 = -t/2(+^-1)-ϵ1= -t/2^-1∏_ℓ=1^2(-z_ℓ).Since R=1 and d=1, we expect two eigenvectors for each value of ϵ. One concludes that theeigenspace of energy ϵ is spanned by thesequencesΦ_z_ℓ,1, ℓ=1,2,if z_1≠ z_2. But, if ϵ=± t, then z_1=z_2=∓ 1, andH∓ t 1 = -t/2 ^-1(-z_1)^2. How can one get two independent solutions in this case? The answer is that, in addition to Φ_z_1,1, the factor(T-z_1)^2contributes another sequence to the kernelof H-ϵ1, namely,Φ_z_1,2 = ∑_j∈Zjz_1^j-1|j⟩. There are two eigenvectors in total, even though there is only one root. Returning to the general case, the sequences <cit.> Φ_z,v =1/(v-1)!∂_z^v-1Φ_z,1 =∑_j∈Zj^(v-1)/(v-1)!z^j-v+1|j⟩,j^(v) ≡ j(j-1)…(j-v+1), j^(0)≡ 1,span the kernel of(T-z)^sfor v=1,…,s. In other words,Φ_z,vis ageneralized eigenvectorof the translational symmetryTof rankvwith eigenvaluez.We refer to eigenvectors withv>1as thepower-law solutions of the bulk equation (solutions with a power-law prefactor). They existbecause translations are not diagonalizable in the full space of sequences(as opposed to the Hilbert space of square-summable sequences), leading to the new quantum number v. The power-law solutions of the bulk equation may be found from theaction ofHon the generalized eigenvectors ofT.For arbitrary internal state |u_x⟩, we have:HΦ_z,x|u_x⟩=1/(x-1)!∂_z^x-1Φ_z,1H(z)|u_x⟩ .Then one can show from Eqs. (<ref>) and (<ref>)that the action ofHon the vector sequenceΨ= ∑_x=1^vΦ_z,x|u_x⟩,where{|u_x⟩}are arbitrary internal states, is given byHΨ=∑_x=1^v∑_x'=1^vΦ_z,x[H_v(z)]_xx'|u_x'⟩ .Here,H_v(z)is anupper triangular block-Toeplitz matrix withnon-trivial blocks [H_v(z)]_xx'≡1/(x'-x)!∂_z^x'-xH(z), 1≤ x≤ x'≤ v.In matrix form, by lettingH^(x)≡∂_z^xH(z), we have H_v(z)= [H^(0)H^(1)1/2 H^(2)⋯ 1/(v-1)! H^(v-1);0⋱⋱⋱⋮;⋮⋱⋱⋱ 1/2H^(2);⋮ ⋱⋱H^(1);0⋯⋯0H^(0) ].We refer toH_v(z)as thegeneralizedreduced bulk Hamiltonian of order v. Notice thatH_1(z)=H(z). In the partial basis Φ_z =[ Φ_z,1 … Φ_z,v ],organized as a row vector, the entries of |u⟩= [ |u_1⟩ … |u_v⟩ ]^T are the vector-valued coordinates ofΨ,Ψ=Φ_z|u⟩ =∑_x=1^vΦ_z,x|u_x⟩. Then, Eq. (<ref>) can be rewritten asHΦ_z|u⟩ =Φ_z H_v(z)|u⟩.Now it becomes clear that forΨto be an eigenvector ofH,the required condition isH_v(z)|u⟩= ϵ|u⟩,which is analogous to the condition derived for the generic casev=1. If a rootz_ℓof Eq. (<ref>) has multiplicitys_ℓ, thenHhas preciselys_ℓlinearly independent eigenvectors corresponding toz_ℓ. This provides a characterization of the eigenstates ofH, which may be regarded as extending Bloch's theorem toH viewed as a linear transformation on the space of allvector-valued sequences, and whose rigorous justification followsfrom Ref. [JPA]: Theorem 2.For fixed, regular ϵ, let {z_ℓ}_ℓ=1^n denote the distinct non-zero roots of Eq. (<ref>), with respective multiplicities {s_ℓ}_ℓ=1^n. Then, theeigenspace of H of energy ϵ is a direct sum ofn vector spaces spanned by generalized eigenstates of Tof the form Ψ_ℓ s= Φ_z_ℓ|u_ℓ s⟩ =∑_v=1^s_ℓΦ_z_ℓ, v|u_ℓ s v⟩, s=1,…,s_ℓ,where the linearly independent vectors{|u_ℓ s⟩}_s=1^s_ℓare chosen in such a way thatH_s_ℓ(z_ℓ)|u_ℓ s⟩ = ϵ|u_ℓ s⟩, and|u_ℓ s⟩ = [ |u_ℓ s 1⟩ … |u_ℓ s s_ℓ⟩ ]^ T.Once the eigenvectors ofHare calculated, the bulksolutions of extended support are readily obtained by projection.Let, forv ≥1,|z,v⟩≡P_1,NΦ_z,v=∑_j=1^Nj^(v-1)/(v-1)!z^j-v+1|j⟩be the projections of generalized eigenvectors ofT. Thenℬ_ ext≡{|ψ_ℓ s⟩, s=1,…,s_ℓ,ℓ=1,…,n} describes a basis of the translation-invariant solutions of the bulkequation, where |ψ_ℓ s⟩ = ∑_v=1^s_ℓ|z_ℓ,v⟩|u_ℓ s v⟩∀ℓ,s.Remark.— The bulk equation bears power-law solutions only at afew isolated values ofϵ<cit.>. However, linearcombinations ofv=1solutions show power-law-like behavior, as soon astwo or more of the roots ofEq. (<ref>) are sufficiently close to each other.Suppose, for instance, that for some valueof energyϵ, two of the roots of Eq. (<ref>)coincide atz_*. For energy differing fromϵby a small amountδϵ, the double rootz_*bifurcates into two roots slightly away from each other, with valuesz_*±δz. The relevant bulk solution space is spanned by |z_*+δ z,1⟩ + |z_*+δ z,1⟩ ≈2|z_*,1⟩, |z_*+δ z,1⟩ - |z_*+δ z,1⟩ ≈2(δ z/z_*)|z_*,2⟩,showing that the second vector has indeed a close resemblance to the power-law solution|z_*,2⟩. Similar considerations apply ifd>1, as it is typically the case in physical applications. Assuming that the relevant bulk solutions at energyϵ+δϵare described by analytic vector functions|ψ(z_*+δz)⟩and|ψ(z_*-δz)⟩, then, from the above analysis, it is clear that for energyϵ, the power-law bulk solution will be proportional tolim_δ z→ 0(|ψ(z_*+δ z)⟩-|ψ(z_*-δ z)⟩) ∝∂_z |ψ(z_*)⟩ .We will make use of this observation for the calculation of power-law solutions in Sec. <ref>. §.§.§ Emergent solutions at regular energies While the extended solutions of the bulk equation correspondto the nonzero roots of Eq. (<ref>), thepolynomialP(ϵ,z)defined in Eq. (<ref>) may alsoincludez_0=0as a root of multiplicitys_0, that is, wemay generally writeP(ϵ,z)=z^dR(H(z)-ϵ1_d)≡ c∏_ℓ=0^n(z-z_ℓ)^s_ℓ,c 0.However, |z=0⟩|u⟩=0 does not describe any state of the system.This observationsuggests that the extended solutions of the bulk equation mayfail to account for all 2Rd solutions we expect for regularϵ. That this is indeed the case follows from a knownresult in the theory of matrix polynomials <cit.>, implying that2Rd=2s_0+∑_ℓ=1^n s_ℓ formatrix polynomials associated to Hermitian Toeplitz matrices <cit.>.Hence, the number of solutions of the bulk equation of theform given in Eq. (<ref>) is ∑_ℓ=1^ns_ℓ = 2Rd- 2s_0.We call the missing 2s_0 solutions of the bulk equation emergent, because they are no longer controlled by Hand (nonunitary) translation symmetry, but rather they appearonly because of the truncation of the infinite lattice down toa finite one, and only if h_R=0<cit.>. Emergentsolutions are a direct, albeit non-generic, manifestationof translation-symmetry-breaking; nonetheless, remarkably,they can also be determined by the analytic continuation ofthe Bloch Hamiltonian, in a precise sense. While full technical detail is provided in Appendix<ref>,the key to computing the emergent solutions is to relate theproblem of solving the bulk equation to ahalf-infiniteHamiltonian, rather than the doubly-infinite Hwe have exploited thus far.Let us define the unilateral shifts T_- =∑_j=1^∞|j⟩⟨ j+1|,T_-^⋆ =∑_j=1^∞|j+1⟩⟨ j|.The HamiltonianH_- ≡1_-⊗ h_0+ ∑_r=1^R(T_-^r⊗ h_r+T_-^⋆r⊗ h_r^†)is then the half-infinite counterpart of H. The correspondinghalf-infinite bulk projector is P_B^- ≡∑_j=R+1^∞|j⟩⟨ j|=T_-^⋆RT_-^R.Suppose there is a state Υ^-, that solves the equationP^-_B(H_–ϵ1_-)Υ^-=0.Then one can check that |ψ⟩=P_1,NΥ^- is a solution of the bulk equation, Eq. (<ref>).Clearly, some of the bulk solutions we arrive at in this way usingH_- will coincide with those obtained from H. These are precisely the extendedsolutions we already computed in Sec. <ref>. In contrast, the emergent solutions are obtainedonly from H_-.Since T_-T_-^⋆=1_-, we may writeP^-_B(H_–ϵ1_-)=T_-^⋆ RK^-(ϵ,T_-), in terms of the matrix polynomial K^-(ϵ,z) ≡ z^R (H(z)-ϵ1_d).Half of the emergent solutions, namely, the ones localized on the left edge, are determined by the kernel of K_s_0^-(ϵ,z_0=0) ≡ K^- (ϵ), with [K^-_v(ϵ,z)]_xx' constructed as in Eq. (<ref>). Explicitly, such a matrix,which was obtained by different meansin Ref. [JPA], takes the form K^-(ϵ)≡ [h^†_R⋯ h_0-ϵ1_d⋯h_R 0⋯0; -0.5em⋱ -1em⋱ ⋱ ⋱⋮; ⋱0; -4em⋱ ⋱⋱ ; h_R;⋱ ⋱⋮;h_0-ϵ1_d;0 -1em⋱ ⋮;⋮⋱ ;0⋯ -2em 0h^†_R ],for systems with fairly large s_0>2R+1. Let { |u^-_s⟩}_s=1^s_0denote a basis of the kernel of K^-(ϵ), with|u^-_s⟩ = [ |u^-_s1⟩ |u^-_s2⟩… |u^-_ss_0⟩ ]^ T.Then,|ψ_s^-⟩=∑_j=1^s_0|j⟩|u^-_sj⟩ ,s=1,…,s_0, are the emergent solutions with support on the firsts_0latticesites, withs_0obeying Eq. (<ref>).We are still missing s_0 emergent solutions for the right edge. They may beconstructed from the kernel of the lower-triangular block matrix K^+ (ϵ) ≡ [K^- (ϵ)]^† =[K^-_s_0(ϵ, z_0=0)]^†. Let { |u^+_s⟩}_s=1^s_0 denote a basis of the kernel of K^+(ϵ), with|u^+_s⟩ = [ |u^+_s1⟩ |u^+_s2⟩… |u^+_ss_0⟩ ]^ T.Then,|ψ^+_s⟩ = ∑_j=1^s_0|N-s_0+j⟩|u^+_sj⟩ s=1,…,s_0, are the emergent bulk solutions associated to the right edge,supported on the lattice sitesN-s_0+1, …, N.Again, for mathematical justifications, see Appendix <ref>.In what follows, we shall denote the spaces spanned by left- and right- localized emergentbulk solutions byℱ_1^-andℱ_N^+,and their bases byℬ^-≡{|ψ_s^-⟩}_s=1^s_0 andℬ^+≡{|ψ_s^+⟩}_s=1^s_0, respectively. §.§.§ Bulk-localized states at singular energies If h_R isnot invertible, there can be at most afinitenumber of singular energy values (usually referred to asflat bands), leading to bulk-localized solutions: these solutions are finitely-supported and appear everywhere in the bulk. Hence,a singular energycannot be excluded from the physical spectrum ofa finite system by way of BCs. In contrast, emergent solutions are alsofinitely-supported but necessarily “anchored” to the edges(and only appearing for regular values of ϵ).Recall that if ϵ is singular, then (H(z)-ϵ1)=0 for anyz. Thus, there exists an analytic vector function,|v(z)⟩≡∑_δ=0^δ_0z^-δ|v_δ⟩,δ_0=(d-1)2Rd,satisfyingH(z)|v(z)⟩= ϵ|v(z)⟩for all z. To obtain|v(z)⟩, one can construct theadjugate matrix of(H(z)-ϵ1_d). (Recall that the adjugate matrix adj(M) associated to a square matrix M is constructedout of the signed minors of M and satisfies adj(M)M= (M)1.) Hence,(H(z)-ϵ1_d)adj(H(z)-ϵ1_d)= (H(z)-ϵ1_d)1_d=0,and so one can useany of the non-zero columns ofadj(H(z)-ϵ1_d), suitably pre-multiplied by a power ofz, for the vectorpolynomial|v(z)⟩. By matching powers ofz, thisequation becomes [ h_R 0 ⋯ 0; h_R-1 h_R ⋱ ⋮; ⋮ ⋱ ⋱ 0; ⋮ ⋱ ⋱ ⋱; h_R^† ⋱ ⋱ ⋱ h_R; ⋱ ⋱ ⋱ ⋮; 0 ⋱ ⋱ ⋮; ⋮ ⋱ ( ⋱ h_R-1^†; 0 ⋯ 0 h_R^† ][ |v_0⟩; |v_1⟩; ⋮; |v_δ_0⟩ ]=0. The idea now is to use the linearly independent solutions ofEq. (<ref>) to construct finite-supportsolutions of the bulk equation. Let us denote such solutions by |v_μ⟩≡[|v_μ 0⟩|v_μ 1⟩… |v_μδ_0⟩ ]^ T, for μ=1,…, μ_0. One can check directly that the finitely-supported sequences Ψ_j μ≡∑_δ=0^δ_0|j+δ⟩|v_μδ⟩, j∈Z, μ=1,…,μ_0,all satisfy (H-ϵ1)Ψ_js=0because |v_μ⟩ obeys Eq. (<ref>). Hence, the statesP_1,NΨ_jμprovide finitely-supportedsolutions of the bulk equation. In addition, as long as2R<j<N-2R-δ_0,the boundary equation is also satisfied trivially, and soall such states become eigenvectors ofH_N+Wwith the singular energy ϵ. This is why singular energies, if present for the infinite system, are necessarily also part of the spectrum of thefinite system and display macroscopic degeneracy of order𝒪(N).Let us further remark that the sequences Ψ_jμ and associated solutions of the bulk equation neednot belinearly independent. To obtain a complete (rather than overcomplete), set of solutions for flat bands, one would requirea technical tool, theSmith normal form<cit.>,which is beyond the scope of this paper. See Ref. [JPA]for details.§.§ The boundary matrix For regular energies, the bulk solutions determine a subspace ofthe full Hilbert space [Theorem 1], whose dimension 2Rd≪ dN for typicalapplications. While not all bulk solutions are eigenstates of the HamiltonianH=H_N+W, the actual eigenstates must necessarily appear as bulk solutions.Hence, the bulk-boundary separation in Eqs. (<ref>), and, in particular,the bulk equation, identifies by way of a translational symmetryanalysis asmall search subspace. In order to find the energy eigenstatesefficiently, one must solve the boundary equation on this search subspace. Since the boundary equation is linear, its restriction to the space of bulk solutions can be represented by a matrix, theboundary matrix<cit.>. The latter is a square matrix that combines our basis of bulk solutionswith the relevant BCs. LetB≡B_ext ∪ℬ^- ∪ℬ^+be a basis forℳ_1,N. Then, building on the previous section, the Ansatz state|ϵ,α⟩≡ |Ψ_ℬ⟩α =∑_ℓ=1^n∑_s=1^s_ℓα_ℓ s|ψ_ℓ s⟩ +∑_s=1^s_0α^+_s|ψ^+_s⟩ +∑_s=1^s_0α^-_s|ψ^-_s⟩ ,represents the solutions of the bulk equation parametrized by the2Rd amplitudes α, where[α≡[α_11 ⋯α_ns_n α_1^+ ⋯ α_s_0^+ α_1^- ⋯ α_s_0^- ]^ T,; |Ψ_ℬ⟩≡[|ψ_11⟩ ⋯|ψ_ns_n⟩ |ψ_1^+⟩ ⋯ |ψ_s_0^+⟩ |ψ_1^-⟩ ⋯ |ψ_s_0^-⟩ ]. ] Moreover, let as before b=1,…,R,N-R+1,…,N label theboundary sites. Then, P_B(H-ϵ1)|ϵ,α⟩=0 and P_∂(H-ϵ1)|ϵ,α⟩=∑_b|b⟩⟨ b|(H_N+W-ϵ1)|Ψ_ℬ⟩α.In particular, the boundary equation is equivalent to the requirement that⟨ b|(H_N+W-ϵ1)|Ψ_ℬ⟩α=0 for all boundary sites. Since ⟨ b|(H_N+W-ϵ1)|Ψ_ℬ⟩≡⟨ b| H_ϵ |Ψ_ℬ⟩denotes a row array of internal states, it is possible to organizethese arrays into the boundary matrix B (ϵ)≡[ ⟨ 1|H_ϵ|ψ_1 1⟩⋯ ⟨ 1|H_ϵ|ψ_n s_n⟩ ⟨ 1|H_ϵ|ψ_1^+⟩⋯ ⟨ 1|H_ϵ|ψ_s_0^-⟩;⋮ ⋮⋮ ⋮; ⟨ R|H_ϵ|ψ_1 1⟩⋯ ⟨ R|H_ϵ|ψ_n s_n⟩ ⟨ R|H_ϵ|ψ_1^+⟩⋯ ⟨ R|H_ϵ|ψ_s_0^-⟩; ⟨ N-R+1|H_ϵ|ψ_1 1⟩⋯ ⟨ N-R+1|H_ϵ|ψ_n s_n⟩ ⟨ N-R+1|H_ϵ|ψ_1^+⟩⋯ ⟨ N-R+1|H_ϵ|ψ_s_0^-⟩;⋮ ⋮⋮ ⋮; ⟨ N|H_ϵ|ψ_1 1⟩⋯ ⟨ N|H_ϵ|ψ_n s_n⟩ ⟨ N|H_ϵ|ψ_1^+⟩⋯ ⟨ N|H_ϵ|ψ_s_0^-⟩ ].By construction,the boundary matrixBis a block matrix of block-size d× 1. In terms of this matrix, Eq. (<ref>) provides the useful identityH|ϵ,α⟩ = ϵ|ϵ,α⟩ +∑_b,s|b⟩ B_bs(ϵ)α_s,ϵ∈R.One may write an analogous equation in Fock space by defining an arrayη_ϵ,α^†≡∑_j=1^N⟨ j|ϵ,α⟩Ψ̂_j^†.Then Eq. (<ref>) translates into[H,η_ϵ,α^†] =ϵ η_ϵ,α^† +∑_b,sΨ̂_b^† B_bs(ϵ)α_s .It is interesting to notice that this (many-body) relation remains true evenif ϵ is allowed to be a complex number. §.§ The generalized Bloch theoremThe bulk-boundary separation of the energy eigenvalue equation shows that actual energy eigenstates are necessarily linear combinations of solutions of the bulk equation. This observation leads to a generalization of Bloch's theorem for independent fermions under arbitrary BCs: Theorem 3 ( Generalized Bloch theorem).Let H=H_N+W denote the single-particle Hamiltonian of a clean system subject toBCs described by W=P_∂ W. If ϵ is a regular energy eigenvalueof H of degeneracy K, the associated eigenstates can be taken to be of the form|ϵ,α_κ⟩=|Ψ_ℬ⟩α_κ, κ=1,…,K,where {α_κ, κ=1,…,K} is a basis of thekernel of the boundary matrix B(ϵ) at energy ϵ. In short, (H_N+W)|ϵ,α⟩=ϵ|ϵ,α⟩ if and only if Bα=0, in which case it also follows from Eq. (<ref>) thatη_ϵ,α^†is a normal fermionic mode ofthe many-body HamiltonianH. From now on, we will refer to energyeigenstates of the form |Ψ_ℬ⟩α_κ as generalized Bloch states. Recall thatHacts onH = ℂ^N⊗ℂ^d, with couplings of finite rangeR.A lower bound onNshould be obeyed, in order for the above theorem to apply.If h_R ≠ 0, sincethere are no emergent solutions nor flat bands, generalized Bloch statesdescribe the allowed energy eigenstates as soon as N>2R, independently of d.If h_R fails to be invertible, we should require that N>2max(s_0,R) to ensure that emergent solutionson opposite edges do not overlap, and are thus independent. Sinces_0≤ Rd, this condition is satisfied for any N>2Rd. In general,N>2R(d+1)always suffices for generalized Bloch statesto describegeneric energy eigenstates <cit.>. We further note that if ϵ isnot an energy eigenvalue,the kernel ofB(ϵ)is trivial. Thus, thedegeneracy of a single-particle energy level coincides with thedimension of the kernel ofB(ϵ). Let ρ(ω) denotethe single-particle density of states. Combining its definition withthe generalized Bloch theorem, we then see thatρ(ω)=∑_ B(ϵ)=0 [ KerB(ϵ)]δ(ħω-ϵ),an alternative formula to the usualρ(ω)=-1/π Im Tr (H_N+W-ħω+i 0^+)^-1,from the theory of Green's functions <cit.>. Another interesting and closely related formula is 𝒵_W= Tre^-β(H_N+W)= ∑_ B(ϵ)=0 KerB(ϵ)e^-βϵ,for the partition function of the single-particle Hamiltonian, with the dependenceon BCs highlighted <cit.> .We conclude this section by showing how, for periodic BCs,one consistently recovers the conventional Bloch's theorem. In this case,the appropriate matrixWreadsW ≡ W_p=∑_r=1^R ( T^N-r⊗ h_r^†+ h.c.),since then one can check thatH_p=H_N+W_p=1_N⊗ h_0+∑_r=1^R (V^r⊗ h_r+ h.c.),in terms of the fundamental circulant matrixV ≡ T+(T^†)^N-1=∑_j=1^N-1|j⟩⟨ j+1|+|N⟩⟨ 1| .Physically, V is the generator of translations (to the left) for a system displaying ring (1-torus) topology. The Bloch states are the states that diagonalize H_p and V simultaneously. Theorem 3 guarantees that we can choose the eigenstates of H_p to be linear combinations oftranslation-invariant and emergent solutions. Thus, we only need to check if these linear combinations includeeigenstates of V. There is no hope of retaining the emergent solutions, because they are localized and too few in number (at most 2Rd) to be rearrangedinto eigenstates of V. The same holds for translation-invariant solutions with a power-law prefactor. Hence, the search subspace that iscompatible with the translational symmetry V is described by thesimplified Ansatz <cit.> |ϵ,α⟩ =∑_ℓ=1^nα_ℓ 1|ψ_ℓ 1⟩ .Now, V|ψ_ℓ 1⟩=z_ℓ|ψ_ℓ 1⟩- z_ℓ(1-z_ℓ^N)|N⟩|u_ℓ s_ℓ 1⟩, and so the generalized Bloch states can only be eigenstates of Vif e^ik_ℓ N=1 withz_ℓ=e^ik_ℓ, and all but one entry in α vanish. That is,|ϵ,α⟩≡|ϵ,k_ℓ⟩=|z_ℓ, 1 ⟩|u_ℓ1, 1⟩.As one may verify,H_p|ϵ,k_ℓ⟩=|z_ℓ, 1 ⟩H(z_ℓ) |u_ℓ1, 1⟩= ϵ|ϵ,k_ℓ⟩,showing that |ϵ,k_ℓ⟩ is indeed compatible with theboundary matrix. Manifestly, |ϵ,k_ℓ⟩ is an eingenstate ofH_p in the standard Bloch form – thereby recoveringthe conventional Bloch's theorem for periodic BCs, as desired. § THE BULK-BOUNDARY ALGORITHMS The results of Sec. <ref> can be used to develop diagonalization algorithms for the relevant class of single-particle Hamiltonians. We will describe two such algorithms. Thefirst treats ϵ as a parameter for numerical search. The second is inspired by the algebraic Bethe Ansatz, as suggested by comparing ourEq. (<ref>) to Eq. (28) of Ref. [ortiz05].§.§ Numerical “scan-in-energy” diagonalization The procedure described in this section is a special instance ofthe Eigensystem Algorithm described in Ref. [JPA], specializedto Hermitian matrices. It employs a search for energy eigenvalues along the real line,and takes advantage of the results of Sec. <ref> to determine whethera given number is an eigenvalue. The overall procedure is schematically depicted in Fig. <ref>.The first part of the algorithm finds all eigenvectors ofHthat correspond to the flat (dispersionless) energy band, if any exists. Two steps are entailed:*Find all real values of ϵ for which(H(z)-ϵ1_d) vanishes for any z.Output these as singular eigenvalues of H. * For each of the eigenvalues found in step (<ref>), find and output a basisof the corresponding eigenspace of H using any conventional algorithm. In implementing step (<ref>) above, one can leverage the analysis of Sec. <ref>.The following part of the algorithm, which repeats until all eigenvectors ofHare found,proceeds according to the following steps:. *Choose a seed value of ϵ, different from those eigenvalues found already. *Find all n distinct non-zero roots of the equation (H(z)-ϵ1_d)=0. Let these rootsbe {z_ℓ, ℓ = 1,…,n},and their respective multiplicities {s_ℓ, ℓ=1,…,n}. *For each such roots, construct the generalized reduced bulkHamiltonian H_s_ℓ(z_ℓ) [Eq. (<ref>)]. *Find a basis of the eigenspace of H_s_ℓ(z_ℓ) with eigenvalue ϵ.Let the basis vectors be {|u_ℓ s⟩, s=1,…,s_ℓ}. The bulk solution corresponding to (ℓ,s) is|ψ_ℓ s⟩ = |z_ℓ,1⟩|u_ℓ s⟩, withΦ_z_ℓ defined in Eq. (<ref>). *If h_R is non-invertible, find s_0 = Rd - ∑_ℓ=1^ns_ℓ/2.Construct matrices K^-(ϵ) as described in Eq. (<ref>),and K^+(ϵ)=[K^-(ϵ)]^†. *Find bases of the kernels ofK^-(ϵ) and K^+(ϵ). Let the basis vectors be {|u_s^-⟩, s=1,…,s_0} and {|u_s^+⟩, s=1,…,s_0}, respectively. The emergent bulk solutions corresponding to each s arefollow from Eqs. (<ref>) and (<ref>). *Construct the boundary matrix B(ϵ) [Eq. (<ref>)]. *If B(ϵ)=0, output ϵ as an eigenvalue.Find a basis {α_κ, κ=1,…, K} of the kernel of B(ϵ). Then a basis of the eigenspace of H corresponding to energyϵ is {|ϵ_κ⟩ =|Ψ_ℬ⟩α_κ, κ=1,…, K}, with |Ψ_ℬ⟩ being defined in Eqs. (<ref>). If all 2dN eigenvectors are not yet found, then go back to step (<ref>). *If B(ϵ) 0, choose a new value of ϵas dictated by therelevant root-finding algorithm <cit.>.Go back to step (<ref>). Some considerations are in order, in regard to the fact thatthe determinant ofB(ϵ)plotted as a function of energyϵmay display finite-precision inaccuracies, that appear as fictitious roots.Such issues arise at thoseϵwhere two (or more) of the roots of Eq. (<ref>) cross as a function ofϵ, due to the non-orthogonality of the basisℬthatresults from the procedure described in Sec. <ref>. Letϵ_*be a value of energy for which this happens,so that the bulk equation bears a power-law solution. Forϵ≈ϵ_*(exceptϵ_*itself), Eq. (<ref>) has two roots that are very close in value, so that the corresponding bulk solutions overlap almost completely.This results in a boundary matrix having two nearly identical columns, with determinant vanishingin the limitϵ→ϵ_*,irrespective ofϵ_*being an eigenvalue ofH(hence, a physical solution).However, if we calculateB(ϵ)exactly atϵ_*, then the basisℬcontains power-law solutions, and accurately indicates whetherϵ_*is an eigenvalue.This also means that the functionB(ϵ)has a discontinuity atϵ=ϵ_*.A simple way to identify those fictitious roots is as follows.Rewrite the polynomial in Eq. (<ref>) as P(ϵ,z) = ∑_r=s_0^2Rd-s_0p_r(ϵ)z^r,which is treated as a polynomial inzwith coefficients depending onϵ(ifs_0changes withϵ, we use thesmallest possible value ofs_0in Eq. (<ref>)).P(ϵ,z)has double roots atϵ_*if and only if thediscriminantD(P(ϵ_*,z)) =0<cit.>. The latter gives a polynomial expression inϵ, of degree𝒪(dR). By finding the roots of this equation, one can obtain all the values ofϵfor which fictitious roots ofB(ϵ)may appear. To check whether these roots are true eigenvalues, one then needs to constructB(ϵ)by including the power-law solutions in the Ansatz. We further note that, while the Ansatz is not continuous at such values ofϵ, the fact that the bulk solution space is the kernel of the linear operatorP_B(H_N+W-ϵ)implies that it mustchangesmoothly withϵ. A way to improve numerical accuracy would be to construct an orthonormal basis (e.g., via Gram-Schmidt orthogonalization) ofM_1,N(ϵ)at eachϵ, and use this basis to construct a modified boundary matrixB̃(ϵ). In practice, one maydirectly compute the new determinant by using B̃(ϵ) =B(ϵ)/√(𝒢 (ϵ)),where𝒢≡⟨Ψ_ℬ | Ψ_ℬ⟩is the Gramian matrix <cit.> of the basis of bulk solutions obtained in steps (<ref>) to (<ref>) of the algorithm, with entries𝒢_s s'≡⟨ψ_s|ψ_s'⟩,s,s'=1,…,2Rd. In fact, it can be checked that the bulk solutions{|ϕ_s⟩≡∑_s'=1^2Rd[𝒢^-1/2]_s's|ψ_s'⟩, s=1,…,2Rd }form an orthonormal basis of the bulk solution space ℳ_1,N. The calculation of the entries of the Gramian is straightforward thanks to the analytic result ⟨ z,1 | z',1 ⟩ = {[ z^*z'-(z^*z')^N+1/1-z^*z'if z'1/z^*; Nifz'=1/z^* ].. In regard to the time and space complexity of the algorithm,the required resources depend entirely on those needed to compute the boundary matrix. For genericϵ, regardless of the invertibility ofh_R,the size ofB(ϵ)is2Rd×2Rd, independently ofN.Calculation of each of its entries is also simple from the point ofview of complexity, thanks to the fact thatH=H_N+Wis symmetrical <cit.>. Accordingly, both the number of steps and the memory space used by thisalgorithm do not scale with the system sizeN, making this approachcomputationally more efficient than conventional methods of diagonalization of generic Hermitian matrices <cit.>.§.§ Algebraic diagonalization The scan-in-energy algorithm can be furtherdeveloped into an algorithm that yields an analytic solution (often closed-form),in the same sense as the Bethe Ansatz method does for a different class of(interacting) quantum integrable systems.The idea is to obtain, for generic values ofϵ, an analytic expression forB(ϵ), since its determinant will then provide acondition forϵto be an eigenvalue, and the corresponding eigenvectors can be obtained from its kernel. As mentioned,for genericϵ, the extended bulk solutions do not include any power-law solutions. This propertycan be exploited to derive an analytic expression forB(ϵ)insuch a generic setting. The values ofϵfor which power-law solutions appear, or the analytic expression fails for other reasons, can be dealt with on a case-by-case basis. By the Abel-Ruffini theorem, acompletely closed-formsolutionby radicals in terms ofϵcan be achieved if the degree inzof the characteristic polynomial of thereduced bulk Hamiltonian is at mostfour. If this is not the case, the roots{z_ℓ}do not possess an algebraicexpression in terms ofϵand entries ofH. The workaroundis then to consider{z_ℓ}as free variables, with the constraint that each of them satisfy the characteristic equation ofH(z).With these tools in hand, the following procedure can be used to find an analytical solution for generic values ofϵ: * Construct the polynomial P(ϵ,z) in Eq. (<ref>), which is a bivariate polynomial in ϵ and z. Determine s_0 usings_0 = 2Rd - deg(P(ϵ,z)), where deg(.) denotes the degree of the polynomial in z. *Assuming that ϵ and z satisfyP(ϵ,z)=0, find an expression for the eigenvector|u(ϵ,z)⟩ of H(z) with eigenvalue ϵ. *Consider variables {z_ℓ, ℓ=1,…,2Rd-2s_0}, each satisfying P_ϵ(z_ℓ)=0. Each of these corresponds to a bulk solution |z_ℓ,1⟩|u(ϵ,z_ℓ)⟩.*If h_R is not invertible, construct matricesK^-(ϵ) and K^+(ϵ)= [K^-(ϵ)^†] [Eq. (<ref>)]. *Find bases for their kernels, each of which contains s_0 vectors. Let thesebe {|u_s^-(ϵ)⟩, s=1,…,s_0} and{|u_s^+(ϵ)⟩, s=1,…,s_0}. These correspond tofinite-support solutions of the bulk equation. *Construct the boundary matrix B(ϵ) ≡ B(ϵ,{z_ℓ})[Eq. (<ref>)].*The condition for ϵ being an eigenvalue of H isB(ϵ,{z_ℓ})=0. Therefore, a complete characterization of eigenvalues is{P(ϵ,z_ℓ)=0,ℓ=1,…,n}, B(ϵ,{z_ℓ})=0.*If deg(P(ϵ,z))≤ 4, substitute for each z_ℓ the closed-form expression of the corresponding root z_ℓ(ϵ).The eigenvalue condition in step (<ref>) simplifies to a single equation, B(ϵ,{z_ℓ(ϵ)})=0. *For every eigenvalue ϵ, the kernel vectorα(ϵ,{z_ℓ}) of B(ϵ,{z_ℓ})provides the corresponding eigenvector of H. In steps (<ref>), (<ref>) and (<ref>), we need to obtain an analytic expression for the basis of the kernel of a squaresymbolic matrix of fixed kernel dimension in terms of its entries.This can be done in many different ways, and often is possible by inspection.One possible way was described in Sec. <ref>in connection to evaluating Ker(H(z)-ϵ1_d)for singular values ofϵ. The above analysis does not hold whenϵsatisfies any of the following conditions: * (H(z)-ϵ1)=0 has one or more double roots.Thisis equivalent to D(P(ϵ,z))=0, as discussed in Sec. <ref>. This is a polynomial equation in terms of ϵ, the roots of which yield allrequired values of ϵ. *The coefficient p_s_0(ϵ) of z^s_0 in P(ϵ,z) vanishes, or equivalently, ϵ is a root of p_s_0(ϵ)=0. *Each entry of |u(ϵ,z)⟩ vanishes. Such points are identified by solving simultaneously the equations⟨ m|u(ϵ,z)⟩ = 0, m=1,…,d and P(ϵ,z)=0,Since a necessary and sufficient condition for these polynomials (in z) to have a common root is that their resultant vanishes <cit.>, we find the relevant values of ϵ by equating the pairwise resultants to zero. * {|u_s^-(ϵ)⟩, s=1,…,s_0} or {|u_s^+(ϵ)⟩, s=1,…,s_0} are linearly dependent. To find such values of ϵ, one may form the corresponding Gramian matrix and equate its determinant to zero. For all the values ofϵthus identified,B(ϵ)is calculated by following steps (<ref>)-(<ref>) in the scan-in-energy algorithm.To summarize, this algebraic procedure achieves diagonalization in analytic form:the upshot is a system ofpolynomial equations, whose simultaneous roots are the eigenvalues,and an analytic expression for the eigenvectors, withparametric dependence on the eigenvalue. § ILLUSTRATIVE EXAMPLES This section contains three paradigmatic examples illustratingthe use of our generalized Bloch theorem, along with theresulting algebraic procedure of diagonalization. §.§ The impurity modelrevisited Let us first reconsider the impurity model of Sec. <ref>. The single-particle Hamiltonianis the corner-modified, banded block-Toeplitz matrix H=H_N+W, with H_N=-t(T+T^†),W= w P_∂.The boundary consists of two sites, so thatP_∂= |1⟩⟨ 1|+|N⟩⟨ N|,for any N>2.Likewise,R=1=d.The first step in diagonalizing H is solving the bulkequation. Since the reduced bulk Hamiltonian H(z)=-t(z+z^-1),P(ϵ,z) = z( H(z)-ϵ) =-t (z^2+ϵ/tz+1).Thus, every value of ϵ is regular and yields two(= the number of boundary degrees of freedom) solutionsof the bulk equation. If ϵ≠± 2t, the solutionsare |z_ℓ,1⟩, with z_ℓ=-ϵ/2t+(-1)^ℓ√(ϵ^2/4t^2-1), ℓ=1,2, withz_1 z_2=1andϵ=-t(z_1+z_2). The special values ϵ=± 2t for whichH_Nyields only one of the two bulk solution have an interpretation as theedges of the energy band. If ϵ=2t, thenH(z)yields only|z_1=-1, 1⟩, whereasif ϵ=-2t, it yields only|z_1=1,1⟩. In order to obtain the missing bulk solution in each case, onemust consider the effective Hamiltonian [Eq. (<ref>)] H_2(z) =-t [ z+z^-1 1-z^-2;0 z+z^-1 ].One may check that H_2(z_1)-ϵ1≡ 0 if ϵ=± 2t, z_1=∓ 1.Thus, the two linearly independent solutions of the bulk equation at these energies are |z_1=1,v⟩,v=1,2,if ϵ=-2t, and |z_1=-1,v ⟩,v=1,2, if ϵ=2t.For the purpose of solving the boundary equation, and hence the full diagonalization problem, it is convenient to organize thesolutions of the bulk equation as|ϵ⟩={[α_1|z_1,1⟩+α_2|z_2,1⟩ϵ≠± 2t; α_1|z_1=-1,1⟩+α_2| z_1=-1,2⟩ϵ = 2t; α_1|z_1=1,1⟩+α_2| z_1=1,2⟩ ϵ=-2t ]. .For comparison with Sec. <ref>, one should think of z_1=e^ik and z_2=e^-ik. Because the Ansatz is naturally brokeninto three pieces, so is the boundary matrix. For instance, when ϵ≠± 2t, direct calculation yields B(ϵ)= [-tz_1^2+(w-ϵ)z_1-t z_2^2+(w-ϵ) z_2;-tz_1^N-1+(w-ϵ)z_1^N -t z_2^N-1+(w-ϵ)z_2^N ].However, from Eq. (<ref>) it follows that-t(z_ℓ+z_ℓ^-1)-ϵ=0,ℓ=1,2.This allows a simpler form to be obtained, by effectively changing the argument of the boundary matrix fromϵtoz_ℓ(ork). The complete finalexpression reads:B(ϵ)= {[ [ t+w z_1 t+w z_2; (z_1t+w)z_1^N (z_2t+w)z_2^N ]ϵ≠± 2t; ; [t-ww;(-1)^N-1(t-w) (-1)^N(N(t-w)+t) ]ϵ = 2t; ; [w+tw;w+t (w+t)N+t ] ϵ=-2t ]. .Notice that ifϵapproaches±2t, the two distinct roots collide atz_1=z_2=∓1, and the boundary matrix becomes, trivially, a rank-one matrix, signalingthe discontinuous behavior anticipated in Sec. <ref>.Furthermore, it follows from Eq. (<ref>) thatthe power-law solution atϵ=±2tmay be written as∂_z(|z_1,1⟩) = |z_1,2⟩. The entries of the second column of the corresponding boundary matrices satisfy⟨b|H_ϵ|z_1,2⟩= ∂_z_2⟨b|H_ϵ|z_2,1⟩|_z_2=z_1, wherez_1is the double root.Thus, the entries in the second column of the boundary matrix forϵ=±2tcan be obtained by differentiating with respect toz_2the second column of the boundary matrix forother (generic) values ofϵ, an observation we will use in other examples as well (see e.g. Sec.<ref>).We now analyze separately different regimes (see also Fig. <ref> for illustration).§.§.§ Vanishing impurity potential If w=0, then B(ϵ=2t) and B(ϵ=-2t) have a trivialkernel; the exoticstates |ϵ =± 2t⟩ cannot possibly arise as physicaleigenvectors. For other energies, we find that the kernel of the boundary matrix B(ϵ)=t [ 1 1; z_1^N+1 z_2^N+1 ] (w=0),is nontrivial only if z_1^N+1=z_2^N+1, in which case we can take α_1=1 and α_2=-1. From Eq. (<ref>), it also follows that z_1z_2=1.Hence, there are 2N+2 solutions,z_1=z_2^-1=e^iπ q/N+1, q=-N-1,-N,…, N.Of the associated 2N+2 (un-normalized) Ansatz vectors|ϵ_q⟩ =|z_1,1⟩-|z_2,1⟩ =2i∑_j=1^N sin(π q /N+1 j )|j⟩,two vanish identically (q=-N-1 and q=0). For q=± 1,…,± N, it is immediate to check that|ϵ_-q⟩=-|ϵ_q⟩. This means that the Ansatz yields exactly N linearly independent energyeigenvectors, of energy ϵ_q=-t(z_1+z_2)=-2tcos(π q/N+1), q=1,…,N.This is precisely the result of Sec. <ref>, wherethe solutions were labelled in terms of allowed quantum numbers k=π q/(N+1), q=1,…,N.According to our general theory, theeigenspacesofHare in one-to-one correspondence with the zeroes of B(ϵ). For this system then, there should be at most N zeroes. The reason we find2N+2 zeroesis due to the above-mentioned (quadratic) change of argumentin the boundary matrix from ϵ to k. Such a change of variables is advantageous for analytic work, and the associated redundancyis always rectified at the level of the Ansatz.§.§.§ Power-law solutions What would it take for|ϵ=±2t⟩to become eigenvectors?The kernel of B(ϵ=2t) is nontrivial only if w=tw=t N+1/N-1.These two values coincide up to corrections of order 1/N, but remember that our analysis is exact for any N>2.Similarly, the kernel of B(ϵ=-2t) is nontrivial only ifw=-tw=-t N+1/N-1.Only one of these conditions can be met: for fixedw, either |ϵ=2t⟩ is an energy eigenstate or |ϵ=-2t⟩ is, but notboth. Let us look more closely at the state at the bottomof the energy band. As we just noticed, this state will be a valid eigenstate for either of the two values of w. Let us pick w ≡ w_N = -t (N+1)/(N-1), since it yields the most interesting ground state. Then,B(ϵ=-2t)=[ w_N+t w_N; w_N+t w_N ],so that one can set α_1=1/(w_N+t), α_2=-1/w_N, and |ϵ=-2t⟩= ∑_j=1^N (1/w_N+t-j/w_N)|j⟩.Notice that⟨j|ϵ=-2t⟩=-⟨N-j+1|ϵ=-2t⟩; that is,the power-law eigenvector of the impurity problem is an eigenstate of inversionsymmetry.§.§.§ Strong impurity potential Lastly, consider the regime where t≪ |w|, for large N.Then, the values ϵ=± 2t are excluded from the physicalspectrum, and the eigenstates of the system can be determined fromB(ϵ)=0. We expect bound states of energy w to leading order and well-localized at the edges, so that 0<|z_1|<1<|z_2|, say, with z_1 (z_2) associated to the left (right) edge. Itis convenient to take advantage of this feature and modify the originalAnsatz to |ϵ⟩=α_1|z_1,1⟩+α_2z_2^-N|z_2,1⟩,so that |z_1,1⟩ (z_2^-N|z_2,1⟩) peaks at the left (right) edge, respectively. The boundary matrix becomes B̃(ϵ)= [t+ z_1 w (t+w z_2)z_2^-N; (z_1t+w)z_1^Nz_2t+w ]≈ [ t+ w z_10;0z_2 t+w ],since |z_1|^N≈ 0≈ |z_2|^-N. Keeping in mind that z_1z_2=1, we see that the kernel ofB̃(ϵ)is two-dimensional forz_1=-t/w=z_2^-1, ϵ_b=-t(z_1+z_2)=w-t^2/w^2,and otherwise trivial.The corresponding energy eigenstates can be chosen to be|ϵ_b,1⟩=∑_j=1^N(-t/w)^j|j⟩, |ϵ_b,2⟩= ∑_j=1^N(-w/t)^j-N|j⟩.Notice that |ϵ_b,2⟩ is the mirrorimage of |ϵ_b,1⟩, up to normalization.The large-N approach to boundary modes exemplified by the preceding calculation can be made systematic, as we will further explain in Sec. <ref>.The remaining (N-2) eigenstates consist of standing waves. They can be computed from the original boundary matrix, approximated for t ≪ |w| asB(ϵ≠ϵ_b)≈w [ z_1 z_2; z_1^N z_2^N ] .This boundary matrix has a nontrivial kernel only ifz_1=z_2^-1=e^iπ s/N-1, s=0,…, 2(N-1)-1,in which case one may choose α_1=z_2, α_2=-z_1. Then, |ϵ_s⟩=∑_j=1^N(z_1^j-1-z_2^j-1)|j⟩= 2i∑_j=2^N-1sin(π s(j-1)/N-1)|j⟩ .Moreover,|ϵ_s⟩=-|ϵ_N-1+s⟩,s=1,…,N-2. Hence, as needed, we have obtained (N-2) linearly independenteigenvectors of energy ϵ_s=-2tcos[π s/(N-1)].The above discussion is further illustrated in Fig. <ref>, where the determinant of the exact boundary matrix is displayed as a function of energy. §.§ Engineering perfectly localized zero-energy modes: A periodic Anderson model Having illustrated the algebraic diagonalization method on a simple impurity model, we illustrate next its usefulness toward Hamiltonian engineering. In this section, wewill design from basic principles a “comb" model, see Fig. <ref>, with the peculiar property of exhibiting a perfectly localizedmode at zero energy while all other modes are dispersive. The zeromode is distributed over two sites on the same end of the comb, withweights determined by a ratio of hopping amplitudes.The starting point is the single-particle Hamiltonian H = H_N=T⊗ h_1+T^†⊗ h_1^†.In order to have perfectly localized eigenvectors at zero energy,the bulk equation must bear emergent solutions. Therefore, we assume thath_1is non-invertible. Let|u^-⟩be in the kernel ofh_1^†.SinceTannihilates|j=1⟩, H(|j=1⟩|u^-⟩) = (T⊗ h_1+T^†⊗ h_1^†)(|j=1⟩|u^-⟩) = T|j=1⟩ h_1|u^-⟩+T^†|j=1⟩ h_1^†|u^-⟩ = 0.Similarly, if|u^+⟩is in the kernel ofh_1, then|j=N⟩|u^+⟩is also in the kernel ofH. Therefore,|j=1⟩|u^-⟩and|j=N⟩|u^+⟩are perfectly localized zero energy modes.A concrete example may be obtained by choosing h_1=-[ t_0 0; t_1 0 ]and h_1^†=-[ t_0 t_1; 0 0 ],whose kernel is spanned by|u^+⟩=[ 0; 1 ]and |u^-⟩=[ -t_1;t_0 ],respectively. This examplecorresponds to a many-body Hamiltonian of two coupledfermionic chains, as illustrated in Fig. <ref>:H =-∑_j=1^N-1(t_0c^†_jc_j+1+ t_1c_j+1^† f_j + h.c.),wherec_jandf_jdenote thejth fermions in the upperand lower chain,t_0denotes intra-ladder hopping in one of the chains, andt_1is the diagonal hopping strength between the two chains of the ladder, respectively.Physically, this “topological comb model” is closely related to the one-dimensional periodic Andersonmodel in its non-interacting (spinless) limit, see Ref. [Dagotto].§.§.§ Zero-energy modes The perfectly localized zero-energy modes in thiscase are|j=1⟩|u^-⟩and|j=N⟩|u^+⟩, that translate, after normalization, into the fermionic operatorsη_1^†=1/√(t_0^2+t_1^2)(t_1c_1^†-t_0f_1^†),η_2^† =f_N^† .The operatorη_2^†trivially describes a zero-energy mode, since it corresponds to the last fermion on thelower chain, that is decoupled from the rest. However,η_1^†corresponds to anon-trivial zero energy mode, localized over the first sites of the two chains.For large values of|t_0/t_1|,η_1^†is localized mostly on thef-chain, whereasfor small values it is localized mostly on thec-chain.Remarkably, such a non-trivial zero-energy mode isrobust against arbitraryfluctuations in hopping strengths, despite the absence of a protecting chiral symmetry.Imagine that in Eq. (<ref>) the hopping strengthst_0,jandt_1, jareposition-dependent. Then,Hmay be written as H = -(t_0,1c^†_1c_2 +t_1,1 c^†_2f_1 + h.c.)+ G,whereGdoes not contain terms involvingc_1andf_1,so that[G,c_1]=0=[G,f_1]. Then it is easy to verify that the expression for the zero-energy mode isobtained fromη_1^†in Eq. (<ref>) after substitutingt_0↦t_0,1andt_1↦t_1,1. We conclude that the zero-energy edge mode is protected by an“emergent symmetry”, that has a non-trivial action only on the sitescorresponding toj=1.Likewise, assume for concreteness thatt_0=±t_1, and consider the inter-chain perturbation described byH_1 ≡μ∑_j=1^N(c^†_j±f^†_j)(c_j± f_j),μ∈R . In this case, the corresponding single-particle Hamiltonian becomes H=1_N⊗ h_0+ T⊗ h_1+T^†⊗ h_1^† with h_0=μ[1 ±1; ±11 ].Nevertheless, the zero-energy mode corresponding to|1⟩|u^-⟩is still an emergentsolution forϵ=0, and can be verified to satisfy the boundaryequation as well.The topological nature of this zero-energy mode is confirmed byits non-trivial Berry phase <cit.> at half-filling.Under periodic BCs, the Hamiltonian in momentum spaceisH_k = -[ 2t_0cos k t_1 e^-ik;t_1 e^ik 0 ],leading to the following eigenvectors for the two bands:|u_m k⟩ = [ -t_0cos k +(-1)^m √(t_0^2cos^2 k + t_1^2); -t_1 e^ik ], m=1,2.Direct calculation shows that the Berry phasehas the non-trivial valueπ(mod2 π), as long ast_10.§.§.§ Complete closed-form solution We now obtain a complete closed-form solution of the eigenvalueproblem corresponding to Eq. (<ref>) (open BCs).The reduced bulk Hamiltonian isH(z)=-[ t_0(z+z^-1)t_1 z^-1; t_1 z 0 ],with the associated polynomial (R=1, d=2)P(ϵ,z)=z^2 [ϵ^2+ϵ t_0(z+z^-1)-t_1^2] .The model has two energy bands with a gap containing ϵ=0,and no chiral symmetry. BecauseHis real, this enforces the symmetry z↔ z^-1 of the non-zero roots ofP(ϵ,z)thatsatisfyz_1 z_2=1. For genericϵ0,there are two distinct non-zero roots and, therefore, two extended bulk solutions.The eigenvector ofH(z)may be genericallyexpressed as|u(ϵ,z)⟩= [ϵ; -t_1 z ]. Using Eq. (<ref>), the number ofemergent bulk solutions is2Rd-2=2=2s_0, one localizedon each edge. AsK^-(ϵ)=h_1^†andK^+(ϵ)=h_1,such solutions are found from their kernels,spanned by|u^-⟩and|u^+⟩, independently ofϵ.The boundary matrixB(ϵ)= [ t_0ϵ-t_1^2z_1 t_0ϵ-t_1^2z_2 0 ϵ t_1; 0 0 0-ϵ t_0; z_1^N+1t_0ϵ z_2^N+1t_0ϵ 0 0; z_1^N+1t_1ϵ z_2^N+1t_1ϵ-ϵ 0 ],whose kernel is nontrivial only ifϵ t_0(z_1^N+1-z_2^N+1)-t_1^2z_1z_2(z_1^N-z_2^N)=0.In this case, sincez_1z_2=1, we may reduce this system to one variable by substitutingz_2=z_1^-1, which then yields the polynomial equationϵ t_0z_1^2N+2-t_1^2z_1^2N+1+t_1^2z_1-ϵ t_0=0.The algebraic system of equations (<ref>) and (<ref>) determine the “dispersing” extended-support bulk modes of the system. When these equations areboth satisfied, the kernel of the boundary matrix is spanned byα = i/2[ z_1^-(N+1) -z_1^N+100 ]^ T ,and the corresponding eigenvectors ofHare given by|ϵ⟩ = iz_1^-(N+1)/2|z_1,1⟩[ ϵ; -t_1z_1 ] - iz_1^N+1/2|z_1^-1,1⟩[ ϵ; -t_1 z_1^-1 ],which, upon substitutingz_1=e^ik, can be recast as <cit.> |ϵ⟩ = ∑_j=1^N|j⟩[ ϵsin k(N+1-j); -t_1 sin k(N-j) ]. To check whether|ϵ⟩in Eq. (<ref>) indeed satisfies the eigenvalue equation, notice that⟨ j|H-ϵ1 | ϵ⟩ ={[ -ϵ⟨ 1|ϵ⟩ + h_1⟨ 2|ϵ⟩ ifj=1; h_1^†⟨ j-1|ϵ⟩ +h_1⟨ j+1|ϵ⟩-ϵ⟨ j|ϵ⟩ if 2≤ j ≤ N-1;h_1^†⟨ N-1|ϵ⟩-ϵ⟨ N|ϵ⟩ ifj=N ]..Using the expression for|ϵ⟩,⟨N|H-ϵ1 | ϵ⟩vanishes trivially, while, forj=1, ⟨ 1|H-ϵ1 | ϵ⟩ = - [ ϵ t_0 sin k(N-1) +ϵ^2 sin kN;0 ],which is seen to vanish from the relationϵ t_0 sin k(N-1) +ϵ^2 sin kN = sin kN [ϵ^2-t_1^2 + 2ϵ t_0 cos k]+[-ϵ t_0 sin k(N+1) + t_1^2 sin kN].The first term on the right hand-side is equal toP(ϵ,e^ik)=0, whereas the second term vanishes due to Eq. (<ref>). Finally, for2≤j ≤N-1, we get⟨ j|H-ϵ1 | ϵ⟩ = - [ sin k(N+1-j)[ϵ^2-t_1^2+2ϵ t_0 cos k];0 ],which equals zero, completing the argument.Next, we find the values ofϵfor which Eq. (<ref>) has a double root. The discriminant ofP(ϵ,z)isD(P(ϵ,z)) = (ϵ^2-t_1^2)^2-4ϵ^2t_0^2,and vanishes forϵ=-t_0±√(t_0^2+t_1^2)andϵ=t_0±√(t_0^2+t_1^2), for which the corresponding double roots arez_1=+1andz_1=-1, respectively. In these cases, the bulk equation may have power-law solutions. While one could construct the reduced bulk HamiltonianH_2(z)to identify these solutions, another quick way to proceed is suggested byEq. (<ref>), as already remarked in Sec. <ref>. A power-law solutionmay now be written as ∂_z_1(|z_1,1⟩|u(ϵ,z_1)⟩) =|z_1,2⟩|u(ϵ,z_1)⟩+ |z_1,1⟩∂_z_1|u(ϵ,z_1)⟩,wherez_1is the double root corresponding toϵ.The first column of the new boundary matrix remains the same as the original one, whileits second column is determined from the derivative of the second column of the original boundary matrix with respect toz_2, computed atz_2=z_1.Forϵ=-t_0±√(t_0^2+t_1^2), we havez_1=1andB(ϵ)=[ t_0ϵ-t_1^2 -t_1^20ϵ t_1;000 -ϵ t_0; t_0ϵ(N+1)t_0ϵ00; t_1ϵ(N+1)t_1ϵ -ϵ0 ].Some algebra reveals thatB(ϵ) 0, so that these values ofϵdo not appear in the spectrum ofHfor any values of parameterst_0,t_1. Similar analysis forϵ=t_0±√(t_0^2+t_1^2)yields the same conclusion.Therefore, there are no power-law solutions compatible with open BCs.We now derive the perfectly localized zero energy modes described in Sec. <ref>.Notice that forϵ=0, the only possible roots ofP(ϵ,z)arez_0=0, and from its degree it follows that there ares_0=2emergent solutions on each edge. In this case,K^-(0) =[ h_1^† 0; 0 h_1^† ],with its kernel spanned by |u_1^-⟩ = [ |u^-⟩ 0 ]^ Tand|u_2^-⟩ = [ 0|u^-⟩ ]^ T.Similarly, the kernel ofK^+(0)is spanned by |u_1^+⟩ = [ |u^+⟩ 0 ]^ Tand|u_2^+⟩ = [ 0|u^+⟩ ]^ T.Thus, the Ansatz forϵ=0consists of all four perfectlylocalized solutions (see Eqs. (<ref>) and (<ref>)). The boundary matrix in this case isB(ϵ=0)=[000 t_1t_0;000t_1^2; -t_1000;0000 ],which has a two-dimensional kernel, spanned by α_1 = [ 0 0 1 0 ]^ T,α_2 = [ 0 1 0 0 ]^ T.The corresponding two zero-energy edge modes are then |ϵ=0, α_1 ⟩ = |1⟩|u^-⟩, |ϵ=0,α_2 ⟩ =|N⟩|u^+⟩ , consistent with the results of Sec. <ref>. The eigenvector|ϵ=0,α_1⟩has support only on the first site of the two band chain.Since|N⟩|u^+⟩= |N⟩[0 1]^T, the eigenvector|ϵ=0,α_2⟩representsthe decoupled degree of freedom at the right end of the chain, as shown in Fig. <ref> (a) and (b). §.§ The Majorana Chain Kitaev's Majorana chain<cit.> is a prototypical model ofp-wave topological superconductivity <cit.>.In terms of spinless fermions, the relevant many-body Hamiltonian in theabsence of disorder and under open BCs readsH_K=-∑_j=1^Nμ c_j^†c^ _j- ∑_j=1^N-1(t c_j^†c^ _j+1-Δc_j^†c_j+1^†+ h.c.),whereμ,t,Δ∈ℝdenote the chemical potential, hopping amplitude, and pairing strengths, respectively.This Hamiltonian, expressed in spin languagevia a Jordan-Wigner transformation, describes the well-known anisotropic XY spin chain, which has a long history in quantum magnetism, including analysis of boundary effectsfor both open BCs and periodic <cit.>. Expressed in the form of Eq. (<ref>), the corresponding single-particleHamiltonian isH_N=1_N⊗ h_0+(T⊗ h_1+ T^†⊗ h_1^†),h_0=[ -μ0;0μ ],h_1= [ -tΔ; -Δt ].Thus, R=1, d=2d_int=2, andh_R=h_1(hence the model) is invertible in the genericparameter regime|t| |Δ|, for arbitraryμ.We have alreadycharacterized in detail both the invertible regime <cit.>and the non-invertible regime <cit.> for generic, regular energy values.While, given the importance of the model, we will summarizesome of these results in what follows, our emphasis here will be on(i) addressing singular energy values, in particular, by directlycomputing compactly-supported eigenstates of flat-bandeigenvectors directly in real space;(ii) uncovering the existence of zero-energy Majorana modes with a power-lawprefactor, emerging in an invertible but non-generic parameter regimerecently discussed in the context of transfer-matrix analysis <cit.>.§.§.§ The parameter regime |t|=|Δ|, μ0 We briefly recall some key steps and results presented in Sec. 5.2 ofRef. [JPA]. For concreteness, we assumet=Δ, but a similar analysismay be repeated for the caset=-Δ. The reduced bulk Hamiltonian in this case isH(z) = [ -μ-t(z+z^-1)t(z-z^-1); -t(z-z^-1)μ+t(z+z^-1) ],with associated polynomial P(ϵ,z)=-z^2[ 2μ t(z+z^-1) + (μ^2 + 4t^2-ϵ^2)].As in the topological comb example, for generic values ofϵthe abovehas two distinct non-zero rootsz_1andz_2, which implies a two-dimensional space of extended bulk solutions and one emergent solution on each edge.Let the two extended solutions be labeled byz_1andz_2=z_1^-1, with|z_1|≤1. Then, we get |u(ϵ,z_ℓ)⟩ =[t(z_ℓ-z_ℓ^-1); ϵ + μ +t(z_ℓ+z_ℓ^-1) ],ℓ=1,2.The two emergent solutions are obtained from the one-dimensional kernels of the matricesK^-(ϵ) = h_1^†andK^+(ϵ) = h_1, which are spanned by |u_1^-⟩ = [1; -1 ]and |u_1^+⟩ = [ 1; 1 ],respectively. Following Eq. (<ref>), the boundary matrix is B(ϵ) =[ 2t^2z_1 +t(ϵ+μ)2t^2z_1^-1 +t(ϵ+μ) 0-μ-ϵ;-2t^2z_1 +t(ϵ+μ) -2t^2z_1^-1 +t(ϵ+μ) 0-μ+ϵ; z_1^N+1[-2t^2z_1^-1-t(ϵ-μ)] z_1^-(N+1)[-2t^2z_1-t(ϵ-μ)]-μ-ϵ 0; z_1^N+1[-2t^2z_1^-1-t(ϵ-μ)] z_1^-(N+1)[-2t^2z_1-t(ϵ-μ)] μ-ϵ 0; ].Our analysis in Ref. [JPA] shows that open BCs do not allow any contributions from the emergent solutions in the energy eigenstates,which are linear combinations of the two extended solutions.The condition forϵto be an energy eigenvalue isB(ϵ)=0,which simplifies to2t z_1+ϵ+μ=±z_1^(N+1)(2t z_1^-1+ϵ+μ).Explicitly, as long asϵ∉𝒮≡{μ±2t,-μ±2t}, thecorresponding eigenstate is |ϵ⟩ = |z_1,1⟩|u(ϵ,z_1)⟩∓ z_1^N+1|z_1^-1,1⟩|u(ϵ,z_1^-1)⟩. The above equation is particularly interesting for zero energy, since it dictates thenecessary and sufficient conditions for the existence of Majorana modes. Forϵ=0, the rootz_1takes valuesz_1 = {[-μ/2t if |μ|<2|t|;-2t/μ if |μ|>2|t| ]..In the large-Nlimit, the factorz_1^N+1in the right hand-side ofEq. (<ref>) vanishes thanks to our choice of|z_1|<1. However, the left hand-side vanishesonly in the topologically non-trivial regime characterized by|μ|<2|t|, giving rise to a localized Majorana excitation.The unnormalized Majorana wavefunction in this limit is characterized by an exactexponential decay (see also Fig. <ref>), namely, |ϵ=0⟩ =(4t^2-μ^2/2μ) ∑_j=1^∞z_1^j|j⟩[1; -1 ]. For the analysis of the non-generic energy values in𝒮, we return to the finitesystem sizeN. For suchϵ,P(ϵ,z)has double roots atz_1=1andz_1=-1, so that the bulk equation has one power-law solution in each case <cit.>.These solutions are compatible with the BCsforcertain points in the parameter space, determined by the condition2tN +μ(N+1) = 0.Explicitly, the eigenstates corresponding to eigenvaluesϵ=±(μ+2t)are then |ϵ=μ+2t⟩ =∑_j=1^N|j⟩[1; -1+2 j/N+1 ] ,|ϵ=- μ-2t⟩ =∑_j=1^N|j⟩[ -1+2 j/N+1;1 ] . §.§.§ The parameter regime |t|=|Δ|, μ=0This regime, sometimes affectionately called the “sweet spot,”is remarkable. Since the analytic continuation of the BlochHamiltonian isH(z)=t [ -(z+z^-1)z-z^-1; -(z-z^-1)z+z^-1 ], one finds that (H(z)-ϵ1_2)=ϵ^2-4t^2. Thus, the energies ϵ=± 2t realize a flat band and its charge conjugate. From the point of view of the generalized Bloch theorem, these two energies are singular. According to Sec. <ref>, theynecessarily belong to the physical spectrum of the Kitaevchainregardless of BCs, each yielding𝒪(N)corresponding bulk-localized eigenvectors. In order to construct such eigenvectors, note that forϵ=±2t, the adjugate ofH(z)-ϵ1_dis the matrixadj(H(z)∓ 2t1_d) = t[z+z^-1∓ 2-z+z^-1; z-z^-1 -z-z^-1∓ 2 ],which immediately providestwo kernel vectors |v_1,±(z)⟩ = [ 1+z^-2± 2z^-1;1-z^-2 ],|v_2,±(z)⟩ = [-1+z^-2; -1-z^-2± 2z^-1 ].In thiscase, we see that the kernel vectors contain polynomials inz^-1of degree2<δ_0=(d-1)2Rd =4(recallEq. (<ref>)). For a suitable range of lattice coordinates js, the compactly-supported sequences Ψ_j1,± = |j⟩[ 1; 1 ]± 2 |j+1⟩[ 1; 0 ] +|j+2⟩[1; -1 ], Ψ_j2,± = -|j⟩[ 1; 1 ]± 2 |j+1⟩[ 0; 1 ] +|j+2⟩[1; -1 ],yield non-zero solutions |Ψ_j μ,±⟩=P_1,NΨ_j μ,±,μ=1,2,of the bulk equation. However, it is not a priori clear how many of these are linearly independent. For example, it is immediate to check that Ψ_j1,±+Ψ_j2,± = ∓(Ψ_j+1,2,±-Ψ_j+1,1,±).In this case, a basis of compactly-supported solutions canbe chosen from the states|Ψ̃_0⟩ =|1⟩[ -1;1 ]if j=0, |Ψ̃_j,±⟩ = |j⟩[ 1; 1 ]± |j+1⟩[1; -1 ]if1≤ j ≤ N-1, |Ψ̃_N⟩ = |N⟩[ 1; 1 ]ifj=N,Out of theseN+1states, the ones corresponding toj=1,…,N-1can be immediately checked to be eigenstates of energy ϵ± 2t<cit.>. In contrast,|Ψ̃_0⟩and|Ψ̃_N⟩arenot eigenstates: they do not satisfy theboundary equation trivially like other states localized in the bulk.We have thus found2N-2eigenstatesof the Hamiltonian,N-1for each bandϵ= ±2t. The two missing eigenstates appear atϵ=0, which is a regularvalue of energy and so it is controlled by the generalized Bloch theorem.For ϵ=0, there are four emergent solutions (two oneach edge), out of which only |ψ^-⟩ = |1⟩[1; -1 ] = -|Ψ̃_0⟩ |ψ^+⟩ = |N⟩[ 1; 1 ] = |Ψ̃_N⟩are compatible with the BCs. Since these solutions are perfectly localized on the two edges, they exist for anyN>2(see also Fig. <ref>). Interestingly, the above states also appeared as solutions of the bulk equation at the singularenergies ϵ=± 2t, and failed to satisfy the BCs at those values of energy.We do not know whether this fact isjust a coincidence or has some deeper significance. §.§.§ Majorana wavefunction oscillations in the regime tΔRecently, it was shown <cit.> that,inside the so-called “circle of oscillations”, namely, the parameter regime( μ/2t)^2 + ( Δ/t)^2 =1 ,the Majorana wavefunctionoscillates while decaying in space. Such oscillations in Majorana wavefunction are not observed outside this circle. This observation has consequences on the fermionic parity of the ground state <cit.>. Because of duality, spin excitations in the XY chainshow a similar behavior in the corresponding parameter regime <cit.>B_z^2=t^2-Δ^2 = J_x J_y. We now analyze this phenomenon by leveraging the analysis of Sec. <ref>. For simplicity, we address directly the large-Nlimit.Clearly, whether a wavefunction oscillates in space depends on the nature of the extended bulk solutions that contribute to the wavefunction. In particular, let|ψ⟩= |z,1⟩|u⟩be one such bulk solution. For a wavefunction to be decayingasymptotically, we must have|z|<1. Further, ifz ∈ℝ, then|ψ_j⟩= z|ψ_j-1⟩implies that the part of the wavefunction associated to this bulk solution simply decays exponentiallywithout any oscillations. On the other hand, ifz ≡|z|e^iϕwith non-zero phase, then a linear combination of vectors|z,1⟩ + |z^*,1⟩ = ∑_j=1^N2|z|^j cos( ϕ j )|j⟩ ,can show oscillatory behavior while decaying. This is precisely the phenomenonobserved in this case. WhentΔ, the reduced bulk Hamiltonian isH(z) = [ -μ-t(z+z^-1)Δ(z-z^-1); -Δ(z-z^-1)μ+t(z+z^-1) ],with associated characteristic equation(z+z^-1)^2(t^2-Δ^2) + (z+z^-1)(2μ t) + (μ^2 + 4Δ^2-ϵ^2)=0.Forϵ=0, the above admits four distinct roots in general, out of which two lie inside the unit circle and contribute to the Majorana mode on the left edge. Whether any of these two roots is complex decides if theMajorana wavefunction oscillates for those parameter values.Notice that the characteristic equation is quadratic in the variableω=z+z^-1. We get the two values ofωto beω_± = -μ t±Δ√(μ^2-4(t^2-Δ^2))/(t^2-Δ^2).Likewise, notice that forμ^2<4(t^2-Δ^2), we get bothω_+andω_-tobe complex, which necessarily means that bothz_1,z_2inside the unitcircle are also necessarily complex. Further, the symmetry of Eq. (<ref>) forces thatz_2=z_1^*. This leads to the oscillatory behavior of the Majorana wavefunction in the regime μ^2<4(t^2-Δ^2), that is,inside the circle defined by Eq. (<ref>). Thus, the spatial behavior of Majorana excitations in this regime is formally similar to the solution of an underdamped classical harmonic oscillator (see Fig. <ref>).Outside the circle, the rootsω_±are real. With some algebra, it can be shown that|ω_±|>2in this regime, which also means that bothz_1,z_2are real roots. This iswhyoscillations are not observed in this parameter regime, in agreement with the results of Ref. [Hegde16]. The Majorana wavefunction in this case resembles qualitatively the solution of a overdamped harmonic oscillator.The situation when the parameters lie preciselyon the circle is particularly interesting. In this case, we find thatω_+= ω_- ≡ω_0 = -4 t/μ. Let us assumet/Δ>0for simplicity. It then followsthatz_1 =z_2= -2(t-Δ)/μ, which rightly indicates appearance of a power-law solution. Let usspecifically analyze the case of open BCs on one end (forN≫1as stated). One of the two decaying bulk solutions is|ψ_1,1⟩= |z_1,1⟩|u(z_1)⟩, where|u(z)⟩ = [ Δ(z-z^-1); μ + t(z+z^-1) ].The other bulk solution is obtained from|ψ_1,2⟩ = ∂_z_1|ψ_11⟩ =z_1^-1|z_1,1⟩[ Δ(z_1+z_1^-1); t(z_1-z_1^-1) ] +|z_1,2⟩[ Δ(z_1-z_1^-1); μ + t(z_1+z_1^-1) ].The relevant boundary matrix, B(ϵ=0) ≡[ B_11(z_1) B_12(z_1); B_21(z_1) B_22(z_1) ],may be computed by relating its second column to the partial derivative ofthe first column atz=z_1as also done previously.Explicitly: [ B_11(z_1); B_21(z_1) ] =[(2t z_1+μ)Δ; -μ t -z_1(t^2+Δ^2)-z_1^-1(t^2-Δ^2) ],[ B_12(z_1); B_22(z_1) ]= [ 2tΔ; -(t^2+Δ^2) +z_1^-2(t^2-Δ^2) ],where we also used Eq. (<ref>) for simplification. Some algebra reveals thatB(0)has a one-dimensional kernel, spanned by the vectorα = [-μ t 2Δ(t-Δ) ]^ T.This leads to the power-law Majorana wavefunction|ϵ=0⟩ = -μ t|ψ_1,1⟩ +2Δ(t-Δ)|ψ_1,2⟩ =8Δ^2(t-Δ)/μ∑_j=1^∞j z_1^j-1 |j⟩[1; -1 ],which decays exponentially with a linear prefactor (see Fig. <ref>). In principle, the existence of such exotic Majorana modes could be probedin proposed Kitaev-chain realizations based on linear quantum dot arrays<cit.>, which are expected to afford tunable control on all parameters. § AN INDICATOR OF THE BULK-BOUNDARY CORRESPONDENCE As stated in the Introduction, a main motivation behind thedevelopment of the generalized Bloch theorem is to elucidatethe bulk-boundary correspondence. In this section, we start presenting an indicator of bulk-boundary correspondence based on the results fromSec. <ref>, generalizing the original definition in Ref. [abc].The indicator is built out of the boundary matrix and, therefore, encodes informationfrom the bulkand the BCs.We will then consider an application of the indicator to studythe Josephson response of ans-wave two-band topological superconductor<cit.>.Interestingly, and to the best of our knowledge, this system provides the first example ofan unconventional (fractional) Josephson effectnot accompanied by afermionic parity switch. We explain the physical reasons behind such a result.§.§ Derivation of the indicator For a system of sizeN, the existence of localized modes at energyϵreflects into a non-trivial kernel of the corresponding boundary matrix, which we now denote byB_N(ϵ)in order to emphasize the dependence onNandϵ. As we increaseNwithout changing the BCs,the energyϵof the bound modes(that is, modes that remain asymptotically normalizable) attains alimiting value. For instance, in topologically non-trivial,particle-hole or chiral- symmetric systems under hard-wall BCs, themid-gap bound modes attain zero energy in the large-Nlimit.This convergence of bound modes and their energies is nicely captured by a modified version of the boundary matrix in the limitN ≫1,which we now construct.Consider a system ofNsites in a ring topology, asshown in Fig. <ref>(a), so as to allow non-zero contribution fromthe matrixw_bb'in the BCs described byW(see Eq. (<ref>)). Let us assumethat the system hosts one or more bound modes near the junction formedby the two ends, which converge in the large-Nlimit to energyϵ. The resulting modes are the bound modes of a bridge configuration that extends to infinity on both sides,and where the boundary region is shown in Fig. <ref>(b). For eachN, we may express the bound eigenstate as in Eq. (<ref>).Such bound states have contributionsonly from those bulk solutions that are normalizable forN≫1. The extended-support solutions corresponding to|z_ℓ|=1are not normalizable, and thereforemust drop out from the Ansatz.Further, while the amplitude of thosecorresponding to|z_ℓ|>1blows up nearj=N, theyremain normalizable in the limit. This becomes apparent once we rescale such solutions byz_ℓ^-N.These rescaled solutions almost vanish atj=1for largeN. Based on these considerations, we propose a modified Ansatz for finiteN,|ϵ,α⟩_N ≡∑_|z_ℓ|<1∑_s=1^s_ℓα_ℓ s|ψ_ℓ s⟩ +∑_s=1^s_-α^-_s|ψ^-_s⟩ +∑_|z_ℓ|>1∑_s=1^s_ℓα_ℓ sz_ℓ^-N|ψ_ℓ s⟩+∑_s=1^s_+α^+_s|ψ^+_s⟩.expressed in terms of up most2Rdamplitudes. The above Ansatz may be used to compute a corresponding boundary matrixB_N(ϵ)in the same way as described in Sec. <ref>. Note thatB_N(ϵ)may not capture the bound modes appearing at finiteNsince, by construction, it does not incorporate contributions from extended support solutions corresponding to|z_ℓ|=1. However,B_∞(ϵ) ≡lim_N→∞ B_N(ϵ)is now well-defined, and describes accurately the presence and exact form of bound modes in the limit. The condition for a non-trivial kernel becomes[B_N^†(ϵ)B_N(ϵ)]=0.Based on this condition, we define the quantity𝒟_ϵ≡log{[B_∞(ϵ)^†B_∞(ϵ)]},as anindicator of bulk-boundary correspondence.This captures precisely the interplay between the bulk properties and the BCs that may lead to the emergence of bound modes, in the sense that,as we parametrically change either or both of the reduced bulk Hamiltonian and the BCs,𝒟_ϵshows a singularity at (and only at) theparameter value for which the system hosts bound modes at energyϵ. Unlike most other topological indicators that are derived from bulkproperties (i.e., in a torus topology), our indicator is constructed from a boundary matrix,that incorporates the relevant properties of the bulk. In cases where thebound modes are protected by a symmetry, this allows for the indicator tobe computed for arbitrary BCs that respect the symmetry, paving the way tocharacterizing the robustness of the bound modesagainst classes of boundary perturbations.An interesting situation is that ofw_bb'=0, in which case the large-Nlimit consists of two disjoint semi-infinite chains.ThenB_∞(ϵ)is block diagonal,B_∞ (ϵ) = [ B_∞^- (ϵ) 0; 0 B_∞^+ (ϵ) ],whereB_∞^-(B_∞^+) may be interpreted as theboundary matrix of a semi-infinite chain, describing the edgemodes at the left (right) edge, respectively.While the indicatorD_ϵof Eq. (<ref>) signals the presence of bound states, it does not convey information about the degeneracyof that energy level, which is nevertheless contained in the boundary matrix. Therefore, it is often useful to also study the behavior of thedegeneracy indicator as a function ofϵ:𝒦_ϵ≡Ker [B_∞(ϵ)]. In practice, the dimension of the kernel is obtained by counting the number of zero singular values ofB_∞(ϵ). Remark.— With reference to the discussion in Sec. <ref>, recall that in numerical computations,B_∞(ϵ)signals fictitious roots whenever the bulk equation has a power-law solution. In such cases, weonce again remedy the issue by resorting to the Gramian. Then the corrected value of the indicator is given by𝒟_ϵ = log{[B_∞(ϵ)^†B_∞(ϵ)]/𝒢 (ϵ)}.Thus, the correct degeneracy of the energy is obtained by counting zero (within numerical accuracy) singular values ofthe matrixB̃_∞(ϵ) = B_∞(ϵ)𝒢 (ϵ)^-1/2.§.§ Application: An s-wave topological superconducting wire The usefulness of the proposed indicator of bulk-boundary correspondence wasdemonstrated in the context of characterizing the Josephson response of a two-bandtime-reversal invariants-wave topological superconducting wire in Ref. [abc].While the calculations reported there employed a simplified Ansatz, including onlyextended-support solutions of the bulk equation,we now validate the analysis by using the complete Ansatz given in Eqs. (<ref>)and (<ref>), and further analyze and interpret our results in terms of fermionic parity switches.The relevants-wave, spin-singlet, two-band superconductor model <cit.> derives its topological nature from the interplay betweena Dimmock-type intra-band spin-orbit coupling and inter-band hybridization terms. Due to the spin degree of freedom in each of the two relevant orbitals, say,candd, the Nambu basis corresponding to an atom at positionjconsists of 8 fermionic operators, that we write as the vectorΨ̂_j^†=[c_j,↑^† c_j, ↓^†d_j,↑^†d_j,↓^†c_j,↑c_j,↓d_j,↑d_j,↓ ].In this basis, the single-particle Hamiltonian under open BCs is given by H_N = 1_N⊗ h_0+(T⊗ h_1+ T^†⊗ h_1^†),h_0 = [ -μ u_cd -iΔσ_y0; u_cd -μ0iΔσ_y;iΔσ_y0μ-u_cd;0 -iΔσ_y-u_cdμ ]= -μτ_z+u_cdτ_zν_x +Δτ_yν_zσ_y, h_1 = [iλσ_x -t00; -t -iλσ_x00;00iλσ_xt;00t -iλσ_x ]= -tτ_zν_x+iλν_zσ_x,where the real parametersμ,u_cd,t,λ,Δdenote the chemical potential,the interband hybridization, hopping, spin-orbit coupling and pairing potential strengths, respectively, andτ_α, ν_α,σ_α,α={x,y,z}, are Pauli matrices in Nambu, orbital and spin spaces. The topological properties of the above Hamiltonian were analyzed in Ref. [swavePRB]. The BdG Hamiltonian is time-reversal invariant, which places it in the symmetry class DIII.The topological phases may thus be distinguished by aℤ_2-invariant,given by the parity of the sum of the Berry phases for the two occupied negative bands inone of the Kramers' sectorsonly<cit.>.For open BCs and for non-vanishingpairing, the system in its trivial phases was found to host zero or two pairs of Majoranason each edge, in contrast to the topologically non-trivial phase supporting one pair ofMajoranas per edge. Similar to the two-dimensional version of the model,one may see that the existence of such Majorana modes is protected by a non-trivial chiralsymmetry, of the formτ_y σ_z. The single-particle HamiltonianH_Nfor open BCs can be exactlydiagonalized as described in Sec. <ref>.In the largeN-limit, the boundary matrixB_∞(ϵ=0)calculatedby using the Ansatz in Eq. (<ref>) yields degeneracyK_0= 0,4,8in the no-pair, one-pair, and two-pair phases, respectively, verifying the bulk-boundarycorrespondence previously established through numerical diagonalization.§.§.§ Josephson responseIn the Josephson ring configuration considered in Ref. [abc], the first and last sites of the open chain are coupled by the samehopping and spin-orbit terms as in the rest of the chain, only weaker by a factorof1/w. A fluxϕis introduced between the two ends via this weak link. In the large-Nlimit, this link acts as a junction,with the corresponding tunneling term in the many-body Hamiltonian being given byH_T(ϕ)=Ψ̂_N^†(wh_1U_ϕ)Ψ̂_1 +h.c., U_ϕ=[e^iϕ/21_40;0 e^-iϕ/21_4 ].The total Hamiltonian is thenH(ϕ) = H_N + H_T(ϕ). It was demonstrated <cit.> that the Hamiltoniandisplays fractional Josephson effect in the topologically non-trivial phase,as inferred from its4π-periodic many-body ground state energy [Fig. <ref>(a)],with the phenomenon being observedonly if the open-chain Hamiltoniancorrespondingly hosts an odd number of Majorana pairs per edge. The physics behind the4π-periodicity was explained in terms of the crossing ofa positive and a negative single-particle energy level happening at precisely zero energyas a function of fluxϕ. The singular behavior resulting at flux valuesϕ=π,3πfrom the indicatorD_ϵ=0(ϕ)computed using both the simplified Ansatz as inRef. [abc] and the complete Ansatz of Eq. (<ref>) is shownin Fig. <ref>(c). The qualitative features are clearly unchanged, indicating that in the large-Nlimit the bound modes formed near the junction are linearcombinationsonly of extended-support solutions, with no contributions from emergent ones. As seen in Fig. <ref>(d), at bothϕ=πandϕ=3πthe junction hosts a total of four Majoranas. §.§.§ Parity switch and decoupling transformation Despite the4π-periodic Josephson response witnessed in the topologically non-trivial phase,it turns out that the ground state fermionic parityremains unchanged for all flux values. In the non-trivial regime of interest, we may focus on the three low-lying energy levels.Specifically, for values ofϕ< π, let|Φ(ϕ)⟩denote the many-body ground state, with energyE_0(ϕ), as inFig. <ref>(a).As we will show,there are two degenerate quasi-particle excitations, say,η_1(ϕ),η_2(ϕ), with small positive energyϵ_0(ϕ). This results in atwo-fold degenerate first excited many-body state, with energyE_1(ϕ)=E_0(ϕ)+ϵ_0(ϕ), and a correspondingeigenspace is spanned by{ η_1^†(ϕ)|Φ(ϕ)⟩,η_2^†(ϕ)|Φ(ϕ)⟩}.The second excited state,η_1^†(ϕ)η_2^†(ϕ)|Φ(ϕ)⟩, is not degenerate and hasenergyE_2(ϕ)=E_0(ϕ)+2ϵ_0(ϕ). Note that this state has the same (even) fermionic parity as the ground state. Atϕ=π, thequasi-particle excitation has exactly zero energy,ϵ_0(π)=0, causing all three energy levels to become degenerate. Asϕcrossesπ,ϵ_0(ϕ)becomes negative. Therefore, forπ<ϕ<3π, we find thatE_2(ϕ)<E_1(ϕ)<E_0(ϕ). The continuationof the stateη_1^†(ϕ)η_2^†(ϕ)|Φ(ϕ)⟩with energyE_2(ϕ)thus becomes the new ground state, whereas the continuation of the original ground state|Φ(ϕ)⟩now attains the maximum energy among these three levels. Since the new ground state has the same parity as the original one, the system shows no parity switch, with a similar analysis holding for the crossover atϕ=3π. We conclude that the absence of a fermionic parity switch originates fromthe twofold degeneracy of the single-particle energy levels.While the system under open BCs is time-reversal invariant, away fromϕ=0,2πthis symmetry is broken by the tunneling termH_T(ϕ).Therefore, Kramer's theorem is not responsible in general for thedegeneracy in the single-particle levels. Instead, we now explain the physical origin ofthis degeneracy in terms of a “decoupling transformation” in real space, thanks to whichthe system in the Josephson bridge configuration is mapped into two decoupled systems in the same configuration, each with half the number of internal degrees of freedom as the original one.Although each of these smaller systemsdoes undergo a parity switch, the total parity being the sum of individual parities remains unchanged. Observe that the HamiltonianH(ϕ)is invariant under the unitary symmetriesŜ_1andŜ_2, defined by the action Ŝ_1:c_↑ (d_↑) ↦ d_↑ (c_↑), c_↓ (d_↓) ↦ -d_↓ (-c_↓), Ŝ_2: c_↑ (d_↑) ↦ ic_↓ (id_↓), c_↓ (d_↓) ↦ ic_↑ (id_↑).We can use the eigenbasis ofŜ_1to decoupleH(ϕ)into two independent Hamiltonians. Consider, for each sitej=1,…, N, the canonical transformation a_jσ≡c_jσ + d_jσ/√(2),b_jσ≡c_jσ - d_jσ/√(2),σ= ↑,↓.and letÛ_1be the unitary change of basis defined byÛ_1: Ψ̂_j^† ↦[ Ψ̂_+,j^† Ψ̂_-,j^†], where Ψ̂_+,j^† ≡ [ a_j,↑^†b_j, ↓^† a_j,↑b_j,↓ ], Ψ̂_-,j^† ≡ [ a_j,↓^† -b_j, ↑^†a_j,↓ -b_j,↑ ]. By lettingΨ̂_±^† ≡[Ψ̂_±,1^† … Ψ̂_±,N^†], the action ofÛ_1then decouplesH(ϕ)according to H(ϕ) ≡H_+(ϕ) + H_-(ϕ)= Ψ̂_+^†H_+(ϕ)Ψ̂_++Ψ̂_-^†H_-(ϕ)Ψ̂_-, whereH_±(ϕ)describes two smaller systems, each in a Josephson ring configuration,with hopping and pairing amplitudes given by h_±,0=[ -μ+u_cdσ̃_z -iΔσ̃_y;iΔσ̃_yμ-u_cdσ̃_z ]=-μτ_z+u_cdτ_zσ̃_z+ Δτ_yσ̃_y, h_±,1 =[± iλσ̃_x -tσ̃_z0;0 ± iλσ̃_x + tσ̃_z ]=± iλσ̃_x-tτ_zσ̃_z, withσ̃_αdenoting Pauli matrices in themodified spin basis. The decoupling transformation in Eq. (<ref>) is close in spirit to the one alreadyemployed under periodic BCs <cit.>. Indeed, it is worth remarking thatΨ̂_+,jandΨ̂_-,jare still time-reversals ofeach other, in the sense that𝒯Ψ̂_+,j^†𝒯^-1= Ψ̂_-,j^†, withTbeing the anti-unitary time-reversal operator forthe system. Because of the tunneling term, however,the two decoupled (commuting) HamiltoniansH_±(ϕ)are related by𝒯 H_+(ϕ)𝒯^-1 = H_-(4π-ϕ).It now remains to show thatH_±(ϕ)have identical single-particle energy spectrum, and therefore lead to the desired degeneracy in the energy levels ofH(ϕ). This follows by examining the symmetries of the single-particle BdG HamiltonianH(ϕ).Corresponding toŜ_1,H(ϕ)has a unitary symmetryS_1=1_N ⊗ν_xσ_z, and thus gets block-diagonalizedinto two blocks,H_±(ϕ), upon the action ofU_1. Similarly, corresponding toŜ_2,H(ϕ)has another unitary symmetryS_2 = i1_N⊗τ_z σ_x. Further,S_1andS_2satisfy the anti-commutation relation{S_1,S_2}=0, which is responsible for the doubly degenerate eigenvalue spectrum <cit.>. In fact, one can also verify directly thatH_+(ϕ)andH_-(ϕ)satisfyŜ_2 H_+(ϕ) Ŝ_2^†= H_-(ϕ). This explains the origin of the double degeneracy of each single-particle energy level, and hence of the absence of fermionic parity switch. § TRANSFER MATRIX IN THE LIGHT OF THE GENERALIZED BLOCH THEOREM Starting with the work in Refs. [HatsugaiPRL]-[HatsugaiPRB], thetransfer matrix has remained the tool of choice for analyticalinvestigations of the bulk-boundary correspondence<cit.> including, as mentioned, recent studies of Majorana wavefunctions in both clean anddisordered Kitaev wires <cit.>. In this section, we revisit the transfer matrix approach to band-structuredetermination in the light of our generalized Bloch theorem. In particular, we show how, insituations where the transfer matrix fails to be diagonalizable, our analysis makes it possible to give physicalmeaning to the generalized eigenvectors by relating them to the power-law solutionsdiscussed in Sec. <ref>.§.§ Basics of the standard transfer matrix method While our conclusions apply more generally to arbitrary finite-range clean models, for concretenesswe refer in our discussion to the simplest settingwhere both approaches are applicable, namely, a one-dimensional chain withnearest-neighbor hopping.We further focus onopen (hard-wall) BCs, as most commonlyemployed in transfer-matrix studies. The relevant single particle-HamiltonianH_Nis then a tridiagonal block-Toeplitz matrix, with entriesh_1^†,h_0andh_1along the three diagonals. Generically,h_1is assumed to be invertible.The starting point of the method entails obtaining the recurrence relation betweeneigenvector components. Specifically, if|ϵ⟩=∑_j=1^N|j⟩|ψ_j⟩is an eigenvector ofHwith energy eigenvalueϵrelative to the usual Hilbert-space factorizationH=H_L ⊗H_I, the components|ψ_j⟩satisfy the recurrence relation h_1^† |ψ_j-1⟩ + (h_0-ϵ1)|ψ_j⟩+ h_1|ψ_j+1⟩ = 0, 2≤ j≤ N-1.In terms of the2d ×2dtransfer matrix t(ϵ) ≡[ 0 1_d;-h_1^-1h_1^† -h_1^-1(h_0-ϵ1) ],the above recurrence relation may be reformulated asP_j,j+1|ϵ⟩ = t(ϵ)P_j-1,j|ϵ⟩,2≤ j≤ N-1,where we have writtenP_j,j+1|ϵ⟩≡[ |ψ_j⟩ |ψ_j+1⟩ ]^T.Thus, P_j+1,j+2|ϵ⟩ = t(ϵ)^jP_1,2|ϵ⟩,0≤ j ≤ N-2,which can be leveraged for obtaining the complete set of eigenvectorsofH_N.We can define|ψ_0⟩,|ψ_N+1⟩by using the relationsP_1,2|ϵ⟩ = t(ϵ)P_0,1|ϵ⟩,P_N,N+1|ϵ⟩ = t(ϵ)P_N-1,N|ϵ⟩,so thatP_N,N+1|ϵ⟩= T(ϵ)P_0,1|ϵ⟩in terms of the matrixT(ϵ) ≡t(ϵ)^N. Hard-wall BCs enforce|ψ_0⟩=0= |ψ_N+1⟩. Substituting these boundary values leads to[ |ψ_N⟩; 0 ]=[ T_11(ϵ) T_12(ϵ); T_21(ϵ) T_22(ϵ) ][ 0; |ψ_1⟩ ],which has a non-trivial solution if and only if T_22(ϵ)=0.Therefore, all values ofϵthat obey the above condition areeigenvalues ofH_N. For each eigenvalue, the corresponding|ψ_1⟩is obtained as the kernel ofT_22(ϵ). In practice,T(ϵ)is calculated by first diagonalizingt(ϵ)by a similarity transformation, and thenexponentiating the eigenvalues along its diagonal <cit.>.As can be appreciated from this example, the standard version of the transfer matrix method relies oninvertibility of certain matrices, although “inversion-free”<cit.> or partially inversion-free <cit.> modifications have also beensuggested. In the standard case, the only prerequisitefor constructingt(ϵ)at each step is the banded structure of the single-particle Hamiltonianand, most importantly, the resulting matrixT(ϵ)is assumed to be diagonalizable.§.§ Connections to the generalized Bloch theorem In order to relate the above analysis to the generalized Bloch formalism, the key observation is tonote that the set of equations in Eq. (<ref>) constitute the complete bulk equation,as described in Sec. <ref>. Consequently, Eq. (<ref>) is satisfied by any bulk solution|ψ⟩∈ℳ_1,N, whereℳ_1,Ndenotes the bulk solution space as usual. It is insightful to recast Eq. (<ref>) in the formt(ϵ)^jP_1,2|ψ⟩ = P_1,2(T)^j|ψ⟩,0≤ j ≤ N-2,suggesting that the action of the transfer matrix in the bulk solution spaceis closely related to the one of the left shiftT. When restricted toℳ_1,N, the above yields the following operator identity: (t(ϵ)-z1_d)^jP_1,2|_ℳ_1,N =P_1,2(T-z1_N)^j|_ℳ_1,N,with z∈C. This relation may be used to establish a direct connection between the basis of the bulk solution space described in the generalized Bloch theorem, and the Jordan structure of the transfer matrix.In the absence of power-law solutions, each bulk solution|ψ_ℓs⟩is annihilated byP_1,2(T-z_ℓ1_N)=P_1,2[P_B(T-z_ℓ1_N)]. In such cases, Eq. (<ref>) reads (t(ϵ)-z_ℓ1_d)P_1,2|ψ_ℓ s⟩ =P_1,2(T-z_ℓ1_N)|ψ_ℓ s⟩=0,implying thatP_1,2|ψ_ℓs⟩is an eigenvector oft(ϵ)with eigenvaluez_ℓ.Naturally, a Bloch wave-like bulk solution corresponds to an eigenvalue on the unit circle, whereas an exponential solution corresponds to one inside or outside the unit circle, in agreement with the literature <cit.>.While, as remarked, the transfer matrix is typically assumed to be diagonalizable, we nowshow that generalized eigenvectors oft(ϵ)are physically meaningful, and in fact related to the power-law solutions of the bulk equation. Letϵbe a value of energy for which power-law solutions are present.We can then generalize our earlier calculation for the eigenvectors of the transfer matrix by notingthat each|ψ_ℓs⟩is annihilated byP_1,2(T-z_ℓ1_N)^s_ℓ, wheres_ℓis the multiplicity of the rootz_ℓas usual. Then, a similar calculation reveals thatP_1,2|ψ_ℓs⟩is a generalized eigenvector oft(ϵ), satisfying(t(ϵ)-z_ℓ1_d)^s_ℓP_1,2|ψ_ℓ s⟩ = 0.Thus,generalized eigenvectors of the transfer matrix are projections ofsolutions with a power-law prefactor. In some non-generic scenarios, they indeed contribute to the energy eigenstates, as we discussed <cit.>.This analysis is vividly exemplified by the parameter regime corresponding to the circle of oscillations in the Majorana chain, Eq. (<ref>), which we found to be associated to a zero-energy power-law Majorana wavefunction. Accordingly, we expectthe corresponding transfer matrix to possess generalized eigenvectorsof rank two, failing to be diagonalizable. Let us verify this explicitly. Except for the pointsμ=0, Δ/t=±1in this regime,the matrixh_1in Eq. (<ref>) is invertible. The transfer matrix is then t(ϵ=0) =1/μ^2[ 0 0 μ^2 0; 0 0 0 μ^2; -4(t^2+Δ^2)-8tΔ-4tμ-4Δμ;-8tΔ -4(t^2+Δ^2)-4Δμ-4tμ; ],whereμ,tandΔsatisfy Eq. (<ref>). It can be checked thatt(ϵ=0)has only two eigenvalues, namely,z_ℓ = -2(t + (-1)^ℓΔ)/μ,ℓ=1,2, each of algebraic multiplicity two, and that both of these eigenvalues have only one eigenvector, given by P_1,2|z_ℓ,1⟩|u_ℓ⟩ = [z_ℓ; (-1)^ℓ z_ℓ;z_ℓ^2; (-1)^ℓ z_ℓ^2 ],hence geometric multiplicity equal to one. Bothz_1,z_2are then defective, makingt(ϵ=0)not diagonalizable. In fact,t(ϵ=0)has one generalized eigenvector of rank two corresponding to each eigenvalue, given by P_1,2|z_ℓ,2⟩|u_ℓ⟩= [ 1;(-1)^ℓ;(2z_ℓ); (-1)^ℓ (2z_ℓ); ]. Returning to the general case, a number of additional remarks are worth making, in regard to points of contact and differences between the transfer matrix approachand our generalized Bloch theorem. First, the eigenstate Ansatz obtained from theanalytic continuation of the Bloch Hamiltonian provides aglobal characterizationof energy eigenvectors (and generalized eigenvectors), as opposed to the local characterization afforded within the transfer-matrix approach, whereby eacheigenvector is reconstructed “iteratively” for any given eigenvalue. Further to that, the generalized Bloch theorem unveils the role of non-unitary representations oftranslational symmetry for finite systems.Perhaps most importantly, the twomethods differ in the way BCs are handled.Clearly, in both approachesit is necessary to match BCs in order to obtain the physical energy spectrum.While open BCs are most commonly used in transfer-matrix calculations,the method has also been applied to relaxed surfaces <cit.> andgeneralized periodic BCs <cit.>, all of which belong to the class of BCs considered in this paper. In this sense,it is tempting to compare Eq. (<ref>) with the condition onthe determinant of the boundary matrix,B(ϵ)=0. However,the class of BCs to which the transfer matrixapproach can be successfully applied is nota priori clear, thuswhether such a condition can be established for as general a class ofBCs as our theorem covers has not been investigated to the best ofour knowledge.From a numerical standpoint, thecomputational complexity of the standard transfer matrix methodfor clean systems (when applicable) is independent of the system sizeN,as is the case of our scan-in-energy algorithm in Sec. <ref>. In those cases where inversion of certain matrices is a difficulty andinversion-free approaches are used <cit.>, the latter also havea comparable computational complexity to our method. Interestingly,all approaches so far that are truly inversion-free rely at some point oranother on the solution of a non-linear eigenvalue problem <cit.>. Thanks to the fact that, as noted, the construction oft(ϵ)inthe generic case relies only on the banded structure ofH_N, bulk disordercan be handled efficiently within transfer-matrix approaches, albeit for a limitedclass of BCs. For general BCs as we consider, it is thus natural tocombine the transfer matrix approach with the bulk-boundaryseparation we have introduced, in order to still find solutions efficiently: the transfer matrix can be employed to find all possible solutions of the bulk equation in the presence ofbulk disorder, and the latter can then be used as input for the boundary matrix, that provides a condition for energy eigenstates.§ DISCUSSION AND OUTLOOK We have formulated a generalization of Bloch's theorem applicable toclean systems of independent fermions on a lattice, subject toBCs that are arbitrary – other than respecting the finite-range natureof the overall Hamiltonian.This generalization, which leverages a reformulation of the problem in terms of corner-modified block-Toeplitz matrices,affords exact, analytical expressions for all the energy eigenvalues and eigenstates ofthe system – which consistently recovers the ones derived from the standard Bloch'stheorem for periodic BCs.As a key component to this theorem, one obtains anexactstructural Ansatz, close in spirit to the Bethe Ansatz, for all (regular) energyeigenstates indispersive bands. This Ansatz is easy to construct since it depends only on the energy eigenvalue and the bulk properties of the Hamiltonian.The individual components of this Ansatz reflect translation invariance in a way we have made precise and are, as such, determined by the analytic continuationof the Bloch Hamiltonian, as shown. Based on the generalized Bloch theorem, we have provided both a numerical and an algebraicdiagonalization algorithm for the class of quadratic Hamiltonians under consideration.For generic energy values, the former is computationally more efficient than existing onesin that its complexity is independent upon the system size; the latter is especially well-suited forsymbolic computation or pen-and-paper solutions, as we explicitly demonstrated by solving inclosed form a number of tight-binding Hamiltonians of interest, under various BCs. With an eyetoward applications in synthetic quantum matter, we have also used the generalized Blochtheorem to engineer a quasi one-dimensional Hamiltonian that support a perfectly localized,robust zero-energy mode, notwithstanding the lack of chiral and charge-conjugation protectingsymmetries.Remarkably, our generalized Bloch theorem predicts the existence, under specific(non-generic) conditions, of edge states that decay exponentially in spacewith apower-law prefactor. Such exotic states were previously believed to ariseonly in systems with long-range couplings. In our framework, their origin may betraced back to the description of the system's eigenstates in terms ofnon-unitaryrepresentations of translation symmetry “outside Hilbert space”– againcapturing the fact that such a symmetry is only mildly broken by the BCs, in aprecise sense.Notably, we have shown how the emergence of zero-energyMajorana modes with a linear prefactor is possible in the paradigmatic Kitaev chainby proper Hamiltonian tuningon the so-called “circle of oscillations”.Their “critical” spatial behavior separates the theoretically observed Majoranawavefunction oscillations inside such a circle from the simple exponential decay outside.Our generalized Bloch theorem makes no prediction about the (singular) energy valueswhich correspond todispersionless, or flat, bands of eigenstates. We have nonetheless provided a prescription for identifying such energy values without diagonalizing the full Hamiltonian, and showed howsuch energy values necessarily enter the physical energy spectrum irrespective of the BCs.In such singular cases, we have further provideda procedure to effectively obtain a (possibly overcomplete) basis ofperfectlylocalized states using an analytic continuation of the Bloch Hamiltonian, and explicitly illustrated such a procedure in the Kitaev's Majorana chainHamiltonian at its sweet spot.Building on our proposal in Ref. [abc], we have rigorously derived and furtherexplored a proposedboundary indicator for the bulk-boundary correspondence. This indicator leverages the other key component to our generalized Bloch theorem, the boundary matrix, and is unique in the sense that, unlike most other indicators in theliterature, it combines information from both the bulk and the boundary. The utility of thisindicator is seen from our analysis of the4π-periodic Josephson effect in a model of as-wave topological superconductor.In the process, we show how, remarkably, the4π-periodicity that distinguishes a topologically nontrivial response isnotaccompanied by a fermionic parity switch in this system. We have provided a physicalexplanation of this behavior by exhibiting a decoupling transformation, which maps therelevant Hamiltonian to two uncoupled “virtual” wires – each undergoing a parity switch.Finally, for systems where no bulk disorder is present, and subject to BCs for which thewell-known transfer matrix approach is also applicable, we have shown how the generalizedBloch theorem may be used to obtain a physical interpretation of the transfer matrix's generalized eigenvectors, in terms of bulk solutions with a power-law prefactor. An explicit example is seen, again, in the semi-infinite Kitaev's chain with open BCs, preciselyin the same circle-of-oscillations parameter regime that hosts power-law zero-energy Majoranamodes. While, in this way, our method may be seen to provide yet another inversion-freealternative to the standard transfer-matrix approach, the connections we have identifiedin this work naturally point to further possibilities for fruitfully combining the two approaches. In particular, since the bulk-boundary separation we proposed remains useful in the presence ofbulk disorder, one may envision a hybrid approach for solving disordered systems subjectto arbitrary BCs, by employing transfer-matrix techniques to handle the resulting bulk equation.The tools we have developed here may serve as the starting point for a number ofadditional studies and applications. As mentioned, in the companion paper <cit.>,we will provide a formulation of the bulk-solution Ansatz and the generalized Bloch theoremfurther accounting for the role played by the transverse momentum (_̨⊥) inhigher-dimensional systems with non-trivial boundaries – as opposed to the single_̨⊥-analysis presented here. We will show that topological power-law modesdiscussed in this paper are not just a feature of one-dimensional systems,and indeed are present in higher dimensions too.Beside exploring the interplay between_̨⊥,the boundary matrix, and the edge states in a number of paradigmaticmodel Hamiltonians, we will also demonstrate how the treatment of one-dimensionalhomogeneous systems can be effectively extended to those of interfaces.From a computational standpoint, we expect that the diagonalization algorithmsemerging from our approach will be useful for large-scale electronic calculationsin both one- and higher- dimensions, possibly in conjunction with perturbativeapproaches for incorporating interactions. Towards a deeper understanding of bulk-boundary correspondence in topologicalinsulators and superconductors, our approach can be instrumental in studyingrobustness against boundary perturbations. It is natural to start by asking how certain symmetries of the system influence thenature of the proposed indicator, or the boundary matrix from which the indicatoritself is derived. This can possibly lead to identifying a symmetry principlewhich dictates the bulk-boundary correspondence, as well as an interpretationat the basic dynamical-system level in terms of stability theory. Likewise, theframework we have developed may also serve as a concrete starting point for rigorously deriving an effective boundary theory for lattice systems.Lastly, while we have focused on fermions in this paper, the general foundation ofour method laid out in Ref. [JPA] is equally valid for bosons and immediatelyapplicable to non-Hermitian effective Hamiltonians with non-trivial boundaries, asoften arising in semi-classical models of open quantum systems in various contexts<cit.>. We plan to explore the correspondinggeneralized Bloch theorems in forthcoming publications, and to ultimately provide extensions to Markovian open quantum systems described by quadratic Lindbladmaster equations. § ACKNOWLEDGEMENTSWe gratefully acknowledge useful discussions with Smitha Vishveshwara. Work at Dartmouth was supported in part by the US NSF through Grant No.PHY-1620541 and the Constance and Walter Burke Special Projects Fundin Quantum Information Science.§ FURTHER DISCUSSION ON ARBITRARY BCS Section <ref> imposes two restrictions on the allowed form of BCs, described byW. The first restrictsthe non-trivial action ofWto the boundary hyperplanes. Since the corresponding single-particle operatorWsatisfies therelation P_BW=0, withP_Bbeing the bulk projector associated toH_N,Wcan be thought of as a corner-modification of the banded block-Toeplitz matrix H_N. The operatorsH_N+Wrepresent boundary value problems in such a way that a changeof BCs is encoded in a change of W. Theintuition behind these ideas comes from finite-differencemethods for solving differential equations. We briefly illuminate this connection here. Consider for concreteness the Schrödinger boundary value problemψ(0)=ψ(L)=0 , (-1/2d^2/dx^2-ϵ)ψ(x)=0x∈(0,L),describing a particle in an infinite one-dimensional potential well.The discretization x↦ x_j=jΔ x, with j=0,1,…,N+1=L/Δ x, reduces this problem to thelattice boundary value problemψ(x_0)=ψ(x_N+1)=0, -1/2ψ(x_j-1)+(1-ϵ)ψ(x_j)-1/2ψ(x_j+1)=0,in terms of the centered second difference approximation to the Laplacian.This set of linear equations is equivalent to the eigenvalue equation (H_N-ϵ1_N)|ψ⟩=0, withH_N=-1/2(T+T^†)+1_N|ψ⟩≡∑_j=1^N|j⟩ψ(x_j).By comparison, the more general BCs α_1ψ(0)+β_1dψ/dx(0^+)=0,α_2ψ(L)+β_2dψ/dx(L^-)=0 ,lead to the lattice boundary value problemα_1ψ(x_0)+β_1ψ(x_1)-ψ(x_0)/Δ x=0 ,α_2ψ(x_N+1)+β_2ψ(x_N+1)-ψ(x_N)/Δ x=0 ,together with Eq. (<ref>). The system of linear equations in Eqs. (<ref>)–(<ref>) is equivalent to the eigenvalue problem (H_N+W-ϵ1_N)|ψ⟩=0, withW=β_1/2(α_1Δ x-β_1)|1⟩⟨ 1| -β_2/2(α_2Δ x+β_2) |N⟩⟨ N| ,a corner modification of the lattice Laplacian H_N. For thespecial case α_1=α_2, β_1=-β_2,we have discussed the exact diagonalization of H_N+Win Sec. <ref>. § ALGEBRAS OF SHIFT OPERATORS Consider the topologically inequivalent manifoldscorresponding to the finite line segment, the circle (of finite or infinite radius),the semi-infinite line, and the infinite line, as illustrated in Fig. <ref>. Given aphysical system whose state space has support on those manifolds,one can define distinct shift (or translation by a distancea) operators acting on the physical states. Certainly, those shift operators encode topologicalinformation that depending on the circumstances may have physical consequences.In the following wewill study the algebra of those shift operators. The subtle difference between the variousshift (or translation) operators is reflected in the fundamental discussions that led tothe modern theory of macroscopic electric polarization in many-body systems in terms of Berry phases <cit.>, and the concomitant definitionof the position operator in extended systems <cit.>.The finite line segment.— This section is based on Ref. [fockpfs], where thematrices we are about to consider appeared with a different physicalmeaning. Consider a line of finite lengthL=N a, written in termsof a characteristic lengtha, typically defined by a periodicpotential or lattice. The left shift operator is given by T=∑_j=1^N-1|j⟩⟨ j+1| , in terms of the orthonormallattice states|j⟩. The lattice state |1⟩ is annihilated byT, T|1⟩=0, and |N⟩ is annihilated by T^†, mirroring the fact that theboundary ofa line segment consists of two points. For states otherthan |1⟩,|N⟩, T and T^† act asordinary translations, to the left or right respectively, i.e.,T |j ⟩= | j-1 ⟩andT^†|j ⟩= | j+1 ⟩.While T can be regarded as the generator ofbulk translations, it is not a unitary transformation. Instead,T^s(T^†)^s+(T^†)^N-sT^N-s=1,s=1,…,N-1,and notice also that T^N=0. The commutator [T,T^†]=|1⟩⟨ 1|-|N⟩⟨ N| captures the extent of translation-symmetry breaking introduced by the BCs. The lattice-regularized position operator X=∑_j=1^Nj|j⟩⟨ j| satisfies the commutation relation[X,T]=-T.While this is formally analogous to [x,e^i p/ħ]=-e^i p/ħ, caremust be exercised with such analogy, precisely because ofissues of definition of the domains of functions where operatorsact upon. The circle.— The other compact one-dimensional manifold is the circle.The standard (periodic) left shift operator in this case is given byV=∑_j=1^N-1|j⟩⟨ j+1|+|N⟩⟨ 1|=T+(T^†)^N-1.No lattice state |j⟩ is annihilated by either V or V^†, because the circle is a manifold with no boundary. One can further check that VV^† =1=V^N . The relation between periodic shifts and the position operatorXis better described in terms ofU≡e^i2π/N X,since then we have the Heisenberg-Weyl relationVU=e^i2π/N UV.This Heisenberg-Weyl algebra is well-known in statistical mechanics in connection to clock models <cit.>, but its relevance totight-binding models appears to have gone unnoticed.The two generators are related by the discrete Fourier transformFasF U F^†= V^†andF V F^†= U, see for example Ref. [pclock] for more details and references. By comparing Eq. (<ref>) to Eq. (<ref>), onesees that the U(1) symmetry of the shift algebra associated tothe line segment is broken to a Z_N symmetry forthe circle. In practice, the full U(1) symmetry is recovered by introducing twisted generalizations of the Heisenberg-Weyl algebra,V_ϕU_ϕ=e^i2π/NU_ϕV_ϕ,V_ϕ^N=e^iϕ1,with U_ϕ and V_ϕ unitary. Their meaning is clear interms of tight-binding models. Twisted Heisenberg-Weyl algebrasdescribe physical problems subject to generalized Born-von-KarmanBCs, needed for example for defining topologicalinvariants such as the Chern number. A representation of these algebrais given byU_ϕ=U, V_ϕ=∑_j=1^N e^iϕ/N |j⟩⟨ j+1| + e^iϕ/N |N⟩⟨ 1| .In statistical mechanics, our twisted Heisenberg-Weyl algebras are connected to chiral Potts models, but this connection seems to be unknown in the literature. The semi-infinite line.— The left and right unilateral shifts T_-, T_-^⋆ were introduced in Sec. <ref>. The commutator [T_-,T_-^†]=|1⟩⟨ 1| captures in some sense the extent of translation symmetry breaking. The lattice position operator X_-^=∑_j=1^∞ j |j⟩⟨ j| satisfies the commutation relations [X_-^,T_-^]=-T_-^, [X_-^,T_-^⋆]=T_-^⋆. The relation T_-^⋆=T_-^† holds if thedomain of these linear transformations is restricted to the Hilbert space of square summable half-infinite sequences. The real line.— The shift operator is T≡∑_j∈Z|j⟩⟨ j+1|, and it is unitary when restricted to the Hilbert space ofsquare-summable sequences, that is, T^-1=T^†. We carefully refrained from restricting T so inSec. <ref>. WithX≡∑_j ∈Z j |j⟩⟨ j| (an unbounded Hermitian operator in Hilbert space), one canshow that[X,T]=-T, [X,T^-1]=T^-1 both in and out of Hilbert space. In summary, the shift operators associated to the finiteand the semi-infinite line segment donot commute with theiradjoints, reflecting the presence of boundary points for these topologies.In contrast, the shift operators defined on the circle and the line V and T do commute with their adjoints (or inverses)and are unitary (or just invertible) – which is why they can representtranslation symmetry. As a consequence, V,V^† can be diagonalizedsimultaneously, and the same goes for T,T^†<cit.>.Their eigenvalues lay on the unit circle due to unitarity. The keydifference between these two types of translation symmetry stems fromtheir interplay with lattice position operators. For all the shiftalgebras but the one associated to the circle, the positionoperators generate U(1) rotations of the shift operators.For the Heisenberg-Weyl algebra, this U(1) symmetry appearsinstead as a family of inequivalent unitary irreducible representations of thedefining relation Eq. (<ref>). § EMERGENT SOLUTIONS AT REGULAR ENERGIES This appendix provides further mathematical detailon the procedure for computing emergent bulk solutions outlined inSec. <ref>. Specifically, we pick up the discussionwhere we left it therein, right after the definition of the matrix polynomialK^-(ϵ,T_-)in Eq. (<ref>).Left-localized emergent bulk solutions.—In analogy to the sequencesΦ_z,vassociated toTin Eq. (<ref>),let us define statesΥ^-_z,1≡∑_j=0^∞z^j|j+1⟩, Υ^-_z,v≡ 1/(v-1)!d^v-1/dz^v-1Υ_z,1^-, v=2,3,….in such a way thatΥ^-_0,v=|j=v⟩ and, also,Υ_z^-|u⟩≡∑_x=1^vΥ_z,x^-|u_x⟩= [ Υ_z,1^- … Υ_z,v^- ][ |u_1⟩; ⋮; |u_v⟩ ].It is then immediate to verify thatK^-(ϵ,T_-)Υ^-_z,1|u_1⟩=Υ^-_z,1K^-(ϵ,z)|u_1⟩.Moreover, using Eq. (<ref>), one also obtainsthe more general relation K^-(ϵ,T_-)Υ_z^-|u⟩=[ Υ_z,1^- … Υ_z,v^- ] K^-_v(ϵ,z)[ |u_1⟩; ⋮; |u_v⟩ ],in terms of the upper-triangular v× v block matrix[K^-_v(ϵ,z)]_xx'= 1/(x'-x)!d^x'-x K^-(ϵ,z)/dz^x'-x,1≤ x≤ x'≤ v.It will be crucial for later use to notice that K^-_v(ϵ,z) is a block-Toeplitz matrix. Both K^-_v(ϵ,z) and H_v(z) are defined by the same formula, recall Eq. (<ref>). The key difference between the two is that K^-_v(ϵ,z) is well-defined also at z=0. So suppose that z_0=0 is a root of P(ϵ,z)of multiplicity s_0>0. Then, one can show using tools fromRef. [JPA], that there are precisely s_0 independent solutions of the equation K_s_0^-(ϵ,z_0=0) |u^-_s⟩=0, s=1,…,s_0.The corresponding emergent bulk solutions are |ψ^-_s⟩=P_1,NΥ^-_0|u^-_s⟩=∑_j=1^s_0 |j⟩|u_sj^-⟩.They are localized on the left edgeover the firsts_0 sites. For Hermitian Hamiltonians, s_0≤ dR necessarily. Right-localized emergent bulk solutions.— Left-localized emergent bulk solutions cannot appear alone;they can only appear in conjunction with a set of right-localizedemergent bulk solutions. The reason is as follows. Consider theunitary, Hermitian operatorU=U^†≡∑_j=1^N|N-j+1⟩⟨ j|⊗1_d ,U^2=1_dN,which implements a mirror transformation of the lattice,by acting trivially on internal states. The transformed Hamiltonian is the Hermitian block-ToeplitzmatrixH_N = UH_N U=1_N⊗ h_0+ ∑_r=1^R(T^r⊗ h_r^†+T^r †⊗ h_r) ,in which the hopping matrices have been exchanged as h_r↔ h_r^†. Therefore, the left-localizedemergent bulk solutions forH_Nare dictated by the matrixK^-(ϵ)with entries[K^-(ϵ)]_ij=[K^-(ϵ)]_ij^†. If |ψ^-⟩ denotes a left-localized emergentsolution for H_N, then 0=P_B(H_N-ϵ)|ψ_s^-⟩= UP_B(H-ϵ)U|ψ_s^-⟩,implying that the stateU|ψ_s^-⟩=∑_j=1^s_0|N-j+1⟩|ũ^-_sj⟩ is an emergent bulk solution for H_N, localized on therightedge. Similarly, the left-localized emergent bulk solutions of H_Nare in one-to-one correspondence with the right-localized emergentsolutions of H_N. This conclusion relies havilyon the commutation relation P_BU=UP_B, which is always necessarily true forclosed systems (Hermitian Hamiltonians), as we considered here.But how can we compute the right-localized emergent bulk solutions directly in terms of H_N? In Sec. <ref>, we answered this question with the help of the matrix K^+(ϵ)≡ K^-(ϵ)^†.We will justify this answer here. Let|ψ_s^-⟩=∑_j=1^s_0|j⟩|ũ^-_sj⟩,s=1,…,s_0, denote the left-localizedemergent solutions associated to H_N, and let|ψ^+_s⟩≡∑_j=1^s_0|N-s_0+j⟩|u^+_sj⟩ = U|ψ_s^-⟩, s=1,…,s_0,denote the corresponding right-localized emergent solutions of H_N, so that |u^+_sj⟩≡|ũ^-_s,s_0-j+1⟩. 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"authors": [
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"Gerardo Ortiz",
"Lorenza Viola"
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],
"primary_category": "cond-mat.stat-mech",
"published": "20170627151451",
"title": "A generalization of Bloch's theorem for arbitrary boundary conditions: Theory"
} |
http://arxiv.org/abs/1706.09061v1 | {
"authors": [
"Ivan Gavrilyuk",
"Volodymyr Makarov",
"Nataliia Romaniuk"
],
"categories": [
"math.NA",
"65L10, 65L12, 65L20, 65L50, 65L70, 34B15"
],
"primary_category": "math.NA",
"published": "20170627220102",
"title": "Super-Exponentially Convergent Parallel Algorithm for a Fractional Eigenvalue Problem of Jacobi-Type"
} |
|
Large-scale Datasets: Faces with Partial Occlusions and Pose Variations in the Wild Tarik Alafif1, Zeyad Hailat2, Melih Aslan3 and Xuewen Chen4 Computer Science Department, Wayne State UniversityDetroit, MI, USA 48120Email: [email protected], [email protected], [email protected], [email protected] 25, 2017 =================================================================================================================================================================================================================================================empty emptyIn this paper we tackle the problem of visually predicting surface friction for environments with diverse surfaces, and integrating this knowledgeinto biped robot locomotion planning.The problem is essential for autonomous robot locomotion since diverse surfaceswith varying friction abound in the real world, from wood to ceramic tiles, grass or ice,which may cause difficulties or huge energy costs for robot locomotion if not considered.We propose to estimate friction and its uncertainty from visual estimation ofmaterial classes using convolutional neural networks, together with probabilitydistribution functions of friction associated with each material.We then robustly integrate the friction predictions into a hierarchical (footstep and full-body) planningmethod using chance constraints, and optimize the same trajectory costs at both levels of the planning method for consistency. Our solution achieves fully autonomous perception and locomotion on slippery terrain, which considers not only friction and its uncertainty, but also collision, stability andtrajectory cost. We show promising friction prediction results in real pictures of outdoor scenarios,and planning experiments on a real robot facing surfaces with different friction.§ INTRODUCTIONLegged and humanoid robot locomotion planning is an important problem for disaster response and service robots. One of the difficulties of this problem is the complexity of general environments and the need to consider several factors such as collision, energy consumption, surface geometry and friction. In this paper we deal with the specific problem of humanoid robot locomotion when environment friction is considered. Our claim is that friction and its uncertainty can be estimated from vision and robustlyintegrated into algorithms for motion planning with contact. We argue that even if the precise coefficient of friction cannot be predicted from vision before touching a surface, priors and accumulated experience associated with surface material or condition (think coefficient of friction tables)can provide a probability distribution of friction. Motion planning with contact can also become prohibitively expensive once multiple factors are considered,such as locomotion cost, collision and friction. In this paper we propose a hierarchical approach to the problem, where a footstep planner optimizes the same cost function as a full-body motion planner by use of anoracle, and considers collision and friction by using simple bounding box collision checks and an“extended footstep planning” <cit.> approach.The contributions of this paper are the following: * We propose a solution to the friction from vision problem using a state-of-the-art deep Convolutional Neural Network (CNN) architecture to predict broad material classes from images, together with known (or learned) distributions of material friction;* We propose a hierarchical planning architecture for biped robots that optimizes the same objective at both levels,and deals with friction, stability, collision and cost to produce full-body trajectories;* We show empirical friction prediction results, as well as planning experiments which show the usefulness and applicability of the approach in complex environments with varying friction.§ RELATED WORKRecent full-body motion planning <cit.> and control <cit.> algorithms for legged robots have started to consider friction by using friction cones in optimization problems. Such methods rely on the fundamental assumption that the coefficient of friction can bepredicted in advance. While in itself a challenging problem, partial evidence from humanvisual perception motivate such an approach to the problem.For example, humans are known to use visual cues to estimate friction, related to surfacetexture <cit.>, shine <cit.> and detection of materials or contaminants (e.g. water) <cit.>. Furthermore, in the human gait literature there is evidence that humans use accumulatedprevious experience to predict friction and adapt walking style before touching slipperyground <cit.>. In this paper, such “accumulated previous experience” is implemented as probabilitydistributions of friction associated with material classes. The term “material” is usedin a broad sense to refer to visually classifiable classes related to material, conditionand context (e.g. “dry metal”, “wet asphalt road”). The use of material classes for prediction here is motivated by a recent study <cit.> which identifies material as one of the most predictive features of both coefficient of friction values and human judgements of friction.Friction estimation work related to this paper includes that of Angelova et. al <cit.>,which predicts the percentage of slip (i.e. lack of locomotion progress) of a rover from terrainclassification and slope. Predictions are made based on non-linear regression of data gathered ona learning stage. Compared to <cit.>, we estimate the coefficients of frictioninstead of slip, we use open segmentation datasets <cit.> to trainmaterial classification, and we decouple the problem from the physical robot.Importantly, our approach allows for sharing material friction data among differentrobots as long as they have similar foot soles. Several approaches exist to the friction-constrained motion planning problem for legged robots. One approach is the non-hierarchical, full-scale trajectory optimizationformulation with implicit contact constraints of <cit.>.While technically elegant and showing promising results, these can still be computationally expensive for online planning. In order to make the problem tractable, full-body motion can be planned aftercontact (or footstep) planning <cit.>, in what is called the contact before motion approach. One common issue with such methods is that contact planners do not take the samefriction or trajectory criteria into account as the subsequent full-body planners. One exception to this lack of consistency between planning levels is theextended footstep planning work of <cit.>,in which learned models are used at the footstep planning level that predict full-body feasibility. The approach can also account for friction constraints by using timing variablesand learned slippage models at the footstep planning level.Still, in <cit.> the costs optimized at the footstep planning level arenot further optimized at the full-body level. In this paper we improve the method byusing an oracle at the footstep planning level which predicts the costs obtained by afull-body trajectory optimizer, thus increasing consistency across planning levels.§ FRICTION FROM VISIONIn this paper we propose to estimate friction of surfaces from visual input by classifying surface material at each image pixel and assuming known (or learned) probabilitydistributions of friction for each material. For convenience we will use the term “friction of a material” to refer to the coefficient of friction between the robot foot sole and a second surface of a given material.We consider a pixel-wise labelling algorithm that, given an input image I with n pixels,provides a probability distribution P(X | θ, I), where X={x_1,...,x_n} are thepixel labels and θ are internal parameters of the algorithm. Each pixel can take one of m possible labels, such that x_k ∈ℒ = { l_1,...,l_m }. Furthermore, let each label be a material associated with a probability distribution function (p.d.f.)of a coefficient of friction p(μ | l_i). Then at pixel k, the conditional p.d.f. of μ isp(μ | θ, I) = ∑_i=1^m p(μ | l_i) P(x_k=l_i | θ, I) . For the results shown in this paper we estimated the friction distributions p(μ | l_i) experimentally,by measuring maximum friction force of the robot foot on several surfaces for each material. We describe the procedure in more detail in Section <ref>.We use a deep convolutional neural network (CNN) to obtain pixel-wise material predictions P(x_p=l_i | θ, I). In particular we use the encoder-decoder architecture of <cit.>, which achieves good results in image segmentation applications and is characterized by a low number of parameters. Its low number of parameters leads to fast inference, which is crucial for robotics. The architecture consists of an encoder network of 13 convolutional layers as in VGG16 <cit.>,followed by a decoder network of 13 layers and a final softmax layer. The output of the last layer of the network (a softmax classifier) is at each pixel a vector ofprobabilities for each class, that is, the probabilities P(x_p=l_i | θ, I) used in equation (<ref>).§ HIERARCHICAL PLANNINGIn this paper we plan full-body robot motion using a contact before motion approach.A footstep planner first searches a stance graph using transition costs provided by an oracle.The stances are then used as constraints in a full-body trajectory optimizer that considers full-body trajectory costs, collisions, joint limits and static stability. The obtained trajectory is finally interpolated and locally adapted for dynamic stability using a ZMP-based method. The oracle basically takes each stance transition and predicts the costs obtained at the end of the whole planning pipeline. This leads to footstep plans which optimize the same criteria as thefull-body planner. See Figure <ref> for a visual representation of the architecture.§.§ Extended footstep planning with an oracleThe footstep planner searches a graph of stances to find a feasible path between the start and goal stance. Each node in the graph is a stance s, which is defined by a set of contacts with the environment. A contact is a tuple (link, position, rotation). A neighbor stance s' either adds or removes a contact with respect to s. In this paper we deal with biped walking only, and hence stances simply transitionfrom double-support to left-foot-contact, to double-support, to right-foot-contact, back to double-support, etc. The advantage of this representation instead of, for example, double-support stances only, is that the swept-volume between consecutive stances can be used by the optimizer to guide a swing leg out of collision.Such an approach is also used by other works focusing on collision detection <cit.>.In this paper we use the extended footstep planning framework of <cit.>.The footstep planner is “extended” because extra parameters associated with stance transitions (e.g. step timing) are computed from the transition itself by a function learned offline.Here we call this function an oracle because it predicts the costs that will be obtained by a subsequent full-body trajectory optimizer.We now briefly describe the footstep planning algorithm. We first discretize the search space by constraining contact positions to a point cloud, and rotations by aligningcontact normals with the environment and constraining the links' yaw orientation to a discrete set of valuesin the global coordinate frame. Then we use an A* variant, ARA* <cit.>, to search the stance graph based on oracle costs. At each state (i.e. stance) of the graph s, a contact is either added or removed togenerate successor stances. Contact removal generates one new stance. On the other hand, adding a new contact consists of doing a range search of points in a radius around the foot in contact. For each of those points, footsteps are placed at all yaw angles and checked for feasibility.We implement feasibility as empirical stance distance limits, as well as foot-foot andCOM-environment collision checking using bounding boxes for the feet and trunk.The feasible stances are added as successors of s.To find an optimal path to the goal state, A* search requires a state transition cost functionc(s,s') and a heuristic cost-to-go function h(s). Similarly to <cit.>, we define the cost asc(s,s') =pmin f̂_cost(s,s',p) subject to P(f̂_RCOF(s,s',p) < μ^(k)) ≥η k=1,...,K , where k is an index of the contacts of s and s', μ^(k) is the friction at these contacts, and p are state transition parameters. Since in this paper we use two full-body posture waypoints per stance at the trajectory optimization level (Section <ref>), we set transition parameters p=(Δ t, Δ t'). These are the time spent from the second waypoint of s until the first waypoint of s', and the time spent from the first waypoint of s' to the second, respectively.The main difference in (<ref>) with respect to <cit.> is thatwe consider uncertainty in the coefficient of friction variable by using chance constraints. RCOF stands for required coefficient of friction and corresponds to the maximum tangential-to-normalforce ratio exerted over the whole trajectory <cit.>.Therefore, the constraints P(f̂_RCOF(s,s',p) < μ^(k)) ≥η in (<ref>)implement robust Coulomb friction at each contact by forcing the inequalities to hold with atleast probability η. The constraints can also be rewritten using the cumulative distribution function of (<ref>) denoted by F_μ^(k)|θ,I,F_μ^(k)|θ,I(f̂_RCOF(s,s',p)) ≤ 1-η .Since each μ^(k) is one-dimensional then F can be inverted and the constraints rewritten in deterministic formf̂_RCOF(s,s',p) ≤ Q_1 - η^(k) ,where Q_1-η^(k) is the (1-η)-quantile of F_μ^(k)|θ,I, which can be computed by an integral of (<ref>) over μ.Regarding the heuristic cost-to-go function of A* search, as in <cit.>, we set it toh(s) = d_xy(s , s^goal) . s, s', pminf̂_cost(s,s',p) / d_xy(s , s') ,where the function d_xy(.,.)computes Euclidean distance on the horizontal plane between two stances(i.e. the distance between left feet and right feet summed). The heuristic (<ref>) is a lower bound on the cost-of-transport times distance, which guarantees that h(s) does not overestimate the total cost to the final stance s^goal(i.e. is admissible, a necessary condition for A* optimality).In this paper, the functions f̂_cost and f̂_RCOF are given by an oracle which predicts the value of f_cost and f_RCOF obtained at the end of the whole planning pipeline. Notice that in this paper, contrary to <cit.>, f_cost is the function that will beoptimized at the full-body trajectory optimization level.We implement f̂_cost and f̂_RCOF as hash tables. The tables are filled offline, by feeding the whole planning pipeline(i.e. trajectory optimization, interpolation, dynamic stabilization) with uniformly distributed samples of (s, s', p) as shown in Figure <ref>. The discrete optimization problems in (<ref>), (<ref>) are then solved for a large number of discretized stances andcoefficient of friction quantiles and stored in new hash tables for fast access to costsand heuristics during search.§.§ Full-body trajectory optimization The full-body trajectory optimizer takes a footstep plan with N stances and produces a full-body trajectory, parameterized by T discrete-time waypoints. Waypoints are full-body robot configurationsq_t ∈ℝ^D, t=1,...,T, where D is the number of degrees-of-freedom consisting of thejoints' angle values and the pose of the robot base. Each stance is associated with 2 full-body postures (at start and midstance) and so T=2N. For convenience we use s_t to refer to the stance associated to q_t.Our optimizer solves the problemq_1,...,q_Tminimize f_cost(q_1,...,q_T) + α f_collision(q_1,...,q_T) subject to f_stance(q_t,s_t) = 0 ∀_t ∈ 1,...,T f_xy(q_t) ∈𝒫_t ∀_t ∈ 1,...,T f_roll(q_t) = 0 ∀_t ∈ 1,...,T A_t q_t ≤ b_t ∀_t ∈ 1,...,T , where q_1,...,q_T are the optimization variables, α is a penalty constant and:* The function f_cost computes the sum of the squared static torques of all joints at all waypoints, as implemented in the trajopt library <cit.>* The function f_collision is a collision cost as proposed by <cit.>.It is the sum of a discrete collision cost computed by the signed distance betweeneach link and all other geometries, and a continuous collision cost computed by the signed distancebetween the swept volume of each link with the environment* The function f_stance(q_t,s_t) computes the pose error of all links in contactas a 6C-dimensional vector where C is the number of active contacts in s_t. This iscomputed as the translation and axis-angle error between the target link pose(given by s_t) and the current link pose (given by q_t)* The function f_xy(q_t) computes the (x,y) coordinates of the COM, and 𝒫_t is the support polygon of s_t. The constraint thus enforces approximate static stability. The support polygon of s_t is computed by the convex hull of the horizontal projection of links in contact and does not include contacts removed in s_t+1* The function f_roll(q_t) computes the rotation around the X axis for the waist link, with respect to the global reference frame.This constraint is necessary as “zero roll” is an assumption of the subsequent dynamic stabilizationmethod (Section <ref>)* A_t, b_t enforce joint angle and velocity limits. We solve problem (<ref>) using the Sequential Quadratic Programming method of <cit.> as implemented in the trajopt library [URL: http://rll.berkeley.edu/trajopt].§.§ Interpolation and stabilizationTo obtain a densely-sampled trajectory for execution on the robot, we interpolate trajectory waypointsusing Hermite cubic splines with derivatives set to zero for smooth contact transitions. The time between two consecutive waypoints q_t is given by the oracle, as we describe in Section <ref>.Since the obtained trajectory is not dynamically stable, we then apply an FFT-basedZMP trajectory compensation scheme <cit.>. The method considers the rigid-body dynamics of the full body and locally adapts COM motion on the horizontal plane using analytic inverse kinematics to iteratively reduce the errorbetween the real and reference ZMP trajectory. We set the reference ZMP trajectory to the interpolated f_xy(q_t), which were used in theoptimization problem (<ref>) and are inside the support polygon at each waypoint. Furthermore, our implementation of the analytic inverse kinematics of the robot WABIAN-2assumes zero roll angle of the waist link with respect to the world reference frame.We include this constraint in the optimization problem (<ref>) for consistency.§ RESULTS§.§ Material segmentation results To train the CNN we first collected 7,791 annotated images from publicly available semantic-segmentation datasets:5,216 from the VOC2010 Context dataset <cit.> and2,575 from the OpenSurfaces dataset <cit.>. We selected all images in the datasets with at least one of the following labels:asphalt, concrete, road, grass, rock, sand, sky, snow, water, carpet, rug, mat, ceramic, tile, cloth, fabric, marble, metal, paper, tissue, cardboard, wood. Due to similarity between some classes at the image and semantic level we joined the labels(asphalt, concrete, road), (carpet, rug, mat), (ceramic, tile), (cloth, fabric) and (paper, tissue, cardboard).The total number of considered classes in the output CNN layer was 14. Sky was only included to avoid classifying it as any of the other materials on outdoor pictures.We used stochastic gradient descent with 0.1 learning rate and 0.9 momentum as in the original SegNet publication <cit.>, and trained the network on an Amazon Elastic Cloud node with a 4GB NVIDIA GPU. We ran a total of 90,000 iterations with a mini-batchsize of 5 (maximum allowed by the GPU). Training was done on 60% of the images, while the other 40% were used as the test set.We obtained a global classification accuracy of 0.7929 and class-average accuracy of 0.4776 on the test set. See Figure <ref> for examples of the (highest probability) material predictions given by the CNN on the test set. The global accuracy is comparable to state-of-the-art performance in semantic segmentation (e.g. <cit.>), and the class-average accuracy is slightly below state-of-the-art (which is around 0.60 <cit.>). We believe one important wayto improve classification accuracy is to improve the dataset itself since, for instance,there is moderate visual similarity between some of the materials such as marble and ceramic,and some materials are lowly sampled (e.g. the lowest sampled materials are snow and sand,present in 173 and 46 images respectively).The material segmentation results in Figure <ref> show an overall good accuracy of the CNN, particularly on wood, grass and sky labels. The figure also shows typical misclassificationssuch as white walls recognized as sky or metal (picture 6), hard snow as rock (picture 7), and some overlap between asphalt/road, ceramic and marble. These are arguably understandable since material labels themselves semantically overlap. However, our approach to the visual frictionestimation problem is such that if there is uncertainty in the material label, then this uncertaintycan be used to weight the friction of the surface through material and friction probabilitydistributions (Section <ref>).§.§ Friction prediction results We empirically measured the coefficient of friction associated with each material label using a force gauge and the robot foot loaded with a 1.5kg mass.The foot is rigid and its sole is covered with a high stiffness soft material for shock absorption and an anti-slippage sheet. We checked whether surfaces were horizontal with a level, then placed the foot and measured maximum friction force with the force gauge. See Figure <ref> for an illustration of the procedure. We took 5 friction measurements on each surface, and used at least 3 surfaces of each material. We fitted a normal distribution to the measurements, obtaining separate parameters μ_i and σ_i^2 for each material, where μ_i is the mean friction of material i,and σ_i^2 the variance. The materials sand, snow, water, cloth, paper were an exception, and since our robot is currently not capable of walking on them (i.e. fall or damage risk is too high) we directly set them toμ_i=0, σ_i^2=0. We similarly set sky's friction to zero as well. See Table <ref> for the parameters of the friction p.d.f. of each material.In Figure <ref> we show the test-set's highest probability material predictionsalong with the (1-η)-quantile of the coefficient of friction which is used in equation(<ref>). We set a typical value of η=0.95. The friction images are darker where friction is higher (μ=1 would be black).Note that ceramic-like surfaces have high predicted friction (pictures 1, 5, 6);beds and jackets have very low friction (pictures 3, 4, 14);grass patches have lower friction than roads (pictures 2, 9, 11, 12, 13);and that water is mostly white - zero friction - (pictures 10, 12).The figure also shows the advantage of using the whole probability distribution of materials (instead of using the highest probability material) to estimate friction.For example in picture 14, the jacket on the ground is classified as cloth and rock depending on the region,but friction is low on most of the object's area since the cloth label still has high probability.§.§ Planning results We prepared a mock-up scenario in the laboratory which demonstrates the capabilities of our planner. The scenario consists of a floor with two areas of different materials. One is made of wood(μ=0.84) and the other is a high-friction flooring resembling ceramic tiles both inappearance and coefficient of friction (μ=1.00).The perception-planning algorithms were run on this scenario, and then a piece of cloth (T-shirt) was laid flat on one of the surfaces to provoke changes in friction and force a different plan. See Figure <ref> for the scenario, segmentation and friction as seen from the robot's camera at the initial condition. Once again, the figures show the advantage of using the full probability distribution of materials given by the CNN. While cloth is the highest-ranking material only in part of the object region, friction is low on a larger region which is highly consistent with object borders.The robot starts in double-support, with one foot on each surface. The goal stance is one meter ahead,also with a foot on each surface. After the robot is placed at the initial state, the perception and planning algorithms run without any human input except the push of a button to execute the planned full-body trajectory open-loop. Trajectory optimization parameters are the collision penalty weight α of equation (<ref>), which is set to 50, and the distance at which the collision penalty starts being applied (for all links except those in contact), which we set to 2.5cm. The obtained full-body trajectory is tracked by position control at the joint level.For these experiments we used the human-sized humanoid robot WABIAN-2 <cit.> customized with aCarnegie Robotics Multisense SL sensor-head. The perception pipeline predicts pixel-wise material label distributions and pixel-wise friction using SegNet <cit.> and equation (<ref>), and combines them with the stereo depth maps computed onboard by the Multisense. It produces friction-annotated point clouds at 2Hz. For collision checking, the point cloud is converted into a mesh using the fast surface reconstruction algorithm of <cit.> as implemented in PCL <cit.>.All perception and planning computation ran on an external PC with network connection to the robot's onboard PC, and we used ROS <cit.> for communication.We show the results of the perception-planning experiments in Figure <ref>. From left to right, we show the material and friction point clouds, the footstep plan, the collision-checking bounding boxes used by the footstep planner and the final planned full-body trajectory after optimization and stabilization. In the first situation there are only wood and ceramic surfaces, but the predicted lower bound of friction of the wood surface is lower than that of the tiles (Q_1-0.95=0.1 vs 0.4). The footstep planner returns a sequence of stances that reduces the amount of times wood is stepped on. This behavior comes naturally from the extended footstep planning approach <cit.>,since walking on low friction ground requires higher stance times (slower motion) and thus more energy cost. Furthermore, note that the trajectory optimization uses all degrees-of-freedom to satisfy the constraints (e.g. trunk roll use is clear in the image sequence, important mainly for the stability constraints), and that the knees are relatively stretched in order to reduce torque consumption but still satisfy stability constraints. Also note that swing leg clearance happens automatically due to the use of collision costs.In the second situation we laid a flat piece of cloth on a ceramic spot used by the previous trajectory.The cloth was correctly classified and its friction was practically zero. The footstep planner returned a trajectory around the cloth and on the wood surface, which led to a slightly longer time and energy cost of the full-body trajectory (63 vs 60 seconds, 5% longer than on the first situation). Note that while the times are long they correspond to 25 stances because of step length limits, and thus the average time per stance is approximately 2.5 seconds.Footstep planning took approximately 20 seconds in the first situation and 10 in the second.The reason for the difference is clear from the scenario: while in the first situation stances onboth surfaces are expanded by A* in order to guarantee optimality, in the second situationno stances are expanded on the surface with cloth since friction zero has infinite cost. Full-body trajectory optimization took approximately 40 seconds and dynamic stabilization 2 seconds. Note that these are for 25-stance, 60 second trajectories, and therefore they should be considerably faster in case planning is done one or two steps at a time.§ CONCLUSIONS AND DISCUSSIONIn this paper we proposed a complete solution to the problem of biped robot locomotion on slippery terrain. We developed both a visual friction estimation algorithm and anobjective-consistent hierarchical planning method which considers trajectory costs,collision, stability and friction.We empirically showed that friction estimates in our algorithm are more consistent withobject/material borders than the highest-probability material label segmentation, whichshows a good integration of segmentation uncertainty into friction estimation. We also showed that the algorithms work for varied terrain and are applicable to planningon a real robot. The algorithms are relevant since not only obstacles but also differentterrain types abound in the real world, and locomotion choices should take them intoaccount - whether for safety or energetic considerations.Regarding the perception problem, we opted to decouple it into (broad sense) materialsegmentation and per-material friction distributions. Even though we obtained thematerial friction distributions manually, these could also be learned over time withlocomotion experience. Alternatively, friction could also be learned from images directly by end-to-end training, for example by initialization of a CNN with the parameters obtained with our architecture.For the context of this paper all surfaces were dry. Wet surfaces could also be included, although from our experience they should be treated as separate material labels(e.g. “dry metal” and “wet metal”) so that the distribution μ|l_i does not become bimodal. Thus, one important detail in this work is the notion of material, which should be takenin a broad sense, as a visually distinguishable terrain class.Importantly, one problem with the proposed perception and planning approach is thatwrong material classifications can lead to there being no solution to the footstepplanning problem. An example of such a situation is when a material the robot cannot walk on,such as water in our case, is mistakenly given very high confidence. Our view is that the solution could be semi-supervision where a teleoperator can correct asegmented region's material label. However, we believe that humans should not directlyannotate COF, since despite their relative ability to adapt gait to slippery ground humanshave difficulties in estimating coefficient of friction values <cit.>.Finally, full-body trajectories in this paper were interpolated after trajectory optimizationat waypoints. 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Behnke, Fast Range Image Segmentation and Smoothing Using Approximate Surface Reconstruction and Region Growing.1em plus 0.5em minus 0.4emBerlin, Heidelberg: Springer Berlin Heidelberg, 2013, pp. 61–73. PCL R. B. Rusu and S. Cousins, “3D is here: Point Cloud Library (PCL),” in 2011 IEEE International Conference on Robotics and Automation, Shanghai, China, May 9-13 2011. ROS M. Quigley, K. Conley, B. P. Gerkey, J. Faust, T. Foote, J. Leibs, R. Wheeler, and A. Y. Ng, “Ros: an open-source robot operating system,” in ICRA Workshop on Open Source Software, 2009. | http://arxiv.org/abs/1706.08685v1 | {
"authors": [
"Martim Brandao",
"Yukitoshi Minami Shiguematsu",
"Kenji Hashimoto",
"Atsuo Takanishi"
],
"categories": [
"cs.RO",
"cs.CV"
],
"primary_category": "cs.RO",
"published": "20170627063853",
"title": "Material Recognition CNNs and Hierarchical Planning for Biped Robot Locomotion on Slippery Terrain"
} |
State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China and Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, ChinaState Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China and Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, ChinaState Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China and Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, ChinaState Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China and Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China [email protected] State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China and Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China Department of Physics, Zhejiang University, Hangzhou 310027, China Institute for Quantum Science and Engineering and Department of Biological and Agricultural Engineering, Texas A&M University, College Station, Texas 77845, USA [email protected] State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China and Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China We measure complete and continuous Wigner functions of a two-level cesium atom in both a nearly pure state and highly mixed states. We apply the method [T. Tilma et al., Phys. Rev. Lett. 117, 180401 (2016)] of strictly constructing continuous Wigner functions for qubit or spin systems. We find that the Wigner function of all pure states of a qubit has negative regions and the negativity completely vanishes when the purity of an arbitrary mixed state is less than 2/3. We experimentally demonstrate these findings using a single cesium atom confined in an optical dipole trap, which undergoes a nearly pure dephasing process. Our method can be applied straightforwardly to multi-atom systems for measuring the Wigner function of their collective spin state.Measurement of complete and continuous Wigner functions for discrete atomic systems Tiancai Zhang December 30, 2023 =================================================================================== § INTRODUCTIONThe Wigner function (WF) <cit.>, originally introduced as a quantum analog of the classical phase-space distribution function, provides a powerful tool to represent quantum mechanics in phase space <cit.>. It is a quasiprobability distribution in that it acts like a probability distribution but can take negative values for some quantum states. The WF is originally designed for describing quantum systems with continuous degrees of freedom. It has been widely used, for example, in quantum optics to facilitate the visualization and tomographic reconstruction of quantum states <cit.>. While it has been successfully applied in continuous variable (CV) systems, the generalizations of the WF to quantum systems with a finite-dimensional Hilbert space have proved challenging. Many efforts have been made along this line, which in general can be divided into two approaches based on the dimension, finite <cit.> or infinite <cit.>, of the phase space, on which the WF is defined. Correspondingly, we refer to these two kinds as discrete and continunous WF, respectively. It remains an open question which approach is better. However, we note that a continuous WF for finite-dimensional systems seems more consistent with the original WF defined for CV systems. Unlike the gradual progress works <cit.>, which have their own restrictions either in the representation space or in the accuracy of representing the state, quite recently an elegant method <cit.> has been proposed for constructing complete and continuous WFs for spin or qubit systems. The method follows the displaced parity operator approach to defining the WF for CV systems <cit.>. The key is, therefore, to find appropriate analogous displacement and parity operators for spin systems. By means of the Bloch sphere representation of the state of a qubit, both the displacement and parity operators have been properly defined satisfying all the requirements of the Stratonovich-Weyl correspondence <cit.>, and hence a complete and continuous WF has been strictly constructed for any two-level systems.Continuous WFs have been measured for a collective spin state of an atomic ensemble <cit.>. In Ref. <cit.>, the WF is reconstructed using the inverse Radon transform implemented by a filtered back-projection algorithm <cit.>. The method employed there does not guarantee positivity of the reconstructed density matrix in the presence of experimental noise <cit.>, which may become a crucial problem for quantitative studies. While Ref. <cit.> adopts the method of Ref. <cit.>, with which a Wigner-like function is defined providing intuitively meaningful pictures, but it only works for systems of definite angular momentum (e.g., the totally symmetric subspace for an atomic ensemble and hence its phase space representation is not complete <cit.>), whereas Ref. <cit.> can handle arbitrary spin systems. Just recently, complete and continuous WFs have been measured for the first time for discrete systems of two Bell states and five-qubit Greenberger-Horne-Zeilinger state <cit.> based on IBM superconducting-qubit quantum processor <cit.>. Though convenient, using such a processor, the measurement of the WF suffers from various imperfections, such as indirect implementation of rotations and detection due to the limited operations that IBM has made available to the user, and considerable noises in the system resulting in imperfect operations and state preparation. Adopting the WF defined in Ref. <cit.>, in this paper we measure complete and continuous WFs of a well controlled truly single two-level cesium atom. Unlike experiments involving a large number of atoms for quantum metrology <cit.>, in which single-atom resolution is unavailable in both control and measurement, in our experiment a single cesium atom is controlled deterministically in a micro-sized dipole trap and undergoes a nearly pure dephasing process. We find that for an arbitrary pure state of a qubit its WF has always negative regions and the negativity vanishes if the purity of an arbitrary mixed state is less than 2/3. We experimentally demonstrate these findings using our system of trapped single atoms. To our knowledge, this is the first time that complete and continuous WFs have been measured for discrete atomic systems and that the evolution of the corresponding WF in a dephasing environment has been demonstrated. § THEORYAny state of a two-level quantum mechanical system can be represented by a point on/in the Bloch sphere. The surface of the Bloch sphere represents all the pure states, whereas the interior corresponds to all the mixed states. Any Hermitian 2× 2 matrix ρ with trρ = 1 can be expressed as ρ=1/2(𝕀 + r⃗·σ⃗ ) <cit.>, where 𝕀 is the identity matrix, r⃗=r e⃗ is the Bloch vector with magnitude r, 0≤ r ≤ 1, and unit vector e⃗=(sinθcosϕ, sinθsinϕ, cosθ), which specifies a point on the surface of the Bloch sphere. θ and ϕ are the polar and azimuthal angle, respectively, θ∈[0,π] and ϕ∈[0,2π). σ⃗ is the 3-element `vector' of Pauli matrices σ⃗=(σ_x, σ_y, σ_z). Thus, ρ can be rewritten asρ(θ,ϕ,r)=1/2[1+r cosθ e^-i ϕ r sinθ;e^i ϕ r sinθ1-r cosθ ].Eq. (<ref>) denotes that any density matrix ρ of a qubit can be characterized by the three parameters (θ, ϕ, r). The purity of the state is defined by ≡ trρ^2=1/2(1+r^2). For pure states with r=1 purity =1, while for mixed states with 0≤ r<1 purity 1/2≤<1. It is evident that the decreasing of r from 1 to 0 corresponds to a decoherence process with the off-diagonal entries of ρ decaying to zero. We wish to simulate the decoherence process as r decreases using our existing two-level atom system with the aim of observing the evolution of the corresponding WF defined in Ref. <cit.>. We notice that in general as r decreases all the entries of ρ vary corresponding to a complicated process that contains both dissipative and dephasing dynamics. However, for the special case of θ=π/2, as r reduces the diagonal entries of ρ are left unchanged, i.e., ρ_11=ρ_22=1/2, and only the off-diagonal entries decay, corresponding to a pure dephasing process. This process can be accurately simulated using our system of single cesium atoms confined in an optical dipole trap. We shall explain this in more detail in the next section. The continuous WF for such a two-level system is defined as <cit.>W_ρ(ξ,χ)= tr[ρΔ̂(ξ,χ)],with the operator Δ̂(ξ,χ) taking the form ofΔ̂(ξ,χ)=1/2[ Î - √(3) ( R̂ σ̂_z R̂^†) ],where Î is the identity operator, σ̂_z can be treated as the parity operator for a qubit, and R̂=e^-i ξ/2σ̂_z e^-i χ/2σ̂_x e^-i Ξ/2σ̂_z is the rotation operator that “displaces" a qubit state along the surface of the Bloch sphere. ξ, χ, and Ξ are the Euler angles and it is known that any target orientation can be realized by composing three elemental rotations, i.e., rotations about the axes of the Bloch sphere. Note that W_ρ(ξ,χ) is a function of only two Euler angles (ξ, χ) because Ξ makes no contribution as e^-i Ξ/2σ̂_z commutes with σ̂_z. Inserting Eq. (<ref>) into Eq. (<ref>), the WF for a generic qubit state ρ(θ,ϕ,r) is therefore obtainedW(ξ,χ;θ,ϕ,r)=1/2π^2{1-√(3) r[ cosθcosχ + sin(ξ-ϕ) sinθsinχ]}, where 1/π^2 is introduced to make the WF normalized over the phase space ξ∈ [0,π] and χ∈ [0, 2π). We note that both ξ and χ have a period of 2π, however, a space of half a period of ξ and a period of χ is enough to determine a WF that contains complete information of the state. It is straightforward to check that W(ξ,χ, r) is in all regions positive when r<1/√(3), or when purity <2/3, since the sum of the two trigonometric terms is bounded by ± 1. This is a general result for a qubit state of arbitrary values of θ and ϕ. Besides, for all pure states (r=1) the WFs always have negative regions and, interestingly, they possess the same minimum value W_ min = 1/2π^2(1-√(3)) ≈ -0.037. In CV systems, the negativity of the WF is typically considered as a nonclassical signature of the state <cit.>. However, in discrete systems things are more complicated because the negativity shows subtle complexities <cit.>. While for more general mixed states, the minimum is only related to r regardless of θ and ϕ, i.e., W_ min = 1/2π^2(1-√(3)r). There exists a critical value of r = 1/√(3) ≃ 0.577 (or of =2/3), below which the negative regions of the WF completely vanish. This is clearly shown in Fig. <ref>. § EXPERIMENTAL SETUP AND PROCEDURESTo make the WF Eq. (<ref>) more closely linked to the actual operations in an experiment, we rewrite it asW_ρ(ξ,χ)=1/2π^2[ 1- √(3)tr(ρ' σ̂_z ) ],where ρ'=R̂_x(-χ) R̂_z(-ξ)ρR̂_z^†(-ξ)R̂_x^†(-χ), and R̂_z(ξ)=e^-i ξ/2σ̂_z and R̂_x(χ)=e^-i χ/2σ̂_x correspond to the rotation about the z and x-axis of the Bloch sphere, respectively. Eq. (<ref>) denotes that the WF of ρ is connected to the expectation value of σ̂_z over the state ρ' that is achieved by performing two sequential rotation operations on ρ. To be more intuitive, we express Eq. (<ref>) in an equivalent formW_ρ(ξ,χ)=1/2π^2[ 1- √(3) (P_0 - P_1) ],where P_0=⟨ 0|ρ'|0 ⟩ and P_1=⟨ 1|ρ'|1 ⟩ are, respectively, the population probability of the two eigenstates |0⟩ and |1⟩. In our system, these two states are embodied by the “clock states" of a cesium atom, i.e., |0⟩≡ |6 S_1/2, F=3, m_F=0⟩ and |1⟩≡ |6 S_1/2, F=4, m_F=0⟩ <cit.>.The experimental setup is depicted in Fig. <ref>(a). Single cesium atoms are repeatedly captured with a blue-detuned “bottle" beam trap <cit.>, which is superposed with a precooled atomic ensemble prepared by a conventional magneto-optical trap (MOT) <cit.>. The “bottle" trap is formed by shining two parallel “donut" 780 nm laser beams with orthogonal polarizations through a group of high numerical aperture (NA) lens. By properly designing the size of the “bottle" trap, no more than one atom at a time could be loaded from the MOT into the trap <cit.>. The trapped atom is cooled to a temperature ∼ 10 μK by polarization gradient cooling. The scattering photons by trapped single atoms are collected and eventually fed to a single photon counting module (SPCM). A microwave is nearly resonant with the 9.2 GHz hyperfine transition of the two “clock states" and is applied to perform the corresponding operation on the qubit. The microwave generator is locked to a commercial Rb atomic clock to stabilize the frequency of the microwave.The sequence of the operations is shown in Fig. <ref>(b). A single trapped atom is initialized to state |1⟩ by optical pumping. Then a microwave pulse is used to prepare the atom into a superposition state |ψ⟩=cosθ/2 |0⟩+e^iϕsinθ/2 |1⟩. In order to verify the state that has been prepared, one needs to do state tomography of the atomic density matrix. This process is of nonnegligible time (about 1 ms) and will make the superposition state evolve into a slightly mixed state with purity close to unity. It has been shown that in such an optical dipole trap the atom suffers from a pure dephasing mechanism <cit.>. This fact has been verified by making state tomography at different decoherence time (see Appendix A). We have explained previously that the only situation corresponding to a pure dephasing process as r reduces is that the initial state should be prepared with θ≃π/2 <cit.>. After the stage of state preparation, the atom evolves through a dephasing channel for a time t, and then the stage of measurement of the WF starts. It is comprised of three sequential operations: two rotations and a detection (see Fig. <ref>(b)). Specifically, a series of microwave pulses are used to implement rotations about the z and x-axis of the Bloch sphere. Rotation of ξ about z-axis can be controlled by the fact that ξ=t Δ, with Δ the detuning of the microwave from the transition frequency of the two eigenstates. While rotation of χ about x-axis can be implemented by acting on a microwave pulse for a time t=χ/Ω_R, with Ω_R the Rabi flopping frequency associated with the two “clock states". Finally, we measure the population probabilities P_0 and P_1 of the states |0⟩ and |1⟩, respectively. To this end, we adopt the method of Ref. <cit.>, i.e., to push the atom in |1⟩ out of the dipole trap by sending another laser beam, whereas the atom in |0⟩ remains trapped. By checking if the atom still stays in the trap, one can discriminate in which state the atom is. After repeating the experiment many times, one then gets the population probabilities P_0 and P_1. Therefore, a value of the WF is achieved according to Eq. (<ref>) for specific rotations of ξ and χ. Repeating the experiments for different values of ξ and χ, a 3D WF W_ρ(ξ,χ) of the state ρ at time t could be measured for the whole phase space. Note that in practice the measured WF is the representation of the state at time t+t_m, with t_m the measurement time which is less than 1 ms (specifically depending on the rotation angle) and much shorter than the atomic coherence time ∼ 17.2 ms (see Appendix B).§ RESULTS AND DISCUSSION Figure <ref> presents the experimental WF for the state ρ_0 prepared at the initial time. The entries of ρ_0 are measured via state tomography and each entry is obtained by the statistic of about 300 rounds of the measurement: ρ_0^11=0.486 ± 0.020, ρ_0^22=0.514 ∓ 0.020, ρ_0^12,21= (-0.033 ± 0.020) ∓ (0.489 ± 0.004) i, corresponding to purity ≃0.981 and r≃0.981. The initial state ρ_0 is of θ ≃(0.509 ± 0.013)π, which is very close to the desired state of θ = π/2. The state ρ_0 (taking average values of its entries) has a unity fidelity with the state ρ(0.509π, 0.521π, 0.981). In Fig. <ref>, each dot with error bar is obtained by the statistic of about 300 times of the measurement and the curves are the theoretical WF of ρ(0.509π, 0.521π, 0.981) for a series of values of ξ. It shows that the experimentally measured values are in good agreement with the theoretical curves. The small difference between the experimental and theoretical WFs is the result of many factors, such as the difference of measurement time of the WF and state tomography (based on which we obtain ρ_0 and plot the theoretical curves), and the non-unity contrast of Rabi flops (about 90% as shown in the figure in Appendix B) which affects the fidelity of rotation operations and thus the accuracy of the measured WF.As the state evolves in the dephasing channel, the state becomes more and more mixed (with a decreasing r) and the phase ϕ will have an increasing fluctuation, leading to an increasing uncertainty of the WF in ξ. In Fig. <ref> (insets), we present experimental WFs of three mixed states at different evolution time. We have measured the WF for a period of ξ at χ=π/2 and then the minimum value will be of high possibility within the range ξ∈ [0, 2π). This is because the initial state ρ_0 of θ≃π/2 guarantees the minimum value be at (or very close to) χ=π/2. In each inset, the corresponding value of r is achieved by the ensemble average of more than 10 times state tomography (each of which yields a value of r) at the same time:r=0.820^+0.104_-0.137 at t=2 ms; r=0.662^+0.091_-0.153 at t=5 ms; and r=0.436^+0.099_-0.154 at t=6.3 ms. The fluctuation of r at the same time is due to the fluctuation of the phase embodied by the considerable differences of the off-diagonal entries at different times of tomography. The insets of Fig. <ref> show clearly that the width (reflecting fluctuation) of the Wigner “stripe" increases with the evolution time as a result of an increasing fluctuation in the phase. The mismatch of the WF at ξ =0 and 2π is due to the nonnegligible time (less than 1 ms) of the z rotation operation. As shown previously, the minimum of the WF is connected to r by W_ min=1/2π^2(1-√(3)r). Despite a considerable fluctuation of r, it is still possible to verify the formula with average values of r and W_ min achieved by many times of measurements. In the insets of Fig. <ref>, the averages of r=0.820, 0.662, 0.436 yield averages of W_ min=-0.021, -0.007, and 0.012, respectively, by the formula. While the averages of more than 10 times measured W_ min are -0.018, -0.006, and 0.014, respectively, which are in good agreement with the values evaluated by the formula. The fit line of W_ min(r) in Fig. <ref> demonstrates the “negative-to-positive" transition of the WF about r≃0.577, or purity ≃ 2/3, almost perfectly verifying the theoretical expectations of Fig. <ref>(b). We note that the measured W_ min at χ=π/2 is actually a bit higher than the “real" W_ min since the initial state ρ_0 is prepared not exactly at θ=π/2. This makes the fit line move upwards a bit, leading to the intersection with W_ min=0 a bit larger than r ≃ 0.577. § CONCLUSIONS We have measured complete and continuous WFs of a single two-level cesium atom in both a nearly pure state and highly mixed states following the method of Ref. <cit.>. We have shown how the WF evolves in a dephasing channel and demonstrated the “negative-to-positive" transition when the purity of the state is about 2/3. Our approach can in principle be applied to measure WFs of any two-level systems, either for a single qubit or for many qubits by implementing identical rotations on each qubit <cit.> still allowing obtaining a visible 3D WF at the price of losing partial information of the state. Furthermore, the demonstration of the WF evolving in a dephasing channel provides a more intuitive phase-space approach to studying fundamental processes in quantum discrete systems, such as the dynamics of decoherence. § ACKNOWLEDGMENT We would like to thank T. Tilma, H. Shen and T. Xia for fruitful discussions. This work has been supported by the National Key Research and Development Program of China (Grant No. 2017YFA0304502), the National Natural Science Foundation of China (Grants No. 11634008, No. 11674203, No. 11574187, and No. 61227902) and the Fund for Shanxi “1331 Project" Key Subjects Construction. § APPENDIX A: STATE TOMOGRAPHY FOR VERIFYING THE NEARLY PURE DEPHASING PROCESSIn our system, a single cesium atom is confined in an optical dipole trap, which undergoes a nearly pure dephasing process <cit.>. In what follows, we further verify this fact by making state tomography at different time in this process. This is necessary since it provides a way for estimating the value of r which is a key parameter in our model and it is also helpful to understand the physics of this process. The pure dephasing nature is characterized by the unchanged diagonal entries and the decaying off-diagonal ones of the density matrix as the state evolves. In Table I, we present density matrices measured at different time in the decoherence channel. We see that in this process the diagonal entries are almost unchanged, about 0.5, with consideration of measurement errors, while the off-diagonal ones may vary significantly and decay with the time. This is a clear signature of (nearly) pure dephasing in such a decoherence process.§ APPENDIX B: ESTIMATION OF THE COHERENT TIME OF THE SINGLE QUBIT BY RAMSEY INTERFERENCEHere we briefly discuss the details of the approach to estimating the coherent time of the qubit in our experiment. The qubit is encoded in the “clock states" of a cesium atom, i.e., |0⟩≡ |6 S_1/2, F=3, m_F=0⟩ and |1⟩≡ |6 S_1/2, F=4, m_F=0⟩. Firstly, the qubit is initialized to state |1⟩ and then a resonant microwave pulse at frequency 9.2 GHz is applied to drive the Rabi flopping. By using single atom Ramsey interferometry <cit.>, the coherent time T^* can be precisely measured. A π/2 pulse is used to prepare the atom into the superposition state (|0⟩+|1⟩)/√(2). After a time t, during which the state evolves freely in the far-off resonance trap <cit.>, a second π/2 pulse is applied and then the state detection is performed. Fig. <ref> shows the Ramsey interference signal of the atom versus the time interval t. The amplitude damping follows an exponential decay and the exponential fitting gives a 1/e decay time of T^* ∼ 17.2 ± 1.9 ms, that is the coherence time of the superposition state embodied in the atom. 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A 72, 023406 (2005).FrontPhys Z. H. Wang, G. Li, Y. L. Tian, and T. C. Zhang, Front. Phys. 9, 634 (2014).note Note that no matter the atom is prepared in the state of any value of θ, the atom suffers only pure dephasing, as shown in Refs. <cit.>. However, only for the special case of θ=π/2, we can model such a process with a simple reduction of r.Kuhr2 S. Kuhr et al., Phys. Rev. Lett. 91, 213002 (2003).Ovchinnikov R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov, Adv. At. Mol. Opt. Phys. 42, 42 (1999).Tuchendler C. Tuchendler et al., Phys. Rev. A 78, 033425 (2008). | http://arxiv.org/abs/1706.08676v4 | {
"authors": [
"Yali Tian",
"Zhihui Wang",
"Pengfei Zhang",
"Gang Li",
"Jie Li",
"Tiancai Zhang"
],
"categories": [
"quant-ph",
"physics.atom-ph",
"physics.optics"
],
"primary_category": "quant-ph",
"published": "20170627053440",
"title": "Measurement of complete and continuous Wigner functions for discrete atomic systems"
} |
^1Raman Research Institute, Bangalore, India; [email protected]; [email protected] We propose a new method to detect off-pulse (unpulsed and/or continuous) emission from pulsars, using the intensity modulations associated with interstellar scintillation. Our technique involves obtaining the dynamic spectra, separately for on-pulse window and off-pulse region, with time and frequency resolutions to properly sample the intensity variations due todiffractivescintillation, and then estimating their mutual correlation as a measure of off-pulse emission, if any. We describe and illustrate the essential details of this technique with the help ofsimulations, as well as real data. We also discuss advantages of this method over earlier approaches to detect off-pulse emission. In particular, we point out how certain non-idealities inherent to measurement set-ups could potentially affect estimations in earlier approaches,and argue that the present technique is immune to such non-idealities. We verify both of the above situations with relevant simulations.We apply this method to observation of PSR B0329+54 at frequencies 730 and 810 MHz, made with the Green Bank Telescope and present upper limits for the off-pulse intensity at the two frequencies. We expect this technique to pave way for extensive investigations of off-pulse emission with the help of even existing dynamic spectral data on pulsars and of course with more sensitive long-duration data from new observations. § INTRODUCTIONIt is the pulsed nature of the emission (as against continuous emission) that made the discovery of pulsars (Hewish et al. 1968) possible. Their average intensities, if were to manifest as continuous emission, are in most cases too weak to be detectable, in presence of possible confusion from other continuous sources. The pulsed emission has been studied in great detail, and has lead to our present understanding of the physical picture of pulsars. However, the question as to whether pulsar radiation indeed has any intrinsic continuous component, in addition to its distinguishing pulsed signature, or if the periodic emission extends well beyond the main/inter-pulse windows, have been issues of much interest since the early days of pulsar studies.There have been several attemptsto detect off-pulse emission from pulsars (as summarized in Table 3 and discussed in Section 5of Basu et al. 2011). Most attempts were primarily aimed at detection of unpulsed emission component of magnetospheric origin (for example, Hugunein et al. 1971; Bartel et al. 1984; Perry & Lyne 1985;Hankins et al. 1993; Basu et al. 2011,2012), which is indeed the focus ofthis paper. In contrast, somewere prompted by, and were aimed to test, the proposition of Blandford et al. (1973) - “existence of ghost supernova remnants around old pulsars". Detection of unpulsed emission of magnetospheric origin is indeed challenging, when based on apparent intensity in the off-pulse region, particularly in presence of a variety of unresolved astronomical sourcesand the resulting confusion. Such contaminants could include pulsar companions, if any, nearby galactic/extragalactic radio sources, and diffuse background emission, in addition to the following sources associated with the pulsar. They may include (e.g. as discussed by Hankins et al. 1993) weak halos (Blandford et al. 1973), remnants of the progenitor supernova, shock structures or synchrotron nebulae, and detectable bow shock. All these contaminations are unavoidable because of finite beam-width of single-dish telescopes and non-negligible side-lobes of interferometers. After several non-detections and some reports of detections that were refuted subsequently, the off-pulse emission has attracted renewed attention with Basu et al. (2011, 2012) reporting detection of off-pulse emission from B0525+21 and B2045-16based on their GMRT observations. It is worth noting, that in their study of 20 pulsars, including B0329+54, B0525+21 and B2045-16, at 2.7 and 8.1 GHz using the NRAO 3-element interferometer, Huguenin et al. (1971) found no significant unpulsed emission, implying an upper limit of 20 mJy within 10 arcsec of the pulsar directions. Much later, Bartel et al. (1984) made observations of pulsars B0329+54 and B1133+16 at 2.3 GHz using Mark III VLBI, and also ruled out continuous emission above their detection limit (2.5 mJy). Soon after, Perry & Lyne (1985), reported their interferometric observations at 408 MHz, on 25 pulsar including B0329+54 and B0525+21, made using 76m MK 1A telescope at Jodrell Bank and the 25m telescope at Defford, with baseline of 127 km. They claimed detection of unpulsed emission from 4 pulsars B1541+09, B1929+10, B1604-00 and B2016+28. However, later it became clear that B1541+09 and B1929+10 are aligned rotators (Hankins et al. 1993; Rathnasree & Rankin 1995), and the unpulsed emission fromB1604-00 and B2016+28 were shown to be from unrelated background sources (Strom & Van Someren Greve 1990; Hankins et al. 1993). The recently reported detections of off-pulse emission (Basu et al. 2011; 2012) from two long period pulsars B0525+21 (3.75 s) and B2045-16 (1.96 s) are based on the imaging mode of GMRT, and also at two frequencies (325 and 610 MHz). Although the authors have discussed some effects that could potentially contaminate off-pulse region with leakage from the emission that is otherwise confined to the main-pulse window, and have attempted some tests based on which they claim absence of such leakage. We consider these tests inadequate to rule out “leakage", since there are a few different aspects, associated with commonly employed receiver setups, that have noticeable potential for undesirably spilling the main pulse contribution across off-pulse region.Ideally, we need a method that is immune to such contamination, as far as possible,while making reliable estimation of possible off-pulse or unpulsed emission intrinsic to pulsar.Owing to their compact size and pulsed emission, pulsars have been an excellent probe of the ISM since their discovery. Primarily, they have revealed the distribution of free electron in the Galaxy, through direct measures of column density (from the observed dispersion) and spatial distribution of electron density irregularities (from scintillations and angular/temporal broadening as a result of scattering). The highly polarized nature of their radiation also allows Faraday Rotation measurements, sampling the magneto-ionic component of the intervening medium, and their pulsed nature facilitates some of the clearest measurements of HI absorption along their sight-lines. Of these, the diffractive scintillation effects are readily observable in pulsar directions thanks to their tiny angular sizes, and become apparent onlyin the cases of some extra-galactic sources having the required compact angular size, such as in early phases of γ-ray burst (GRB) afterglow sources(see for example, Frail et al. 1997; Macquart & de Bruyn 2006).The diffraction induced chromatic modulation of intensity, when combined with relative motions, translates to intensity variations across time and frequency. Similarly it is only the pulsed nature of the radio pulsars which makes the dispersion effect measurable, and also to reveal the temporal broadening due to scattering. However, the latter can be probed indirectly via other manifestation of scattering (such as decorrelation scales in frequency and/or angular broadening), even in the case of continuous sources. The camaraderie between the pulsars and the interstellar medium is indeed reciprocal. For example, ultra-high angular resolution probe of pulsar emission is made possible by the ISM acting as a lens. This was first pointed out by Lovelace (1970) and has been followed-up by many (e.g. Cordes & Wolszczan 1988; Pen et al. 2014; and references therein). Here, the diffractive/refractive effects due to large scale irregularities are considered as providing interstellar interferometric measurements capable of resolving even magnetospheric emission regions of pulsars. The refractive effects leading to multiple imaging manifest themselves as fine-scale corrugations or drift patterns within scintles in the dynamic spectra resulting from diffractive scintillations (e.g. Wolszczan and Cordes 1987, Gupta et al. 1994,1999).In this paper, we present a technique which advantageously uses such interstellar-scale telescope for search and detection of unpulsed emission, if any, from pulsars. Our technique (described in Section 2) is based on diffractiveinterstellar scintillation (DISS)and its correlated imprint on the pulseintensity and any off-pulse emission intrinsic to the pulsar, and has the potential for providing more reliable measurement of intrinsicoff-pulse/unpulsed emission, without needing conventional interferometric measurements, i.e. which are possible even with single-dish observations.In Section 3, we demonstrate sensitivity of our technique usingsimulated dynamic spectra over wide band, and assess its immunity to various known sources of contamination in the off-pulse region.Discussion of one such potential contaminant is given in Appendix A.The details of the DISS simulation are presented in the Appendix B. In Section 4,we illustrate application of our technique to real data, using observations on B0329+54 at two radio frequencies.We summarize the main conclusions of our paper in Section 5.§ SCINTILLATION-BASED TECHNIQUE FOR SEARCH/DETECTION OF UNPULSED EMISSION FROM PULSARS In this section, we present a new technique based on diffractive interstellar scintillation (DISS) and assessment of correlation between dynamic spectra for the pulse and off-pulse intensities. It effectively renders measurements with fine angular resolution offered by interstellar diffraction to distinguish pulsar emission region from sources of confusion, even in close proximity to the pulsar.This DISS correlation criterion effectively and readily discriminates against all discrete anddiffuse radio emissions on angular scales larger than that of the pulsar magnetosphere, since they will be devoid of DISS imprint in their dynamic spectra,let alone show any correlation with pulse intensity variations. Any confusing compact source, unresolved by the observing telescopes, and compact enough to show DISS, will show a dynamic spectral signature, i.e., the scintillation pattern, significantly different from that associated with the pulsar emission. In fact, differences between the scintillation patterns associated with even the different components within the pulse profile have been probed to assess spatial separation, if any, between the apparent sites of emission (Cordes et al. 1983). If indeed, a pulsar has a component of intrinsic emission that is unpulsed/continuous, we expect its intensity modulation due to interstellar scintillation to be closely related to, if not matching, that of the pulsed component.For the desired correlation to exist between the diffractive scintillation spectra of intensities in the two longitude regions, the spatial transverse separation between the associated emission regions should ideally be well within the equivalent spatial resolution of the interstellar aperture/interferometer at work. As Cordes et al. (1983) have already noted, the spatial scale S_d of the diffraction pattern in the observer plane also (reciprocally) defines the associated spatial resolution at the source distance. A suitable data set for implementation of our technique is, in general, an appropriately sampled data cube of intensity I(ν,t,ϕ), as a function of rotational longitude ϕ, radio frequency ν and time t, and over wide frequency and time spans of, say, Δν_BWand Δ t_obs, respectively.The two dynamic spectra, I_on(ν,t) and I_off(ν,t), to be tested for mutual correlation, are to be constructed for the apparent average intensity across (i) an appropriate number of bins spanning or within the pulse window, and (ii) a chosen set of bins or longitude range in the off-pulse region which is well-separated from the pulse window. All of the (dynamic) spectra here are assumed to be alreadycorrected for any non-uniformity in spectral response of the observingsystem within the observed band.[An estimate of the requirednormalized spectralgain response (G(ν)), to be used for dividing all the observed spectra,can be made by averaging the observed off-pulse spectra over the entire time span of observation to first obtain a mean uncalibrated spectrum<S_off>(ν), and then normalizing it with band-averaged intensity S̅_off,such that G(ν) = <S_off>(ν)/S̅_off.]Sensitivity in the estimation of correlation depends on the signal-to-noise ratio in estimation of the two dynamic spectra, and the degrees of freedom provided by the richness in the dynamic spectra, quantifiable to the first order in terms of number of scintles. Naturally, scintillation dynamic spectra obtained from longer duration observations with wide spectral coverage are desired, if not essential. The dynamic spectral resolutions in time and frequency, say, δ t_res and δν_res, respectively,need to be adequately finer than the respective decorrelation scales (t_s and ν_d), which together characterize the average size of scintles. The dynamic spectra, therefore, are to be smoothed optimally to reduce the uncertainty in estimation of the intensity variations due to scintillation, without washing out details in the ISM induced diffractive variation of interest.In practice, the dynamic spectra are not free of (additive) random noise in estimating intensity at each pixel in the time-frequency plane, but the magnitude of this noise is expected to be largely consistent with the system temperature and the integration employed (quantified by the relevant time-bandwidth product). Thus, in general, I_on(ν,t) = I^p_on(ν,t) + U_on(ν,t), and I_off(ν,t) = I^p_off(ν,t) + U_off(ν,t), where the U_on and U_off represent random noise (with zero mean and standard deviations σ_on and σ_off respectively), which is uncorrelated from pixel to pixel, and contaminates the respective underlying pulsar dynamic spectra I^p_on and I^p_off.These delta-correlated noise contributions, of magnitude σ^2_on and σ^2_off, will be clearly noticeable as such at zero-lag in the respective auto-correlation functions, ACF_on and ACF_off, of the dynamic spectra, on top of the the otherwise smoothly varying auto-correlations of I^p_on and I^p_off, respectively. Hence, the zero-lag auto-correlation of the underlying intensity variation is estimated routinely by interpolation from correlations at adjacent lags.The average cross-correlation between the intensity variations in two dynamic spectra, I_on(ν,t) and I_off(ν,t), defined asCC(0,0) = <δ I_on(ν,t) δ I_off(ν,t)>at zero lags, is to be assessed for significance against uncertainties, where δ I_yy(ν,t) = I_yy(ν,t)-<I_yy> for yy state (on or off), and <x> indicates ensemble average of x across the span of (ν,t). The uncertainty in the estimated correlation, in the best case (i.e. dynamic spectra free of noise and other undue contaminants), will be dominated finally by the finiteness of available scintle statistics. In case of detection of significant correlation, the off-pulse emission intensity as fraction η of on-pulse intensity can be estimated asη = <δ I_on(ν,t) δ I_off(ν,t)>/<(δ I_on(ν,t))^2>= CC(0,0)/AC_on(0,0) - σ^2_onwhere AC_on(0,0) is the average zero-lag auto-correlation of (on-)pulse intensity variations, which includes the variance σ^2_on of the delta-correlated noise U_on(ν,t).In the discussion so far, the apparent intensity fluctuations across the dynamic spectrum for the on-pulse region are, in an ideal case, assumed to be primarily a manifestation of the interstellar scintillations across the observing band. However, a finite but small part of these may be due to 1) variations in the system noise, including the sky noise (other than that from the pulsar), in addition to 2) the contribution from aliased spectral range, if any. The former additive contributions equally affect the dynamic spectrum for the off-pulse region, and may undesirably contribute to the apparent correlation between the dynamic spectra. It is therefore important that the on-pulse dynamic spectrum I_on(ν,t) is obtained after subtraction of I_off (ν,t) from the corresponding spectrum for the on-pulse region. The version of I_off(ν,t), to be used for subtraction here, should be for intensity averaged over the entire off-pulse region, as far as possible. In case of any genuine unpulsed intensity with correlated variations with those for on-pulse region, the suggested subtraction would result in an under-estimation of η by an amount η^2. On the other hand, even intrinsic variations in the off-pulse region that are uncorrelated between the two dynamic spectra would be unduly subtracted from the on-pulse dynamic spectrum, and would introduce a negative bias in η estimate. The magnitude of such bias is given by the ratio of variance σ^2_U of these uncorrelated variations in the off-pulse spectrum to that for variations in the on-pulse spectrum (i.e. σ^2_U/(AC_on(0,0) - σ^2_on)). In any case, the negative bias will be limited to η^2_max, where η_max is as defined later in the Equation 6. The advantage of thus removing any common unpulsed intensity variations, either due to sky or system, from I_on(ν,t), in terms of obtaining a more reliable estimate of η, overwhelms the undesirability of the the mentioned bias, which is expected to be insignificant any way.Of course, any intrinsic variability in the pulsar intensity wouldleave an unavoidable (multiplicative) imprint in the dynamic spectrum.The spectral scales of intrinsic variability are expected to be much wider than those associated withinterstellar scintillation. There is no a priory basis yet for expecting the possible unpulsed component, if any,to have correlated intrinsic variability. Hence, in general, any independent intrinsic variability of intensities in the two regions would reduce the net cross-correlation, and in any case, increase the uncertainty in the estimation of the unpulsed intensity. Fortunately, any pulse-to-pulse variations in intrinsic intensity areexpected to average out, with suitable temporal smoothingof the dynamic spectrum (P≪Δ t_res< t_s).Any residual variation, on time scales shorter than Δ t_res, would be indistinguishable from the random uncertainty in estimation of the dynamic spectral elements. The combined magnitude of these fluctuations would be readily apparent in the auto-correlation function across the first few time-lags, as the delta-correlated contribution. In comparison, the auto-correlation due to scintillation-induced intensity variations is expected to decorrelate on a relatively longer time-scales (t_s).The expected implicit linear inter-relationship between the patterns (after removing the respective mean values),assessed through formal cross-correlation, can be modeled explicitly as followsδ I_off(ν,t) = η δ I_on(ν,t) + U(ν,t)where, the first term on the right-side is the best-fit model, and U(ν,t) isthe apparent deviation orthe part of observed off-pulse dynamic spectrum that is uncorrelated in time and frequency with δ I_on(ν,t), with its nominal mean <U(ν,t)>=0, and other quantities as defined earlier. The uncorrelated part U(ν,t) includes also any measurement uncertainties in I_off and also the model ηI_on. In the above formulation, as in the Equation 2, η is a measure of the ratio δ I_off/δ I_on.The uncertainty σ_η in its estimate can be expressed asσ_η=σ_<U>/√(<(δ I_on(ν,t))^2>)where σ_<U> is the reduced uncertainty in the mean of U(ν,t), and is related to standard deviation in U(ν,t) asσ_<U> = σ_U √(1/N_eff)where N_eff is the effective size of the ensemble. The <U> and σ_U are, in practice, computed using all of the N samples available in the dynamic spectral array, including U(ν,t). The total number of points N in these arrays is equal to N_ν0N_t0, where N_ν 0 is the number of spectral channels andN_t0 number of time bins/sections in the dynamic spectrum. However, since all the points/pixels in the dynamic spectrum are not independent, particularly when the random measurement noise is much smaller than the intensity variations due to scintillation. Hence, in such cases, N_eff is often much smaller than N, and represents rather the number of independent samples in the dynamic spectrum.We have used the number of scintles as defining N_eff, so that our uncertainty estimate σ_η corresponds to worst-case error. The definition of number of scintles, as given in Cordes & Lazio (1991), is N_eff=N_t× N_ν, where N_t=1+κ (Δ t_obs/t_s), N_ν=1+κ (Δν_BW/ν_d), where κ is an empirically obtained number (we can call it filling factor), lies in range 0.1-0.5. If N_eff for one spectrum is different from that for the other, we use the geometric mean of the two N_eff values.For dynamic spectra spanning long durations, explicit attention would be needed to examine if they are affected by possible slow variations in pulse intensity within the span, due to intrinsic variations and/or originating from extrinsic reasons, including refractive scintillations and any instrumental gain variations that remain to be corrected. The correlation scales across frequency for these are expected to be generally wide. Hence, any contamination in the off-pulse region, as mentioned above, is likely to be modulated the same way, resulting in spurious correlation corrupting the correlation of interest. It may become necessary therefore to either estimate slow modulation, and correct at least the on-pulse dynamic spectra accordingly, or estimating the correlation or η using dynamic spectra of shorter spans at a time, repeating the analysis for each of such sections separately, and then computing a weighted average of η, combining independent estimates made using subsets of data. Before proceeding further, we wish to draw attention to a particular ready utility of the dynamic spectra of the apparent intensity variations in the on-pulse and off-pulse regions. We argue that, regardless of the details of contamination, and the presence or the lack of correlation between the two dynamic spectra, it is possible to define a hard upper-limit for the unpulsed intensity, asη_max = √(<δ I_off(ν,t)^2>/<(δ I_on(ν,t))^2>)= √(AC_off(0,0) - σ^2_off/AC_on(0,0) - σ^2_on)where AC_off(0,0) is the average zero-lag auto-correlation of observed intensity variations in the off-pulse region, which includes the variance σ^2_off of the delta-correlated noise U_off(ν,t).When AC_off(0,0) ≫σ^2_off, fractional uncertainty (1σ) in η_max would be 1/√(2N_eff).However, even when δ I_off(ν,t) appears to consist ofonly delta-correlated noise, i.e. η_max≈ 0, the uncertainly would at best be limited to σ_off/√(N<(δ I_on(ν,t))^2>). §.§ Implications of relative location of possible off-pulse emission region In general, the apparent emission in the off-pulse region would be a combination of the intrinsic and confusing sources of continuous emission, and the discussed correlation would be correspondingly partial, but providing a measure of the intrinsic component (spatially confined within the transverse scale S_d).Given the form of spatial distribution of electron density irregularities in the ISMas detailed in Appendix B, this spatial resolution scale S_d,same as the diffraction pattern scale, is given by the following relation (Armstrong et al. 1995). S_d=[8π r_e^2λ^2C_N^2zf(α)/(α+1)]^-1/αwhere, r_e is the classical radius of electron, λ is the radio wavelength, α = β - 2, andz is the effective propagation distance throughthe ISM.[ For uniformly distributed scattering, z would correspond to the distance to the pulsar. For Kolmogorov turbulence, β=11/3 (α=5/3) and the numerical value of the function f(α) is ≈1.12.]The ISM parameters in the above equation are not directly measurable, although can beestimated.[The diffractive scintillation time-scale t_s (decorrelation time) is directly related to S_d, where t_s = S_d/V, but the equivalent velocity V of the medium relative to the pulsar sight-line is not independently known in most cases. On the other hand, the associated angular scatter broadening θ, which ultimately limits the resolution in imaging observations, also relates to the above spatial scale, as θ = λ/2π S_d.] A related and more readily measurable quantity is the scintillation decorrelation bandwidth ν_d, or alternatively the temporal scatter broadening τ of the pulse, where τ = zθ^2/2c ≈ 1/2πν_d, and c is the speed of light. Thus, S_d can be estimated from ν_d measurement, as[The form of this expression is consistent with that in Equation 13 of Cordes et al. (1983).] S_d = √(zλ^2 ν_d/4cπ) = r_f√(ν_d/2ν)where r_f(=√(zλ/2π)) is the Fresnel scale, and ν is the observing radio frequency. A positive result in our proposed correlation testwould not only conclusively confirm the claimed detections, but would constrain the apparent size and spatial separation, at the so-called “retarded emission time" (Cordes et al. 1983), between the corresponding emission regions, and the level of correlation would provide clues on the relative spatial separation. A negative result, on the contrary, would not necessarily imply absence of unpulsed emission, unless the resolution scale S_d is large enough to cover the entire spatial extent within which emission can be considered as intrinsic to the pulsar. Considering the maximum separation between relevant emission regions to be the so-called light-cylinder radius r_L (=cP/2π), the above requirement implies that r_L≲ S_d, where P is the pulsar period[ For the spin periods in the range 1.4 ms-11.8 s, the range of light-cylinder radius corresponds to ∼10^4-10^8m.]. This condition can also be expressed ascP/2π≲ r_f√(ν_d/2ν), assessment of which would require an estimate of z, in addition to that of the decorrelation bandwidth ν_d. Although independent distance measurement is desired, z estimated from dispersion measure would also render useful for the present purpose. It is worth emphasizing that the above stated condition is not model dependent, i.e. independent of β.For a given pulsar, i.e. given P, z and C_N^2, the observing frequency ν can be chosen suitably, to see if the above condition can be met. The condition is more likely to be satisfied in cases of higher frequency probe of scintillation patterns for relatively nearby short-period pulsars.[ The underlying basic dependencies, as in the Equation 7, imply that the spatial resolution scale S_d broadens with increasing radio frequency (ν) and with decreasing integrated scattering measure C_N^2z. This diffraction pattern scale, in the weak scattering regime at adequately high frequencies, would of course saturate to its upper limit, that is the Fresnel scale r_f, equal then to the refractive scale at its lower limit. ]In any case, if any intrinsic unpulsed emission were to originate within the angular scale S_d/z around the pulsar, we expect to find the expected correlation signature. Although such cross-correlation (at zero-lag) is expected to fall significantly and rapidly, as exp[ - (Δ S/S_d)^(β -2) ] (Cordes et al. 1983), with increasing separation Δ S. However, if the separation, even if large (i.e. many times S_d), happens luckily to be near parallel (within angle (S_d/Δ S)^c, for S_d≤Δ S), then again significant cross-correlation would be expected, but now at time-lag Δ t ∼ t_sΔ S/S_d), if the scattering transfer function can be considered as essentially frozen over those time scales. The above considerations necessitate exploration of the discussedcorrelation over a range of lags in both time and frequency, as we do in our tests and analysis to follow.§ TESTS WITH SIMULATED DYNAMIC SPECTRA: ASSESSMENT OF SENSITIVITY AND IMMUNITY Here, we illustrate application of our technique to simulated scintillation dynamic spectra, and assess its performance, in terms of ability to reliably estimate off-pulse/unpulsed intensityintrinsic to the pulsar,and immunity to potential contaminants in the off-pulse region. As mentioned in Section 2, and illustrated in Appendix A, one of the subtle contamination of the off-pulse region could come from genuine main-pulse signal itself, if it is not adequately filtered out from the spectral regions beyond the observing band.These aliased contributions (from possible image bands relevant to heterodyning, and regions inadequately attenuated by band-defining filter before digitization) appear at longitudes that are, in general, offset from the main-pulse window (see Figure A1),depending on dispersion measure and frequency separation.Fortunately, the scintillation-induced intensity pattern would significantly differ for spectral separations larger than the decorrelation scales, ν_d, particularly when ν_d≪Δν_BW, and even the overall shape and sizes of the scintles (characterized by the decorrelation scales, ν_d and t_s), themselves vary systematically with ν, more rapidly with decreasing frequency. Any aliased contribution from other bands will have their own different scintillation-induced imprint, and hence, is not expected to contribute to any significant net correlation. This forms the basis of our expectation for potential immunity of our scintillation correlation method against aliasing-induced contribution which disguises as off-pulse emission, and we assess it by using simulated dynamic spectrum over a spectral range several (7) times the nominal bandwidth of observation. A detailed description of our simulation of diffractive scintillation is presented in Appendix B, and resultant dynamic spectrum spans 115.5 MHz (7x16.5 MHz) centered at 270 MHz, and covers a duration of about 1000 t_s seconds. The time and spectral sampling here is ∼ t_s/4 seconds and 64.45 kHz (∼ν_d /20), respectively.This simulated dynamic spectrum, is treated as directly corresponding to an on-pulse intensity pattern. A small section (∼ 180 t_s; or 3 hr, if t_s = 60 s) of this pattern is shown in Figure 1, sampled across 1792 spectral channels and 700 time bins (out of the simulated duration spanning 4000 time bins).The central spectral region, of 16.5 MHz width, is treated as the observing band, and the associated scintillation pattern is assumed to directly simulate an observed on-pulse dynamic spectrum. As an example, Figure 2 presents a zoomed portion over a short duration (∼ 25 t_s, or say, 25 minutes),where the scintle scales in both the dimensions are clearly discernible.Dynamic spectrum corresponding to the off-pulse region, on the other hand, is constructed by appropriately superposing the intensity variation simulated across the seven bands, following different assumed levels of genuine unpulsed (off-pulse) emission, and those of aliasing from contaminating bands, if any.For completeness, enabling a range of assessments, we consider the following three kinds of off-pulse dynamic spectra, as having (a) only genuine unpulsed emission, (b) genuine unpulsed emission plus contamination from aliasing, and (c) no genuine unpulsed component, but with only aliased contributions.In (a) and (b), the dynamic spectral contribution as due to a genuine unpulsed emission is readily obtained from the on-pulse dynamic spectrum, suitably scaled by an assumed factor η. Unless mentioned otherwise, η is assumed to be 0.01. We assume, for simplicity, that the aliased bands, contributing in (b)and (c), are attenuated by also the same factor (i.e. 0.01 or -20 dB), The off-pulse dynamic spectra, across the same span (16.5 MHz, with 256 channels), wherein any aliased contribution from other bands is added together, with or without band-flips, as appropriate. An off-pulse dynamic spectrum thus simulated for the case (c), only with aliased contribution from only two bands adjacent to the observed band (i.e. immediate upper and lower band) on either side, is shown in Figure 3 (depicting a similarly zoomed section as in Figure 2).The Figure 4 presents the auto-correlations, and the cross-correlation maps computed using the the respective dynamic spectra shown in Figure 2 and Figure 3.The results in Table 1 are for the following two distinct cases of simulated off-pulse dynamic spectrum; namely, for η_true = 0.01 and0. In each case, the aliasing-induced contamination from adjacent spectral bands is explicitly included (with chosen attenuation), for aliasing-order (AO) ranging for 1 to 3, making a total of six versions of simulated off-pulse spectra. These are separately used along with a common on-pulse dynamic spectrum to estimate η in each case. For example, an aliasing-order “k" corresponds to a spectral span of k.Δν_BW on either side of the observing band as being the source of contamination.The estimates of uncertainty in η depend of the N_eff, which is computed following the same procedure as described in Section 2. In each case, having dynamic spectra that are same for the on-pulse, but differently constructed for the off-pulse region, the N_eff is computed based on the decorrelation scales seen in the latter (i.e. off-pulse) spectra. The decorrelation scales as seen to effectively broaden with the number of independent spectral patterns contributing to the constructed off-pulse dynamic spectra, as would be expected. The simulated intensities in the dynamic spectra are essentially exponentially distributed. These distributions would approach to Gaussian, when additive measurement uncertainties are significant. The overall positive bias in the estimates of η computed from the entire span is understood as due to the slow modulation of pulse intensity (owing to refractive scintillations) which is shared by the contaminants of the off-pulse dynamic spectrum. When the suggestion made in an earlier section is followed, i.e. η is estimated separately for each of the shorter spans, and such estimates combined appropriately, the average η estimate is largely free of such bias, without loss of sensitivity. This can be appreciated from the comparison of the results presented in Table 1.[ The 6 models correspond to 2 sets, with and without genuineunpulsed emission component,in each of the three aliasing-orders. Two estimates of η are presented for each model; A: using the entire span together, and B: using eight sub-spans separately for η estimation, and the weighted average of such estimates computed. The latter is largely free of the bias corrupting the former estimates. See the main text for details.] rrrrrr 6 0pcThe estimated η,for the 6 models (2 cases each).AO N_eff η±σ_η N_eff η±σ_η (η_true=0.01) (η_true=0) 1 A 628 0.0099±0.0007 7360.0014± 0.0005 1 B 0.0111±0.00060.0011±0.0005 2 A 609 0.0139± 0.00096090.0043±0.0008 2 B0.0106 ±0.00080.0007 ± 0.00073 A 5170.0137± 0.0012511 0.0049± 0.001 3 B 0.0104 ±0.0010.0009±0.0009 The η estimates in all considered cases are consistent with their respective model/assumed values η_true within the mentioned uncertainty. The N_eff changes systematically with aliasing-order, indicating possible increase in the decorrelation scales (ν_d and t_s).The correlation maps in Figure 5 are presented to illustrate the implication of the relative location of the region corresponding to the intrinsic unpulsed/off-pulse emission, for a location offset of 0.1R_LC. These are a result of our extended simulations, to directly obtain special versions of off-pulse dynamic spectra (see the text at the end of Appendix B), and enable us to examine modification of cross-correlation signature for different magnitudes and specific orientations (||, ⊥ and 45^o to V) of location offsets (for unpulsed emission source) within the light cylinder. The corresponding correlation maps(such as in Figure 5),for different magnitude and orientation of the location offset, indeed show the expected qualitative correspondence(in terms of shift and/or reduction of the correlation peak,as discussed in Section 2.1) in all of the specific cases we simulated.The above tests with simulated data demonstrate the sensitivity of our technique in reliably searching/estimating possible intrinsic unpulsed emission using pulsar dynamic spectra, and confirm its desirable immunity to possible contaminants of off-pulse dynamic spectra.§ ILLUSTRATION OF OUR TECHNIQUE: A CASE STUDY WITH DATA ON B0329+54 We now apply our technique to data fromobservation on pulsar B0329+54, made using multi-band receiver system (MBR; Mann et al. 2013) with the Robert C. Byrd Green Bank Telescope (GBT), on July 25, 2009.From among many pulsars observed in ten well separated bands simultaneously, we have chosen B0329+54 based on sensitivity considerations, given that it is one of the brightest pulsars known. Width of each band Δν_BW is 16.5 MHz, and time-span of the observation Δ t_obs is 1 hour. Other considerations include choice of the frequency band, as well as the resolutions in time and frequency, with which we can expect to see scintillation features in the the on-pulse dynamic spectrum. These choices depend largely on the decorrelation bandwidth and timescale, which are given by ν_d(kHz)=59C_-4^-1.2λ^-4.4D^-2.2 and t_s(s)=149C_-4^-0.6λ^-1.2D^-0.6v_7^-1, respectively (Romani et al 1985[Romani et al. (1986) has some typographical errors in these expressions; a wrong exponent of C_-4 in the expression for ν_d, and v instead of v_7 the expression for t_s(s). ]). Adequately fine sampling of the scintles, and ensuring as large a number of scintles as possible within the spectro-temporal span, require that δν_res≪ν_d≪Δν_BW and δ t_res≪ t_s≪Δ t_obs. Based on these criteria, the data at 730 MHz and 810 MHz are found suitable, while other data at lower and higher frequencies did not meet these criteria. For B0329+54, at 730 MHz, the estimated ν_d and t_s are ∼100 kHz to ∼1800 kHz and 970 s to 240 s, respectively, at 810 MHz, the corresponding ν_d and t_s are ∼200 kHz to ∼2900 kHz and 1100 s to 270 s, respectively,for C_N^2 ∼3×10^-5 and ∼3×10^-4 m^-20/3. Available measurements by Stinebring et al. (1996) and Wang et al. (2008) at 610 and 1540 MHz, respectively, imply ν_d and t_s to be about 750 kHz and 450 s, respectively, at our lower frequency. For comparison, our estimated decorrelation scales (from correlation analysis such as shown in Figure 8) of about 1 MHz, 360 s and 1.3 MHz, 400 s, at 730 and 810 MHz, respectively, are largely consistent with the above mentioned values, within the uncertainties. Our dynamic spectra have frequency resolution of 64.45 kHz, and time-resolution of 18 s, with 200 time bins across 3600 s.The recorded raw voltage time sequences corresponding to a bandwidth of 16.5 MHz were analyzed to obtain dynamic spectra with frequency resolution of 64.45 kHz (i.e., across 256 channels). After suitable corrections for dispersion and gain compression,[ When signal levelsare even slightly larger than the limit within which a radio receiver has linearresponse, output power becomes less than that expected from its linear response, amounting to reduction in the gain.“Gain compression" refers to this reduction in gain, or non-linear response. In the context of dispersed pulsar signals, if not corrected, such a situationcan cause a dip proportional to the pulse intensity, with a spread in longitude corresponding to the dispersion delay across the observed bandwidth. This effect can contaminate off-pulse region significantly, in casesof bright and high dispersion measure pulsars.] if any, dynamic spectra for various choices of ranges in the longitude were obtained separately. Figures 6 and 7 show these pairs of dynamic spectra, in which the spectral or time ranges affected by radio frequency interference have been removed.For our data, after removing bad time-sections and RFI channels, we have performed the dynamic spectral correlations, to estimate various quantities mentioned above, including the measure of correlation η. More specifically, we estimate N_t, N_ν, N_eff, η and σ_η, along with η_max, as listed in Table 2. rrr 3 0pc Results from the dynamic spectral cross-correlation analysis of scintillation data for the on- and off-pulse regions. The estimates of N_t and N_ν assume a conservative value of the filling factor κ=0.2.ν 730 MHz 810 MHz N_t ∼ 5 ∼ 4N_ν ∼ 3 ∼ 3N_eff14.412.7σ_η 0.0017 0.0021η±σ_η 0.0002 ± 0.0017 0.0013 ± 0.0021η_max 0.0045 0.0066 As is apparent from these estimates, the off-pulse emission can be said to be less than about 0.5% of the main pulse flux density (corresponding a 3-σ limit). The present reduction in the uncertainties in η is moderate, in comparison with the hard limit η_max, and is consistent with the relatively small statistics (i.e. N_eff).At this stage, our presentation of these results is mainly as an illustration, but with improved spectro-temporal span of the data (i.e. largeN_eff), we can expect a significant refinement in these estimations and their uncertainties.As can be seen readily from the1-d plots corresponding to the auto-correlations shown in Figure 8, the random measurement noise de-correlates at non-zero lags, and the sharp drop in the auto-correlation with respect to its value at zero-lag provides a ready estimate of its relative contribution (i.e. σ^2_on or σ^2_off), which is to be discounted while estimating decorrelation bandwidth or time-scale.From the cross-correlation map (the bottom plot in Figure 8), and in general, the level of cross-correlation, or the lack of it, can be assessed across the respective lags, and interpreted either in terms of upper limits on the unpulsed emission intensity, or possible separation of the associated emission region from that of the main-pulse emission. § DISCUSSION AND CONCLUSIONSThe new technique proposed here, for searching forintrinsic off-pulse/unpulsed radiation for pulsars, is inspired by the expectation that such emission originating fromapparent location(s) matching toor in the vicinity of that of the pulsed component (compared at their respective retarded emission times) would also carry a scintillation imprint similar to that measurable for the pulse intensity.Needless to say that a systematic search for unpulsed emission at a range of frequencies, with appropriate spectro-temporal resolutionand spans, will naturally be rewarding. On the other hand, for data sets at sufficiently nearby frequencies, a combined estimate (or upper limit) for η can be obtained. For example, such an upper limit (3σ) for unpulsed intensity of B0329+54 would be 0.4%, based on the data at the two frequencies.Since our method, although a truly high angular resolution probe, is based on longitude-resolved dynamic spectral information, it can be expected to be applied to most of the archived observations on pulsars, made from single-dish and synthesis telescopes, in addition to future observations, as long as the scintles, or the refractive fringing when present, are at least Nyquist sampled. In fact, it would not be surprising to see a massive initiative to search of the continuous/unpulsed emission in the near future, using existing data and new observations.Although the relevant correlation would reduce exponentiallywith increasing apparent angular offset Δ S/z (Cordes et al. 1983)of the source to be searched, opportunity to detect significant peak in the correlation map (at a non-zero lag in time) is to be expected when the orientation of the offset is along the pattern velocity (or within an alignment margin of (S_d/Δ S)^c), as illustrated in our simulations. Interestingly, given the reported pulsar spin-velocity alignment, for young pulsars in particular (Johnston et al. 2005), the location offset along latitudinal direction is likely to be along the pattern velocity, when the scintillation speed is dominated by pulsar motion. Regardless of the possible apparent location of the region responsible for unpulsed emission, if any, within the light-cylinder, it is unlikely that the associated key morphology (say, w.r.t. rotation axis) would differ significantly from pulsar to pulsar. This combined with the expected variety in the orientation of the apparent velocity of the diffractive scintillation pattern (again viewed with respect to the pulsar spin-axis) and in S_d (depending on sight-line and frequency), suggests that there would be adequate number of known pulsars offering conducive situation for the proposed probe to be rewarding, in either detecting their elusive continuous emission, or ruling it out to the extent possible. In summary, the existence of unpulsed emission from pulsars is yet to be fully established, let alone be understood. 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Although, in practice, a variety of non-idealities in spectral filtering are possible[ The key filtering stages include, a) image-band rejection before heterodyneor mixing stage, and b) band-defining before digitization (at Nyquist rate) of signal either at baseband or that located around a chosen centerfrequency (where harmonic or band-pass sampling at Nyquist rate is employed).], for illustration purpose, we consider a simple case of the band-defining filter having a non-zero response in adjacent spectral ranges on either side of the intended band of observation. It is easy to see that the contribution from a dispersed pulse in these out-of-band spectral regions would be aliased in the Nyquist sampled band of interest. On dedispersion, these aliased contributions from the foldedbands would not only spill out side the pulse window, but can systematically spread across a large part of the off-pulse longitudes, and could disguise as off-pulse emission. For an illustrative example of this effect, lets assume that the filter function over the desired band (of width = BW) is flat, as corresponding to a perfect rectangular filter, but this non-ideal filter offers finite, though high, attenuation in other spectral regions.In our simulation, we consider the out-of-band region on either side to be 3 times wider than BW, and relative attenuation to be 20 dB, such that the filter response F(ν) as function of frequency ν is given byF(ν)={[ 1, |ν-ν_c|≤ 0.5BW ,;0.01, 0.5BW≤|ν-ν_c|≤ 3.5BW ,; 0,otherwise;].where, ν_c is the center frequency of observation.Assuming a train of Gaussian pulses from a pulsar, the intensity pattern across time and frequency can be expressed asI(ν,t)=I_0exp{-((t-τ(ν))-nP)^2/(2σ^2)}where, P is the pulsar period, σ is the standard width of the Gaussian, n is number of periods defining the time span of the simulated sequence. The dispersion delay at frequency ν, with respect to that at a reference frequency ν_ref, is given byτ(ν)=4.15×10^3DM[1/ν^2-1/ν_ref^2] (s)where DM is the dispersion measure in pc cm^-3 and the frequencies are in MHz. A dynamic spectrum containing dispersed pulses was simulated assuming BW= 16 MHz centered at ν_c= 300 MHz, σ = 0.04 s, and for pulsar parameters similar to that of B0329+54 (DM = 26.77 pc cm^-3, P= 0.714472578 s). The top panel of Figure A1 (or 8) shows the dedispersed pulse sequence when no aliasing of sky signal occurs from outside of the desired band of bandwidth BW (a case of ideal filtering). The middle panel shows a similarly obtained sequence, but now including aliasing (of remaining small level of sky signal)from the adjacent bands as described above. Note that the dispersed pulse contribution from the aliased adjacent bands would be at a much lower level, but will make its appearance in the off-pulse region. These contributions will occupy different longitude spreads after dedispersion, depending on the alias order, in addition to the DM and P. Significant contamination in both the on-pulse and the off-pulse regions, as result of aliasing, is apparent from the difference between the sequences in the top two panels, as shown in the bottom panel. Here, we have deliberately used a lower ν_c (=300 MHz), than those for the data we present (i.e. 730 MHz or 810 MHz), so that the mentioned contamination is more pronounced.§ B: SIMULATION OF DIFFRACTIVE INTERSTELLAR SCINTILLATION (DISS) AND DYNAMIC SPECTRAThe two main steps in simulation of DISS dynamic spectra, using a thin-screen approximation, are: (i) generation of a random phase screen following an assumed spatial distribution of electron density irregularities in the intervening ISM, which modifies the emerging wave-front, and (ii) calculation of resultant intensity, of the received signal at the observer's location, as a function frequency and time.As mentioned already, the most accepted 3D spatial power spectral description of the turbulent ISM is a power-law spectrum across spatial frequency q, ranging from q_min to q_max, (Armstrong et al. 1995) P_3N(q_x,q_y,q_z)=C_N^2 q^-β; q=√(q_x^2+q_y^2+q_z^2)where, C_N^2 is the level of turbulence, and the power-law index β=11/3 for the Kolmogorov turbulence. Armstrong et al. (1995) have given evidence of the validity of Equation B1 for q_min=10^-12 m^-1 < q < q_max = 10^-6 m^-1, and have derived the typical turbulence strength C_N^2∼10^-3 m^-20/3, but the C_N^2 can deviate significantly from this typical value, depending on direction and distance to the source. A convenient way to study propagation effects due to this 3-D distribution of refractive index irregularities in the ISM is to model the modification of the incident wave-front by an equivalent thin phase-changing screen, located between the source and the observer.We use this thin screen approximation (see Lovelace 1970; Romani et al. 1986) for our present simulations, wherein the equivalent 2-D spatial power spectrum (P_2ϕ) of the phase deviation ϕ is given byP_2ϕ(q_x,q_y) =2π z(λ r_e)^2 P_3N(q_x,q_y,q_z=0)=2π z(λ r_e)^2C_N^2(√(q_x^2+q_y^2))^-βwhere, z is the distance to the source, r_e=2.82×10^-15 m is the classical radius of the electron, and λ is the wavelength of the propagating radiation. §.§ Generation of Equivalent Thin Phase-Screen using FFT-Based TechniqueThe spatial power spectrum of the equivalent 2-D thin screen and the associated distribution of the random phase deviation ϕ(x,y) across that screen in transverse plane (x,y) have the following Fourier relationshipϕ(x,y)=∫_-∞^+∞∫_-∞^+∞g(q_x,q_y)√(P_2ϕ(q_x,q_y))exp[-j2π(xq_x+yq_y)]dq_xdq_yi.e., ϕ(x,y) is the (inverse) Fourier transform of the product g(q_x,q_y)√(P_2ϕ(q_x,q_y)), where g(q_x,q_y) is a Hermitian-symmetric complex Gaussian variable representing zero-mean white noise process, with unity variance (Johansson & Gavel 1994). We obtain ϕ(x,y) distribution using the above relation, employing FFT technique for computational ease.The discrete form needed for simulation, of the Equation B3, for a square screen, ϕ, made-up of N× N grid points, isϕ=2π(2π)^-β/2(NΔ r)^-1+β/2√(2π z(λ r_e)^2C_N^2)[ℱ𝒯^-1{g M_0}]where, Δ r is spatial sampling interval, g is a N× N matrix (the procedure to obtain it is explained in the following) and M_0 is also a N× N matrix whose elements are M_0(i,j)=[(i-N_c)^2+(j-N_c)^2]^-β/4, where the origin is defined at N_c = N/2 + 1, and the contribution at zero spatial frequency is set to zero, i.e. M(N_c,N_c)=0. The recipe for getting the Gaussian random matrix g is as follows: * generate a complex matrix (say) M of size N× N, whose elements are a+j b, where, j=√(-1), a and b are independent Gaussian random numbers with zero mean and unity variance.* obtain the 2-D discrete Fourier transform of `M', which will be the required matrix g (g=ℱ𝒯^-1{M}).§.§ Electric Field Distributions and Resultant Intensity in the Observer's PlaneHaving generated ϕ(x,y) (i.e., the discrete ϕ, Equation B4), the electric field distribution at the thin screen can be given byE_s(x,y)=exp(-j ϕ(x,y)).In the thin screen approximation, the ray optics is applicable and so the electric field received at any point (ξ,η) on the observer plane, can be represented by the Fresnel-Kirchhoff integral (Born & Wolf 1980)E_O(ξ,η)= e^-jπ/2e^j2π z/λ/2π r_f^2∫∫exp[jϕ(x,y) +j(x-ξ)^2+(y-η)^2/2r_f^2]dxdywhere, r_f=√(λ z/2π). This integral can be either calculated from 2-D numerical integration or methods using Fourier transform. We have used angular spectrum method to calculate the electric field, i.e., via following relation (in discrete form)E_O=ℐℱ𝒯{ℱ𝒯{h(x,y)} ℱ𝒯{exp(-jϕ)}}where, h(x,y) is called transfer function. By use of the above method to get E_O, the spatial sampling interval at the observer's plane will be the same Δ r as that of phase distribution of the thin phase screen. So corresponding to each value of input frequency/wavelength, we will have phase distribution ϕ of the thin screen (in discrete form a matrix of size say N× N) and the electric field E_O at the observer's plane (again a matrix of size N× N). From this matrix E_O, we select a spatial 1D cut, say E_O(r), which may be an arbitrarily chosen row or column, and obtain E_O(ν,r) by varying only ν in uniform steps over the range of interest. The spatio-spectral description of observed intensity is trivially obtained as I_O(ν,r) = |E_O(ν,r)|^2. This I_O(ν,r) can be translated into I_O(ν,t), i.e. the dynamic spectrum, by assuming a velocity V_trans along r for the intensity pattern in the observer's plane, which depends on the relative transverse velocities of the pulsar, the scattering medium and the observer.§.§.§ Our Simulation Parameters and ResultsDiffractive effects correspond to spatial frequency range q∼10^-8 m^-1 to 10^-6 m^-1 of the ISM irregularities (Stinebring 1996; Rickett 1988; Narayan 1988; Wang, Manchester 2008 ). We have used a square (scattering) screen so sampling interval, say Δ r, in x and y directions are equal, i.e. Δ x=Δ y =Δ r. The Nyquist sampling criteria demands Δ r=1/(2q_max). What size of the phase screen will be suffice for simulation of DISS ? The observer receives radiation from a cone of half-angle θ_S≈ r_ref/z≈ r_mp/z≈λ/(2π S_d), (Cordes 1986) where, r_ref is refractive length scale, r_mp is multi-path propagation length scale (strong scintillation), S_d is diffractive length scale, z is distance from the observer to the thin screen and λ is the wavelength of the radiation from pulsar. So to properly simulate DISS, the phase screen size r_max should be at least ∼ r_mp≈ r_f^2/S_d in each of the two dimensions. Hence the required number of grid points N across r_mp, for a N× N matrix describing the screen, would be N ≥ r_max/Δ r = 2q_maxr_f^2/S_d Thus for the case of our data on B0329+54, where ν=810 MHz andz=1.44 kpc, r_mp∼ 10^11 m. To satisfy the criteria Δ r≤ 1/(2q_max), the required N∼2^18 [N=10^11/(5×10^5)∼ 2×10^5 ∼2^18] ! To generate a phase screen of this overwhelmingly large dimension, of order 2^18× 2^18, and the subsequent Fourier analysis involving bigger dimension (i.e., 2^19×2^19) is not only computationally intensive (even with use of FFTs), but well beyond the readily available computing resources.However, we note the red nature of the underlying spatial spectrum of phase variation P_2ϕ∝ q^-β (β is +ve). The associated structure function for phase ϕ, at a given scale r can be expressed as D_ϕ(r) = (1^c)^2 (r/S_d)^(β -2), given that for the diffractive (or the coherence) scale S_d,the phase structure function is 1 radian^2 (Armstrong et al. 1995). Since contribution to phase fluctuations from smaller spatial scales is expected to decrease rapidly, for the relevant values of β, we consider revision of the sampling scale Δ r, such that D_ϕ(Δ r) ≤Δϕ^2_min, for the desired small phase variation Δϕ_min that is to be sampled duly. With this criterion, and recalling Equation 8, we express the required grid dimension as N ≥(ν/ν_d)/(Δϕ_min)^2/(β -2) For example, with Δϕ_min = 0.1 radian, and β = 11/3, N ≥ 16 (ν/ν_d). Choosing a suitable ratio (ν/ν_d), say 200, the requirement of N ≥ 3200 appears feasible with computational constraints, i.e. without needing supercomputers, and more importantly, without compromising significantly on the details of the phase screen.In our present simulations, we have used ν_d∼ 1.35 MHz, to keep reasonable correspondence with the discussed observations, but use a relatively smaller ν of 270 MHz. We use N=4096, so that the screen and the diffraction patterns are sampled with adequate details (corresponding to Δ r=5×10^5 m and r_max∼ 2×10^9)[It is worth noting that now the Fresnel scale r_f and z are artificially small and C_N^2 is correspondingly large, as a result of the spatial dynamic range we have chosen.], and the resultant dynamic spectrum suffices for demonstrating the key aspects of our technique. It is worth pointing out that now the Fresnel scale r_f and z are artificially small and C_N^2 is correspondingly large, as a result of the spatial dynamic range we have chosen. However, the scale of direct relevance to us, that is the diffraction pattern scale S_d corresponds to typical 4 samples across r, implying (t_s/4) as the sampling interval in the dynamic spectrum. The overall time scale, and t_s, can be defined by choice of the velocity V_trans, if required. In any case, the simulated time span corresponds to ∼ 1000 t_s.For completeness, to examine the sensitivity of the correlation technique to the relative location off-pulse emission region, we extended our simulations to obtain the off-pulse dynamic spectra separately from that for the pulsed component. Using the common description for the phase pattern, we added a suitable extra phase-gradient to it for simulating new phase screen, corresponding to the location offset, and used the resultant intensity pattern for constructing off-pulse dynamic spectra. The magnitude and direction of the location offset were varied, and the resultant cross-correlation maps were examined (see Figure 5). | http://arxiv.org/abs/1706.08896v2 | {
"authors": [
"Kumar Ravi",
"Avinash A. Deshpande"
],
"categories": [
"astro-ph.HE",
"astro-ph.GA",
"astro-ph.IM"
],
"primary_category": "astro-ph.HE",
"published": "20170627145245",
"title": "Scintillation based search for off-pulse radio emission from pulsars"
} |
^1State Key Laboratory of Surface Physics, Department of Physics, and Laboratory of Advanced Materials, Fudan University, Shanghai 200433, China ^2Center for Correlated Electron Systems, Institute for Basic Science, Seoul 08826, Korea ^3Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea ^4Rutgers Center for Emergent Materials and Department of Physics and Astronomy, Rutgers University, Piscataway New Jersey 08854, USA ^5Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, ChinaWe present the ultra-low-temperature thermal conductivity measurements on single crystals of the prototypical charge-density-wave material 1T-TaS_2, which was recently argued to be a candidate for quantum spin liquid. Our experiments show that the residual linear term of thermal conductivity at zero field is essentially zero, within the experimental accuracy. Furthermore, the thermal conductivity is found to be insensitive to the magnetic field up to 9 T. These results clearly demonstrate the absence of itinerant magnetic excitations with fermionic statistics in bulk 1T-TaS_2 and, thus, put a strong constraint on the theories of the ground state of this material.Heat transport study of the spin liquid candidate 1T-TaS_2 Y. J. Yu,^1,† Y. Xu,^1,† L. P. He,^1 M. Kratochvilova,^2,3 Y. Y. Huang,^1J. M. Ni,^1 Lihai Wang,^4 Sang-Wook Cheong,^4 Je-Geun Park,^2,3 and S. Y. Li^1,5,* December 30, 2023 =================================================================================================================================================================The quantum spin liquid (QSL), where strong quantum fluctuations obstruct long-range magnetic order even down to the absolute zero temperature, is one of the most elusive and exotic quantum state of matter <cit.>. In the QSLs, Mott physics plays a significant role in localizing electrons and forming S = 1/2 spins, as has been manifested in the study of high-temperature superconductors <cit.>. Experimentally, triangular-lattice organic compounds κ-(BEDT-TTF)_2Cu_2(CN)_3 <cit.> andEtMe_3Sb[Pd(dmit)_2]_2 <cit.>, together with kagome-lattice ZnCu_3(OH)_6Cl_2 <cit.> and Cu_3Zn(OH)_6FBr <cit.>, are typical examples of Mott-assisted QSL candidates. Richer physics, together with further complications, would be brought to the game, if this novel physics is to play on the stage of the charge-density-wave (CDW) state of a transition-metal dichalcogenide (TMD).1T-TaS_2 is a layered material, and the only correlation-driven insulator discovered among TMDs <cit.>. As for the charge degree of freedom, 1T-TaS_2 features a number of peculiar CDW phases. Upon cooling, it turns into a metallic incommensurate CDW (ICCDW) phase below 550 K, a textured nearly commensurate CDW (NCCDW) phase below 350 K, and finally enters a commensurate CDW (CCDW) phase below 180 K <cit.>. The low-temperature CCDW phase is characterized by a √(13)×√(13) structure described as star-of-David clusters <cit.>. There is one unpaired electron per David-star due to energy gaps induced by the periodic lattice distortion <cit.>. At the same time, the electron correlation effects set in and localize this electron, leading to a Mott insulating state with S = 1/2 spins arranged on an ideal triangular lattice <cit.>. This is one of the few model spin configurations that may harbor the exotic QSL state, and exactly the one proposed by Anderson in his resonating-valence-bond model <cit.>.The possibility of the realization of a QSL in 1T-TaS_2 has been proposed recently in Ref. <cit.>. By analyzing the existing data of this material, it was argued that 1T-TaS_2 should be considered as a QSL, either a fully gapped Z_2 spin liquid or a Dirac spin liquid <cit.>. The muon spin relaxation (μSR) and nuclear quadrupole resonance (NQR) experiments have been performed on 1T-TaS_2 single crystals <cit.>. No long-range magnetic order was detected from 210 K down to 70 mK by μSR. On the other hand, the NQR experiments reveal a gapless QSL-like behavior in part of the CCDW phase, from 200 K to T_f = 55 K. Below T_f, a novel quantum phase with amorphous tiling of frozen singlets emerges out of the QSL <cit.>. Meanwhile, another group performed polarized neutron diffraction and μSR measurements on 1T-TaS_2 <cit.>. Their results indicate the presence of the short-ranged magnetic order below 50 K, and support the scenario that an orphan S = 1/2 spin moment is localized at the center of the David-star <cit.>.To find out what is the true ground state of bulk 1T-TaS_2, it is essential to know the details of the low-lying elementary excitations. Ultra-low-temperature thermal conductivity measurement has proven to be a powerful technique in the study of low-lying excitations in QSL candidates <cit.>. Taking the spin-1/2 triangular-lattice Heisenberg antiferromagnets as example, the thermal conductivity result implied a possibility of a tiny gap opening in κ-(BEDT-TTF)_2Cu_2(CN)_3 <cit.>, while highly mobile gapless excitations with fermionic statistics exist in EtMe_3Sb[Pd(dmit)_2]_2 <cit.>. For the QSL candidate YbMgGaO_4, which has been studied extensively recently, no significant contribution of thermal conductivity from magnetic excitations was observed <cit.>.In this Rapid Communication, we report the ultra-low-temperature thermal conductivity measurement on a high-quality 1T-TaS_2 single crystal down to 0.1 K. No significant contribution from magnetic excitations is detected at zero magnetic field. Furthermore, the thermal conductivity is found to be insensitive to magnetic fields up to 9 T. The absence of κ_0/T at all fields unambiguously demonstrates that no fermionic magnetic excitations with itinerant character exist in 1T-TaS_2. We shall discuss the implications of our findings on the ground state of bulk 1T-TaS_2.The high-quality 1T-TaS_2 single crystal was grown by the chemical vapor transport method <cit.>. The x-ray diffraction (XRD) measurement was performed on the 1T-TaS_2 sample by using an x-ray diffractometer (D8 Advance, Bruker). The single crystal with a large natural surface was cut to a rectangular shape of 3.25 × 0.72 × 0.1 mm^3. The large natural surface (3.25 × 0.72 mm^2) was determined to be the (001) plane by XRD, as shown in Fig. 2(a). A standard four probe method was used for both resistivity and thermal conductivity measurements. Contacts were made directly on this natural surface with silver paint. The resistivity was measured in a ^4He cryostat from 300 to 1.5 K. The thermal conductivity was measured in a dilution refrigerator, using a standard four-wire steady-state method with two RuO_2 chip thermometers, calibrated in situ against a reference RuO_2 thermometer. Magnetic fields were applied perpendicular to the large natural surface.As shown in Fig. 1(a), 1T-TaS_2 crystallizes in the CdI_2-type trigonal structure belonging to the P-3m1 space group <cit.>. It has a layered structure, in which each atomic layer is composed of one Ta layer sandwiched between two S layers in an octahedral arrangement <cit.>. Within the CCDW phase, 13 Ta atoms form a fully interlocked David-star cluster, where 12 peripheral Ta atoms shrink towards the central Ta atom. Such a deformation leads to the formation of a √(13)×√(13) triangular superlattice <cit.>, as illustrated in Fig. 1(b).The temperature dependence of the resistivity ρ(T) for the 1T-TaS_2 single crystal in zero magnetic field is plotted in Fig. 2(b). A sharp increase indicative of the occurrence of the transition from the NCCDW phase to the CCDW phase can be clearly seen around 180 K, below which the resistivity exhibits an insulating behavior. All of these features are consistent with previous resistivity measurements on 1T-TaS_2 <cit.>. The inverse resistivity ratio ρ(1.5 K)/ρ(295 K) is about 130, which is comparable with those measured previously <cit.>.Figure 3(a) presents the in-plane thermal conductivity of the 1T-TaS_2 single crystal at H = 0 T. In a solid, the contributions to thermal conductivity usually come from various quasiparticles, such as phonons, electrons, magnons, and spinons. For 1T-TaS_2, the thermal conductivity from electrons (κ_e/T) at 1.5 K is estimated to be 6.13 × 10^-5 mW K^-2 cm^-1 according to the Wiedemann-Franz law κ_e/T = L_0/ρ(1.5 K), with the Lorenz number L_0 = 2.45 × 10^-8 W Ω K^-2 and ρ(1.5 K) = 399.8 mΩ cm. The electron contribution becomes smaller upon further cooling and is negligible at ultra-low temperature, due to the insulating behavior of the resistivity. Therefore, the thermal conductivity at very low temperature can be fitted by κ/T = a + bT^α-1, in which the two terms aT and bT^α represent the contributions from fermionic magnetic excitations (if they exist) and phonons, respectively <cit.>. Because of the specular reflections of phonons at the sample surfaces, the power α in the second term is typically between 2 and 3 <cit.>. The fitting of 0 T data below 0.35 K gives the residual linear term κ_0/T ≡ a = 0.005 ± 0.002 mW K^-2 cm^-1 and α = 2.69. Considering our experimental error bar ± 5 μW K^-2 cm^-1, the κ_0/T of 1T-TaS_2 at zero field is essentially zero. Note that EtMe_3Sb[Pd(dmit)_2]_2 has a value of κ_0/T as big as 2 mW K^-2 cm^-1 <cit.>. The in-plane thermal conductivity of the 1T-TaS_2 single crystal in magnetic fields (H = 0, 4, and 9 T) applied along the c axis is plotted in Fig. 3(b), with the three curves almost overlapping on top of another. The same fitting process is performed, giving κ_0/T = -0.002 ± 0.009 mW K^-2 cm^-1 and κ_0/T = 0.008 ± 0.005 mW K^-2 cm^-1 for H = 4 and 9 T, respectively. The three κ_0/T values are plotted in Fig. 3(c). One can see that magnetic field barely has any effect on the thermal conductivity of 1T-TaS_2 up to 9 T.Now we would like to discuss the implications of our thermal conductivity results on the proposal of 1T-TaS_2 being a QSL. Theoretically, all known QSLs can be classified in terms of a spectrum of gapless spinons (or their absence) and the nature of the emergent gauge fields to which they couple <cit.>. Various kinds of exotic models have been proposed in the study of various QSL candidates <cit.>. A systematic analysis of whether these models can be applied to 1T-TaS_2 is beyond the scope of this work, and we only discuss the feasibility of these models in the light of our experimental data on the low-energy spin excitations. Generally, a finite residual linear term κ_0/T represents the contribution to κ from fermionic magnetic excitations in the zero temperature limit, i.e., the spectrum of the fermionic magnetic excitations is gapless. This might come from a spinon-Fermi surface or nodes in the momentum space. For 1T-TaS_2, the former one has been ruled out <cit.>, because of the tiny linear term γ (∼ 2 mJ mol^-1 K^-2) observed in specific heat <cit.>. For the latter one, the most common case is a U(1) Dirac spin liquid <cit.>. In such a state, nodal fermionic spinons at the Dirac points would still result in a finite κ_0/T, and the thermal conductivity would be enhanced by a magnetic field <cit.>. This is incompatible with our results that the κ_0/T is negligible at all fields and the thermal conductivity is insensitive to magnetic field. It seems that any gapless QSL scenarios, whether gapless everywhere or only at nodes in the momentum space, are not consistent with our data. Note that there are also some exotic scenarios with nodal bosonic excitations <cit.>. The contribution to the thermal conductivity from these nodal excitations exhibits a power-law temperature dependence (∼ T^δ). However, unlike nodal fermionic excitations, for which the power-law exponent δ is 1, the δ value for nodal bosonic excitations is unknown in advance, so that it is hard to be separated from the phonon contribution.However, there is another possibility that might reconcile our data with the gapless QSL scenarios. For the low-temperature phase (T ≤ T_f = 55 K) of 1T-TaS_2, the NQR shows a broad distribution of 1/T_1 values with a stretched exponent p < 1 (p ≈ 0.5), implying a highly inhomogeneous magnetic phase at all Ta sites <cit.>. We note that similar spectral broadening and stretched exponent behavior have been observed in another triangular-lattice QSL candidate κ-(BEDT-TTF)_2Cu_2(CN)_3 <cit.>. The thermal conductivity measurement on κ-(BEDT-TTF)_2Cu_2(CN)_3 also gives a negligible κ_0/T, which was argued to come possibly from the localization of the gapless spin excitations due to the inhomogeneity <cit.>. This might also be the case for 1T-TaS_2. The low-temperature phase (T ≤ T_f) still exhibits a gapless behavior for the low energy fractional excitations, according to Ref. <cit.>, but these gapless excitations can be localized so that they cannot conduct heat.Next we turn to the gapped QSL scenarios. A fully gapped Z_2 spin liquid was suggested in Ref. <cit.>, which is a state with gapped spinons together with gapped visons. For κ-(BEDT-TTF)_2Cu_2(CN)_3, an alternative explanation of the negligible κ_0/T is that the spin excitations are gapped <cit.>. The total thermal conductivity of κ-(BEDT-TTF)_2Cu_2(CN)_3 is the sum of a phonon contribution term and a magnetic part with an exponential temperature dependence, indicating the existence of a gap (Δ∼ 0.46 K) in the spin excitation spectrum <cit.>. A field-induced gap closing was also observed in κ-(BEDT-TTF)_2Cu_2(CN)_3 for magnetic fields higher than ∼ 4 T <cit.>. For 1T-TaS_2, however, we found that a similar fitting procedure does not work. Having said that, we caution that our data do not contradict the gapped QSL scenario. Indeed, if the magnitude of the gap is sufficiently large, the magnitude of the exponential term in the total thermal conductivity would be too small to be discerned at such low temperature, so that the total thermal conductivity is dominated by the phonon term. And the gap, if it exists, cannot be closed by a magnetic field up to 9 T. For example, an unusually large exchange interaction J ≈ 0.13 eV (∼ 1500 K) has been derived from the susceptibility data <cit.>. The spin gap is estimated to be above 200 K in Ref. <cit.>. In fact, in most cases a large gap is more common than a small gap (compared to its J) as possibly in κ-(BEDT-TTF)_2Cu_2(CN)_3 <cit.>.Of course, our results do not entirely rule out the possibility that the ground state of bulk 1T-TaS_2 may not be a QSL. The low-temperature phase (T ≤ T_f) proposed by the NQR experiment is a highly unusual one, featuring frozen singlets, pseudogap in the spinon density of states, and a high degree of local disorder <cit.>. In Ref. <cit.>, the neutron diffraction and μSR results provide evidence for the existence of a short-range-ordered state. To what extent do the behaviors of these states resemble those of a QSL is an open question and requires future scrutiny. As stated in Ref. <cit.>, it might be more interesting to look for a QSL ground state in ultra-thin crystals of 1T-TaS_2.In summary, we have measured the thermal conductivity of a 1T-TaS_2 single crystal down to 0.1 K. No residual linear term of thermal conductivity was observed at zero field. The thermal conductivity is found to be insensitive to a magnetic field up to 9 T. These results provide evidence for the absence of itinerant magnetic excitations obeying fermionic statistics in 1T-TaS_2. Our results set strong constraints on the nature of its ground state and, thus, of its theoretical description.This work is supported by the Ministry of Science and Technology of China (Grant No: 2015CB921401 and 2016YFA0300503), the Natural Science Foundation of China, the NSAF (Grant No: U1630248), the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, and STCSM of China (No. 15XD1500200). 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"authors": [
"Y. J. Yu",
"Y. Xu",
"L. P. He",
"M. Kratochvilova",
"Y. Y. Huang",
"J. M. Ni",
"Lihai Wang",
"Sang-Wook Cheong",
"Je-Geun Park",
"S. Y. Li"
],
"categories": [
"cond-mat.str-el",
"cond-mat.mtrl-sci",
"cond-mat.supr-con"
],
"primary_category": "cond-mat.str-el",
"published": "20170626184503",
"title": "Heat transport study of the spin liquid candidate 1T-TaS2"
} |
Codebook Based Hybrid Precoding for Millimeter Wave Multiuser Systems Shiwen He, Member, IEEE, Jiaheng Wang, Senior Member, IEEE, Yongming Huang, Member, IEEE, Björn Ottersten, Fellow, IEEE, and Wei Hong, Fellow, IEEE Manuscript received Jan. 06, 2017; revised Apr. 14, 2017;accepted Jun. 24, 2017. S. He is with the State Key Laboratory of Millimeter Waves, School of Information Science and Engineering,Southeast University, Nanjing 210096, China. (Email: [email protected]). J. Wang is with the National Mobile Communications Research Laboratory, School of Information Science and Engineering, Southeast University, Nanjing 210096, China. (Email: [email protected]). Y. Huang (Corresponding author) is with the National Mobile Communications Research Laboratory, School of Information Science and Engineering, Southeast University, Nanjing 210096, China. (Email: [email protected]). B. Ottersten is with the Interdisciplinary Centre for Security Reliability and Trust (SnT), University of Luxembourg, Luxembourg, and also with the Royal Institute of Technology (KTH), Stockholm, Sweden. (e-mail: [email protected]; [email protected]). W. Hong is with the State Key Laboratory of Millimeter Waves, School of Information Science and Engineering, Southeast University, Nanjing 210096, China. (Email: [email protected]). Received *** ; accepted *** ===============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================In millimeter wave (mmWave) systems, antenna architecture limitations make it difficult to apply conventional fully digital precoding techniques but call for low cost analog radio-frequency (RF) and digital baseband hybrid precoding methods. This paper investigates joint RF-baseband hybrid precoding for the downlink of multiuser multi-antenna mmWave systems with a limited number of RF chains. Two performance measures, maximizing the spectral efficiency and the energy efficiency of the system, are considered. We propose a codebook based RF precoding design and obtain the channel state information via a beam sweep procedure. Via the codebook based design, the original system is transformed into a virtual multiuser downlink system with the RF chain constraint. Consequently, we are able to simplify the complicated hybrid precoding optimization problems to joint codeword selection and precoder design (JWSPD) problems. Then, we propose efficient methods to address the JWSPD problems and jointly optimize the RF and baseband precoders under the two performance measures. Finally, extensive numerical results are provided to validate the effectiveness of the proposed hybrid precoders.Hybrid precoding design, millimeter wave communication, energy efficient communication, successive convex approximation, power allocation. § 1. INTRODUCTION The proliferation of multimedia infotainment applications and high-end devices (e.g., smartphones, tablets, wearable devices, laptops, machine-to-machine communication devices) causes an explosive demand for high-rate data services. Future wireless communication systems face significant challenges in improving system capacity and guaranteeing users' quality of service (QoS) experiences <cit.>. In the last few years, various physical layer enhancements, such as massive multiple-input multiple-output (MIMO) <cit.>, cooperation communication <cit.>, and network densification <cit.> have been proposed. Along with these technologies, there is a common agreement that exploiting higher frequency bands, such as the millimeter wave (mmWave) frequency bands, is a promising solution to increase network capacity for future wireless networks <cit.>.MmWave communication spans a wide frequency range from 30 GHz to 300 GHz and thus enjoys much wider bandwidth than today's cellular systems <cit.>. However, mmWave signals experience more severe path loss, penetration loss, and rain fading compared with signals in sub-6 GHz frequency bands. For example, the free space path loss (FSPL) at 60 GHz frequency bands is 35.6 dB higher than that at 1 GHz <cit.>. Such a large FSPL must be compensated by the transceiver in mmWave communication systems. Fortunately, the very small wavelength of mmWave signals enables a large number of miniaturized antennas to be packed in small dimension, thus forming a large multi-antenna system potentially providing very large array gain. In conventional multi-antenna systems, each active transmit antenna is connected to a separate transmit radio frequency (RF) chain. Although physical antenna elements are cheap, transmit RF chains are not cheap. A large number of transmit RF chains not only increase the cost of RF circuits in terms of size and hardware but also consume additional energy in wireless communication systems <cit.>. Therefore, in practice, the number of RF chains is limited and much less than the number of antennas in mmWave systems.For ease of implementation, fully analog beamforming was proposed in <cit.>, where the phase of the signal sent by each antenna is manipulated via analog phase shifters. However, pure analog precoding (with only one RF chain) cannot provide multiplexing gains for transmitting parallel data streams. Hence, joint RF-baseband hybrid precoding, aiming to achieve both diversity and multiplexing gains, has attracted a great deal of interest in both academia and industry for mmWave communications <cit.>. El Ayach et al in <cit.> exploited the inherent sparsity of mmWave channels to design low-complexity hybrid precoders with perfect channel state information (CSI) at the receiver and partial CSI at the transmitter (CSIT). Alkhateeb et al further investigated channel estimation for multi-path mmWave channels and tried to improve the performance of hybrid precoding using full CSIT <cit.>. Note that the hybrid precoding designs in <cit.> assume that either perfect or partial CSIT is available. In practice, while using partial CSIT may degrade system performance, perfect CSIT is often difficult to obtain in mmWave communication systems, especially when there are a large number of antennas. The RF-baseband hybrid precoders in <cit.> were designed to obtain the spatial diversity or multiplexing gain for point-to-point mmWave communication systems. It is well known that multiuser communications can further provide multiuser diversity <cit.>. In <cit.>, the authors proposed a RF precoder for multiuser mmWave systems by matching the phase of the channel of each user also under the assumption of perfect CSIT. Later, a low-complexity codebook based RF-baseband hybrid precoder was proposed for a downlink multiuser mmWave system <cit.>. Note that both <cit.> and <cit.> assumethat the number of users equals the number of RF chains. In mmWave multiuser systems, it is very likely that the number of the served users per subcarrier will be less than that of RF chains. Therefore, it is necessary to study more flexible hybrid precoding designs for multiuser mmWave communication systems.The existing RF-baseband hybrid precoding designs focus on improving the spectral efficiency of mmWave communication systems <cit.>. On the other hand, accompanied by the growing energy demand and increasing energy price, the system energy efficiency (EE) becomes another critical performance measure for future wireless systems <cit.>. In mmWave communication systems, although reducing the number of RF chains can save power consumption, the RF-baseband hybrid architecture requires additional power to operate the phase shifting network, the splitter, and the mixer at the transceiver <cit.>. Therefore, it is also necessary to investigate the RF-baseband hybrid precoding for improving the system EE. Recently, following the idea in <cit.>, an energy efficient hybrid precoding method was developed for 5G wireless communication systems with a large number of antennas and RF chains <cit.>. Differently, in this paper, we propose a codebook based hybrid precoding method that uses the effective CSIT to design the RF-baseband precoders.In this paper, we study the RF-baseband hybrid precoding for the downlink of a multiuser multi-antenna mmWave communication system. The hybrid precoding design takes into account two hardware limitations: (1)the analog phase shifters have constant modulus and a finite number of phase choices, and (2) the number of transmit RF chains is limited and less than the number of antennas. The design goal is to maximize the sum rate (SR) and the EE of the system. We introduce a codebook based RF precoding design along with a beam sweep procedure to reduce the complexity of the hybrid precoder and relieve the difficulty of obtaining CSIT. The contribution of this paper are summarized as follows. * We investigate joint optimization of the RF-baseband precoders in multiuser mmWave systems under two common performance measures, i.e., maximizing the SR and the EE of the system. * Considering the practical limitation of phase shifters, we propose a codebook based RF precoder, whose columns (i.e., RF beamforming vectors) are specified by RF codewords, and then transform the original mmWave system into a virtual multiuser downlink multiple input single output (MISO) system. * We propose a beam sweep procedure to obtain effective CSIT with less signaling feedback by utilizing the beam-domain sparse property of mmWave channels. * Based on the codebook based design, we are able to simplify the original RF-baseband hybrid precoding optimization problems into joint codeword selection and precoding design (JWSPD) problems. * We propose an efficient method to address the JWSPD problem for maximizing the system SR. * We also develop an efficient method to address the more difficult JWSPD problem for maximizing the system EE. * Finally, extensive numerical results are provided to verify the effects of the proposed codebook based hybrid precoding design. It is shown that the proposed method outperforms the existing methods and achieves a satisfactory performance close to that of the fully digital precoder. The remainder of this paper is organized as follows. The system model and optimization problem formulation are described in section 2. Section 3 introduces a codebook based mmWave RF precoding design with beam sweep. An effective joint codewords selection and precoder design method is proposed for SRmax problem in section 4. In section 5, an effective joint codewords selection and precoder design method is developed for EEmax problem. In section 6, numerical evaluations of these algorithms are carried out. Conclusions are finally drawn in section 7.Notations: Bold lowercase and uppercase letters represent column vectors and matrices, respectively. The superscripts (·)^T, and (·)^H represent the transpose operator, and the conjugate transpose operator, respectively. tr(·), ·_2, |·|, ·_ℱ, (·) and (·) denote the trace, the Euclidean norm, the absolute value (element-wise absolute if used with a matrix), Frobenius norm, the real and imaginary operators, respectively. X≥Y and X≤Y denote an element-wise inequality. A≽0 denotes matrix A is a semidefinite positive matrix. 1_N× N and1_N denote respectively N× N matrix with all one entries and N× 1 all-one vector. A(m,n) represents the (m th,n th) element of matrix A and diag(A) stands for a column vector whose elements are the diagonal element of the matrix A.ℝ and ℂ are the real number field and the complex number field, respectively. log(·) is the logarithm with base e. The function floor(x) rounds the elements of x to the nearest integers less than x. mod(,) is the modulo operation. υ_max^(d)(A) is the set of right singular vectors corresponding to the d largest singular values of matrix A.§ 2. PROBLEM STATEMENT §.§ A. System ModelConsider the downlink of a mmWave multiuser multiple-input single-output (MISO) cellular system as shown in Fig. <ref>, where the BS is equipped with M transmit antennas and S RF chains and serves K≤ S single-antenna users. Different from conventional multi-antenna communication systems, e.g., <cit.>, where the numbers of antennas and RF chains are equal, in mmWave systems the number of antennas could be very large and it is expensive and impractical to install an RF chain for each antenna, so in practice we often have S≤ M.To exploit the full potential of mmWave system with a limited number of RF chains, we consider an RF-baseband hybrid precoding design, in which the transmitted signal is precoded in both the (digital) baseband domain and the (analog) RF domain. Specifically, the system model can be expressed asy=HFGs+n,where s^T=[s_1,⋯,s_K] with s_k∼𝒞𝒩(0,1) being the transmitted signal intended for the kth user, y=[y_1,⋯,y_K]^T with y_k being the received signal of the kth user, H^H=[h_1,⋯,h_K] and h_k∈C^M contains the channel coefficients between the BS and the kth user, and n∼𝒞𝒩(0, σ^2I_K) is an additive white gaussian noise (AWGN) vector with independent identically distributed (i.i.d.) entries of zero mean and variance σ^2. In (<ref>), G∈C^S× K is a baseband precoder that maps s to the S RF chains, and F∈C^M× S is a RF precoder using analog circuitry, e.g., the analog phase shifting network. Due to the implementing limitation, the elements of F are often required to have a constant modulus and only change their phases <cit.>. Then, given the RF precoder F, the baseband precoder G, and the instantaneous CSI h_k, ∀ k∈𝒦≜{1,2,⋯,K}, the signal-to-interference-plus-noise ratio (SINR) of the kth user is_k=|h_k^HFg_k|^2/∑_l=1,l≠ k^K|h_k^HFg_l|^2+σ^2,where g_k denotes the kth column of G. §.§ B. Channel Model In this paper, the channel between the BS and each user is modeled as a narrowband clustered channel based on the extended Saleh-Valenzuela model that has been widely used in mmWave communications <cit.>. The channel coefficient vector h_k is assumed to be a sum of the contributions of N_cl scattering clusters, each of which includes N_ray propagation paths. Specifically, h_k can be written as <cit.>h_k=√(M/N_clN_ray)∑_m_p=1^N_cl∑_n_p=1^N_rayα_m_p,n_pa(ϕ_m_p,n_p,θ_m_p,n_p),where α_m_p,n_p is a complex Gaussian random variable with zero mean and variance σ_α,m_p^2 for the n_pth ray in the m_pth scattering cluster, and ϕ_m_p,n_p(θ_m_p,n_p) is its azimuth (elevation) angle of departure (AoD). a(ϕ_m_p,n_p,θ_m_p,n_p) is the normalized array response vector at an azimuth (elevation) angle of ϕ_m_p,n_p(θ_m_p,n_p) and depends on the structure of the transmit antenna array only. The N_ray azimuth and elevation angles of departure ϕ_m_p,n_p and θ_m_p,n_p within the cluster m_p follow the Laplacian distributions with a uniformly-random mean cluster angle of ϕ_m_p and θ_m_p, respectively, and a constant angular spread (standard deviation) of σ_ϕ and σ_θ, respectively <cit.>.In particular, for an M-element uniform linear array (ULA), the array response vector is given by <cit.>a_ULA(ϕ)=√(1/M)[1,e^j2π/λ_sdsin(ϕ),⋯, e^j(M-1)2π/λ_sdsin(ϕ)]^T,where λ_s is the signal wavelength, and d is the inter-element spacing. For uniform planar array (UPA) in the yz-plane with M_1 and M_2 elements on the y and z axes respectively, the array response vector is given by <cit.>a_UPA(ϕ,θ)=√(1/M_1M_2)[1,⋯,e^j2π/λ_sd(m_psin(ϕ)sin(θ)+n_pcos(θ)),⋯, e^j2π/λ_sd((M_1-1)sin(ϕ)sin(θ)+(M_2-1)cos(θ))]^T,where the antenna array size is M_1M_2 and 0≤ m_p< M_1(0≤ n_p< M_2) is the y(z) indices of an antenna element. §.§ C. Problem Formulation The goal of this paper is to design proper RF-baseband hybrid precoders for the mmWave communication system. For this purpose, we consider two common performance measures: the system sum rate (SR) and the system energy efficiency (EE). The problem of maximizing the system SR (SRmax) is formulated as:max_F,G ∑_k=1^KR_k,s.t.R_k=log(1+_k)≥γ_k, ∀ k∈𝒦,F∈ℱ_RF, FG_ℱ^2≤ P.The problem of maximizing the system EE (EEmax) is formulated as: max_F,G ∑_k=1^KR_k/ϵ∑_k=1^KFg_k_2^2+Q_dyn,s.t.R_k=log(1+_k)≥γ_k, ∀ k∈𝒦,F∈ℱ_RF, FG_ℱ^2≤ P. In the above two problems, ℱ_RF is the set of feasible RF precoders, i.e., the set of M× S matrices with constant-modulus entries, γ_k is the target rate of the kth user, P is the maximum allowable transmit power, ϵ≥ 1 is a constant which accounts for the inefficiency of the power amplifier (PA) <cit.>. Q_dyn is the dynamic power consumption, including the power radiation of all circuit blocks in each active RF chain and transmit antenna, given byQ_dyn=g̈_0(P_RFC+MP_PS+P_DAC) +P_sta,where g̈=[g_1_2,⋯, g_S_2]^T with g_m denoting the mth row of G, and the ℓ_0-(quasi)norm g̈_0 is the number of nonzero entries of g̈, i.e., g̈_0=|{t: g_t_2≠ 0}|. P_RFC, P_PS, and P_DAC denote the the power consumption of the RF chain, the phase shifter (PS), and the digital-to-analog converter (DAC) at the transmitter, respectively. P_sta=M(P_PA+P_mixer)+P_BB+P_cool, where P_PA, P_mixer, P_BB, and P_cooldenote the power consumption of the PA, the mixer, the baseband signal processor, and the cooling system, respectively[The proposed framework in the paper can be readily extended to include the power consumption at the receivers.].The formulated problems (<ref>) and (<ref>) are challenging due to several difficulties, including the constant-modulus requirement of F∈ℱ_RF, the coupling between G and F, the nonconvex nature of the user rates and the QoS constraints, and the fractional form of the objective (in problem (<ref>)). Another practical difficulty is the CSIT, which requires in general each user to estimate a large number of channels and feed them back to the BS. Throughout this paper, we assume that the set of user target rates is feasible. In the following, we will address these difficulties and propose efficient precoding designs.§ 3. CODEBOOK BASED MMWAVE PRECODING DESIGN WITH BEAM SWEEPING In the mmWave system, the RF precoder is optimized in the analog domain and required to have a constant modulus. Unlike the digital baseband signal that can be precisely controlled, the RF signal is hard to manipulate and a precise shift for an arbitrary phase is prohibitively expensive in the analog domain. Therefore, in practice, each element of the RF precoder F usually takes only several possible phase shifts, e.g., 8 to 16 choices (3 to 4 bits), while the amplitude change is usually not possible <cit.>. To facilitate the low complexity implementation of the phase shifter, the RF precoder is often selected from a predefined codebook, which contains a limited number of phase shifts with a constant amplitude.An RF codebook can be represented by a matrix, where each column specifies a transmit pattern or an RF beamforming vector. In particular, let F∈ℱ_𝒞ℬ be an M× N predesigned codebook matrix, where N is the number of codewords in the codebook F, and ℱ_𝒞ℬ denotes the space of all M× N constant-modulus RF precoding codewords. There are different RF codebooks, such as the general quantized beamforming codebooks and the beamsteering codebooks.A q-bit resolution beam codebook for an M-element ULA is defined by a codebook matrix F, where each column corresponds to a phase rotation of the antenna elements and generates a specific beam. A q-bit resolution codebook that achieves the uniform maximum gain in all directions with the optimal beamforming weight vectors is expressed as <cit.>F(m,n)=1/√(M)j^4(m-1)(n-1)-2N/2^q, ∀ m∈ℳ, ∀ n∈𝒩,where j denotes the square root of -1, i.e.,j=√(-1), ℳ={1,⋯,M}, 𝒩={1,⋯,N}.The codebooks in IEEE 802.15.3c <cit.> and wireless personal area networks (WPAN) operating in 60 GHz frequency band <cit.> are designed to simplify hardware implementation. The codebooks are generated with a 90-degree phase resolution and without amplitude adjustment to reduce the power consumption. In this case, the (m,n)th element of the codebook F is given by (<ref>), ∀ m∈ℳ, ∀ n∈𝒩.F(m,n)=1/√(M)j^floor(4(m-1)(mod((n-1)+N/4,N))/N).Note that when M or N is larger than 4, the codebooks obtained from (<ref>) result in the beam gain loss in some beam directions, due to the quantized phase shifts per antenna element with a limited 2-bit codebook resolution.In practice, discrete Fourier transform (DFT) codebooks are also widely used as they can achieve higher antenna gains at the beam directions than the codebooks in IEEE 802.15.3c. The entries of a DFT codebook are defined asF(m,n)≜1/√(M)e^-j2π(m-1)(n-1)/M, ∀ m∈ℳ, ∀ n∈𝒩.The DFT codebooks generated in (<ref>) do not suffer any beam gain loss in the given beam directions for any M and N. For mmWave systems, an efficient DFT codebook based MIMO beamforming training scheme was proposed in <cit.> to estimate the antenna weight vectors (AWVs).In Fig. <ref>, we show the polar plots of array factor for two 3-bit resolution codebooks using (<ref>) and (<ref>), and a 2-bit resolution codebook using (<ref>). It can be observed that compared to the 2-bit resolution codebook in IEEE 802.15.3c generated according to (<ref>), the 3-bit resolution beam codebook generated according to (<ref>) and the DFT codebook provide a better resolution and a symmetrical uniform maximum gain pattern with reduced side lobes. Adopting an RF codebook dramatically redeuces the complexity of computing the RF precoder. Indeed, given an RF codebook F, the optimization of the RF precoder F in (<ref>) and (<ref>) is then equivalent to selecting S codewords (columns) from the RF codebook (matrix) F. Moreover, instead of obtaining directly the exact CSIT, we can obtain the equivalent CSIT via a beam-sweep procedure <cit.>. Specifically, during the beam-sweep procedure, the BS sends training packets from each direction defined in the RF codebook F, and the users measure the received signal strength and estimate the effective channel across all directions. Then, each user provides the beam-sweep feedback to the BS, indicating the received signal strength and the effective channel of each direction, i.e., h_k^Hf_n, where f_n is the nth codeword (column) of the RF codebook (matrix) F. Such a beam-sweep procedure is shown in Fig. <ref>. Through the beam sweeping, the original system can be viewed as a virtual multiuser MISO downlink system, as illustrated in Fig. <ref>, where the BS is equipped with N virtual antennas (i.e., codewords) and the channel coefficient between the BS and the kth user is h_k^eff=F^Hh_k, ∀ k∈𝒦. It is well known that a mmWave channel equipped with a directional array usually admits a sparse property in the beam domain <cit.>. That is, the effective channel may be near zero for most codewords f_n in the RF codebook F. As a result, the effective channel coefficient vector h_k^eff is a sparse vector, implying that we only need to feedback a few nonzero effective channel coefficients to the BS. Therefore, by using a RF codebook along with the beam sweeping, the burden of obtaining CSIT in the mmWave system can be relieved.Now, the hybrid precoding design becomes the joint optimization of the RF codeword selection and the baseband precoder. We show that this twofold task can be incorporated into the baseband precoder optimization. Specifically, instead of using the original S× K baseband precoder G, we introduce an expanded baseband precoder G∈C^N× K with size of N× K. Let g_m denote the mth row of G. Then, by multiplying the RF codebook F with G, i.e., FG, the mth codeword in the RF codebook F is selected if and only if g_m is nonzero or equivalently g_m_2≠ 0. Consequently, the original RF-baseband hybrid precoding design problem (<ref>) can be reformulated into the following joint codeword selection and precoder design (JWSPD) SRmax problem: max_G ∑_k=1^K R_k,s.t. R_k=log(1+_k)≥γ_k, ∀ k∈𝒦,∑_k=1^KFg_k_2^2≤ P, g̈_0≤ S, where g_k denotes the kth column of G, g̈=[g_1_2,⋯,g_N_2]^T, the SINR of the kth user is given by_k=|h_k^HFg_k|^2/∑_l=1,l≠ k^K|h_k^HFg_l|^2+σ^2.In (<ref>), the constraint g̈_0≤ S guarantees that the number of the selected codewords is no larger than the number of the available RF chains. Problem (<ref>) represents a sparse formulation of the baseband precoder design as g̈ has up to S≤ N nonzero elements. It also implies that the baseband precoder G is a sparse matrix.Similarly, problem (<ref>) can be reformulated into the following JWSPD EEmax problem: max_G ∑_k=1^KR_k/ϵ∑_k=1^KFg_k_2^2+P_dyns.t.R_k=log(1+_k)≥γ_k, ∀ k∈𝒦,∑_k=1^KFg_k_2^2≤ P, g̈_0≤ S, where P_dyn=g̈_0(P_RFC+M P_PS+P_DAC)+P_sta. Let m_l be the row index of the lth nonzero row vector of G for l=1,⋯, g̈_0 with m_1⩽⋯⩽ m_g̈_0. Without loss of generality, we can let the lth row vector of the baseband precoder be the g_m_l and the lth phase shifter network steer vector be the m_lth codeword in the RF codebook F for the lth RF chain. Then, the remained S-g̈_0 RF chains with the corresponding phase shifter networks can be turned off to save power.So far, we have simplified the original RF-baseband hybrid precoding design into the JWSPD optimization problem. However, problems (<ref>) and (<ref>), although there is only one (matrix) variable G, are still difficult, due to the nonconvex objective, the nonconvex QoS constraint, and the ℓ_0-(quasi)norm constraint g̈_0⩽ S.§ 4. JOINT CODEWORD SELECTION AND PRECODER OPTIMIZATION FOR SRMAX PROBLEM In this section, we consider first the JWSPD SRmax problem (<ref>), which, unfortunately, is NP-hard as a result of the nonconvex (sum rate) objective and the ℓ_0-(quasi)norm constraint. Hence, finding its globally optimal solution requires prohibitive complexity, so in practice an efficient (probably suboptimal) solution is more preferred. In what follows, we will provide such an efficient solution. §.§ A. Joint Codeword Selection and Precoder Design for SRmax problem To address the joint codeword selection and precoder design (JWSPD) in (<ref>), we first introduce some auxiliary variables α_k, β_k, ∀ k∈𝒦, τ, κ, and χ. Let log(1+α_k)≥β_k and _k≥α_k, ∀ k∈𝒦. After some basic operations, (<ref>) can be rewritten into the following equivalent form: min_{g_k, α_k, β_k} -∑_k=1^Kβ_ks.t. 1+α_k≥ e^β_k,∀ k∈𝒦,_k≥α_k, _k≥γ_k, ∀ k∈𝒦,∑_k=1^KFg_k_2^2≤ P, g̈_0≤ S, where γ_k=e^γ_k-1. It can be easily proven that the constraints (<ref>) and _k≥α_k, ∀ k shall be activated at the optimal solution <cit.>. The difficulty lies in (<ref>) and (<ref>), as (<ref>) and g̈_0≤ S are nonconvex constraints. To overcome these difficulties, we first move the constraint g̈_0≤ S into the objective as follows:min_{g_k, α_k, β_k} -∑_k=1^Kβ_k+λg̈_0s.t. 1+α_k≥ e^β_k,∀ k∈𝒦, ∑_k=1^KFg_k_2^2≤ P_k≥α_k, _k≥γ_k, ∀ k∈𝒦, where λ is a group-sparsity inducing regularization <cit.> to control the sparsity of the solution, i.e., the larger λ is, the more sparse solution of (16) is.Therefore, one can always choose a λ large enough such that the constraint g̈_0≤ S is satisfied.Then, we use the convex ℓ_1,∞-norm squared to approximate the nonconvex ℓ_0-(quasi)norm[It is worth pointing out that the RF chain constraint g̈_0≤ S cannot be simply replaced by g̈_p≤ S with p⩾ 1, since it is unknown whether ℓ_0-norm ≥ ℓ_p-norm or ℓ_0-norm < ℓ_p-norm, which may result in a violation of the RF chain constraint.]. In this way, problem (<ref>) is approximated as: min_{g_k, α_k, β_k} -∑_k=1^Kβ_k+λG_1,∞^2s.t. 1+α_k≥ e^β_k,∀ k∈𝒦, ∑_k=1^KFg_k_2^2≤ P_k≥α_k, _k≥γ_k, ∀ k∈𝒦, where G_1,∞=∑_n=1^Nmax_k|g_k(n)| is as the ℓ_1,∞-norm of the matrix G. Note that G_1,∞^2 in (<ref>) can be rewritten as follows:G_1,∞^2 =(∑_n=1^Nmax_k|g_k(n)|)^2=∑_n_1=1^N∑_n_2=1^N((max_k|g_k(n_1)|) (max_k|g_k(n_2)|))=∑_n=1^N∑_m=1^Nmax_i,j∈{1,⋯,K}|X_i,j(n,m)|,where X_i,j=g_ig_j^H, ∀ i,j. Note that X_i,j=g_ig_j^H, ∀ i,j if and only if X_i,j≽0 and (X_i,j)=1, ∀ i,j. Thus, problem (<ref>) can be relaxed to min_{X_i,j, α_k, β_k} -∑_k=1^Kβ_k+λG_1,∞^2,s.t. 1+α_k≥ e^β_k,∀ k∈𝒦, ∑_k=1^K(FX_k,k)≤ P,_k≥α_k, _k≥γ_k,X_k,k≽0, ∀ k∈𝒦,(X_i,j)=1, ∀ i,j, where F=F^HF, and_k=(H_kX_k,k)/∑_l=1,l≠ k^K(H_kX_l,l)+σ^2where H_k=F^Hh_kh_k^HF, ∀ k∈𝒦. The relaxed problem (<ref>) is still difficult as it is still nonconvex. Nevertheless, note that X_i,j, ∀ i≠ j only appear in the objective (<ref>), it is easy to have the following results which can help us simplify (<ref>).Let {X̆_i,j, ᾰ_k, β̆_k} be the optimal solution of (<ref>), then the inequalities |X̆_i,j|⩽|X̆_i,i|, ∀ i≠ j hold. For brevity, let X_k=X_k,k, ∀ k∈𝒦 and define Z(n,m)=max_k∈𝒦|X_k(n,m)|, ∀ m,n. Considering that the rank one constraint is nonconvex <cit.>, we obtain a tractable formulation form of problem (<ref>) by dropping the nonconvex constraints (X_k)=1, ∀ k∈𝒦. According to Theorem <ref>, problem (<ref>) can be relaxed to: min_{X_k, α_k, β_k}, Z -∑_k=1^Kβ_k+λ(1_N× NZ)s.t. 1+α_k≥ e^β_k,∀ k∈𝒦, ∑_k=1^K(FX_k)≤ P,_k≥α_k, _k≥γ_k, ∀ k∈𝒦,X_k≽0, Z≥|X_k|, ∀ k∈𝒦. To address the nonconvex constraints (<ref>), we transform it into the following problem (<ref>), at the top of this page, by introducing auxiliary variables ψ_k, ϕ_k, ∀ k∈𝒦, τ, κ, and χ, min_{X_k, α_k, β_k, ψ_k, ϕ_k},Z -∑_k=1^Kβ_k+λ(1_N× NZ),s.t. ψ_k^2⩽(H_kX_k), X_k≽0, 1+α_k≥ e^β_k,∀ k∈𝒦∑_l=1,l≠ k^K(H_kX_l)+σ^2⩽ϕ_k, ∀ k∈𝒦, ∑_k=1^K(FX_k)≤ P,∑_l=1,l≠ k^Kγ_k(H_kX_l)+γ_kσ^2⩽(H_kX_k),ψ_k^2/ϕ_k⩾α_k,∀ k∈𝒦,[ Z(n,m)-(X_k(n,m))(X_k(n,m));(X_k(n,m)) Z(n,m)+(X_k(n,m)) ]≽0, ∀ k∈𝒦,m,n. The difficulty of solving (<ref>) lies in (<ref>), as the constraints ψ_k^2/ϕ_k⩾α_k, ∀ k are nonconvex. To overcome this difficulty, we exploit the SCA method <cit.> to approximate the inequality ψ_k^2/ϕ_k⩾α_k, ∀ k by its convex low boundary asψ_k^2/ϕ_k≥Φ_k^(I)(ψ_k,ϕ_k) ≜ 2ψ_k^(I)/ϕ_k^(I)ψ_k -(ψ_k^(I)/ϕ_k^(I))^2ϕ_k, ∀ k∈𝒦,where the superscript I denotes the Ith iteration of the SCA method. Note that Φ_k^(I)(ψ_k,ϕ_k) is in fact the first order of ψ_k^2/ϕ_k around the point (ψ_k^(I), ϕ_k^(I)). Thus, the approximate convex problem solved at iteration I+1 of (<ref>) is given by: min_{X_k, α_k, β_k, ψ_k, ϕ_k},Z -∑_k=1^Kβ_k+λ(1_N× NZ),s.t. (<ref>), (<ref>), (<ref>),Φ_k^(I)(ψ_k,ϕ_k)⩾α_k, ∀ k∈𝒦,∑_l=1,l≠ k^Kγ_k(H_kX_l)+γ_kσ^2⩽(H_kX_k),∀ k∈𝒦, which can be solved efficiently via a modern convex solver such as MOSEK <cit.>. For conciseness, let Ξ^(I) denote the set of all variables in problem (<ref>) at the Ith iteration. Algorithm <ref> outlines an iterative procedure for finding a solution to problem (<ref>) (or equivalently (<ref>)) with a fixed λ, where τ denotes the objective of problem (<ref>). Problem (<ref>) consists of a linear objective function, K(M^2+1) positive-semidefinite constraints, 5K linear inequality constraints, and one convex constraint. It can be solved via convex optimization methods, such the interior point method <cit.>. The interior point method will take 𝒪(√(KM)log(ϵ)) iterations, where the parameter ϵ represents the solution accuracy at the algorithm's termination. In each iteration, the complexity of solving (<ref>) is ((M^6+64)K^3+6K^2M^2) <cit.>. The optimal solution returned at theIth iteration is also feasible for the problem at the (I + 1)th iteration, as a result of the approximation in (<ref>). Hence, Algorithm <ref> yields a nondecreasing sequence. Since the objective of problem (<ref>) is bounded under the limited transmit power, the convergence of Algorithm <ref> is guaranteed <cit.>. In addition, following the similar arguments in <cit.>, it can be proved that Algorithm <ref> converges to a Karush-Kunhn-Tuker (KKT) solution of problem (<ref>) <cit.>. To obtain a good initial point Ξ^(0) for Algorithm <ref>, one can solve the problem (<ref>) which was extensively studied in <cit.>.min_{X_k}, Z (1_N× NZ)s.t.(<ref>),(<ref>), X_k≽0, ∀ k∈𝒦. Let Ξ^λ denote the optimal solution to problem (<ref>) with fixed λ. By definition, the nonzero diagonal entries of Z^λ correspond to the selected virtual antennas (codewords). If an entry of Z^λ is zero, then the corresponding entry in all X_k^λ, ∀ k must be zero. Let L^λ be the number of nonzero diagonal entries of Z^λ. Then, the effective channel of the kth user is an L^λ× 1 vector h_k=F^Hh_k where the columns of F are the L^λ selected codewords from the RF codebook F. Thus, the analog precoder F is obtained as F=F^H.§.§ B. Sparse Parameter for SRmax problem In the previous subsection, we have introduced a turnable sparse parameter λ to control the sparsity of the solution of the JWSPD optimization. In this subsection, we investigate how to choose a proper λ to satisfy the RF chain constraint g̈_0≤ S. Note that in (<ref>), a larger λ makes the entries of Z (as well as X_k, ∀ k∈𝒦) more sparse, implying that less RF chains are used. On the other side, to maximize the system SR and guarantee the target rate requirement of each user, one cannot force all entries of X_k, ∀ k∈𝒦 to be zero. Thus, the sparse parameter λ has to be properly chosen to balance maximizing the system SR and minimizing the number of the selected virtual antennas (codewords).It is not difficult to find that the system SR increases with the number of the RF chains. Therefore, the task of find the minimum λ such that the RF chain constraint g̈_0≤ S is satisfied can be accomplished by the classical one-dimension search methods, such as the bisection method <cit.>. For completeness, the algorithm used to find the proper sparse parameter λ such that g̈_0≤ S is summarized in Algorithm <ref>, where Λ^λ and τ^λ denote respectively the set of the solution of (<ref>) and the value of ∑_k=1^Kβ_k with λ, Λ^T and τ^T denote respectively the set of the temporary solution of (<ref>) and the temporary value of ∑_k=1^Kβ_k with λ. Note that the initialization of Algorithm <ref> can also be finished by solving (<ref>). §.§ C. Refined Solution for SRmax Problem Recall that in the previous subsections, the ℓ_0(quasi)-norm has been approximated by the mixed ℓ_1,∞-norm squared to obtain a tractable solution. In addition, due to dropping the nonconvex rank constraint in (<ref>), the solution X_k, ∀ k∈𝒦 obtained by solving (<ref>) may not be rank one. Thus, the solution provided by (<ref>) has to be refined to fit the original problem (<ref>). For this purpose, after obtaining an approximate solution to (<ref>), we propose to solve a size-reduced SRmax problem as the last step, omitting the antennas corresponding to the zero diagonal entries of the approximated sparse solution Z. The size-reduced SRmax problem is given by: max_{g_k} ∑_k=1^KR_k,s.t. _k≥γ_k, ∀ k∈𝒦,∑_k=1^KFg_k_2^2≤ P, where R_k=log(1+_k), and _k is given by_k≜h_k^Hg_k_2^2/∑_l=1,l≠ k^Kh_k^Hg_l_2^2+σ^2.Similarly, the size-reduced SRmax problem (<ref>) can be equivalently reformulated as: max_{g_k,α_k, β_k, ϕ_k} ∑_k=1^Kβ_k,s.t. 1+α_k⩾ e^β_k, ∀ k∈𝒦, ∑_k=1^KFg_k_2^2≤ P, h_k^Hg_k_2^2/ϕ_k≥γ_k, ∀h_k^Hg_k_2^2/ϕ_k≥α_k, ∀ k∈𝒦,∑_l=1,l≠ k^Kh_k^Hg_l_2^2 +σ^2⩽ϕ_k, ∀ k∈𝒦. Similar to the problem (<ref>), (<ref>) is also a nonconvex problem due to the constraints in (<ref>). For (<ref>), we have the following the convex low boundary:h_k^Hg_k_2^2/ϕ_k≥Φ_k^(I)(g_k,ϕ_k) ≜2((g_k^(I))^Hh_kh_k^Hg_k)/ϕ_k^(I) -(h_k^Hg_k^(I)_2/ϕ_k^(I))^2ϕ_k, ∀ k∈𝒦,where I denotes the Ith iteration. Thus, the constraints in (<ref>) can be approximated as:Φ_k^(I)(g_k,ϕ_k)≥γ_k,Φ_k^(I)(g_k,ϕ_k)≥α_k, ∀ k∈𝒦.Consequently we can obtain a stationary solution to (<ref>), by solving the following series of convex problems:max_{g_k,α_k, β_k, ϕ_k} ∑_k=1^Kβ_k, s.t. (<ref>), (<ref>), (<ref>).Such an iterative procedure is outlined in Algorithm <ref>, where Ξ^(I) and τ^(I) denote the set of the solution and the objective value of problem (<ref>) at the Ith iteration, respectively. The convergence property of Algorithm <ref> is similar with that of Algorithm <ref>. The computational complexity of Algorithm <ref> is about 𝒪(M^4K^4) <cit.>. In the next,we investigate how to obtain a good initial point for Algorithm <ref>. Let g_k=√(q_k)g_k, ∀ k∈𝒦[It is easy to find that (<ref>) is a weighted sum power minimization problem which can be regarded as an extension ofthe conventional power minimization problem.]. We propose to use the solution of the following problem as the initial point:min_{q_k, g_k}∑_k=1^Kq_kg_k^HF^HFg_k s.t. _k≥γ_k, g_k_2^2=1, ∀ k∈𝒦.We can show that problem (<ref>) is dual to the following virtual uplink problem <cit.>:min_{p_k, g_k}σ^2∑_k=1^Kp_k s.t. _k≥γ_k, g_k_2^2=1, ∀ k∈𝒦,where g_k can be regarded as the combiner of the dual uplink channel, p_k has the interpretation of being the dual uplink power kth user in the virtual uplink, and _k is given by_k≜p_kh_k^Hg_k_2^2/∑_l=1,l≠ k^Kp_lh_l^Hg_k_2^2+g_k^HF^HFg_k.Furthermore, when the optimal solutions of problems (<ref>) and (<ref>) are obtained, we have ∑_k=1^Kq_kg_k^HF^HFg_k=σ^2∑_k=1^Kp_k. It was shown in <cit.> that the solution {g_k} of (<ref>) is given byg_k^*∝((∑_l=1,l≠ k^Kp_lH_l+F^HF)^-1H_k).Thus, the algorithm used to solve (<ref>) is summarized in Algorithm <ref> with provable convergence <cit.>. To find {q_k} in terms of {g_k} that is obtained from the virtual uplink channel, i.e., (<ref>), we note that the SINR constraints in (<ref>) must be all actived at the global optimum point. Soq_k=∑_l=1,l≠ k^Kq_lγ_k/h_k^Hg_k_2^2h_k^Hg_l_2^2+σ^2γ_k/h_k^Hg_k_2^2, ∀ k∈𝒦.Thus, we obtain a set of K linear equations with K unknowns {q_k}, which can be solved asq=ΨGq+σ^2Ψ1_K,where q=[q_1,⋯,q_K]^T, Ψ=diag{γ_1/h_1^Hg_1_2^2,⋯, γ_K/h_K^Hg_K_2^2}, G(k,k)=0 and G(k,l)=h_k^Hg_l_2^2 for k≠ l. Defining an extended power vector q=[q^T,1]^T and an extended coupling matrixQ=[ ΨG Ψ1_K; 1/P_maxa^TΨG 1/P_maxa^TΨ1_K ].where P_max=σ^2∑_k=1^Kp_k, a^T=[a_1,⋯,a_K], a_k=g_k^HF^HFg_k, ∀ k. According to the conclusions in <cit.>, we can easily obtain the optimal power vector q as the first K components of the dominant eigenvector of Q, which can be scaled such that its last component equals one. The solution for {q_k}, combined with that for {g_k}, gives an explicit solution of the beamforming vector {g_k} via an virtual uplink channel. Once the beamforming vector {g_k} is obtained, the baseband beamforming vector g_k is obtained, as g_k=[{g_k}^T,0_(S-L^λ),1^T]^T. In fact, the remaining S-L^λ RF chains with the corresponding phase shifter networks can be turned off to improve the system EE.§ 5. JOINT CODEWORD SELECTION AND PRECODER OPTIMIZATION FOR EEMAX PROBLEM In this section, we consider the EEmax problem (<ref>), which is more difficult than the SRmax problem. Indeed the objective in (<ref>) is given by a more complex fractional form, and the ℓ_0-(quasi)norm appears not only in the constraint but also in the denominator of the objective. To find the globally optimal solution to (<ref>) requires an exhaustive search over all∑_l=L_Min^S(N l) possible sparse patterns of g̈, where L_Min⩽ S is the minimum number of the selected RF chains that can achieve the target rate requirement of each user under the power constraint. Unfortunately, for each pattern of g̈, (<ref>) is an NP-hard problem. Thus, we seek a practical and efficient method to address the EEmax problem (<ref>). §.§ A. Joint Codeword Selection and Precoder Design for EEmax problem Similarly, we first use the convex squared ℓ_1,∞-norm to approximate the nonconvex ℓ_0-(quasi)norm in the power consumption term P_dyn. Then, we also introduce a turnable sparse parameter λ≥ 0 as a group-sparsity inducing regularization to control the sparsity of the solution so that the RF chain constraint (<ref>) can be temporarily omitted for fixed λ. By doing so, problem (<ref>) can be relaxed as:max_{X_i,j} ∑_k=1^K R_k/ϵ∑_k=1^K(FX_k,k)+P_dyn(λ),s.t. _k≥γ_k, ∀ k, ∑_k=1^K(FX_k,k)≤ P,X_k,k≽0, ∀ k∈𝒦, where the nonconvex (X_i,j)=1, ∀ i,j constraints are dropped, and the dynamic power consumption is given byP_dyn(λ)=f(λ)∑_n=1^N∑_m=1^Nmax_i,j∈{1,⋯,K}|X_i,j(n,m)|+P_sta,where f(λ)=P_RFC+M P_PS+P_DAC+λ. Note that X_i,j, ∀ i≠ j, only appear in the power consumption item P_dyn(λ). Therefore, similar to Theorem <ref>, we have the following result.Let X̆_i,j, ∀ i,j∈𝒦 be the optimal solution of (<ref>), then X̆_i,i, ∀ i∈𝒦 with X_i,j=0, ∀ i≠ j, i,j∈𝒦 is also the optimal solution of (<ref>). First, we prove that the inequalities |X̆_i,j|⩽|X̆_i,i|, ∀ i≠ j hold. Suppose that there is one pair of indices (i_0,j_0), i_0≠ j_0 and (n_0,m_0) such that |X̆_i,j(n,m)|⩽|X̆_i,i(n,m)|, ∀ i≠ j,n, m except for |X̆_i_0,j_0(n_0,m_0)|≥|X̆_i,i(n_0,m_0)|, ∀ k. Let X_i,j, ∀ i,j∈𝒦 be another solution obtained by letting X_i,j(n,m)=X̆_i,j(n,m), ∀ i, j, n, m except for X_i_0,j_0(n_0,m_0)=0. Note that X_i,j,∀ i≠ j only appear in the constraints (<ref>). Thus, X_i,j, ∀ i,j∈𝒦 is a feasible solution to problem (<ref>) and satisfies the following inequality (<ref>).P̆_dyn(λ) =f(λ)(|X̆_i_0,j_0(n_0,m_0)|+ ∑_n=1^N∑_m=1^N_(n,m)≠(n_0,m_0)max_i|X̆_i,i(n,m)|)+P_sta>P_dyn(λ)=f(λ)∑_n=1^N∑_m=1^Nmax_i|X_i,i(n,m)|+P_sta.Note that X̆_i,j and X_i,j, ∀ i,j∈𝒦 achieve the same user rate. Combining the objective of problem (<ref>) and (<ref>), we can obtain a better objective by using X_i,j, ∀ i,j∈𝒦 than using X̆_i,j, ∀ i,j∈𝒦, which is a contradiction. Therefore, we have |X̆_i,j|⩽|X̆_i,i|, ∀ i≠ j.Note that X_i,j, ∀ i≠ j, only appear in the power consumption item P_dyn(λ). Combining |X̆_i,j|⩽|X̆_i,i|, ∀ i≠ j with (<ref>), one can easily see that the power consumption item P_dyn(λ) dose not change by setting X_i,j=0, ∀ i≠ j. Consequently, X̆_i,i, ∀ i∈𝒦 with X_i,j=0, ∀ i≠ j are still optimal.Theorem <ref> also indicates that we can simplify problem (<ref>) by setting X_i,j=0, ∀ i≠ j without any loss of optimality. Hence, similar to the transformation between (<ref>) and (<ref>), (<ref>) is equivalent to max_{X_k}, Z ∑_k=1^K R_k/ϵ∑_k=1^K(FX_k)+P_dyn(Z,λ),s.t. _k≥γ_k, ∀ k∈𝒦, ∑_k=1^K(FX_k)≤ P,X_k≽0, Z≥|X_k|, ∀ k∈𝒦, where P_dyn(Z,λ)=f(λ)(1_N× NZ)+P_sta. Introducing auxiliary variables α_k, β_k, ψ_k, ϕ_k, ∀ k∈𝒦, τ, κ, and χ, (<ref>) can be equivalently rewritten as max_{X_k, α_k, β_k, ψ_k, ϕ_k},Z, τ, κ, χ χ,s.t. τ^2/κ⩾χ,ψ_k^2/ϕ_k⩾α_k, ∀ k∈𝒦∑_k=1^Kβ_k⩾τ^2, (<ref>), (<ref>), (<ref>)ϵ∑_k=1^K(FX_k) +P_dyn(Z,λ)⩽κ∑_l=1,l≠ k^Kγ_k(H_kX_l)+γ_kσ^2⩽(H_kX_k),∀ k∈𝒦 Similarly, the difficulty of solving (<ref>) lies in (<ref>), as the two constraints in (<ref>) are nonconvex. Thus, we exploit the SCA method <cit.> to approximate the two inequalities in (<ref>) by two convex constraints. By replacing (<ref>) with the convex lower bounds at the Ithiteration, problem (<ref>) can be approximated by the following convex program: max_{X_k, α_k, β_k, ψ_k, ϕ_k, μ_k},Z, τ, κ, χ χ,s.t. (<ref>), (<ref>), (<ref>), Ψ^(I)(τ,κ)⩾χ,Φ_k^(I)(ψ_k,ϕ_k)⩾α_k, ∀ k∈𝒦, where Ψ^(I)(τ,κ)≜ 2τ^(I)/κ^(I)τ -(τ^(I)/κ^(I))^2κ. Thus, problem (<ref>) can be solved via the similar procedure as described in Algorithm <ref>. §.§ B. Sparse Parameter for EEmax problemSimilarly, a larger λ leads to a more sparse solution to the (approximated) EEmax problem (<ref>), which corresponds to less RF chains used. On the other side, λ cannot be infinite, which would lead to a zero solution and contradict the task of maximizing the system EE. Hence, λ has to be properly chosen. However, unlike to the SRmax problem (<ref>) or the total power minimization problem with RF chain constraints <cit.>, the system EE is not monotonic with respect to the number of RF chains or the sparse parameter λ. Indeed, the system EE is a piecewise function with respect to the sparse parameter λ, as illustrated in Fig. <ref> and Table <ref>. Consequently, the bisection method cannot be used to optimize λ <cit.>.To address the above issue, we devise a dynamic interval compression method to search a suitable λ. Specifically, let 𝒜_Min be the set of the indices of the L_Min selected virtual antennas (codewords). Let L_Max be the number of the virtual antennas (codewords) achieving the maximum EE by ignoring the available RF chain constraint, which correspondes to λ=0, and 𝒜_Max be the set of the indices of the L_Max selected virtual antennas (codewords) in this case. Considering that the allowable number of RF chains is a discrete value but the sparse parameter λ is continuous, we introduce the following definition.For any small positive number ϵ and ∀ L∈{L_Min,⋯, L_Max-1}, λ_L^key is called a breaking point if the optimal solutions Z^λ_L^key-ε and Z^λ_L^key of (<ref>) have L+1 and L nonzero diagonal entries, respectively. Let Ξ^λ be the solution of (<ref>)with ∀λ∈[λ_L+1^key,λ_L^key) and L, L+1∈{L_Min, L_Min+1,⋯, L_Max}. Then, Z^λhas also L+1 nonzero diagonal entries and the inequality χ^λ⩽χ^λ_L+1^key holds. Following the definition of the breaking point λ_L+1^key, it is easy to see that Z^λ has also L+1 nonzero diagonal entries. If χ^λ>χ^λ_L+1^key, recalling λ_L+1^key⩽λ, then we have (<ref>),χ^λ_L+1^key=(τ^λ_L+1^key)^2/κ^λ_L+1^key =(τ^λ_L+1^key)^2/ϵ∑_k=1^K(FX_k^λ_L+1^key)+f(λ_L+1^key)(1_N× NZ^λ_L+1^key)+P_sta<χ^λ=(τ^λ)^2/κ^λ =(τ^λ)^2/ϵ∑_k=1^K(FX_k^λ)+f(λ)(1_N× NZ^λ)+P_sta<(τ^λ)^2/ϵ∑_k=1^K(FX_k^λ)+f(λ_L+1^key)(1_N× NZ^λ)+P_sts,which contradicts the fact that Ξ^λ_L+1^key is the optimal solution to problem (<ref>) with fixed λ_L+1^key. Thus, the conclusions given in Theorem <ref> are proven. According the definition of the breaking point and the non-monotonic property of the system EE with respect to λ, one shall find the values of all breaking points. Let Z^λ_i be the solution of problem (<ref>) with fixed λ_i, i=1,2. Let Z^λ be the solution of problem (<ref>) for ∀λ∈[λ_1,λ_2]. Theorem <ref> implies that if Z^λ_1 and Z^λ_2 have the same number of the nonzero diagonal entries, Z^λ_1, Z^λ_2, and Z^λ have the same number of the nonzero diagonal entries. Based on this result, we propose a one-dimension dynamic interval compression method, which is summarized in Algorithm <ref>, to find a suitable sparse parameter λ and obtain the corresponding codewords. Note that in Algorithm <ref>, 𝒜^λ denotes the set of the indices of the selected virtual antennas (codewords) with fixed λ and ϱ^λ is calculated as∑_k=1^Klog(1+tr(H_kX_k)/∑_l=1, l≠ k^Ktr(H_kX_l)+σ_k^2)/ξ∑_k=1^Ktr(X_k) +L^λ(P_RCF+M P_PS+P_DAC)+P_sta.The initialization of Algorithm <ref> can also be obtained by solving (<ref>) and letting other constraints to be activated. In addition, λ_U should be large enough such that the number of the active RF chains equals to L_Min. According to Theorem <ref>, if two intervals in the intervals set ℐ have an intersection in the intervals set ℐ, they shall be combined into one interval, for example [100,150] and [150,200] are combined to [100,200].§.§ C. Refined Solution for EEmax problem Due to the introduction of the mixed ℓ_1,∞-norm squared for the selection of the RF chains in the previous subsections, the energy efficient beamforming vector cannot be directly extracted from the solution of (<ref>), i.e., {X_k}. Therefore, we need to construct the reduced-size channel h_k=F^Hh_k according to the codewords selected by Algorithm <ref>. Thus, the reduce-sized EEmax problem is given by max_{g_k} ∑_k=1^KR_k/ϵ∑_k=1^KFg_k_2^2+P_dyn^*,s.t. _k≥γ_k, ∀ k∈𝒦, ∑_k=1^KFg_k_2^2≤ P, where P_dyn^λ_L^key=L^λ_L^key(P_RFC+ M P_PS+P_DAC)+P_sta. Problem (<ref>) can be formulated as: max_{g_k,α_k, β_k, ϕ_k},τ, χ, κ χ,s.t. ∑_k=1^Kβ_k⩾τ^2, τ^2/κ⩾χ,ϵ∑_k=1^KFg_k_2^2+P_dyn^*⩽κ,(<ref>), (<ref>), (<ref>). Similarly, instead of directly solving (<ref>), we resort to solving the following convex approximated problem max_{g_k,α_k, β_k, ϕ_k},τ, χ, κ χ,s.t. ∑_k=1^Kβ_k⩾τ^2, Ψ^(I)(τ,κ)⩾χ, (<ref>), (<ref>), where I denotes the Ith iteration, and Ψ^(I)(τ,κ) is given byΨ^(I)(τ,κ)≜ 2τ^(I)/κ^(I)τ -(τ^(I)/κ^(I))^2κThus, problem (<ref>) can be solved in a similar manner as described in Algorithm <ref>. § 6. NUMERICAL RESULTS In this section, we present numerical results to demonstrate the performance of our developed RF-baseband hybrid precoding design. A uniform linear array with antenna spacing equal to a half wavelength is adopted, and the RF phase shifters use quantized phases. The predesigned codebook F is the DFT codebook. The propagation environment is modeled as N_cl=6 with N_ray=8 for each cluster with Laplacian distributed angles of departure. For simplicity, we assume that all clusters are of equal power, i.e., σ_α,m_p^2=σ_α^2, ∀ m_p <cit.>. The mean cluster angle of ϕ_m_p is uniformly distributed over [-π, π), and the constant angular spread of AoD σ_ϕ is 7.5^o. P_RFC=43 mW, P_PA= 20 mW, P_DAC= 200 mW, P_PS = 30 mW, P_mixer= 19 mW, P_BB=300 mW, and P_cool=200 mW <cit.>. The noise power spectrum density is σ^2=1. For fairness, all simulated precoding designs use the same total power constraint and the signal-to-noise ratio is defined as SNR=10log_10(P/σ^2). The inefficiency factor of power amplifier ϵ is set to unit and the stop thresholdis ζ=10^-3. In all simulation figures, the simulated EE of the system is given by∑_k=1^Klog_2(1+h_k^HFg_k_2^2/∑_l=1,l≠ k^Kh_k^HFg_l_2^2+σ^2)/ϵ∑_k=1^KFg_k_2^2+P_dyn^*.We compare the performance of the proposed strategy to the optimal fully digital precoder with one RF chain per antenna, whose EE is calculated as∑_k=1^Klog_2(1+h_k^Hg_k_2^2/∑_l=1,l≠ k^Kh_k^Hg_l_2^2+σ^2)/ϵ∑_k=1^Kg_k_2^2+M(P_RFC+P_DAC+P_PA)+P_BB+P_cool. In our simulation scenario, fully digital precoding denotes using Algorithm 3 to solve the SRmax problem, where each antenna connects with an independent RF channel at the BS. Fully analog beamforming is achieved by selecting the best codeword from the codebook via beam training and setting the baseband precoder as an identity matrix with uniform power allocation between users. OMP SRmax Hybrid Precoding uses the orthogonal matching pursuit method <cit.> to obtain the RF-baseband precoders based on the solution of the fully digital SRmax problem. Fig. <ref> and Fig. <ref> show the SR performance of various hybrid precoding designs as well as the fully digital precoder. The results are obtained by averaging over 1000 random channel realizations. The target rate of the kth user is set to be zero. Numerical results show that the proposed hybrid precoding design achieves the highest SR among several hybrid RF-baseband precoders. This is because the proposed hybrid RF-baseband precoding design provides a more flexible way to achieve the beam diversity gain. One can see that the fully analog beamforming method results in the worst SR performance, indicating that the inter-user interference cannot be effectively suppressed. Fig. <ref> illustrates the change of the objective χ of (<ref>) versus an increasing λ for two random channel realizations. The target rate of the kth user is set to be the rate achieved by randomly selecting S analog codeword from codebook F and using the baseband precoder as G=P/Kυ_max^(K)(HF). Simulation results show that there exist indeed breaking points of λ, i.e., the number of selected RF chains keep unchanged within a range of λ but suddenly changes at some points. One can observe that the number of selected RF chains decreases with an increasing value of λ. Within a certain interval of λ, the number of selected RF chains keep the same but the objective χ of (<ref>) decreases when λ increases. Table <ref> lists the set of the indices of the selected RF chains corresponding to the channel realization used in Fig. <ref>. One can observe that the same set of the RF chains (or codewords) are selected for any λ∈[λ_L+1^key,λ_L^key). This observation is consistent with the result in Theorem <ref>, which has been used in Algorithm <ref>. Fig. <ref> and Fig. <ref> illustrates the SR and EE of various mmWave precoding designs and the fully digital precoding design, respectively. The results are obtained by averaging over 1000 random channel realizations. The target rate of kth user is set to be the rate achieved by selecting randomly S analog codeword from codebook F and defining the baseband precoder as G=P/Kυ_max^(K)(HF). It is observed that using the DFT codebook in the proposed EEmax precoder is better than using the 802.15.3c codebook in terms of the EE performance, while the two codebooks lead to the similar SR performance. Compared to the fully digital precoder, the proposed SRmax/EEmax hybrid precoders have certain system SR performance loss as the RF-baseband hybrid architecture may not fully exploit the multi-path diversity gain. The circuit power consumption of the hybrid architecture increases with the number of phase shifter and the number of mixers, which are determined by the number of transmit antennas and the number of RF chains. Therefore, the system EE performance of the hybrid precoding is also determined by the number of transmit antennas and the number of RF chains. One can see that different configuration of the number of antennas, RF chains, and users leads to different system EE performance. For example, for the configuration of M=N=32, S=4, and K=2, the EE performance of the hybrid precoders is better than that of the fully digital precoder. Besides, the fully digital precoder leads to a much higher hardware cost (M=32 RF chains versus S=8 RF chains), which is critical for mmWave communication systems using GHz bandwidth and even higher sampling rates. § 7. CONCLUSIONS In this paper, we considered the design of the hybrid RF-baseband precoding for the downlink of multiuser multi-antenna systems with the aim to maximize the system SR and the system EE. We developed a codebook based RF precoding method and obtained the channel state information via a beam sweep procedure. Exploiting the codebook based design, we simplified the complicated hybrid precoders optimization problems to JWSPD problems. Then, efficient methods were developed to address the JWSPD problems for maximizing the SR and EE of the system. 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"authors": [
"Shiwen He",
"Jiaheng Wang",
"Yongming Huang",
"Bjorn Ottersten",
"Wei Hong"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170627034839",
"title": "Codebook Based Hybrid Precoding for Millimeter Wave Multiuser Systems"
} |
On the Light Massive Flavor Dependence of the Large Order Asymptotic Behavior and the Ambiguityof the Pole Mass [ December 30, 2023 =================================================================================================================Modjtaba Shokrian Zini [[email protected]],[1]1, Zhenghan Wang [[email protected]; [email protected]],[2]2We provide a mathematical definition of a low energy scaling limit of a sequence of general non-relativistic quantum theories in any dimension, and apply our formalism to anyonic chains.We formulate [cnj4.3]Conjecture 4.3 on conditions when a chiral unitary rational (1+1)-conformal field theory would arise as such a limit and verify the conjecture for the Ising minimal model M(4,3) using Ising anyonic chains.Part of the conjecture is a precise relation between Temperley-Lieb generators {e_i} and some finite stage operators of the Virasoro generators {L_m+L_-m} and {i(L_m-L_-m)} for unitary minimal models M(k+2,k+1) in [cnj5.5]Conjecture 5.5.A similar earlier relation is known as the Koo-Saleur formula in the physics literature <cit.>. Assuming [cnj4.3]Conjecture 4.3, most of our main results for the Ising minimal model M(4,3) hold for unitary minimal models M(k+2,k+1), k≥ 3 as well. Our approach is inspired by an eventual application to an efficient simulation of conformal field theories by quantum computers, and supported by extensive numerical simulation and physical proofs in the physics literature. § INTRODUCTION Quantum field theory (QFT) is arguably the best experimentally tested model of Nature, yet its mathematically rigorous formulation is far from clear as exemplified by the Yang-Mills existence and mass gap millennium problem (recently Seiberg articulated that QFT is not even mature physically <cit.>).Besides the intrinsic beauty of a mathematical formulation, a mathematical definition will provide the missing foundation for proving the conjecture that all theoretically physical QFTs can be efficiently simulated by quantum computers. The circuit model of quantum computing is based on quantum mechanics, but it was stated explicitly as a conjecture in <cit.> that QFTs would not provide extra computational power beyond quantum mechanics as suggested by the efficient simulation of (2+1)-topological quantum field theories (TQFTs) <cit.>.Quantum estimate of scattering probabilities in massive scalar quantum field theories also supports such an extended quantum polynomial Church thesis <cit.> if the convergence of lattice models to the continuum is addressed mathematically.Our approach to a potential quantum simulation of rational (1+1)-conformal field theories (CFTs) would follow closely the efficient quantum simulation of TQFTs in <cit.> using the functorial definitions of TQFTs and CFTs.An important difference between TQFTs and CFTs is that, while TQFTs being realized as gapped quantum systems, CFTs represent universality classes of gapless critical phases.Our program seems to be a first attempt towards a quantum simulation of gapless QFTs mathematically.While the efficient simulation of CFTs is not addressed in detail in this paper, the approach to CFTs here is inspired by this eventual application as outlined in the last section.How to represent CFTs on quantum computer is already a challenging problem.The main issues that we are addressing in this paper are the algorithmic convergence of finite lattice theories to the continuum limit with an emphasis on the convergence of observables as an algebra, and hidden localities of CFTs with respect to both space and energy as required by a quantum simulation.Another closely related motivation is to study CFTs based on the significant new insights of TQFTs from their applications to topological phases of matter. From this angle, our paper is a second possible answer to the question how to recover a chiral CFT or vertex operator algebra (VOA) from a TQFT or modular tensor category (MTC) (another one in <cit.>).Our anyonic chain approach to chiral CFTs should encompass all other formulations such as VOAs and local conformal nets (LCNs) as anyonic chains (ACs) are finite versions of microscopic models of CFTs, and we succeed in proving that the algebra of local observables in VOAs and LCNs for unitary minimal models (assuming [cnj4.3]Conjecture 4.3 beyond Ising) can be recovered from our formalism.CFTs and TQFTs are exemplars of QFTs that both have rigorous mathematical formulations and important mathematical and physical applications such as in the monster Moonshine conjecture, critical statistical mechanics models and topological phases of matter, respectively.CFTs and TQFTs are closely related to each other first by the bulk-edge correspondence as in the fractional quantum Hall effect, and secondly their algebraic data, VOAs and MTCs respectively, are conjectured to be Tannaka-Krein dual to each other (see <cit.> and the references therein).The bulk-edge correspondence and Tannaka-Krein duality suggest the possibility that VOAs can be reconstructed from MTCs as a generalization of the reconstruction of compact groups from symmetric fusion categories, though the correspondence is far from one-to-one. Locality is a salient feature for any physical QFT.Since TQFTs are low energy effective theories, their locality is not intrinsic and usually hidden.For example, in the Witten-Chern-Simons (WCS) modeling of the fractional quantum Hall liquids, the WCS theory is an effective description for the emergent anyons, it follows that locality of WCS TQFTs should be derived from that of the underlying electron systems.The simulation of TQFTs in <cit.> uses a hidden locality given by pairs of pants decomposition of the space surfaces.Similarly, there are no intrinsic infinite degrees of freedom for a CFT to define locality.Anyons are modelled by simple objects in unitary MTCs. ACs are the anyonic analogues of quantum Heisenberg spin chains investigated purely as an academic curiosity <cit.>.ACs' conceptual origin can be traced back at least to Jones' Baxterization of braid group representations and his idea of generalized spin chains regarding "spins" as something each with a large algebra of observables at sites and being tensored together with generalized tensor products such as Connes fusion (see section 4 of <cit.>.) In the scaling limit, ACs are exactly solvable but not known to be rigorously solvable mathematically <cit.>.We reverse the logic in this paper to regard ACs as localization of CFTs, thus provide a space locality for VOAs.Our philosophy, as inspired by algorithmic discrete mathematics, is that instead of using ACs to approximate VOAs, VOAs serve as good approximations of sufficiently large finite ACs in their low energy spectrum.Our limit of a sequence of quantum theories {(𝒲_n,H_n)} will be dictated by both space and energy localities.The Hilbert space 𝒲 of a quantum theory has two important bases: the basis encoding the spacial locality, and the basis of energy eigenstates of H. The two bases will be referred to as space basis and energy basis, respectively.Operators can be local with respect to one of the two bases, but there is a tension of locality with respect to both bases.To define a limit of the sequence of quantum theories {(𝒲_n,H_n)}, embedding the Hilbert space 𝒲_n into 𝒲_n+1 is the first step.Which locality of space and energy is preserved by the embedding leads to different notions of limit.We will construct the scaling limit of a sequence of quantum theories {(𝒲_n,H_n)} from their low energy behaviors when the lattice sizes go to zero, therefore we preserve energy locality.Preservation of space locality will lead to the thermodynamic limit. Besides the Hilbert space and Hamiltonian, another essential feature of any quantum theory is the algebra of observables. Since our quantum theories are non-relativistic, time needs to be addressed separately.The algebra structure of observables encodes consecutive measurements as multiplication, hence somewhat reflects time in the limit.As noted above, our formulation of scaling limit will have everything that can be computed using some limit of physical objects. Compared to other well-established formulations of chiral CFTs such as VOAs following Wightman's axioms,and LCNs, our scaling limit results in a much bigger set of observables. In fact, we will show, in the case of Ising anyonic chain, the resulting observables contain a subset corresponding to smeared fields (or Wightman's) observables ϕ(f), a subset corresponding to bounded observables of LCN and a subset corresponding to observables in the VOA M(4,3). We conjecture the same holds for all unitary minimal models M(k+2,k+1) for k≥ 3.An important desideratum of our scaling limit is finitely complete and accessible in the sense that any sequence that should have a limit indeed has one in the scaling limit, and anything in the scaling limit is a limit of some sequence. So the theory in the limit should be completely describable by the sequence of finite theories and there should be no extra object that is not some limit of finite objects.Our scaling limits VOA 𝒱=⊕_n=0^∞𝒱_n should be regarded as computable using the AC approximations.Philosophically, such VOAs 𝒱=⊕_n=0^∞𝒱_n from ACs categorify computable integral sequences such that each vector space 𝒱_i serves as a categorification of the integer dim𝒱_i. § PRELIMINARIES AND OUTLINE OF MAIN RESULTS First we recall the notion of a VOA, which we regard as the mathematical definition of a chiral CFT (χCFT), along with Wightman's observables, local conformal nets (LCNs), and finally, anyonic chains (ACs). Then we outline our results.While the results on Ising ACs are interesting, our most important contribution of the paper is a framework for addressing the reconstruction of CFTs from MTCs, and the potential simulation of CFTs by quantum computers. Note that full CFTs can be constructed from a nice χCFT with a choice of an indecomposible module category over its representation category <cit.>.In the following, CFTs will mean χCFTs, but there are a few cases where a CFT can be interpreted either as a chiral or full one.A VOA is mathematical axiomatization of the chiral algebra of a CFT.The vertex operator Y(a,z) implements the state-operator correspondence of CFTs.They are the field operators which insert the state a at a space-time point z=0 with a small neighborhood locally parameterized by z.VOA is our preferred framework for our discussion on CFTs. Other frameworks for CFTs such as LCNs will also be discussed at times. The notations and definitions for VOA follow closely those of <cit.>. §.§ Vertex operator algebra Let ℕ_0 be the set of non-negative integers.Consider an ℕ_0-graded ℂ-vector space 𝒱=⊕_n=0^∞𝒱_n, where the weight spaces 𝒱_n satisfy 𝒱_n<∞, equipped with a linear map called the vertex operator, Y(·,z):𝒱→End(𝒱)[[z,z^-1]], Y(v,z)=∑_n∈ℤ v_nz^-n-1,wherev_n ∈End(𝒱) are called the mode operators of v. The mode operators satisfyv_nu=0,for allv,u ∈𝒱 andnsufficiently large.As a different notation, which will be motivated later, for a homogeneous vector v in some weight space with weight wt v, we can shift the index to obtainY(v,z)=∑_n∈ℤy(v)_nz^-n-wt v, where y(v)_n=v_n+ wt v-1.Further, there are two distinguished vectors, the vacuum 1 ∈𝒱_0 and the conformal or Virasoro vector ω∈𝒱_2. The vacuum vector satisfies Y(1,z)=id_𝒱 and the creation property holdsY(v,z)1=v+…∈𝒱[[z]],giving the operator-state or field-state correspondence lim_z → 0 Y(v,z)1=v when we replace the indeterminate z with a complex number. That is why we may sometimes use the expression “conformal field” which is the field associated to the conformal vector ω and we may also sometimes use the word field while actually meaning the vector (this will be clear from the context). On the other hand, the Virasoro vector ω gives us the modes and fieldω_n+1=y(ω)_n=L_n, Y(ω,z)=∑_n ∈ℤ L_nz^-n-2,where the L_ns generate the Virasoro (lie) algebra with relations[L_n,L_m]=(n-m)L_n+m+c/12n(n^2-1)δ_n+m,0·id_𝒱,∀ m,n ∈ℤ,where the constant c is called the central charge (also called rank 𝒱). The grading of 𝒱 is the spectral decomposition of L_0, so L_0v=n v for any homogeneous v ∈𝒱_n. A homogeneous vector v is quasi-primary if L_1v=0 and it is primary if L_nv=0, ∀ n>0. We also have the translation propertyd/dzY(v,z)=Y(L_-1v,z),where the left side is the formal derivative of a Laurent series. Finally, for all a,b ∈𝒱, there exists k ∈ℕ_0 such that(z_1-z_2)^k[Y(a,z_1),Y(b,z_2)]=0 (locality condition).Evidently, we are defining products of vertex operators using formal series. This finishes the description of vertex operator algebra. The tuple (𝒱,Y,1,ω) with the above properties is called a vertex operator algebra (VOA).There are some immediate implications of the above axioms. For any two homogeneous vectors u,v ∈𝒱,wt (y(v)_nu)=wt u-n. The locality axiom implies (the Jacobi or) the Borcherds identity,which can be formulated as 0.9Res_z_1-z_2(Y(Y(a,z_1-z_2)b,z_2)(z_1-z_2)^p ι_z_2,z_1-z_2(z_2+(z_1-z_2))^q)= 0.9Res_z_1(Y(a,z_1)Y(b,z_2)ι_z_1,z_2(z_1-z_2)^pz_1^q)-Res_z_1(Y(b,z_2)Y(a,z_1)ι_z_2,z_1(z_1-z_2)^pz_1^q),for all p,q∈ℤ. In the above expression, Res_zf(z) is the coefficient of z^-1 in f(z). ι_z_1,z_2f(z_1,z_2) is the series expansion of f(z_1,z_2) in the domain |z_1|>|z_2|. As an equivalent formulation:∑_j=0^∞pj (a_q+jb)_p+k-jc=∑_j=0^∞(-1)^jqja_p+q-jb_k+jc -∑_j=0^∞(-1)^j+qqjb_q+k-ja_p+jc,a,b,c ∈𝒱, p,q,k ∈ℤ.The next objects to discuss are the admissible modules of a VOA. A module has a structure similar to that of a VOA and some compatibility properties with the VOA. We will focus on irreducible modules.An irreducible admissible module (A,Y_A) for a VOA (𝒱,Y,1,ω), is an ℕ_0-graded vector space A with a linear mapY_A(·,z): 𝒱→End(A)[[z,z^-1]], Y_A(v,z)=∑_n ∈ℤv_n^Az^-n-1,where v_n^A are the mode operators of v and ω_1^A=L_A,0 gives the weights in the module, which are eigenvalues of L_A,0, with the following difference ∀ a ∈ A_n, L_A,0a=(α+n)a,for some unique α depending onA.The unique conformal or highest weight α gives the grading A=⊕_n ∈ℕ_0 A_n.A_0 is called the top-level and A_n the n-th level of module A.Lastly, there is an analogous notation of y(v)_n for a homogeneous vector v ∈𝒱,Y_A(v,z)=∑_n∈ℤy_A(v)_nz^-n-wt v, where y_A(v)_n=v_n+ wt v-1 and for any two homogeneous vectors u,v ∈𝒱,wt (y_A(v)_nu)=wt u-n. The vertex operator and the modes L_A,n of Y_A(ω,z) satisfy all the axioms of a VOA (A should be seen as a representation of a VOA), except the creativity property. Locality holds and more importantly for us, Borcherds identity also holds in this case with the obvious necessary change c ∈ A. The sub(/super)script A will be dropped from the operators involved as it will be clear from the context.In this paper, the weight spaces need to be finite dimensional. Moreover, the condition C_2-co-finiteness is imposed on the VOAs. This means the space C_2=span{u_-2v|u,v ∈𝒱} has finite co-dimension C_𝒱=𝒱/C_2. Then, a result <cit.> on the growth of the dimension of the weight spaces of an irreducible module A followsA_n ≤ ( A_0) · e^2π√(C_𝒱n/6),where C_𝒱=𝒱/C_2. This at most exponential growth is necessary if an approach to simulation requires a truncation of energy up to some N, where one can not afford more than polynomially many qubits to be used to simulate the vector space.Finally, the character for a module A is defined aschar(A)=Tr_A(q^L_0-c/24)=∑_n ∈ℕ_0(A_n)q^n+h-c/24.An important class of VOAs consists of the unitary minimal models (UMMs) introduced in the next section. A UMM 𝒱 satisfy many properties such as being CFT-type, i.e. V_0=ℂ1, or in other words only the vacuum has energy zero. Also, 𝒱 is rational, i.e. every admissible 𝒱-module is a direct sum of irreducible 𝒱-module. Last but not least, 𝒱 is unitary (as explained below). As a convention, the expression CFT or chiral CFT or full CFT, will be referring to a VOA with the described properties.For us, a unitary VOA has some a positive definite hermitian form (·,·)_𝒱:𝒱×𝒱→ℂ with respect to which one can define adjoint of mode operators. Specifically, L_n^†=L_-n. One can similarly define unitary modules: a positive definite (·,·)_A: A× A →ℂ with respect to which L_n^†=L_-n. Using the hermitian form, one can define a norm in the obvious way and get the completion of a graded unitary module (which includes 𝒱 itself). They will be represented by A. For a complete definition of a unitary VOA, see <cit.>.Next, to describe a full CFT, we will focus only on full diagonal CFTs. The description here will not be completely elaborate as full CFTs are not discussed much in our work. Only the essential concepts will be mentioned in simple terms. Consider a chiral CFT with the restrictions imposed earlier. The idea is to take any irreducible module coupled with the contragredient module (which, assuming e.g. unitarity, is isomorphic to the module itself) and consider the Hilbert space it gives after completionℋ=⊕_irreducible modules𝒜_i⊗𝒜_i',where 𝒜_i' is the contragredient module of 𝒜_i. The contragredient module 𝒱' is defined as the linear functionals that vanish except on finitely many of the weight spaces, in other words𝒱'=⊕_n ∈ℕ_0𝒱_n',and it can be given a 𝒱-module structure. In the case of minimal modules, this means another isomorphic (as 𝒱-module) copy of the module itself. The vacuum vector 1=1_L⊗1_R and the conformal vector ω=ω_L⊗1_R+1_L⊗ω_R are defined in the obvious way using the left (and right) vacuum 1_L(1_R) and left (and right) conformal vector ω_L(ω_R). The Virasoro mode operators are defined accordingly as𝕃_n=L_n+L_n,where the first term is the n-th Virasoro mode for the chiral copy and the second, for the antichiral copy. Primary fields are accordingly defined as those a ∈ℋ that satisfy 𝕃_n a=0,∀ n>0.Introducing the analog of vertex operator Y(·,·) for the full CFT requires us to describe what intertwiners are, but we will only need to introduce the conformal field, which is𝕐(ω,(z,z))=∑_n ∈ℤ L_nz^-n-2+L_nz^-n-2. §.§ Unitary minimal models and Ising CFT A special class of VOAs are the highest weight representations of the Virasoro algebra with central charge c<1 that are unitary. These highest weight representations can be completely characterized by their central charge,which form a discrete series c=1-6/(k+1)(k+2) for k≥ 2. These VOAs M(k+2,k+1), called the unitary minimal models (UMMs), can be constructed as cosets SU(2)_k× SU(2)_1/SU(2)_k+1 and have central charge c=1-6/(k+1)(k+2) for k≥ 2. We will refer to M(k+2,k+1) as the UMM at level k (of SU(2)_k). They have finitely many irreducible modules determined by their conformal weightsh_r,s=((k+1)r-(k+2)s)^2-1/4(k+1)(k+2), 1≤ r≤ k+1, 1≤ s ≤ k.Hence, due to the symmetry h_k+2-r,k+1-s=h_r,s, there are k(k+1)/2 many irreducible modules. In particular, the Ising CFT has central charge 1/2 corresponding to level k=2 (see <cit.> for more on minimal models, and <cit.> and particularly the notes <cit.> for the Ising CFT).The chiral Ising CFT has 3 irreducible modules with conformal weights h_1,1=0 (the VOA _0 itself with the vacuum field 1), h_2,1=1/2 (the module _1/2 corresponding to the free Fermionic field ψ), h_3,1=1/16 (the module _1/16 corresponding to the spin field σ). In the Ising CFT, the fusion rules are as follows:_1/2⊗_1/2 = _0 _1/16⊗_1/16 = _1/2⊕_0 _1/2⊗_1/16 = _1/16 and of course anything fused with _0 becomes itself. The Fermionic algebra is used to generate the Hilbert spaces _i. The Hilbert spaces _0,_1/2 are generated by the Fermionic modes {Ψ_n-1/2}_n ∈ℤ satisfying the anticommutative canonical relations (ACR){Ψ_k,Ψ_k'}=δ_k+k',0,and their actions satisfy the conjugacy relationΨ_k=Ψ_-k^†.The third Hilbert space _1/16 is generated by {Ψ_n}_n∈ℤ which are also another version of the Fermionic algebra where, this time, the modes are indexed by integers and they satisfy the same properties:{Ψ_k,Ψ_k'}=δ_k+k',0, Ψ_k=Ψ_-k^†.The first algebra generates _0 and _1/2 by acting on the vacuum 1. Indeed, the vectors {Ψ_-k_r…Ψ_-k_11| k_1<…<k_r, k_i ∈ℕ-1/2},with corresponding weight ∑ k_i, give an orthonormal basis for _0 ⊕_1/2, hence giving the characterq^-c/24∏_n=1^∞ (1+q^n-1/2).As a matter of convenience, the factor q^-c/24=q^-1/48 will sometimes get dropped. Obviously, the part of which has powers of q in ℕ-1/2 corresponds to _1/2 and the rest with powers of q in ℕ_0 corresponds to _0.The second algebra {Ψ_n}_n∈ℤ generates _1/16 in a similar way: the orthonormal basis{Ψ_-k_r…Ψ_-k_1|1/16⟩| 0<k_1<…<k_r, k_i ∈ℕ},where |1/16⟩ is the highest weight vector, or the vector at the top level satisfying L_0|1/16⟩=1/16|1/16⟩. The corresponding weight is naturally ∑ k_i. Notice that|1/16⟩ is sent to a scalar multiple of itself by Ψ_0. The character ischar(_1/16)=q^1/16-1/48∏_n=0^∞(1+q^n).Although not mentioned, but from the above description, it is clear what the hermitian form should be. The formulae for L_ns are well-known <cit.> and will be derived in the [A.1]appendix.As a final note, the Ising full CFT isℋ=_0_0 +_1/2_1/2 +_1/16_1/16,with the corresponding operators 𝕃_n which will be derived using the formulae for L_ns. §.§ Wightman's observables and local conformal nets In addition to VOA, we will also work with observables coming from LCNs and Wightman's axioms. One of the objectives of this work is to obtain the fields in the scaling limit and prove that products of fields are also in the scaling limit, hence obtaining a “scaling limit of algebras”. Only the observables (or fields) in each framework will be defined. As we shall see, observables are related to the fields Y(a,z) we have been using so far. For this section, the definitions and facts follow those of <cit.>. So far, the observables or fields that are point-like have been described; the insertion of the field is exactly at a point. Other types of observables that can be derived formally from these are called smeared field operators or Wightman's observables. Taking some function f ∈ C^∞(S^1), define formallyY(a,f):=∮ Y(a,z)f(z)z^wt a dz/2π i z=∑_n ∈ℤf̂_ny(a)_n,where f̂_ns are the Fourier coefficients of f. As f is smooth, it is known that its Fourier coefficients will be rapidly decreasing:∀ k , ∃ N_k such that ∀ |n| ≥ N_k|f̂_n| ≤1/n^k. In order to have truly a linear operator defined on 𝒱 (before taking its completion), an energy bound on the mode operators is needed||y(a)_nb|| ≤ C_a(|n|+1)^r_a||(L_0+1)^s_ab||,∀ b ∈𝒱where the constants C_a,r_a,s_a>0 dependent on a, and the norm is given by the unitary structure. If the above inequality holds, then one easily observes that ||Y(a,f)b||≤ C_a||f||_r_a||(L_0+1)^s_ab||,where the r_a-norm of f is defined as||f||_r_a=∑_n |f̂_n|(|n|+1)^r_a.As it will be observed in [4]section 4, the correlation function F^0 using the smeared formalism can be defined as0.95F^0((a_1,f^(1)),…,(a_k,f^(k)),u,v):=(u,Y(a_1,f^(1))Y(a_k,f^(k))v), f^(i)∈ C^∞(S^1)where the fields a_i satisfy an energy bound and u,v ∈𝒱. The correlation function for smeared fields on a full CFT is defined similarly, but we will defer that to [4]section 4 since this will be used in a very restricted case.For the UMMs, all y(a)_n are energy bounded. The most important field for us is the conformal smeared field Y(ω,f) which is denoted by L(f)=∑_n ∈ℤf̂_nL_n. For all UMMs, it can be shown that L_n satisfies the energy bound||L_nb|| ≤√(c/2)(|n|+1)^3/2||(L_0+1)b||,giving as a result||L(f)b||≤√(c/2) ||f||_3/2||(L_0+1)b||. The next observables are the ones coming from the LCN picture of CFT. Only the most relevant features of this framework shall be discussed. Denote by ℐ the family of proper intervals of S^1. A net 𝒜 of von Neumann algebras on S^1 is a map that associates a von Neumann algebra 𝒜(I) ⊂ℬ(ℋ) for some fixed Hilbert space ℋ. These nets should be local in the sense that for I_1,I_2 ∈ℐ with I_1 ∩ I_2=∅,[𝒜(I_1),𝒜(I_2)]={0},and they satisfy isotony, I_1 ⊂ I_2 𝒜(I_1) ⊂𝒜(I_2).In the case of UMMs, and more generally unitary Virasoro VOAs, taking the Hilbert space to be any irreducible module of conformal weight h,𝒜(I)={e^iL(f)|f ∈ C^∞(S^1), supp(f) ⊂ I}”(see <cit.>).So 𝒜(I) is the double-commutant of the algebra generated by the unitary operators e^iL(f) associated to functions with support inside I. The double-commutant theorem implies that the strong (or weak limit) of the algebra generated by e^iL(f)s is also 𝒜(I), a fact that will be used later in [4.2]section 4.2. §.§ Anyonic chains Though ACs are closely related to and inspired by spin chains, there are some fundamental differences between them.The most salient difference touches on the trade-off between explicit locality and unitarity in QFTs.Spin chains implement locality explicitly by attaching local state spaces to each site, while the Hilbert spaces of ACs do not have such explicit tensor product decomposition.In general, it is harder to obtain unitary interacting exactly solvable spin chains with CFT scaling limits, while such examples of ACs are ubiquitous <cit.>.This phenomenon is related to the localization of braid group representations, where finite order unitary R-matrices are very rare <cit.>.This section follows the exposition of anyonic chains (ACs) in <cit.>. An AC is a periodic or open (with boundary condition) chain, along which pairwise interactions occur between quasi-particles (the anyons). e.g. the generalized spin j anyons of SU(2)_k. The chain is usually presented along a straight path if it is non periodic and as a loop if it is periodic. We will also put the nonperiodic chain along the upper half-circle ([fig1]Figure 1) as this picture will be used in [4.2]section 4.2 to relate the AC to LCN.The channel between each two anyons provides the means for fusion. A boundary condition (a,b) means x_1=a and x_L=b. Each admissible fusion path has to satisfy the fusion rules of SU(2)_k:j_1 ⊗ j_2=|j_1-j_2| ⊕ (|j_1-j_2|+1) ⊕…⊕min{j_1+j_2,k-j_1-j_2}.All admissible fusion paths form an orthogonal basis of the Hilbert space Hom(x_1⊗ x_L,j^⊗ (L-1)) where the inner-product comes from the diagram calculus of the unitary MTC SU(2)_k. An important observation is that generally for all j and k, the resulting Hilbert spaces do not have a tensorial structure, though the case of Ising AC does have one.We specialize to the case j=1/2, where the spin-1/2 chain is given a Hamiltonian. The motivation of all these settings could be seen as a generalization of the Heisenberg model <cit.>. In the Heisenberg model, there exist a spin-spin nearest neighbor interaction given by the term S⃗_i.S⃗_i+1=P_i^1-3/4I_i=-P_i^0+1/4I_i,where P_i^s is the projection onto the total spin s channel of two spins S⃗_i and S⃗_i+1. This leads to the following Hamiltonian H=J∑_j P_j^0, where J determines if the chain is antiferromagnetic (J=-1) or ferromagnetic (J=1). In order to generalize this, we first need to define the projection onto the total spin using the so called F-move in [fig2]Fig. 2.The next step would be to project onto the desired fusion which is 0—the vacuum- and go back to the previous basis of fusion path by applying the inverse of the F-move (<cit.>,<cit.>):H=∑_i=1^L-1F_i^-1P_i^0F_i,where the antiferromagnetic coupling J=-1 has been chosen in order to obtain UMMs in the scaling limit. In the case of spin-1/2 chain, letting d=2cos(π/k+2), the quantum dimension of 1/2, F_i^-1P_i^0F_i=-1/dX_iH=-1/d∑_i=1^L-1 X_i.The operators X_i satisfy the following relations <cit.>:X_i^2=dX_i, X_iX_i±1X_i=X_i, [X_i,X_j]=0,for|i-j|>1.These are the same operators e_i of the Temperley-Lieb (TL) algebra. Thus,H=-1/d∑_i=1^L-1 e_i.For the special case of Ising anyonic chain, if non-periodic, there are several possibilities (a,b) for the boundaries as a,b ∈{0,1/2,1}. For example, the chain (1/2,1/2) has odd length L=2n+1 due to the fusion rules and the Hamiltonian is H=-1/√(2)∑_i=1^2n-1 e_i. However, the periodic chain has always even length 2n.Back to the general case, the operator e_i=e[i] acts non-trivially on the i-th particle according to its neighbor particlese_i|j_i-1j_ij_i+1⟩=∑_j_i'(e[i]_j_i-1^j_i+1)_j_i^j_i'|j_i-1j_i'j_i+1⟩,where (e[i]_j_i-1^j_i+1)_j_i^j_i' is determined by the S-matrix entries of the MTC(e[i]_j_i-1^j_i+1)_j_i^j_i'=δ_j_i-1,j_i+1√(S_j_i^0S_j_i'^0/S_j_i-1^0S_j_i+1^0), S_j^j'=√(2/k+2)sin(π(2j+1)(2j'+1)/k+2).In brief, from the MTC point of view, one can think of the (open) AC as a diagram inside Hom(x_1 ⊗ x_L, (1/2)^⊗ (L-1)) on which e_j with the above entries act. Diagrammatically, the TL algebra acts by annihilating (cap) and then creating (cup) an adjacent pair of spin-1/2 particles. Then H can be defined as above. Numerical experiments suggest that the scaling limit of the ACs of SU(2)_k give chiral or full CFT (for open boundary condition or periodic chains, respectively). These results are outlined in <cit.>, and show that depending on the boundary condition, we obtain different chiral CFTs, i.e. different irreducible modules of UMMs with central charge c=1-6/(k+1)(k+2). As emphasized before, this happens for the antiferromagnetic chain, and it is expected that one obtains the parafermion CFT with central charge c=2(k-1)/k+1 for the ferromagnetic chain.Exact diagonalization numerically solves the anyonic chain model by finding the excitation spectra. Conformal dimensions of the predicted CFT limit are extracted from the energy levels for a length L chain given by E=E_1L+2π v/L(-c/12+h_L+h_R)+O(1/L^2). The scaling limit CFT is stable under symmetry-preserving perturbation. More precisely, by symmetry, we mean the topological symmetry that the periodic chain has. One can imagine a loop <cit.> inside the chain and repeatedly use the F-move until it gets removed. As demonstrated in <cit.>, this provides a topological symmetry and any perturbation preserving such symmetry will not change the scaling limit.As a final note, an important connection between AC model and the RSOS lattice model provides a physical proof that the scaling limits of ACs are CFTs. One can show that the Hamiltonian derived from the logarithmic derivative of the transfer matrix, coincides with the AC Hamiltonian <cit.>. This lattice model has been studied for a long time and the literature has similar numerical results for this model (see <cit.>, <cit.>, and the references in <cit.>).While there is no doubt that the two approaches are equivalent in the end, mathematically it seems easier to obtain CFTs as scaling limits in the AC approach.As comparison, we recall the recovery of Ising CFT in the 2d classical Ising model <cit.>.One aspect of the difference is manifested in the order of enforcing Jones-Wenzl projectors in SU(2)_k:in the ACs, the Jones-Wenzl projector p_k+1 is implemented first, whereas in spin chains the projector, which is non-local, will be implemented at a later time. §.§ Outline of main results In this paper, we provide a mathematical definition of a low energy scaling limit of a sequence of general non-relativistic quantum theories in any dimension, and apply our formalism to ACs.Similar ideas for defining related scaling limits for lattice models and spin chains appeared earlier in the physics and mathematics literature (see the next section).Of utmost importance to our future applications are the rate of convergence to the scaling limit, and the recovery of all algebras of local observables in the scaling limit.We emphasize those points for the scaling limits of the Ising ACs. We formulate [cnj4.3rep]Conjecture 4.3 reproduced below on conditions when a chiral unitary CFT would arise as such a limit and verify the conjecture for the Ising minimal model M(4,3) using Ising ACs.Part of the conjecture is a precise relation between Temperley-Lieb generators {e_i} and finite versions of Virasoro generators {L_m+L_-m} and {i(L_m-L_-m)} for UMMs M(k+2,k+1).Our approach is supported by extensive numerical simulation and physical proofs in the physics literature.In [2.]section 2, we define the low energy (strong) scaling limit of quantum theories — Hilbert spaces 𝒲_n with Hamiltonians H_n and algebras of observables 𝒜_n— as a Hilbert space 𝒱 with Hamiltonian H, and address the issues that come up with our definition. We define the scaling limit of observables O_nO when O_n's low energy behavior converges to that of O.Those O's defined on 𝒱 generate the vector space 𝒜 of observables on 𝒱.On an important related issue, we propose a definition of locality with respect to both space and energy. As an example, intuitively, local energy observables are those that do not shift the energy by more than a constant. We explore the (space and energy) local operators in [4]section 4 in greater details ([thm4.1]Theorem 4.1 and [thm4.2]Theorem 4.2).In [3]section 3, we obtain the scaling limits of Ising ACs with all kinds of boundary conditions. Proving the limits is a rather computationally involved procedure, where the same technique (<cit.>) is applied to each case. While the variation of details from case to case is small, the details of the proofs are necessary for our future discussion. Part of the proofs for some cases has been done in the physics literature with different or similar approaches (see <cit.> as an example), but we could not locate a mathematically rigorous proof for all Ising ACs in the literature using one consistent method, and with explicit estimate of the convergence rate for the limits. Furthermore, we need to set up notations and use some details of the proofs in later sections.In the end, we find enough reasons to provide a thorough and detailed mathematical proof of the scaling limit of all Ising ACs in one place, estimate the rate of convergence, and show how one can obtain the Virasoro algebra generators in the scaling limit. The final result is a long [thm3.1]Theorem 3.1 which motivates the conjecture below:4.3 For any unitary minimal model VOA 𝒱=𝒱_c,0 and a chiral representation 𝒱_c,h, there is a sequence of quantum theories (𝒲_n, H_n, 𝒜_n) with strong scaling limit (𝒱_c,h,L_0) such that for each Virasoro generator L_m, we have a sequence L_m(n) ∈𝒜_n with the following properties: * L_m(n) is a space local observable such that the hermitian operators aL_m+aL_-m∈𝒜_n^H for any complex number a, where 𝒜_n^H is the generating set for 𝒜_n consisting of hermitian observables.* L_m(n) shifts the energy no more than |m| for each n.* There are positive constants d_ω,g_ω,e_ω such that when L_m(n) restricted to energy at most n^d_ω, it has the following approximation by L_m|_n^ d_ω with remainder R_n^m:L_m=L_m|_n^d_ω+O(1/n^g_ω)+R_n^m,and the operator norm of the remainder R_n^m is bounded by O(n^e_ω).In [4]section 4, using the results in [3]section 3, we obtain the observables of different types in the scaling limit. Obtaining the action of the VOA on its modules is the next step. As the VOA is generated by L_ns applied to the primary fields, the ability to obtain the Virasoro generators as scaling limits is of utmost importance. In other words, realizing the smeared conformal field Y(ω,f) should be the top priority. Then, we would need to get the smeared primary fields. For UMMs, the VOAs are generated from only the conformal vector ω applied on the vacuum (the only primary field). Thus, obtaining Y(ω,f) is close to obtaining all the operators Y(a,f) for all fields a and that is due to the Borcherds identity. Assuming the above conjecture for UMMs, most theorems in [4]section 4 notably [thm4.8]Theorem 4.8 mentioned below, hold for all UMMs as well (the exact results are mentioned in [rmk8]Remark 8 and [rmk9]Remark 9). So although our discussion is for the Ising ACs, everything below about the conformal field and Y(a,f)s is conjectured to hold for higher level UMMs as well.Consider a non periodic Ising chain placed on the upper half-circle S^1_+, what is the “finite version” ω of ω? Informally, the answer is ω=e (the TL operator) where subtleties are explained below. For example, we will show that for any function f with a Fourier series where coefficients of sin(nθ)s are zero, we have (informally)∫_S^1_+ f(e^iπ j/n)e_jY(ω,f)=∮ Y(ω,z)f(z)z^2 dz/2π i z,where the integral on the left is an integral over a “finite” space, in other words, a summation. Hence, as ω can be regarded either as a vector or a field, so does e_j, which can be seen as a diagram or as an operator (stacking up diagrams). Here Y (the vertex operator) is the analog of the stacking at infinity.For the opposite situation, i.e. the functions with a Fourier series where coefficients of cos(nθ)s are zero, we could consider this as a derivative of a series with nonzero coefficients for only cos(nθ). In this case, we are basically looking at the derivative of Y(ω,z) or in other words d/dzY(ω,z)=Y(L_-1ω,z). So it is necessary to find the corresponding operator for the finite version of L_-1ω which should be the derivative (as it is the interpretation of L_-1) of e_j. The first candidate that comes to mind is [e_j,e_j+1] and informally, we havei∫_S^1_+ f(e^iπ j/n)[e_j,e_j+1]Y(ω,f)=∮ Y(ω,z)f(z)z^2 dz/2π i z.For general functions f, we need a linear combination of e_j and [e_j,e_j+1] to get Y(ω,f). Once this is achieved, an application of the Borcherds identity gives all Y(a,f)s and the action of the VOA on its module can be recovered as an algebra in the scaling limit (SL-algebra): 4.8 The set of operators {Y(a,f) |a ∈𝒱, f ∈ C^∞(S^1)}⊂𝒜 generate an SL-algebra.The importance of this result lies in the required recovery of operators as an algebra in the scaling limit, not just as isolated operators generating a vector space as explained in the beginning. Next goal is the algebra of observables in LCNs. We define finite versions of local conformal nets in the obvious way by thinking of intervals in the interval set ℐ_+ on the upper-half circle and the algebra of observables on them. Then, we show that there is a net of bounded observables 𝒜_b obtained in the scaling limit exactly matching the one in an LCN, i.e. 𝒜_lcn, at least on those intervals touching the boundary of the upper half-circle:4.13 We have 𝒜_b(I)=𝒜_lcn(I ∪ j(I)) for I ∈ℐ_+, with j(I) being I's reflection in the lower half-circle and |I ∩∂ S^1_+|=1 We also propose a method to recover the point-like fields ([thm4.14]Theorem 4.14). In that case, we are unable to show that they form an algebra in the scaling limit. In fact, the sequence of operators that we are proposing to identify them in the scaling limit is probably not the suitable one. We will discuss these more in the relevant section.In the final section, conjectures and problems that needs to be addressed to fully recover all the structures of CFTs in the scaling limit are listed. Lastly, we make an attempt to formulate the problem of simulating CFTs using quantum computers.§.§ Previous works We discuss briefly some prior works in the literature on the mathematically rigorous definition of a scaling limit in the quantum mechanics approach, and the recovery of algebras of observables. As previously mentioned, there is a vast literature on the subject of scaling limits in statistical mechanics <cit.>, and substantial progress has been made in the case of Ising model proving the correlation functions in the limit are conformal invariant (see <cit.> and the references therein). Statistical mechanics approach could also provide techniques with which one could compute the conformal weights present at the scaling limit without actually diagonalizing the Hamiltonian <cit.>. A recent program to construct CFTs from subfactors is in <cit.>, where the inductive limit of Hilbert spaces is clearly discussed based on planar algebras, which have the same Hilbert spaces of states as ACs (spin chains in these papers are better interpreted as generalized spin chains as in <cit.>). Our work focuses on the quantum mechanics approach to scaling limits of ACs enriching the inductive limits <cit.> with explicit Hamiltonians and algebras of local observables.A scaling limit of spin chains close to our Ising AC was analyzed earlier in <cit.> starting with the idea of how to take the scaling limit of the Hamiltonians of the chains and also obtain the Virasoro modes L_n from Fourier transforms of the TL generators e_i. More recently, in the first paper of the series <cit.> on the 𝔤𝔩(1|1) (free) model, the authors proposed a potentially rigorous definition for the scaling limit <cit.>, obtained operators like our L_ms and computed their commutators to check their convergence to the commutators of the Virasoro modes. Such computations are commonly pursued after one obtains some operators L_mL_m and have been done in different models both rigorously and numerically (<cit.>,<cit.>). We go beyond the convergence of commutators and further pin down the conditions necessary ([cnj4.3]Conjecture 4.3) to prove the same theorems for higher level UMMs. In the third paper of the series (<cit.>), the authors gave a rigorous definition of scaling limit based on their previous ideas while working on the scaling limit of JTL algebra (with d=0) as it acts on a 𝔤𝔩(1|1) periodic spin-chain model (the scaling limit is the c=-2 Logarithmic CFT—symplectic fermions theory).Even though the context and the type of model (on Logarithmic CFTs) are quite different from ours (unitary CFTs), our definitions closely mirror theirs.But there are some differences due to our different motivation, emphasis and applications.As defined in <cit.>, our scaling limit is also dictated by the low energy behavior of Hamiltonians, Hilbert spaces, and observables.In <cit.>, the primary focus is on the algebraic scaling limit of JTL_N's action (π_𝔤𝔩(JTL_N)). However, we focus on the analytic side of scaling limits motivated by our goal of simulating CFTs as we need to know how computations in the finite stages converge. Especially, the unitary evolution and correlation functions involve unbounded operators for which we desire a clear description on how they are obtained in the scaling limit. In fact, even when restricted to the bounded observables, not all bounded operators can be obtained through the algebraic approach (for example the unitary operators e^iL(f)). Related to this, the analytic approach provides a more direct picture at how the LCNs emerge ([4.2]section 4.2) since we still keep the JTL operators e_i as our operators of interests and mostly, do not switch to fermionic fields. This enables us to obtain theorems with proofs general enough for higher UMMs assuming [cnj4.3]Conjecture 4.3. First, our definition of scaling limit of a sequence of theories but also a sequence of operators is very much in the same spirit as the one provided in <cit.>. As defined in <cit.>, we consider the limit of the low energy behavior of these objects (Hamiltonians, Hilbert spaces, observables) and take some limit on top of the first limit to construct the corresponding object at the scaling limit. What the difference is, is what kind of scaling limit one should focus on. In <cit.>, we see the primary focus on the algebraic scaling limit of JTL_N's action (π_𝔤𝔩(JTL_N)) and some final comparison to the analytic scaling limit (which is our focus) in <cit.>.In order to analyze the scaling limit of π_𝔤𝔩(JTL_N), the authors first provide the scaling limit of the Hilbert spaces with the help of the embeddings of the Clifford algebra 𝒞_N <cit.> which is generated by the fermion operators satisfying ACR relations. Then in <cit.>, an identification of π_𝔤𝔩(JTL_N) is established with the enveloping algebra of a lie algebra called 𝔖_N <cit.>, formed by bilinear fermionic operators. As a corollary, the authors prove that the successive embeddings of 𝒞_2N↪𝒞_2(N+2) restricts to an embedding of π_𝔤𝔩(JTL_2N)↪π_𝔤𝔩(JTL_2(N+2)) (the parity issue is not of interest to us). Of course, these embeddings are trivial with respect to the fermion operators while highly nontrivial in terms of JTL generators. Then, just like the Clifford algebra 𝒞 can be easily obtained through some colimit, by taking the colimit of the sequence …↪π_𝔤𝔩(JTL_2N)↪π_𝔤𝔩(JTL_2(N+2))↪…we can obtain an algebraic scaling limit of the JTLs which will be the enveloping algebra U𝔖_∞ of the algebraic scaling limit 𝔖_∞ of 𝔖_Ns. The natural question is: how does this compare to a more analytic scaling limit of JTL? One quick observation is that U𝔖_∞ is generated by finite sums of bilinear fermion operators. Hence, they are all bounded observables. But there are many operators of interest in CFT, notably the Virasoro modes which are unbounded operators. These unbounded operators are given if we take a sequence of operators like ∑ e_j inside π_𝔤𝔩(JTL_2N) which are not a successive embedding of a single operator.As explained in <cit.>, one can consider the algebra 𝔖 generated by the Virasoro modes (and some other operators called the interchiral modes which will be discussed later in [5.2]section 5.2), which we can get through the analytic scaling limit of π_𝔤𝔩(JTL_2N). Then one can show that 𝔖≅𝔖_∞ and so U𝔖≅ U𝔖_∞ where the overline is the completion.This is also what one would expect. In fact, we can see that if by some means, we achieve to prove that there exist some algebraic scaling limit for JTL in the anyonic chains, then U𝔖_∞^w would be the subset of linear operators inside 𝒜 (see [dfn6]Definition 6). Indeed, if we want to compute the scaling limit of any sequence of operators O_n ∈𝒜_n (the algebra of observables generated by JTL), we can as well consider all of them as a sequence inside U𝔖_∞ (the hypothetical algebraic scaling limit of 𝒜_n). Then one can easily observe that the first sequence has a linear operator defined on the VOA as an analytic scaling limit if and only if the sequence inside U𝔖_∞ has a weak limit. The algebraic approach, and algebraic-numerical techniques <cit.>, has been used to obtain more information about the algebraic structure of the Hilbert space and the algebra of observables in the scaling but to our knowledge, a mathematically rigorous procedure has been applied mainly for free models like 𝔤𝔩(1|1). Recently, emergence of conformal symmetry has been numerically investigated using the Koo-Saleur generators (KSGs) <cit.>.To compare our version of KSGs with those of <cit.>,first recall our notation 𝕃_n=L_n+L_n. Our counterparts of the KSGs are operators 𝕃_n±𝕃_-n on the ACs that give us 𝕃_n±𝕃_-n in the scaling limit. On the other hand, using a different diagonalization of the Hamiltonian in <cit.> (same as that in <cit.>), the authors found their KSG operators, different from ours, in the AC notation to beH_n=-N/2π∑_j=1^2N e^2n(j+1)π/2Ne_j, which converge to L_n+L_-n. Taking the sum and difference of H_n and H_-n respectively, we obtain 𝕃_n+ 𝕃_-n from the sum and (L_n-L_-n) -(L_n-L_-n) from the difference, which does not have a counterpart in our version.The difference stems from different diagonalizations of the same Hamiltonian, which illustrates the potential importance of connecting maps in our definition of scaling limits. In <cit.>, the diagonalization of the Hamiltonian is accomplished by constructing creation and annihilation operators from the usual Fourier transforms of the Majorana operators. While in our version, the creation and annihilation operators are obtained as sin() and cos() transforms for the left and right moving sectors, which impliesthat going from one diagonalization to the other requires a mixing of the right and left moving sectors of the full CFT. It follows that the scaling limit of H_n from our diagonalization will have an interchiral part which mixes left and right moving sectors which is clearly different from H_n=L_n+L_-n. The method in <cit.> works well numerically, and for the NS sector_0_0+_1/2_1/2, the resulting scaling limit (see e.g. <cit.> for a proof) gives rise to a full CFT isomorphic to ours by a not necessarily local isomorphism that connects the two different sets of creation and annihilation operators.Finally, while not directly related, the paper <cit.> serves as a conceptual inspiration for our work and the techniques introduced there address analytic problems of similar nature to ours.§ SCALING LIMIT OF QUANTUM THEORIES It is commonly believed that QFTs are low energy effective theories such as WCS TQFTs are the low energy effective theories for two dimensional fractional quantum Hall liquids.In this section, we define mathematically a low energy limit of a sequence of quantum theories.Our formalism is closely related to the definition of topological phases in <cit.> and ideas in <cit.>.We start with the definition of quantum theories by imagining quantum theories that describe a collection of interacting quantum particles. The theories considered have a discrete energy spectrum in the scaling limit like all CFTs. Notice this is different than the energy spectrum given by the primary fields. In the context of CFTs, there are non unitary Virasoro representations with continuous spectrum of primary fields, while still having a discrete energy spectrum in each sector. The definition below is for a finite dimensional theory, with the next sections defining what the scaling limit (infinite dimensional) is. A quantum theory is (𝒲, H, 𝒜) where * 𝒲 is the Hilbert space of states,* H is the Hamiltonian and hermitian,* 𝒜 is the algebra of observables,We can also add a number of notions to the definition above. For example, The space information of the system can be thought of a graph G, which is usually the 1-skeleton of a triangulation of the space. In the following text, G is always a chain. There are also different notions of locality based on the basis we choose. As an example, considering the space information given the graph G, the Hamiltonian H is r-local for some constant r>0 if H=∑_i=1^p H_i such that each local hermitian term H_i is trivial outside the ball B_r(v_i) of distance r at some vertex v_i of G. If p=1, then H is ultra r-local. There will also be a notion of energy-local operators as discussed later in this section. §.§ Low energy limit of quantum theories The first part of a limit theory is a Hilbert space and a Hamiltonian, which are constructed from the low energy spectra of a sequence of quantum theories (𝒲_n,H_n) with strictly increasing dimensions.Assume a sequence of quantum theories (𝒲_n,H_n) with H_n's eigenvalues being ordered as λ_1^(n)≤…≤λ_d(n)^(n), where d(n)=dim(𝒲_n). The Hilbert spaces 𝒲_n decompose into the corresponding one-dimensional eigenspaces𝒲_n=E_λ_1^(n)⊕⋯⊕ E_λ_d(n)^(n). Denote by 𝒲_n^M the Hilbert space 𝒲_n restricted to energies at most M, i.e. 𝒲_n^M=⊕_λ_i^(n)≤ ME_λ_i^(n). Assume the following set of properties (P) * λ_i=lim_n →∞λ_i^(n) exists for all i ∈ℕ with the convention λ_i^(n)=0 for i>d(n), and lim_i→∞λ_i=∞,* (connecting maps) for all M > λ_1 where M ≠λ_j for all j, there exist connecting unitary maps ϕ_n^M: 𝒲_n^M →𝒲_n+1^M for all n>N_M for some N_M depending on M,* (extension) ϕ_n^M is an extension of ϕ_n^M' when M ≥ M', i.e. ϕ_n^M|_𝒲_n^M'=ϕ_n^M'.Consider the sequence (𝒲_n^M,ϕ_n^M) with M > λ_1 and M≠λ_j. We note that this sequence eventually stabilizes due to the existence of unitary maps for large enough n.The reason for M ≠λ_j for all j in the first property is that energies oscillating around their limit points would make the stabilization of the low-energy spectrum impossible for a cut-off M=λ_j. From now on, any cut-off will be implicitly assumed to be not equal to any λ_j.Taking the colimit of the sequence (𝒲_n^M,ϕ_n^M) gives a finite dimensional vector space, called 𝒱^M, along with the unitary maps ρ_n^M : 𝒲_n^M →𝒱^M. It follows that𝒱^M has a natural Hilbert space structure. Further, due to the first property, one can easily see that for all M∈ (λ_j,λ_j+1), the space 𝒱^M is the same as 𝒲_n^M will have the same dimension (for large enough n). The space 𝒱^M can be also conveniently called 𝒱^λ_j. So there are only countably many different 𝒱^Ms. Next we add the following property to P on the convergence of H_n^M, the restriction of H_n to 𝒲_n^M:* (convergence) The push-forward of H_n^M on 𝒱^M given by ρ_n^M converges to some operator H^M:ρ_n^MH_n^M(ρ_n^M)^-1→ H^M.Obviously, H^M will be hermitian. Furthermore, the above property is equivalent to the following diagram “commuting up to ϵ_n^M in the norm operator”, which goes to zero as n →∞:𝒲_n^M rρ_n^M[swap]dH_n^M 𝒱^M dH^M 𝒲_n^M rρ_n^M 𝒱^M . The construction of the scaling limit (𝒱,H) of the sequence is not hard from here. We spell out the formal details although future constructions in the case of Ising will be much more straightforward.Properties of the colimit imply that the set {(𝒱^M,H^M)}_M > λ_1 is unique up to unique isomorphism. We would like to construct these spaces in such a way that {(𝒱^M,H^M)} are restrictions of a single Hilbert space and its Hamiltonian (𝒱,H). Due to the extension property, N_M≥ N_M' for M≥ M'. The existence of connecting maps for n>N_M ensures that one can build 𝒱^M using the orthonormal basis0.93{(v_i^(N_M+1),ϕ_N_M+1^M(v_i^(N_M+1)),ϕ_N_M+2^M(ϕ_N_M+1^M(v_i^(N_M+1))),…)}_i=1^𝒱^Mwhere v_i^(N_M+1) is an orthonormal basis of 𝒲_N_M+1^M. Each sequence represents an actual vector v_i^M and addition is component-wise and inner product is given by the inner product on any component, which due to isometry of connecting maps gives the same number. To be really precise, we will have to take v_i^M not exactly as that sequence, but as the colimit of that sequence and every sequence that is a truncation of the sequence from the left. This choice will soon become clear. Now consider making the above construction (inductively) for the sequence M=λ_1',λ_2',… where λ_j'∈ (λ_j,λ_j+1) in such a way that for M=λ_j' for j>1, the orthonormal basis used in the previous case is extended. As an example, assuming λ_1<λ_2<λ_3, for M=λ_2', we take the orthonormal basis provided by M'=λ_1'on 𝒲_N_M+1^M' which is {ϕ_N_M^M'(ϕ_N_M-1^M'( … (ϕ_N_M'+1^M'(v_i^(N_M'+1))) … )) }_i=1^𝒱^M'and extend it to an orthonormal basis for 𝒲_N_M+1^M. Notice that the colimit will be having the vectors {v_i^M'}_i=1^𝒱^M' which are the colimit of the sequences starting by the vectors of basis of 𝒲_N_M+1^M' mentioned above; this is why we had to consider truncations from the left as the sequence does not start with basis of 𝒲_N_M'+1^M'. It is formal diagram chasing, using the extension property of the connecting maps, that this construction gives a well-defined colimit 𝒱^M. By the choice of λ_j's, these 𝒱^Ms are {𝒱^λ_j}_j which means we have obtained 𝒱^M for all possible cut-off M. This is also compatible with the action of H^M as defined in the convergence property. This means for the set {(𝒱^M,H^M)}_M>λ_1, the embedding 𝒱^M'→𝒱^M is by identity for M' ≤ M and the following diagram commutes𝒱^M'r[swap]dH^M' 𝒱^M dH^M 𝒱^M'r 𝒱^M .Taking a second colimit of the set {(𝒱^M,H^M)}_M > λ_1 leads to the desired scaling limit (𝒱,H) where {(𝒱^M,H^M)} are restrictions of (𝒱,H). To see this, the (unique) operator H, which would make the diagram below commute for all M, is the desired Hamiltonian. Notice the embedding 𝒱^M →𝒱 is by identity since 𝒱 is a union of all 𝒱^Ms as they have a nested structure.𝒱^Mr[swap]dH^M 𝒱dH 𝒱^Mr 𝒱. Since two colimits are taken to obtain the scaling limit (similar to the construction in <cit.>), the above process is called the double colimit construction, allowing the following definitionGiven a sequence of quantum theories (𝒲_n,H_n) with given connecting maps ϕ_n^M satisfying properties [P](P), the scaling limit (𝒱,H) is the result of the double colimit construction. This limit will be written as (𝒲_n,H_n)(𝒱,H). We emphasize that as long as the connecting maps are specified the scaling limit process is unique up to unique isomorphism due to the nature of colimit. From now on, whenever a sequence of quantum theories is given with a scaling limit, implicitly, there is a given set of connecting maps. We do not discuss the issue of uniqueness any further and for a relevant example, we refer to the previous discussion in [1.6]section 1.6 on different diagonalization in the case of the Ising full CFT.Notice that 𝒱 is separable but not complete, so not yet a Hilbert space. The completion of 𝒱 will be 𝒱. For notational easiness, The scaling limit will be written as (𝒱,H) with the understanding that one needs to take a completion whenever the context requires so.We would like to think of the scaling limit as the result of stacking up the low energy spectra of H_ns, and the double colimit construction indeed fulfills this expectation. LetE_λ_1 be the eigenspace of the limit Hamiltonian H corresponding to λ_1 and λ_k be some larger eigenvalue of H for some k. Choose some M such that λ_1<M<λ_k, then the above construction builds E_λ_1 from the spaces 𝒲_n^M with large enough n, which contains all the vectors whose energy converges to λ_1 in the limit. The same holds for the other eigenspaces. Although our definition does not assume an embedding of the whole space 𝒲_n into 𝒲_n+1, we expect this to be the case for all physical models. Indeed, scaling limit should be after all a physical process in which a whole system is embedded into another one when some new particles are added.Our discussions in [4]section 4 will be based on this assumption, hence the need for a more refined definition:(𝒲_n,H_n) have a strong scaling limit (𝒱,H) if in addition to properties P, for all n and M, the connecting maps ϕ_n^M are the restriction up to energy M of an isometryϕ_n : 𝒲_n ↪𝒲_n+1,for large enough n. Given the above, the colimit of the sequence of embeddings 𝒲_n ↪𝒲_n+1 gives 𝒱 directly.Usually, the chosen basis for 𝒲_n closely relates to a notion of space, and locality in this space basis is supposed to represent locality in space. Finding the embedding ϕ_n, though an isometry, is nottrivial based on this basis. In the scaling limit, the space embedding is not the “trivial” embedding, in contrast to the thermodynamical limit <cit.>. In the scaling limit, the energy embedding is the trivial one as shown in the definition. As a result of this trivial energy embedding, the space local operators in 𝒲_n, like e_is in ACs, are generally space non-local when their actions are push-forwarded. This will become clearer in next few sections.Finding the “energy basis” requires an understanding of the energy local degrees of freedom (EL-DOFs), which comes from an exact diagonalization of the Hamiltonian. Even numerical exact diagonalization is very limited for interacting models. In the Ising AC case, exact diagonalization analytically gives us the creation and annihilation operators, which are the EL-DOFs.This, in turn, provides us the energy basis, which allows us to construct the scaling limit at each energy eigenspace. For all the models with known CFT limits, only free theories have mathematical descriptions of their EL-DOFs so far (see <cit.> for some recent examples). Another difficulty with a scaling limit is the description of observables in the limit. In the scaling limit, a “space” description of the operators in the limit is hard to find. For example, if we look at any observable in a CFT, the description which allows us to compute with, is in terms of mode operators, which are more naturally described as energy shifting operators while their space action is obscure. Indeed, the Y(a,z)s are considered to be space local observables, yet their description is a Fourier series of mode operators ∑ a_nz^-n-1. This, along with the fact that the chosen basis for 𝒲_n is closely related to the notion of space and not energy, complicates the process of finding a description of observables in the scaling limit. In the case of ACs, the e_is are space local operators. Therefore, having a general definition of Fourier transform on the e_is is essential, especially one that relates to the mode operators L_n. Alternatively, one will have to find and work with some space description of Y(a,z).§.§ Scaling limit of observables Given a sequence of quantum theories {(𝒲_n,H_n,𝒜_n}_n=1^∞ with the scaling limit (𝒱,H), by definition, Hilbert spaces 𝒲_n have strictly increasing dimensions,Hamiltonians H_n, and algebras of observables 𝒜_n. Recall that the algebra 𝒜_n is generated by an underlying real vector space of hermitian observables called 𝒜_n^H, and H_n ∈𝒜_n^H. In the examples of ACs, the space 𝒜_n^H is spanned by {e_j,i[e_j,e_j+1]}. This choice of generating set is motivated on one hand from including the local terms of interaction of the system, and on the other hand to recover the Virasoro algebra in the scaling limit; see [thm4.2]Theorem 4.2 and [rmk12]Remark 12. To build the observables of 𝒱 from the observables in 𝒲_n, the low energy behavior of the observables has to be taken into account.Let O_n ∈𝒜_n be any sequence of observables. For a given M and u,v ∈𝒱^M, denote by u_n,v_n ∈𝒲_n^M the vectors (ρ_n^M)^-1u,(ρ_n^M)^-1v, which are defined for sufficiently large n. The scaling limit of O_n is a partially-defined (defined on a subset of 𝒱×𝒱) sesquilinear form O(·,·), where O(u,v) is defined as lim_n →∞(u_n,O_nv_n) when it exists. We will denote the scaling limit by O_nO.The idea is that the operator O is constructed to exactly store the information in the expectation values of O_n.Consider the set of sesquilinear forms (which will also be called operators) in [dfn5]Definition 5. Define * 𝒜: the set of observables in [dfn5]Definition 5,* 𝒜: the vector space of sesquilinear forms that are scaling limit of observables in 𝒜_n and defined on 𝒱×𝒱,* 𝒜^H: the real vector space consisting of all hermitian operators defined on 𝒱 which are scaling limits of hermitian observables in 𝒜_n^H. One can ask whether 𝒜 generates 𝒜?This is true for Ising and any other model with what would be called an algebraic scaling limit (defined after the next remark). The algebraic scaling limit gives a copy of each observable of 𝒜_n inside 𝒜. Therefore, any operator O_nO ∈𝒜 can be seen as an operator obtained as scaling limit of the copies of O_n inside 𝒜, implying that 𝒜 in a sense generates 𝒜. Another simple observation is that (again by some standard diagonal argument) 𝒜 is closed under the obvious“weak limit”. In fact, we can consider the semi-norms ||·||_n on 𝒜 which is defined by ||O||_n=||P^nOP^n|| where P^n is the restriction up to energy λ_n, and ||P^nOP^n|| is the usual norm of a linear operator (linear, as it is nonzero only on a finite-dimensional space). Then, it is not hard to see that 𝒜 is a Fréchet space with respect to these (separated) countably many semi-norms providing the scaling limit metric d_SL. Indeed, to show completeness of 𝒜, assuming operators O^(n)∈𝒜 forming a Cauchy sequence, one can easily construct their limit O. But to prove this limit is in 𝒜, we need a sequence O_i^(n_i)∈𝒜_i having scaling limit O. This sequence is constructed by a standard diagonal argument from the sequences O^(n)_i ∈𝒜_i giving O^(n)s. Adding to the above remark, in the case of a strong scaling limit, assume there also exist embeddings τ_n: 𝒜_n ↪𝒜_n+1 compatible with the embeddings ϕ_n, i.e. ϕ_n ∘ O_n = τ_n(O_n)|_ϕ_n(𝒲_n),∀ O_n ∈𝒜_n. Then scaling limit becomes convergence in the metric d_SL. In fact, the closure with respect to d_SL, of the colimit of the sequence of embeddings 𝒜_n ↪𝒜_n+1 is precisely 𝒜, which we could call analytic scaling limit. The colimit can also be called the algebraic scaling limit and it contains a copy of each O_n by the sequence O_n,τ_n(O_n),τ_n+1(τ_n(O_n)),… which by algebraic construction (or scaling limit as τ_n is compatible with ϕ_n) gives a copy of O_n ∈𝒜 defined on 𝒱. By going through the definitions, we have the same picture presented in [rmk2]Remark 2. The embeddings τ_n exist in the case of study in <cit.>. A similar theorem can be established for the Ising ACs as both algebras are simply the even algebra generated by Dirac operators. We conjecture that it holds for higher level anyonic chains. When one looks at the different set of observables in the different frameworks for unitary CFTs, there is always an underlying set of hermitian observables generating the whole set. Indeed, as proved in <cit.>, the hermitian fields (more strongly, hermitian quasi-primary fields) generate the VOAs. As for LCNs, since the algebra corresponding to an interval I is a Von Neumann algebra, it is trivially true that it can be generated by hermitian observables. But does 𝒜^H generate 𝒜 in any way? We do not know the general answer.We wish to identify some subsets of 𝒜 that may be algebras. Since some operators are not linear, it is not clear how one can have an algebraic structure. Note that by definition, there might not be a linear operator which gives the sesquilinear form O. But if such an operator exists, it will be called O as well.In some cases, these operators can be almost linear. Consider for example the case of Y(a,z) in VOAs. Their expectation values are defined, while none of them is actually defined on the VOA. They are almost linear operators since there is a grading of the VOA. For an almost linear observable, one can formally set Ov=∑_i v_i where v_i are the well-defined degree λ_i component of Ov, and for any u ∈𝒱_i, we have O(v,u)=(v_i,u). This motivates us to call all operators inside 𝒜 almost linear operators. If the formal sum is always finite, O is a linear operator.The definition for the product of such operators is exactly in the same spirit of the correlation function(u,Y(a_1,z_1)… Y(a_k,z_k)v). Given almost linear observables O^(1),…,O^(k)∈𝒜, we define their product as a partially-defined sesquilinear form F by using the formal sum interpretation. If the result is absolutely convergent for some u,v ∈𝒱,F(u,v):=(u,O^(1)… O^(k)v),then the above is considered to be well-defined.We discuss one basic obstacle to get an algebraic structure by an example; observables O_n that have a significant mix of the low and high energy states. For example, the two sequences below where v^(i)_n ∈ E_λ_i^(n) are pull-back of some v^(i)∈ E_λ_i:* O_n,1=v_n^(1)(v^d(n)_n)^†+v^d(n)_n(v_n^(1))^†,* O_n,2=0.Both sequences converge to zero while being quite different. The significant (non-decaying) mix of low-high energy states in the O_n,1smanifests itself not in the expectation values of the observables at low energies, but the higher powers of the observables. Looking at the expectation values of powers, O_n,2^kO^k=0 whileO_n,1^2v^(1)(v^(1))^†≠ 0. Next example shows that just the decay of this low-high energy mix is not enough: * O_n,1=v_n^(1)(v_n^(1))^†+2^d(n)v^d(n)_n(v^d(n)_n)^†,* O_n,2=v_n^(1)(v_n^(1))^†+ ∑_i=1^d(n)1/i^2(v_n^(i)(v_n^(1))^†+v_n^(1)(v_n^(i))^†).It is not hard to check that O_n,1^kO_1^k, whereO_1=v^(1)(v^(1))^†, andO_n,2^kO_2^k, where O_2=v^(1)(v^(1))^†+ ∑_i=1^∞1/i^2(v^(i)(v^(1))^†+v^(1)(v^(i))^†).Both sequences would be regarded as well-behaved but the first one has a significant high-high energy mix while the second one has a decaying low-high energy mix. One can check that ||O_n,1O_n,2v_n^(1)|| ↛||O_1O_2 v^(1)||, i.e.(v_n^(1),O_n,2O_n,1O_n,1O_n,2v_n^(1)) ↛(v^(1),O_2O_1O_1O_2v^(1)).So O_n,2O_n,1O_n,1O_n,2 does not have O_2O_1O_1O_2 as a scaling limit. The reason behind this is an imbalance between the low-high energy mix decay rate and the rate of high-high energy mix. We note that it is possible to have a collection of observables with high-high energy mix, which is even increasing, and yet have an algebra, as will be shown in the case of Virasoro operators L_nL_n. For the discussion of algebra structures in scaling limit, a natural definition isGiven a set of almost linear observables {O^(i)}_i ∈ I, and the algebra of operators generated by this set. If this algebra is inside 𝒜, we call the resulting algebra a scaling limit algebra (SL-algebra).There could be many overlapping and yet different and maximal sets of observables forming an SL-algebra. Some of these are special in the sense that each observable has a nice sequence associated to:Given an SL-algebra as in [dfn8]Definition 8, assume each O^(i) is associated a sequence O^(i)_nsuch that for any i_1,i_2,…,i_k,lim_n→∞(u,O^(i_1)_n⋯ O^(i_k)_nv)=(u,O^(i_1)⋯ O^(i_k)v), ∀ u,v ∈𝒱.Then the algebra generated by {O^(i)}_i ∈ I is called a strong SL-algebra. For example, in the case of the Ising model, {L(f)| f ∈ C^∞(S^1)} gives a strong SL-algebra. The above definition assumes a strong property which is sometimes not easy to show; in [4]section 4, it is shown that {Y(a,f)| f ∈ C^∞(S^1)} gives only an SL-algebra.§.§ Locality in scaling limit First we review the terminology when it comes to the meaning of local observables. In LCNs (or more generally for QFTs in Haag Kastler's axioms), a local net A of Von Neumann algebras refers to the locality axiom: If I_1 and I_2 are spacelike separated, then elements in A(I_1) and A(I_2) commute. So local is used for the net when it satisfies the locality axiom. But also elements inside the local observables algebra 𝒜(I) are called local observables <cit.>.For the VOA or more generally for Wightman's axioms in QFT, observables are (primary) fields or distribution of operators Φ and limits of observables localized at a point x, Φ(x) <cit.>. In addition, there are local smeared fields Φ(f) with functions f having support in some region O <cit.> (if f is a test function, then Φ(f) is “almost local”). We also have a similar locality axiom: Let Φ_1 and Φ_2 be two observables and functions f_1 and f_2 be space-like separated in their supports, then [Φ_1(f_1), Φ_2(f_2)]= 0. The conclusion is that there is a notion of locality in all frameworks as an axiom and the elements of the sets satisfying those axioms are called local observables.Our definition of locality (in space and in energy) turns out to be more restrictive.§.§.§ Energy-local observables One goal in this work is to find out the constraints on observables in the scaling limit that will force them to be a specific type of observables (Wightman's, bounded as in LCN or point-like fields). Locality is one of these fundamental constraints.We propose a definition of energy local operators without using any explicit knowledge of the EL-DOFs. Therefore, it might not be the most refined definition. Still, our notion of energy locality, which is intrinsic, together with space locality put enough constraint on operators so that they are easier to work with (see [thm4.1]Theorem 4.1).All smeared operators Y(a,f) where f has finite Fourier series do not shift the energy of any eigenvector by more than a constant. This is a motivation for the definition of energy locality and to analyze energy local observables in general. The sequence (O_n)_n is Λ-energy local for Λ∈ℕ, if for any n and for all u∈ E_λ_i,v ∈ E_λ_j with |i-j|>Λ, and any sequences u_n ∈E_λ_i^(n),v_n ∈E_λ_j^(n) with u_nu, v_nv,(u_n,O_nv_n)=0.Any observable O ∈𝒜 which is the scaling limit of such a sequence is also called a Λ-energy local observable. It turns out that any almost linear observable which is Λ-energy local is a linear operator, as the formal sum Ov is a finite sum with no more than 2Λ terms. The important observation isThe set of all Λ-energy local observables for all Λ forms a strong SL-algebra. Consider Λ_i-energy local observables O^(i),1≤ i≤ k and corresponding sequences (O_n^(i))_n. Note (u,O^(k)… O^(1)v) is well-defined; indeed, if v ∈ E_λ_t for some t, then every multiplication by some O^(i) makes a vector in a space enlarged by adding Λ_i to the energy level. This means taking projections onto ℒ=⊕_i=t-∑Λ_i^t+∑Λ_iE_λ_i called P_ℒ, all operators O^(i) in the product can be replaced with the linear operator P_ℒO^(i)P_ℒ without changing the result.Similarly for the corresponding expectation values (u_n,O^(k)_n… O^(1)_nv_n), everything is also happening in a finite dimensional Hilbert space. In fact, the limit can be taken with restriction to 𝒲_n^M \𝒲_n^M', with λ_t+∑Λ_j<M<λ_1+t+∑Λ_j and λ_t-1-∑Λ_j<M'<λ_t-∑Λ_j, which is a finite dimensional Hilbert space stabilizing for large enough n and becoming isometric to ℒ. This means for large enough n, we might as well assume that all operators O^(i)_n are acting on ℒ, by using the connecting maps followed by the projection P_ℒ like the previous case. In this setting, we have a sequence of operators weakly convergent, but all acting on a finite dimensional Hilbert space. This implies norm convergence and the convergence of their product as a (∑Λ_j)-energy local operator. This is our first example of an algebraic structure which is preserved under the scaling limit. One can ask whether it is truly necessary for a constant Λ to be present in order to define energy locality. One might think of the possibility to enlarge the set of all Λ-energy local observables to include those operators that are scaling limits of Λ(n)-energy local observables where Λ(n) is a function of n. The motivation for this modification again comes from the smeared operators Y(a,f) where f has infinite Fourier series. Any product of these operators is defined on the VOA ([thm4.8]Theorem 4.8), so it is possible that they form a strong SL-algebra. They are not energy local by themselves, but it is clear that the higher shift of energies happen with ever smaller magnitude which depends on the Fourier coefficients |f̂_n|, a rapidly decaying sequence. There is also another motivation. In quantum computation, a space local operator is defined to be a sum of operators, each acting on no more than O(log(n)) particles for a system with n particles. But this is a discrete way of characterizing locality and equivalently, one could define space locality as an action that has exponential decay when one gets away from a specific particle. A similar picture exists for Y(a,f)s. The extent to which an operator can be called energy local could therefore be more than just shifting the energy by a constant. But we need to keep in mind that no matter how one extends this definition, the algebraic structure has to be preserved under the scaling limit. This issue will be explored further for the Ising AC. §.§.§ Space-local observables Another property of the smeared operators is that they are considered to be space-local. In order to have a notion of space, some notion of adjacency for particles in 𝒲_n is needed. In the case of anyonic chains, the notion of space locality is clear.The r-space local operators in ACs are a sequence of operators O_n ∈𝒜_n that are the sum of r-ultra space local operators. An r-ultra space local operator acts on r many of adjacent particles. A typical example is the 3-ultra space local operator e_i. Notice the difference between space-locality in our sense and locality in quantum computation. In quantum computation, a sequence of observables like O_n=e_1e_⌊n/2⌋∈𝒜_n is considered to be local, while it is clearly not space-local. Therefore, space locality is a stronger locality than the one in quantum computation.Still, the picture we hope to obtain for Y(a,f) in finite settings is that of a quantum system with a large number of equidistant particles, and some ultra space local operator a, which is supposed to be the finite version of a, applied with weight f on each particle and constantly many of its close neighbors. Informally,∑_j=1^n f(e^i2π j/n)a_j,will have the scaling limit Y(a,f). This will be explored in [4]section 4.§ SCALING LIMIT OF ISING ANYONIC CHAINS The main theorem of the section will be written in its entirety as a reference for the next sections. The proof will be given in the [A.1]appendix. We shall use the notations in [1.4]section 1.4, especially (a,b) which will be used to denote the Hilbert space given by the anyonic chain with the two ends of the chain being a and b. 1- The following strong scaling limits hold, up to some scalings of the Hamiltonians (explained below) * 𝒲_n=(1/2,1/2), H_n=-∑_j=1^2n-1e_j. Then (𝒲_n,H_n)(_0+_1/2,L_0). * 𝒲_n=(0,0) or (1,1), H_n=-∑_j=2^2n-2e_j. Then (𝒲_n,H_n)(_0,L_0). * 𝒲_n=(0,1) or (1,0), H_n=-∑_j=2^2n-2e_j. Then (𝒲_n,H_n)(_1/2,L_0). * 𝒲_n=(1/2,1) or (1/2,0), H_n=-∑_j=1^2n-2e_j. Then (𝒲_n,H_n)(_1/16,L_0). * 𝒲_n be the periodic chain of size 2n, and H_n=-∑_j=1^2ne_j. Then(𝒲_n,H_n)(_0_0+_1/2_1/2+_1/16_1/16,L_0+L_0)if n is even. Furthermore, the rate of convergence of each scaling limit is O(1/n) while we have restriction of energies up to O(√(n)). 2- For the corresponding higher Virasoro generators action, with the same rate of convergence as above, given a fixed m ≠ 0, we have (up to some scalings) * -∑_j=1^2n-1cos(m(j+1/2)π/2n+1)e_jL_m+L_-m, i∑_j=1^2n-2sin(m(j+1)π/2n+1)[e_j,e_j+1]i(L_m-L_-m) * -∑_j=2^2n-2cos(m(j+1/2)π/2n-1)e_jL_m+L_-m, i∑_j=2^2n-3sin(m(j+1)π/2n-1)[e_j,e_j+1]i(L_m-L_-m) * -∑_j=2^2n-2cos(m(j+1/2)π/2n-1)e_jL_m+L_-m, i∑_j=2^2n-3sin(m(j+1)π/2n-1)[e_j,e_j+1]i(L_m-L_-m) * -∑_j=1^2n-2cos(m(j+1/2)π/2n)e_jL_m+L_-m,i∑_j=1^2n-3sin(m(j+1)π/2n)[e_j,e_j+1]i(L_m-L_-m) * -∑_j=1^2ncos(2m(j+1/2)π/2n)e_j𝕃_m+𝕃_-m i∑_j=1^2nsin(2m(j+1)π/2n)[e_j,e_j+1]i(𝕃_m-𝕃_-m)If m ≤√(n), we have a rate of convergence of O(1/n) for energies up to √(n).Notation and L_m^c,s identities. For the Hamiltonians, assuming an n which will always be obvious from the context, we choose the notation L_0^c as a scaling of it, which has scaling limit L_0. The notations and scalings for the case 1(a), i.e. _0+_1/2, areL_0^c=α_n^cH_n + β_n^0,c1 L_0,where α_n^c=(2n+1)√(2)/8π and β_n^0,c∈ℝ. For the higher Virasoro generators, the first observable is O_n^c (superscript c because of cos), and the second O_n^s with the following similar notation and identities for 2(a)L_m^c+L_-m^c/2=α_n^cO_n^c+β_n^m,c1L_m+L_-m/2, i(L_m^s-L_-m^s)/2=α_n^sO_n^s+β_n^m,s1i(L_m-L_-m)/2,where α_n^s=(n+1/2)(√(2))^2/8π and the scalars β_n^m,c,β_n^m,s∈ℝ. Similarly for the full CFT, 𝕃_m^c+𝕃_-m^c and i(𝕃_m^s-𝕃_-m^s) can be defined. It will turn out that such a splitting is possible so that L_± m^c and L_± m^s, has scaling limit L_± m. The proof of the above theorem is provided in the [A.1]appendix and one can easily recover the scaling factors by following the proof. We will only need the rate of growth of these scaling factors which will be at best O(n^2) and α_n^s,α_n^c do not depend on m while β_n^m,c and β_n^m,c do. § SCALING LIMIT ALGEBRAS IN TEXT We would like to obtain the observables of each of these three types and prove they form an SL-algebra: * Wightman's observables or smeared fields Y(a,f),* LCN observables O ∈𝒜(I).* VOA observables or fields Y(a,z),It is not hard to show that they are all in 𝒜 as a vector space, i.e. all in a single framework. This fact tells us two things known before. First, that they are all physical as they describe some computable convergent sequence. And second, although they are all related and each one is believed to store all the information of the CFT by itself, by definition of scaling limit, they have to be in our set of observables simultaneously.We will first obtain (a), as a result, recover the observables of (b), and lastly, some comments will be made on (c). The nonperiodic chains or in other words the chiral cases will be handled first. Due to its simplicity, only the case 𝒱=_0+_1/2 will be analyzed, but all theorems can be similarly stated for the other chiral cases. At the end, there will be some comments on similar results for the full CFT. §.§ Wightman's observables §.§.§ Smeared vertex operator Y(a,f) We will try to identify when hermitian observables of the form η1+∑_j t_je_j &η1+i∑_j t_j [e_j,e_j+1]that are already space-local, are also energy-local. A trigonometric interpolation of the t_js with cos(m(j+1/2)π/2n+1) or sin(m(j+1)π/2n+1) is performed. Afterwards, previous results can be used to write down the observable in terms of (L_m^c+L_-m^c) or i(L_m^s-L_-m^s), where L_± m^c,L_± m^s are the operators with scaling limit L_± m.For the observable O_n=η_n1+∑_j=1^2n-1 t_je_j, a trigonometric interpolation using cos(m(j+1/2)π/2n+1) for 0 ≤ m ≤ 2n-2 givest_j= α_n^c∑_m=0^2n-2 a_mcos(m(j+1/2)π/2n+1)O_n=γ_n1+a_0L_0^c+∑_m=1^2n-2 a_mL_m^c+L_-m^c/2,where γ_n is some multiple of identity. Next, suppose O_n does not shift the energy more than some given Λ.An analysis of L_m^c formula given in [eq32](32) and [eq33](33), shows two distinct parts0.99(∑_k+m ≤ 2ncos((k+m/2)π/2n+1) Ψ_k+mΨ_k^†-∑_k+m>2ncos((k+m/2)π/2n+1)Ψ_2(2n+1)-k-mΨ_k^†).The first part provides an energy shift of exactly -m. The second part provides an energy shift of 2(k-(2n+1))+m when k+m > 2n+1 (if k+m=2n+1, since Ψ_2n+1=0, that term is irrelevant). This energy shift is between (-m,m) and it has the same parity as m. The same holds for L_-m.After the appropriate relabelling Ψ_k →Ψ_n/2+1-k (explained after [eq17](17)), the term Ψ_-(n+1/2)Ψ_-(n+3/2) provides an energy shift of -(2n-2) and it is only in L_2n-2^c due to the observation in the previous paragraph. Since O_n is energy local (1,O_nΨ_n-1/2Ψ_n-3/21) = 0 implying a_2n-2=0.It is easy to see how inductively each a_m is zero; for a_2n-3, taking the term Ψ_-(n-1/2)Ψ_-(n-5/2) leading us to the similar conclusion a_2n-3=0 and so on.The case η1+i∑ t_j[e_j,e_j+1] can also be done by using the trigonometric interpolationt_j=α_n^s ∑_m=0^2n-3 b_msin(m(j+1)π/2n+1).By mixing both cases (L_m^c+L_-m^c) and i(L_m^s-L_-m^s), we getO_n is a Λ-energy local observable made from a linear combination of e_j and [e_j,e_j+1]s and the identity if and only if it is of the formO_n=γ_n 1+ a_0L_0+∑_m=1^Λ(a_mL_m^c+ib_mL_m^s)+∑_m=1^Λ(a_mL_-m^c-ib_mL_-m^s),where a_m,b_m ∈ℝ. An operator L_m is desired which has scaling limit L_m so that expressions like ∑f̂_mL_m∑f̂_mL_m can be used where f̂_m=a_m+ib_m ∈ℂ. Dealing with a_mL_m^c+ib_mL_m^s every time can become inefficient and the choice below resolves this issueL_m:=(L_m^c+L_m^s/2+L_-m^c-L_-m^s/2) ∀ m ≠ 0, L_0=L_0^c.The above is a definition for an operator for which L_mL_m and it satisfies the properties for convergence as it inherits those from the two operators. Indeed, L_-m^c-L_-m^s/2 when restricted to √(n) energy, will be an operator with a norm at most O(1/n) and so will become part of the error of the approximation. The rest of the operator acting on energy higher than √(n) will join that of L_m^c+L_m^s/2.Notation. O|_E denotes the restriction to energy at most E, i.e. OP^E, and O|_>E:=O(1-P^E).Notation. From now on, n will not be used for the virasoro mode operators, but for the sequence index which will be related to the size of the chain 2n+1. For exampleL_m=L_m|_√(n)+O(1/m)+R_n^m,where R_m^n=L_m|_>√(n).We can now state our first result for the scaling limit of observables.The energy local scaling limit of the sequence of hermitian observables 𝒜_n^H spanned by e_j,i[e_j,e_j+1] and the identity as a real vector space is {L(f)+γ1 |fhas finite Fourier series, γ∈ℝ}.One can remove the space local condition as all observables in 𝒜_n^H are space-local. Assume a sequence of Λ-energy local operatorsO_n=γ_n1+∑_j=-Λ^Λf̂_j^nL_j,where f̂_-j^n=f̂_j^n and O_nO. To show that O=L(f)+γ1 for some function f with finite Fourier series, restrict O_n to some energy M>2Λ, O_n|_M=γ_n1+∑_j=-Λ^Λf̂_j^nL_j|_M.According to the properties of L_js, for large enough n, O_n|_M=γ_n1+∑_j=-Λ^Λf̂_j^nL_j|_M+f̂_j^nO(1/n).Since O_n|_M has a limit in the operator norm to O|_M, f̂_j^ns must have a limit. To prove that, we compute the inner product below for the vacuum 1:(L_-Λ|_M1,O_n|_M1)=f_-Λ^n ||L_-Λ1||+ (L_-Λ1,(∑_j f̂_j^nO(1/n))1)→ (L_-Λ|_M1,O1),where |_M is dropped as it is no longer needed. Notice all the errors O(1/n) corresponding to L_j give at most |j| energy shift. This mean only the errors corresponding to L_±Λ have to be handledf_-Λ^n ||L_-Λ1||+(L_-Λ1,(f_Λ^nO(1/n)+f_-Λ^nO(1/n))1).f_-Λ^n ||L_-Λ1|| can be exactly computed and is of orderf_-Λ^n Λ^3/2. The rest can have norm at most O(1/n)|f_-Λ^n| as f_Λ^n=f_-Λ^n. It is easy to see from here that in order for the above to have some limit,f_-Λ^n must have some limit f_-Λ.Next step is to subtract f_Λ^nL_Λ+f_Λ^nL_-Λ from O_n and repeat the procedure. For the special case of j=0,γ_n1+f_0^nL_0 can be seen to give the same conclusion. Denoting lim_n→∞f̂_j^n= f̂_j, lim_n→∞γ_n=γ,O=γ1+∑_j=-Λ^Λf̂_jL_j.By [thm2.1]Theorem 2.1, we have a strong SL-algebra.We would like to have our theorems as general as possible. For UMMs, higher level ACs <cit.> is conjectured to give the same results as in [thm3.1]Theorem 3.1, implying the above theorem for UMMs. But a relaxed version of that theorem for UMMs would still give us the results in this section:For any UMM VOA 𝒱=𝒱_c,0 and chiral representation 𝒱_c,h, there is a sequence of quantum theories with strong scaling limit (𝒱_c,h,L_0) such that for each L_m, we have a sequence L_m ∈𝒜_n with the following properties: * It is a space local observable with hermitian operators aL_m+aL_-m∈𝒜_n^H.* It shifts the energy no more than |m|.* Restricted to energy at most n^d_ω it has the following approximation by L_m|_n^ d_ω with the rest being R_n^m:L_m=L_m|_n^d_ω+O(1/n^g_ω)+R_n^m,where d_ω,g_ω are positive constants.* Its norm is bounded by O(n^e_ω) for some constant e_ω.It should be noted that the second and third item above have a meaning after the “push-forward” of the map L_m acting on 𝒱_c,h is assumed. This is done by the natural embedding ρ_n: 𝒲_n ↪𝒱_c,h from the strong scaling limit and the map ρ_nL_m(ρ_n)^-1 which acts on the copy of 𝒲_n inside 𝒱_c,h and extended by zero on the orthogonal complement. This “push-forward” will be implicitly assumed whenever it is necessary. Also, notice that this is not the “natural” embedding but it will work for our purposes (in Ising, the natural embedding, is described in the [A.1]appendix). The last assumption is true for the Ising chain as L_m^c and L_m^s are after all a sum of 2n terms of e_js which have norm order one. Taking the scaling factors α_n^c and β_n^m,c,β_n^m,s and their norm into account||L_m ||≤O(n^2).Assuming the above conjecture, the [thm4.2]Theorem 4.2 is true for all UMMs with the exception that the statement should change to: the scaling limit of space energy local contains the set {L(f)|ffinite Fourier series}. Therefore, except for [thm4.1]Theorem 4.1, [thm4.2]Theorem 4.2 and [thm4.15]Theorem 4.15, all other theorems in sections [4.1]4.1 and [4.3]4.3 will hold the way they are stated for all UMMs. For all theorems in [4.2]section 4.2, the stronger [cnj5.5]Conjecture 5.5 which tells us exactly how to recover the higher Virasoro modes for UMMS has to be assumed.The theorems below will be proved using the Ising AC, but by replacing some of the powers by appropriate constants (d_ω, etc), the results hold for UMMs assuming [cnj4.3]Conjecture 4.3. It is conjectured that all the VOAs we care about (as described in [1.1]section 1.1) satisfy energy boundedness <cit.>. A generalization of the [cnj4.3]Conjecture 4.3 to all chiral CFTs which satisfy energy boundedness is possible. Sequences in the same fashion of the Virasoro modes have to exist for all elements inside a minimal quasi-primary hermitian field generator set of the VOA. Then, all theorems in section [4.1]4.1 and [4.3]4.3 except [thm4.1]Theorem 4.1, [thm4.2]Theorem 4.2 and [thm4.15]Theorem 4.15 can be recovered. In UMMs, the generator is only ω and in WZW models, the currents corresponding to the Lie algebra 𝔤 (see <cit.> for a numerical demonstration and also for W-algebra currents see <cit.>). Notation. Set L(f)_≤ m=∑_|j| ≤ mf̂_jL_j and similarly for L(f). Similarly define L(f)_> m and L(f)_> m. Also set||f||_s^≤ E=∑_|i|≤ E |f̂_i|(|i|+1)^s,and |f|^≤ m:=∑_|i| ≤ m|f̂_i|.We wish to show that the choice of the “natural” sequence corresponding to L(f) gives a strong SL-algebra. Some lemmas are needed.We haveL(f):=∑_j=-∞^∞f̂_jL_j ∈𝒜_n^H, for allf ∈ C^∞(S^1)Note that f̂_js are rapidly decreasing. Also, from [rmk7]Remark 7,||L_j|| ≤ O(n^2).The estimation does not depend on j. This gives an absolute convergence to an operator with norm bounded by |f|O(n^2). On the other hand, for each j, we have f̂_jL_j+f̂_-jL_-j∈𝒜_n^H implying L(f) ∈𝒜_n^H.The next step to establish a strong SL-algebra is to proveL(f)L(f). The result shown here on the convergence behavior of L(f) will be useful in the next theorem. Take any k ∈ℕ and consider N_f,(10k)^3+1 for which |f̂_j|<1/j^(10k)^3+1 for all j>N_f,(10k)^3+1. For n large enough such that √(n)>N_f,(10k)^3+1, using [eq1](1),||L(f)_>√(n)||=||∑_|j|>√(n)f̂_jL_j|| ≤ O(n^2)∑_|j|>√(n) |f̂_j| ≤ O(n^2)∫_√(n)^∞1/x^(10k)^3+1dx < O(n^2)(10k)^3+1/n^(10k)^2=O(n^-(10k)^2+2).An same estimate for L(f)_>√(n) via energy bounds is the next step:||f||_3/2^>√(n)<2∑_j>√(n)^∞(j+1)^3/2/j^(10k)^3+1<∫_√(n)^∞1/x^(10k)^3-10k+1=O(n^-(10k)^2+1),therefore||L(f)_>√(n)v|| <O(n^-(10k)^2+1)||(L_0+1)v||.Next, given a vector v ∈𝒱 and the embedding 𝒲_n ↪𝒱,(L(f)-L(f))v= (L(f)_≤√(n)-L(f)_≤√(n)) v + (L(f)_>√(n)-L(f)_>√(n)) vThe two estimations above imply that the second part vanishes. For the first part,L(f)_≤√(n)=L(f)_≤√(n)|_√(n)+O(|f|_j ≤√(n)/n)+R(f),where R(f)=L(f)_≤√(n)|_>√(n). Since v has finite energy, for large enough n, R(f)v=0 and L(f)_≤√(n)|_√(n)v=L(f)_≤√(n)v. This implies ||(L(f)-L(f))v||→ 0, which is indeed a stronger result than L(f)L(f). The set {L(f)|f ∈ C^∞(S^1)} gives a strong SL-algebra with corresponding sequence L(f) to each L(f). For the vacuum 1 and 1_n=(ρ_n)^-11, the statement implies (1_n, ∏_j=1^k L(f^(j)) 1_n) → (1,∏_j=1^k L(f^(j))1).Proving the above is enough as this can be done similarly for any two vectors u,v ∈𝒱. The fact that the right side is defined is shown in <cit.>. We will prove the above by using triangle inequality after estimating the intermediate terms|(1,∏_j=1^t-1 L(f^(j)) (L(f^(t))-L(f^(t)))∏_j=t+1^k L(f^(j))1)|, 1 ≤ t ≤ k,where the embedding ρ_n is used implicitly. For each 1≤ j ≤ t, L(f^(j))=L(f^(j))_≤√(n)+L(f^(j))_> √(n). Denote y_t=∏_j=t+1^k L(f^(j))1 and let y_t=y_t^1+y_t^2, where the first vector is inside 𝒱^(k-t)√(n)⊂𝒱^k√(n) of vectors with energies at most k√(n), defined asy_t^1=∏_j=t+1^kL(f^(j))_≤√(n)1.To estimate the norm of ||y_t|| and ||y_t^i||s, the norm of the two operators decomposing L(f^(j)) has to be bounded from above. Equation [eq2](2) gives ||L(f^(j))_> √(n)|| <O(n^-(10k)^2+2). As for ||L(f^(j))_≤√(n)||, there are two different estimations. One will be used to find an upper bound for ||y_t^1|| and the other to bound ||y_t^2||.For y_t^1, as the product is applied on the vacuum, consider the restriction of each of those operators to energy ≤ k√(n). By energy bounds||L(f^(j))_≤√(n)|_k√(n)|| = ||L(f^(j))_≤√(n)|_k√(n) + O(|f^(j)|_≤√(n)/n) || ≤ 2𝒞_ω||f^(j)||_3/2^≤√(n)(k√(n)+1)=O(√(n)),for large enoughn.The second estimate is coming from [eq1](1)||L(f^(j))_≤√(n)|| ≤ O(n^2).By using [eq5](5),||y_t^1|| ≤ O((√(n))^k-t)<O(√(n)).To estimate ||y_t^2||, let us take the expansion of ∏_j=t+1^kL(f^(j))1=∏_j=t+1^k (L(f^(j))_≤√(n)+L(f^(j))_> √(n))1and consider those terms that have at least one L(f^(j))_> √(n) in them. Those will be the ones contributing to y_t^2. Hence, as there are 2^k-t-1 such terms,||y_t^2|| < (2^k-t-1) O(n^-(10k)^2+2)O((n^2)^k-t)) ≤ O(n^-(10k)^2+2k+2).The estimates for ||y_t^1||,||y_t^2|| give||y_t|| < 2||y_t^1|| < O(√(n)).Letx_t:=1^†∏_j=1^t-1 L(f^(j))&max_t=1,…,k||x_t||=p &max_t=1,…,k||(L_0+1)x_t^†||=q.It can be shown that (<cit.>)p=max_t=1,…,k||x_t|| ≤ r||(L_0^k+1)1||,where r depends on f^(j)s. p depends on k, f^(j)s, and the degree of vector v (which is chosen to be the vacuum here). Obviously, there is no dependence on n. We can derive a bound on q using the above. Let us approximate|x_t(L(f^(t))-L(f^(t)))(y_t^1+y_t^2)|.Decomposing (L(f^(t))-L(f^(t))) as in [lem4.5]Lemma 4.5:(L(f^(t))_≤√(n)-L(f^(t))_≤√(n))+ (L(f^(t))_>√(n)-L(f^(t))_>√(n)).For the second part, using the estimates [eq9](9) for ||y_t||, [eq2](2) on ||L(f^(t))_>√(n)||, and finally [eq3](3) for ||x_tL(f^(t))_>√(n)||=||L(f^(t))_>√(n)x_t^†||,|x_t(L(f^(t))_>√(n)-L(f^(t))_>√(n))y_t|< pO(n^-(10k)^2+2)O(√(n))+ qO(n^-(10k)^2+1)O(√(n))0.For the first part, considering the approximation of Ls for energies up to k√(n),L(f^(t))_≤√(n)-L(f^(t))_≤√(n)= -L(f^(t))_≤√(n)|_>k√(n)+O(|f^(t)|_≤√(n)/n)+R(f^(t)),The first term L(f^(t))_≤√(n)|_>k√(n) annihilates y_t^1 as the vector is inside 𝒱^k√(n). As for its action on y_t^2, instead of taking the norm of that multiplication, one can apply the energy bound on the left multiplication by x_t and due to the smallness of ||y_t^2||, it is easy to see that it vanishes when n →∞.The second term, which is the only term where our approximation gets somewhat tight, when acting on y_t, has to compete with its norm. The estimation [eq7](7) tells us that the result is bounded by O(√(n)/n) which still goes to zero.Finally, the last term is the higher energy term L(f^(t))_≤√(n)|_> k√(n). When acting on y_t^1, this will give zero. Then, one can use the bound on the norm of L(f^(t))_≤√(n) (recall that this is a bounded operator like L(f^(t)) with norm O(n^2)). As ||y_t^2|| is much smaller, this will vanish as well.We note that 𝒜^H contains more than just the strong SL-algebra above:We have {L(f)|||f||_3/2<∞}⊂𝒜^H which contains {L(f) | f ∈ C^∞(S^1)} as a maximal strong SL-algebra. For the maximality part, one has only to estimate the norm of L_0^kL(f)1 for any k ∈ℕ. This would imply that the Fourier series of f must be rapidly decreasing and therefore f ∈ C^∞(S^1).It is also clear, by some analysis easier than [lem4.5]Lemma 4.5, that any L(f), with ||f||_3/2<∞, is obtainable as a sequence by choosing (e.g.) O_n=L(f)_≤log(n). The rest was done in the previous theorem.The next theorem generalizes to all fields. We list three facts <cit.> * In a UMM, the descendants of ω span the VOA.* Due to the Virasoro algebra identities, all descendants of ω can be obtained only by applying operators L_n(n≥-2).* In a UMM, all fields are energy bounded:||Y(a,f)v|| ≤ C_a||f||_r_a ||(L_0+1)^s_av||{Y(a,f) |a ∈𝒱, f ∈ C^∞(S^1)}⊂𝒜 generates an SL-algebra. The algebra is also local but in the quantum computation sense of locality (QC-locality) where product of a constant number of e_is far apart from each other still counts as local. But it must be observed that if a field is space local, then its derivative (Y(L_-1a,z)=[L_-1,Y(a,z)]) is also space local. It is not hard to show using the same approach in <cit.> that||Y(a_1,f^(1))… Y(a_k,f^(k))v|| ≤α||(L_0+1)^∑ s_a_iv||,where α depends on f^(j)s and k. Thus the set Y(a,f) has all products defined on 𝒱. We will proceed by induction. Choose a basis with descendants. Then, for each field L_i_r… L_i_1ω=a an induction will be performed on r. Hence, assume hypotheses have been shown to hold for the field b and we wish to prove the same for a=L_-2b; the Borcherds identity shows that this is the hardest case and L_rb for r≥ -1 are easier and will be described later.For the field a, we want to obtain operators y_E(a)_m for energy E and mode m with the following hypotheses:There exist d_a such that for all E ≤ n^d_a there are operators y_E(a)_m satisfying * y_E(a)_m is generated by the e_is QC-locally; i.e there is some constant p_a such that y_E(a)_m is p_a QC-local and p_a is independent of m and E.* ∃ v_a<d_a such that for any m and n^v_a≥ |m|, y_E(a)_m|_E provides an energy shift at most K_a(E+|m|) for some constant K_a ≥ 1.* y_E(a)_m has norm at most n^e_a where e_a depends on a.* There exist a constant g_a>0 such that y_E(a)_m=y(a)_m|_E+O(1/n^g_a)+R_E,n^a,m, for n^v_a≥ |m|where y(a)_m|_E is the restriction of y(a)_m in the VOA to energy at most n^d_a but acting on 𝒲_n via pushback. O(1/n^g_a) should be regarded as the error in the approximation of y_E(a)_m|_E by y(a)_m|_E, and it has norm at most O(1/n^g_a). Finally, the last term is R_E,n^a,m=y_E(a)_m(1-P^E).Notice the last hypothesis implies the same for restriction of energy to any E' ≤ E since projection to energy E' has norm at most 1 and the rest will mix with R_E,n^a,m. Further, the base of induction ω is essentially done. For E ≤ n^1/4 and any √(n)≥ |m|, as y_E(ω)_m=L_m provides an energy shift of at most |m| for any mode m, in other words, at most 1 × (E+|m|).Suppose the hypotheses are true for b and a=L_-2b. In [1.1]Borcherds identity, putting p=0 and q=-1, and some index shifting givesy(a)_m=y(L_-2b)_m=∑_j=0^∞(L_-2-jy(b)_m+j+2+y(b)_m-j+1L_j-1),which is an infinite sum but when restricted to energy E≤ n^d_a, where d_a will be determined, the summation above will be finite and summed to some j. Indeed, as shown in <cit.>, consider the projection y(a)_mP^E. Then the first term is always zero when E-(m+j+2)<0 and the second term is always zero when E-(j-1)<0. So both are zero when E<j+max{-1,m+2}. Hencey(a)_m|_E= ∑_j=0^E-max{-1,m+2}(L_-2-jy(b)_m+j+2P^E+y(b)_m-j+1L_j-1P^E).Putting redundant projections in the middle of the operators leads to y(a)_m|_E=0.86∑_j=0^E-max{-1,m+2}(L_-2-j|_E+K_b(E+|m+j+2|)y(b)_m+j+2|_E+y(b)_m-j+1|_E+|j-1|L_j-1|_E).This will be important as the last induction hypothesis for ω and b will be applied separately. Based on the above identities, our choice for y_E(a)_m will be y_E(a)_m= ∑_j=0^E-max{-1,m+2}(L_-2-jy_E(b)_m+j+2+y_E+|j-1|(b)_m-j+1L_j-1)where we recall that y_E(ω)_j=L_j for all E. The first hypothesis obviously holds as p_a≤ p_w+p_b for all m. One can easily see why only QC-locality can be proved.For the second hypothesis, assume v_a<d_a and smaller than v_b (<d_b). This allows us to apply the hypothesis on y_E(b)_m+j+2, i.e. to haven^v_b≥ (2n^v_a+n^d_a+4) ≥ (|m|+|j|+2).This will be one of the restrictions on v_a and d_a. At the end of the argument, choosing v_a<d_a << d_b,g_b,v_b will be shown to be enough. Given E ≤ n^d_a, the hypothesis for b implies that y_E(b)_m+j+2 and y_E+|j-1|(b)_m-j+1 will provide an energy shift at mostK_b(|m+j+2|+E) and K_b(|m-j+1|+E+|j-1|)) which added to the energy shift of the Virasoro operators is at most|j+2|+K_b(|m+j+2|+E) and K_b(|m-j+1|+E+|j-1|)+|j-1|. Since |j| ≤ E+|m|+2, there is a constant K_a such that the energy shift is at most K_a(E+|m|).The third hypothesis is easy to check as this rough estimate for all m holds|y_E(a)_m| < n^d_a+v_a+3n^e_b+e_ω. Implying that e_a can be chosen d_a+v_a+e_b+e_ω+3.For the last hypothesis, from the equations [eq12](12) and [eq13](13), the sum of the approximations given by the hypothesis for ω and b is y(a)_m|_E. But what about the other terms? Consider L_-2-jy_E(b)_m+j+2 which is the product(L_-2-j|_E+K_b(E+|m+j+2|)+O(1/n^g_ω)+R_E+K_b(E+|m+j+2|),n^ω) (y(b)_m+j+2|_E+O(1/n^g_b)+R_E,n^b),where the superscript for Rs indicating the mode is dropped as it is clear from the context. In each parenthesis, the first two terms are L_-2-j|_E+K_b(E+|m+j+2|) and y_E(b)_m+j+2|_E respectively. The term that contributes to y(a)_m|_E is preciselyL_-2-j|_E+K_b(E+|m+j+2|)y(b)_m+j+2|_E.What contributes to R_E,n^a,m is also clear(L_-2-j|_E+K_b(E+|m+j+2|)+O(1/n^g_ω)+R_E+K_b(E+|m+j+2|),n^ω)R_E,n^b,which is indeed an operator with restriction to energy higher than E.Further, due to energy restrictionsR_E+K_b(E+|m+j+2|),n^ω (y(b)_m+j+2|_E+O(1/n^g_b))=0.The only terms remaining should contribute to O(1/n^g_a): * O(1/n^g_ω)O(1/n^g_b)* O(1/n^g_ω)y(b)_m+j+2|_E* L_-2-j|_E+K_b(E+|m+j+2|)O(1/n^g_b)We need to show that while choosing g_a appropriately. Also, in addition to this analysis, one has to analyze the product y_E+|j-1|(b)_m-j+1L_j-1 for all 0 ≤ j ≤ E-max{-1,m+2}.But E ≤ n^d_a and further, |m| ≤ n^v_a. It is not hard to see that although there are so many terms contributing to what should be O(1/n^g_a), by choosing v_a,d_a small enough compared to g_b,g_ω, the approximation will be of the form O(1/n^g_a) with norm at most O(1/n^g_a).The first term O(1/n^g_ω)O(1/n^g_b) is O(1/n^g_a+g_ω) with the obvious bounded norm. As mentioned in the last paragraph, this will show up many times and so, there is one restriction here on d_a,v_a.As for the second term, y(b)_m+j+2|_E is a bounded operator with norm at most C_b(|m+2+j|+1)^r_b(E+1)^s_b≤ C_b(n^d_a+2n^v_a+4+1)^r_b(n^d_a+1)^s_b.In other words, the norm is bounded byO((n^d_a+2n^v_a+5)^r_b(n^d_a+1)^s_b/n^g_ω).Hence small enough d_a,v_a can deliver the desired result. The story for the third term is similar, energy bound is used for L_-2-j|_E+K_b(E+|m+j+2|) C_b(|2+j|+1)^3/2(E+K_b(E+|m+j+2|)+1) ≤ C_b(n^d_a+n^v_a+4+1)^3/2(n^d_a+K_b(2n^d_a+2n^v_a+4)+1).The norm is bounded byO((n^d_a+n^v_a+5)^3/2(n^d_a+K_b(2n^d_a+2n^v_a+4)+1)/n^g_b)which is another restriction on how small d_a,v_a have to be.One can handle the product y_E+|j-1|(b)_m-j+1L_j-1 in a similar way. The induction is finished when a=L_-2b.If a=L_rb for any r>-2, then the Borcherds identity would be(L_rb)_m=∑_j=0^∞ (-1)^j r+1j(L_r-jb_m+j-(-1)^r+1b_r+1+m-jL_j-1)and the treatment of this case is easier since the summation is finite; for j>r+1≥0 we have r+1j=0. All the properties described in the induction can be proved here as well. Therefore, the induction is fully proved.It remains to show that {Y(a,f) | a ∈𝒱, f ∈ C^∞(S^1)} generates an SL-algebra. Using the properties in the induction hypotheses, it can be seen that the proof is nothing but a more involved version of [thm4.6]Theorem 4.6.To get the product ∏_j=1^k Y(a_j,f^(j)) in the scaling limit, the operatorsY_E_j(a_j,f^(j))=∑_m f̂^(j)_my_E_j(a_j)_m ∈𝒜_nhave to be chosen where E_js need to be determined carefully by taking into account the constants in the energy bound inequalities for all a_js, and also all other constants, notably d_a_js and g_a_js, so that we can use the approximation provided by the last hypothesis. It is clear that the choice of E_js will not be universal and depends on the product. They will also not be equal due to the second hypothesis and will be very small compared to all other constants.The reason we could not obtain smeared fields as a strong SL-algebra generating set is the dependence of the energy shift on the energy itself. If somehow all vectors were obtained by only applying L_r, r > -2 (because of the finite sum) or if we knew that the base of induction L_m shifts the energy exactly by m, this issue would not be present. One would wish to get the hermitian fields giving self-adjoint Y(a,f), as a scaling limit of hermitian observables generated by the e_is. Descendants of even degree of ω are hermitian if and only if they are quasi-primary. Also, as <cit.> demonstrates, quasi-primary hermitian fields generate (not span) any unitary VOA. Further, we could not find any exact formula or general description of these fields. But a generating set of quasi-primary hermitian fields can exist which have a corresponding sequence coming from (a generating set formed by hermitian observables, i.e.) 𝒜_n^H. For UMMs, that generating set is {ω}, for which there is a corresponding sequence from 𝒜_n^H. §.§ Local conformal nets observables In this section, bounded operators in the LCN framework are recovered. Recall that for UMMs, the observables algebra on an interval I is given by {e^iL(f)|supp(f) ⊂ I }” <cit.>.From results of the previous section, the following is immediate The sequence of observables below give a strong SL-algebra:e^iL(f) e^iL(f).This is a direct application of the Trotter-Kato approximation theorem (see e.g. <cit.>) on [thm4.6]Theorem 4.6. The fact that the scaling limit is a strong SL-algebra is simply due to the uniform boundedness of the operators involved, all being unitary.All operators in {e^iL(f)}” are in 𝒜 giving a strong SL-algebra. As all algebras here are generated by self-adjoint operators, we will be considering only self-adjoint operators (this will make applying Kaplansky's density theorem easier). Consider a sequence of self-adjoint operators O^(i) in the algebra generated by e^iL(f)s with a strong limit to a self-adjoint bounded operator O. Each O^(i) has a corresponding sequence of self-adjoint (O^(i)_n)_n with scaling limit O^(i) which can be thought of replacing any e^iL(f) in O^(i)'s expression by e^iL(f). From these sequences, by a standard diagonal argument, one can get a sequence O_n with scaling limit O.As long as O^(i)s are uniformly norm bounded, there is the possibility of having a sequence O_n that is uniformly bounded, giving a strong SL-algebra as in the previous theorem. This includes the case where O is in the norm-operator closure of the algebra generated by {e^iL(f)}. So the C^*-algebra can be recovered. Then, Kaplansky's density theorem does the rest: one can apply it on the sequence O^(i), such that it becomes uniformly bounded by ||O|| and then apply the same theorem on each sequence associated to O^(i) so that they become in turn uniformly bounded by ||O^(i)||. Hence, all observables in LCN form a strong SL-algebra. The next question is whether there exist some definition of the algebra 𝒜_n(I) and how the bounded scaling limit would compare to the LCN, called 𝒜_lcn(I). As we shall see, the anyons must be on the upper half-circle as in [fig1]Figure 1.Consider the upper half-circle S^1_+ with its two points on the boundary. The set of intervals ℐ_+ are the connected sets in one of the following forms: * Open intervals I inside S^1_+ for which ∂ I ∩∂ S^1_+=∅,* Closed-open intervals I where |I ∩∂ S^1_+| = |∂ I ∩∂ S^1_+|= 1,* S^1_+. On these sets, the following nets of observables are definedGiven I ∈ℐ_+, 𝒜_n(I) is generated by e_js where [jπ/2n+1,(j+1)π/2n+1] ∈ I and the identity.The definition can be seen to imply [𝒜_n(I_1),𝒜_n(I_2)]={0} which is locality. Isotony is obvious, i.e. I_1 ⊂ I_2 𝒜_n(I_1) ⊂𝒜_n(I_2).Consider the set of self-adjoint bounded linear operators O in the scaling limit of the algebra of observables 𝒜_n(I) such that there exist a self-adjoint sequence O_n ∈𝒜_n(I) with bounded norm andρ_n(O_n(ρ_n)^-1(u)) → Ou, ∀ u ∈𝒱,i.e. there is sequence with strong SL convergence to O or the strong-operator convergence in 𝒱. Define 𝒜_b(I) as the von Neumann algebra generated by the set.Locality is the reason behind the above definition. Consider two sequences of operators x_nx and y_ny which are self-adjoint and commuting. In order to ensure [x,y]=0, it can be easily observed that the weak-limit offered by scaling limit is not enough and we need at least a strong type of that limit (which is the above definition). But that could not be enough as x_ny_nξ→ xyξ for ξ∈𝒱 can not be necessarily true yet:(x_ny_n-xy)ξ=x_n(y_n-y)ξ+(x_n-x)yξ.The first and second part of the above summation are not guaranteed to go to zero unless x_ns are uniformly bounded and x_n → x in the strong-operator topology (of 𝒱 as yξ∈𝒱). It turns out that the strong SL convergence (which is strong-operator convergence in 𝒱) and norm boundedness are in some way equivalent to convergence in the strong-operator topology (in 𝒱). One direction is clear and the other is the application of Kaplansky's density theorem to get such a sequence with norms uniformly bounded. The definition above imposes these properties and 𝒜_b(I) can be seen to satisfy locality and isotony. In fact similar to the procedure carried out in [cor4.10]Corollary 4.10, it can be seen to be have a sequence associated to any of its elements which are norm bounded and converge strongly to that element. Therefore, it is a strong SL-algebra.How does this “net” compare to 𝒜_lcn(I)? Denote by j(I) the reflection of the interval I with respect to the x-axis where j:z →z̅.Given a function f=∑f̂_me^imθ∈ C^∞(S^1) with supp(f) ⊂ I ∪ j(I), and f̂_m=a_m+ib_m, define e(f)=0.9α_n^c∑_j=1^2n-1 f_c(π(j+1/2)/2n+1)e_j +iα_n^s ∑_j=1^2n-2f_s(π(j+1)/2n+1)[e_j,e_j+1]+(∑_m=-∞^∞ a_mβ_n^m,c+b_mβ_n^m,s)1which is inside 𝒜_n(I) (for large enough n), andf_c(θ)=f(θ)+f(-θ)/2∈ C^∞(S^1_+), ∀ e^iθ∈ S^1_+, f_s(θ)=f(θ)-f(-θ)/2∈ C^∞(S^1_+), ∀ e^iθ∈ S^1_+.we have e(f)L(f). In fact, e(f)=L(f) and this implies the theorem (using [thm4.6]Theorem 4.6). To show that equality, the formula for L_m givesf̂_mL_m+f̂_-mL_-m=a_mL_m^c+ib_mL_m^s+a_-mL_-m^c+ib_-mL_-m^swhere f̂_m=f̂_-m. Next, the identities for L_m^c,s gives L(f)=α_n^c∑_j=1^2n-1 c_je_j + iα_n^s∑_j=1^2n-2s_j[e_j,e_j+1]+(∑_m=-∞^∞ a_mβ_n^m,c+b_mβ_n^m,s)1,where c_j=∑_m=-∞^∞a_mcos(m(j+1/2)π/2n+1), s_j=∑_m=-∞^∞b_msin(m(j+1)π/2n+1).But f_c(θ) and f_s(θ) are precisely the cos() and sin() part of the Fourier series of f. Therefore, the above is precisely e(f). As a corollary, by definition, {e^iL(f)}”⊂𝒜_b(I) for supp(f) ⊂ I ∪ j(I).This hints to the relation between 𝒜_b(I) and 𝒜_lcn. Assume I touches the boundary of upper half-circle. Then, I ∪ j(I) is some connected interval in the circle and so {e^iL(f)|supp(f) ⊂ I ∪ j(I)}” = 𝒜_lcn(I ∪ j(I)). By the corollary above,𝒜_lcn(I ∪ j(I)) ⊂𝒜_b(I).But due to Haag duality for the conformal net 𝒜_lcn and locality for 𝒜_b, for the complement of I, called J, in S^1_+,𝒜_b(J) ⊂𝒜_b(I)' ⊂𝒜_lcn(I ∪ j(I))'=𝒜_lcn(J ∪ j(J)) ⊂𝒜_b(J) .Therefore, one recovers exactly, no more and no less, the LCN by taking the bounded scaling limit.𝒜_b(I)=𝒜_lcn(I ∪ j(I)) for I ∈ℐ_+ with |I ∩∂ S^1_+|=1.The above theorem is true for all UMMs assuming [cnj4.3]Conjecture 4.3. §.§ Vertex operators TEXT In [thm4.8]Theorem 4.8, y(a)_m was found to be in the scaling limit using QC-local operator. Therefore, Y(a,z) should also be in the scaling limit as an almost linear operator. In fact, Y(a,z) is the weak limit of a sequence Y(a,f) where f shrinks to the δ Dirac function. Then, according to [rmk3]Remark 3, Y(a,z) ∈𝒜. Here, we wish to construct a concrete sequence for the observable. Y(a,z) ∈𝒜 as an almost linear operator. Choose the sequence of observablesO_n=∑_|m| < g_alog(n)/2log(|z|)y_log(n)(a)_m z^-m-wt awhere wt a is the degree of a. Take u,v ∈𝒱. Without loss of generality, assume u,v are homogeneous with weight difference -s. For (u,Y(a,z)v), only the term (u,y(a)_sz^-s-wt av) is nonzero. Thus, (u,∑_s< |m| < g_alog(n)/2log(|z|)y_log(n)(a)_m z^-m-wt av)must go to zero. Once this is proved, the rest is the sum (u,∑_ |m| <sy_log(n)(a)_m z^-m-wt av)is a finite sum of operators for which the scaling limit is known. By [thm4.8]Theorem 4.8, we need to compute(u, (y(a)_m|_log(n)+O(1/n^g_a)+R_log(n),n^a)z^-m-wt av).For n where log(n)>wt v, the term R_log(n),n^av is zero. Similarly for m ≠ s, the term (u,y(a)_m|_log(n)v) is zero. It remains to showlim_n →∞ O(1/n^g_a)(∑_s<|m|<g_alog(n)/2log(|z|)z^-m) = 0.Due to symmetry of the summation, assume |z|> 1, and the summation is not small only for positive powers. In that case, the summation has norm at most |z|^g_alog(n)/2log(|z|)+1/|z|-1 which vanishes when divided by n^g_a. We discussed the following intuition on the scaling limit of smeared field∫ f(e^iπ j/n)e_jY(ω,f)=∮ Y(ω,z)f(z)z^2 dz/2π i z.Informally, one could think of this smooth function being a Gaussian distribution which goes to the δ Dirac function at some point corresponding to angle θ. In that case, one would expect to gete_θ Y(ω,e^iθ).Of course, with e_js, the “cos()” part appears in the scaling limit. For the other part, the bracket [e_j,e_j+1] must be used.Finding some ultra local operator in 𝒜_n giving us the field operator in the scaling limit would be a “proof” that the field operator Y(w,z) should not only be called a local observable, but an ultra local observable. Unfortunately, the natural guess does not work.Notations.v_x^c is the vector with entries (cos(m(x+1/2)π/2n+1))_0≤ m≤ 2n and v_x^s=(sin(m(x+1)π/2n+1))_1≤ m≤ 2n. Define β_n^c as the infinite vector with entries β_n^m,c for all m≥ 0 and similarly define β_n^s. Extend v_x^c and v_x^s by zeros to have infinite entries for them as well.We do not have O_n=α_n^c||v_x^c||^2e_x+iα_n^s||v_x^s||^2[e_x,e_x+1]+(β_n^c.v_x^c+β_n^s.v_x^s)1 Y(ω,z)z^2,where z=e^iθ and we pick the unique 1 ≤ x≤ 2n-1 such that θ∈ [xπ/2n+1,(x+1)π/2n+1].Notice sometimes x can only be chosen for large enough n as θ may be close to the boundaries. O_n is exactly the expression for e_x and [e_x,e_x+1] one obtains by considering the [identities]L_m^c,s identities of [thm3.1]Theorem 3.1. That is why we believe this should be the first candidate for convergence to Y(ω,z)z^2. By the [identities]L_m^c,s identities,O_n=∑_m=0^2ncos(m(x+1/2)π/2n+1)L_m^c+L_-m^c/2+i∑_m=1^2nsin(m(x+1)π/2n+1)L_m^s-L_-m^s/2.In other words, the following should not holdO_n=∑_m=-2n^2n(cos(m(x+1/2)π/2n+1)+isin(m(x+1)π/2n+1))L_m ∑ e^imθL_m.It is clear that any finite sum up to some M for O_n goes to ∑_|m| ≤ Me^imθL_m. Restrict to some finite energy M from right and left. Notice the scaling limit is not supposed to be a linear operator so restriction needs to be made from both sides. It will be shown that the approximations to high Virasoro modes give something other than zero. This will be shown for the cos() part (in other words, the real part) of the summation which is provided by the operators L_m^c+L_-m^c. The sin() partL_m^s+L_-m^s (complex part) can be done similarly.One can easily observe from the formula of L_m^c+L_-m^c ([eq32](32) and [eq33](33)), that for M<m<2n+1-2M, the restriction from left and right is exactly zero for large enough n. This is easy to observe by considering the picture of the half-circle having the momenta on it.Now notice that for 2n+1-2M≤ |m| ≤ 2n, the formula for L_m^c+L_-m^c gives many fermion pairs which are distinct for different m. Indeed, after the restrictions, any term Ψ_k'Ψ_k should have both n+1-M ≤ k,k' ≤ n+M and as Ψ_k appears only once for each k, there are around O(M) possibilities. Also, the difference between k and k' for each one of these pairs is exactly 2n+1-m implying that the pairs are not repeated by different ms. Each of these terms will have a coefficient of order O(n); Indeed there is a scaling provided by the α_n^c and further, there is also the coefficient 2cos((k+m/2)π/2n+1) which is close to 2 as m/2π/2n+1 is close to π/2 and n+1-M ≤ k ≤ n+M. Therefore, although the part |m|≤ M of the summation cause no problem, there are terms corresponding to 2n+1-2M≤ |m| ≤ 2n that blow up in norm. Hence, the scaling limit is certainly not an almost linear operator which is defined on 𝒱×𝒱 and the cos() part of Y(ω,z)z^2 is not the scaling limit.For the sin() and L_m^s case, with a similar reasoning, the 2n+1-2M≤ |m| ≤ 2n part of the summation does not blow up in norm but it is non-zero and due to the divergent coefficient sin(m(x+1)π/2n+1), does not have a limit and the sin() part of Y(ω,z)z^2 is not the scaling limit.As all problems emerge from the m close to the both ends of the summation, i.e. -2n and 2n, one could take the observable O_n as follows with the desired scaling limit∑_m=-2n+log(n)^2n-log(n)(cos(m(x+1/2)π/2n+1)+isin(m(x+1)π/2n+1))L_m ∑ e^imθL_m.In fact, any function f(n) ∞ instead of log(n) would work to avoid the discussed issues. But this is not the nice ultra local expression in terms of just e_x and [e_x,e_x+1] we desired for O_n. It is unknown whether there exists ultra local observable O_n with O_nY(ω,z)z^2. § CONJECTURES AND FUTURE DIRECTIONS§.§ On scaling limit of anyonic chains In this section, we provide a list of problems that need to be addressed for a clearer picture of the structures in the scaling limit relevant to CFT.After [dfn6]Definition 6, it was asked whether or not 𝒜^H generates 𝒜 and whether that in turn generates 𝒜. If the scaling limit is a CFT, such a conjecture becomes reasonable:The observables 𝒜 are “generated” by some means, like closure with respect to some topology, from the set of observables of the form Y(a,f). The smeared fields Y(a,f) were obtained as QC-local operators.There is a spanning set 𝒮={a}_a ∈𝒱 of the VOA such that for any a ∈𝒮, the smeared field Y(a,f) is space-local. One obstacle to space locality is the absence of commutators in the Borcherds identity. Otherwise, the terms involving products of far apart e_js would disappear. If Y(a,f) can be expressed in terms of commutators of Y(ω,f) with the Virasoro generators, the above conjecture would be true. Therefore, the obstacle may just be some simple lemma that is missing.Closely related to conformal invariance, is the scaling limit of the product of unitaries e^iL(f) and the smeared field operators L(f).Prove that the algebras in [cor4.10]Corollary 4.10 and [thm4.6]Theorem 4.6 together generate a (strong) SL-algebra. As a remark on the emergence of conformal invariance in the scaling limit, notice that due to [cor4.9]Corollary 4.9, for e^iL(g)∈𝒜_lcn(I), we havee^iL(f)e^iL(g)e^-iL(f) e^iL(f)e^iL(g)e^-iL(f).Further, the scaling limit above is itself in 𝒜_lcn(exp(f)(I)) and it is this fact that is the reason for conformal covariance (expressed below) in the LCN as e^-iL(f)s generate the local algebras:U(γ)𝒜_lcn(I)U(γ)^†=𝒜_lcn(γ(I)), γ∈Diff_+(S^1).But it is easy to show that 𝒜_n(I) can also be generated by e^iL(f)s. Due to [cor4.9]Corollary 4.9 and [thm4.13]Theorem 4.13, this essentially implies that, loosely speaking, the two sets e^iL(f)𝒜_n(I)e^-iL(f) and 𝒜_n(exp(f)(I)) become the same in the scaling limit (at least for I and exp(f)(I) satisfying condition of [thm4.13]Theorem 4.13). Therefore, conformal invariance emerges in the scaling limit. This may not be satisfying as it is not clear whether the group of operators e^iL(f) is the natural choice for the group (sequence of groups acting on the anyonic chains) that should recover the action of Diff_+(S^1) in the scaling limit (see <cit.> for a different candidate, the Thompson's group).We could not get all types of observables as algebra. The point-like field operators Y(a,z) were just given as a vector space in the scaling limit.Field operators Y(a,z) form a (strong) SL-algebra whenever their product is defined on 𝒱×𝒱.The techniques used in <cit.> may be useful to prove this as it involves some kind of truncations of the field operators. Finally, it would be useful for quantum simulation if one could obtain the field operators as ultra local operators because the product of ultra local hermitian operators can be simulated efficiently <cit.> on a quantum computer.Due to the numerical results on higher level ACs, it was conjectured ([cnj4.3]4.3) that the theorems of [4]section 4 are true for higher level UMMs. Here, we emphasize the identities giving us the Virasoro algebra The Hilbert space and the Virasoro algebra action of every chiral UMM with central charge c=1-6/(k+1)(k+2) is obtainable as a scaling limit of some SU(2)_k AC with some suitable boundary condition and the same theorems proved in [4]section 4 hold for them. Also denoting by O_n^c and O_n^s the followingO_n^c=-∑_j=1^2n-1cos(m(j+1/2)π/2n+1)e_j, O_n^s=i∑_j=1^2n-2sin(m(j+1)π/2n+1)[e_j,e_j+1],we have operators L_± m^c,L_± m^s L_± m satisfying the properties in [cnj4.3]4.3 and,L_m^c+L_-m^c/2=α_n^cO_n^c+β_n^m,c1L_m+L_-m/2, i(L_m^s-L_-m^s)/2=α_n^sO_n^c+β_n^m,s1i(L_m-L_-m)/2, where α_n^c,α_n^s,β_n^m,c, and β_n^m,s are suitable scaling factors. In <cit.>, there is a very similar conjecture although different. These will be compared in [A.2]appendix subsection 2. The size of the chain was assumed to be 2n-1. The chain size depends on the boundary condition which needs to be adjusted accordingly. One could then define the operators L_ms as done before. Proving the above by direct diagonalization (as done for non-interacting theories), seems to be hard. As mentioned in [1.6]section 1.6, one could hope to consider the commutators of the above observables and show that they satisfy similar relations as the Virasoro algebra. But note that by taking commutators, terms appear with non-vanishing norm and yet, what should be, vanishing in the scaling limit. These terms make it harder to recover the Virasoro algebra relations. §.§ Intertwiners and full CFTs Consider a rational VOA 𝒱 with irreducible modules A,B,C and corresponding conformal weights h_A,h_B,h_C. An intertwiner of type CAB is a map 𝒴(·,z): A →End(B,C)[[z,z^-1]], 𝒴(a,z)=∑_n ∈ℤa_mz^-τ-m, where τ=h_A+h_B-h_C. It has the following notation for homogeneous a ∈ A_k𝒴(a,z)=∑_n∈ℤy(a)_nz^-n-k-τ,and it satisfies similar axioms as the vertex operator. Intertwiners are part of the fundamental features of a CFT as they describe the fusion rules. As an example, the fusion rules in [1.2]section 1.2 for the Ising model correspond to three different free fermionic fieldsψ_1/2^0(z)=∑_n ∈ℤ z^-(n-1)Ψ_n-1/2 ψ_1/2^0(z)^†=ψ_0^1/2(z)=∑_n ∈ℤ z^-nΨ_n-1/2 ψ_1/16^1/16(z)=∑_n ∈ℤ z^-nΨ_n,where ψ_i^j : _i →_j. Therefore, the natural question is how they emerge in the scaling limit and what the right framework to discuss them in finite settings should be. As an example, we could define some set of maps from 𝒲_n^1 _1/2 to 𝒲_n^2 _0 as the real vector space of observables 𝒜^h,1/2→ 0_n={∑_m=-n+1/2^n-1/2f̂_m-1Ψ_m-1/2 | f̂_m=f̂_-m∈ℂ},and then define 𝒜^i → j_n, the algebra generated by odd numbers of Majorana operators (as in [A.1]Appendix subsection 1), and also its closure 𝒜^i → j_n. Still, this example clearly requires a very good idea of the finite versions of the primary fields. This is a hard problem in general (see <cit.> as an example).Although we will not provide an answer for this question,independent of what the framework should be, one can try to find the fermionic free fields in the scaling limit as a QC-local operator. There are three types of free fermionic field ψ_i^j as described in [1.2]section 1.2.As an example, one could use the basis provided in <cit.> to localize the modes of the field ψ_1/2^0(z)+ψ_0^1/2(z) on Hilbert spaces 𝒲_n _0+_1/2. Then, using the Borcherds identity (for intertwiners),𝒴(a,f)=∮𝒴(a,f)z^kdz/2π i z,where it can be seen that τ=0,can be obtained similar to what was done in [thm4.8]Theorem 4.8. Another important subject we did not discuss, was finding the algebra of observables for full CFTs. First, note that the scaling limit of the periodic anyonic chain should be assumed to be the full CFT on the torus. This changes the definition of the expectation values that one needs to measure in order to claim that a set generates an SL-algebra. The correlation functions for the torus are traces taken over the whole Hilbert space <cit.>. The second issue is the presence of interchiral observables <cit.> for which there is no counterpart in the VOA picture. These observables can be obtained by using ∑sin()e_j and ∑cos()[e_j,e_j+1] as shown in the [A.2]appendix. Still, with similar techniques as in [4]section 4, it can be shown that the trace of the observables in finite spaces corresponding to the conformal field and the interchiral observable converge to what we expect:Tr_𝒱((𝕃(f^(1))+𝕃(g^(1)))… (𝕃(f^(k))+𝕃(g^(k)))r^𝕃_0), 0<r<1,where r is the diameter of the torus, f^(i),g^(i)∈ C^∞(S^1), 𝕃(f^(i)) is the smeared field for the conformal field and 𝕃(g^(i)) is the smeared field for the interchiral observable. §.§ Simulation of CFTs by quantum computers The motivation of our work was an efficient quantum simulation of CFTs. What insights have we gained from this work? The first step is to define the problems that we want to solve. In each case, there will be local observables for which we ask their expectation values to be efficiently computed in polynomial time with respect to the inputs. By computing, we always mean approximating in polynomial time up to an error inverse polynomial with respect to the inputs. Informally speaking, we could also say that we are simulating efficiently (some of) the local observables themselves. Therefore, it is important to find out what those efficiently simulatable local observables are. For example, in quantum computation, in the context of many problems like the simulation of the unitary evolution <cit.>, the efficient k-Local Hamiltonians are a sum of polynomially many k-ultra local operators.When it comes to locality, quantum computation has its own precise definition. A fundamental aspect of the definition of locality is that explicitly or implicitly, there is a sequence of operators O_n that are the sum of ultra-local operators acting on at most O(1) many particles. In this way, one could distinguish between local and nonlocal (sequence of) operators. This idea can not be applied directly to CFTs, simply because a CFT is a single Hilbert space with no sequences attached naturally to any observable. Since we believe that CFTs must be efficiently simulatable by a quantum computer, there are two paths for defining locality in CFTs. The first is to declare a subset of observables in CFTs to be ultralocal depending on what kinds of problems one wants to solve, and then show that one can simulate them efficiently using a quantum computer. The other, which is less problem oriented, is to associate a sequence that “quantifies” locality (just like in finite dimensions where we have 2-local, 3-local, etc.) for each observable.In this work, the second strategy has been followed since the beginning. This strategy, as shown in [4]section 4, by using a rigorous definition of locality inspired by quantum computation, demonstrates how locality in finite dimensions translates into that of infinite dimension. It also reconciles to a great extent with what mathematical physicists and physicists think of the notion of local observables, although not being exactly the same.But there is no guarantee that this is the right path for the problems ahead and in fact, we will also point out the disadvantages of the anyonic chain approach when it comes to tackling these problems. §.§.§ Unitary evolution of CFTs The first problem is the unitary evolution of CFTs. In TQFTs, using the functorial approach, this problem has been shown to be in BQP <cit.>. The important observation made is that the unitary evolution is a representation of the mapping class group and the mapping class groups is generated by braids and Dehn twists. Those are operators for which one can have a local expression. We seek the same picture in CFTs.In CFTs, using the functorial approach, the unitary evolution is guided by unitary maps called U(γ) and simulating|(1,U(γ)1)|is the goal, where 1 is the vacuum and U is a positive-energy projective unitary representation of Diff_+(S^1) with γ a diffeomorphism in the Lie group. It is well-known (<cit.>) that the representation U corresponds to a unitary positive energy representation of the Virasoro algebra. By a result of <cit.>, simulating the above quantity is the same as simulating|(1,∏_j=1^k e^iL(f^(j))1)|,where f^(j)∈ C^∞(S^1) and γ=exp(f^(1))∘⋯∘exp(f^(k)). Loosely speaking, if in TQFT, the complexity of an evolution (a cobordism in the functorial point of view) arises at ultra local locations where braids happen, in CFT due to the continuous picture, one has to look at the diffeomorphism γ infinitesimally, hence the decomposition of U(γ) to finite products of e^iL(f)s.Theoretically, the above quantity is obviously computable as long as functions f^(j)s are computable. But when there is the issue of efficiency, one needs to make sure to ask the right question. Ideally speaking, one needs to know what nature does efficiently and ask whether a quantum computer can do that efficiently as well. In other words, what is the set ℱ of operators e^iL(f) that can be considered in [eq14](14)? One important observation is that in the same problem for other theories (TQFTs or usual quantum computation), the analog of the set ℱ has always been given by a “local generating” set. In this case, the natural candidates are the L_ns and the fact that they are scaling limit of sums of e_is, which are themselves the generators in the similar TQFT problem, is another evidence.For example, the operator e^if_0L_0 which is the evolution by the Hamiltonian corresponding to the constant function f ≡ f_0 is certainly one of the operators in ℱ. And in general the Virasoro operators L_n are thought to be local and e^i(f̂_nL_n+f̂_-nL_-n) corresponding to the function f=f̂_ne^inθ+f̂_-ne^-inθ must be in ℱ. Therefore, it is reasonable to ask a finite combination of these to be simulated efficiently. This means e^iL(f)∈ℱ for f having finite Fourier series. The next question is which functions with infinite Fourier series can also be considered for the simulation problem. An analogy in quantum computation, would be to think of a hermitian matrix H that may be nonlocal and acts on certain qubits but the norm of its action has an exponential decay away from those qubits. This translates to a unitary operator which is nonlocal but has an action exponentially close to identity except in some centers of action. The Fourier coefficients f̂_n are rapidly decaying ∀ k , ∃ N_k such that ∀ |n| ≥ N_k|f̂_n| ≤1/n^kbut the rate of this decay or equivalently, what the rate of growth of N_k should be is unclear. Perhaps, an exponentially decaying f̂_n or a polynomial growth for N_k is the answer. Finding the exact form of dependence of the rate of convergence of the scaling limit in [cor4.10]Corollary 4.10 on N_k will help to answer this question.So far, we can safely assume that the set ℱ has all operators corresponding to functions with finite Fourier series. We have the following definition for the CFT unitary evolution problem.(CFT UNITARY EVOLUTION) Consider functions f^(1),…,f^(k) with finite Fourier series and coefficients nonzero up to n_1,n_2,…,n_k all given as inputs, find an approximation up to given error ϵ, of the following quantity|(1,∏_j=1^k e^iL(f^(j))1)| The conjecture in the same spirit of TQFT, isCFT UNITARY EVOLUTION is in BQP, i.e. there is a polynomial time quantum algorithm with respect to the inputs, namely {n_j}_j ∪{(f̂^̂(̂ĵ)̂)_l}_j,l∪{k,1/ϵ}. Generically, the problem is BQP-complete. It is worth mentioning that in contrast to TQFT, where unitary evolution on the vacuum is trivial (as it is a one dimensional space in the case of the sphere), in CFT due to the existence of descendents provided by L_-m for m>1, we have a nontrivial problem in an infinite dimensional Hilbert space being the vacuum sector of the highest-weight representation with central charge c. Of course, one can generalize the above to other sectors and this is part of the next problem.The AC approach provides evidence for which operators have to be in ℱ. It also provides insights as to what operators in the scaling limit can be called local, which is a fundamental aspect of a theory from a computational point of view. Further, it gives a discretized picture of what unitary evolution looks like in a CFT. Consider many particles on a chain where it is allowed to have fusion between nearby particles with certain penalties for the undesired (nontrivial) fusion. The value of the penalties is what gives the function f. If we let the system evolve in this setting, the unitary evolution guided by those constraints is e^iL(f). If f is the constant function, then it is the usual Hamiltonian.But there is no guarantee that the AC approach is the right one for this question. In fact, in order to be able to approximate [eq14](14) using ACs, a proof for which AC gives the VOA in the scaling limit is needed. Furthermore, a bigger obstacle could be proving the BQP-completeness, as the expression for approximating the operators e^iL(f) are exponentials of weighted sum of all e_is and it is hard to build specific unitary operators using these. Lastly, one needs to prepare the vacuum which could be a hard problem (see <cit.> for the case of Ising).There are other possible approaches to this problem. First, one could look for combinatorial realization of the action of Virasoro generators. Path representations of the states of the Hilbert space and how the Virasoro algebra acts on these paths can be analyzed. We refer to <cit.> for nonunitary models minimal models M(2,q) with q odd where the action of every Virasoro generator is obtained, and <cit.> for unitary minimal models where actions of higher Virasoro generators is not known yet. Another possible approach to prove that the problem is in BQP, would be to use large enough tensor power of the Fock space which contains many interacting models as subtheories (including all 𝔰𝔲(2)_k WZW models and all minimal models; see <cit.> for a list). This free theory is essentially (_0+_1/2)^2 and can be modelled using an ultra local realization of the Dirac operators Ψ_k,Ψ_ks (as localized in <cit.>). We could then derive a local expression for the Virasoro generators of any subtheory using the Dirac operators. Then, by some energy truncation and taking the scaling limit, one should compute the rate of convergence and show that it is inverse polynomial with respect to the energy truncation. This could provide a faster convergence than AC; indeed, CFT has quantized energy but continuous spacetime. This approach focuses on the energy local degrees of freedom as the local basis for quantum computation, instead of the space local degrees of freedom as in AC. A third approach would be to first derive an exact expression for the quantity |(1,U(γ)1)|, and then try to simulate it. This exact expression can be obtained for all free models <cit.> but no such closed formula is known for higher minimal models. We hope to pursue these in future works.§.§.§ Correlation functions of CFTThe second problem is in fact a generalization of the first. It is the simulation of the correlation functions of CFT which is very much like in TQFT and simulation of Jones polynomial <cit.> (which is the value (<cit.>) of the TQFT correlation function); 2n fields are inserted, denoted by n cups (inserting two fields dual to each other) as a TL diagram, a unitary evolution is applied and the probability of getting back to same state is measured (denoted by n caps). The normalization in the case of Jones polynomial is also the norm of the state given by the n cups. The direct analogy in CFT would be (not a chiral but) a full diagonal CFT where the cup inserting the dual pair is 𝕐(a⊗ a',z,z) inserting the pair a ∈ A and a' ∈ A' (the contragredient module of A). But as will shown, it is not obvious that the similar quantity can be defined. Here, we make an attempt for the definition. The proof that the quantity is well-defined is not given. In future works, we will aim to fill the gaps in the arguments. Let us start with the chiral formulation.We will assume a nice VOA: unitary, CFT-type, rational (including C_2-co-finite). The goal would be to approximate the following point-like chiral correlation function C_chiral^p efficiently:C_chiral^p=|(𝒴_n(a_n,γ(z_n))…𝒴_1(a_1,γ(z_1))1,U(γ)𝒴_n(a_n,z_n)…𝒴_1(a_1,z_1)1)|/||𝒴_n(a_n,z_n)…𝒴_1(a_1,z_1)1||. ||𝒴_n(a_n,γ(z_n))…𝒴_1(a_1,γ(z_1))1||,where |z_i|=1 are on the unit circle arranged as 0<(z_1)<…<(z_n)<π, and a_i∈ A_i are primary fields in the irreducible modules and 𝒴_i are of type B_i-1A_iB_i with irreducible modules B_i. We note that fields inserted at z_i move to γ(z_i) by conformal covariance of primary fields and therefore, this is where we should measure the amplitude of getting back the same configuration.There are multiple issues with this definition. First, it is not clear that the numerator or denominator exist. For the numerator to exist, it makes sense to impose the condition of intertwiners having energy bounds. Indeed, notice that one can first perform the evolution by U(γ) and then insert the fields at γ(z_i) (due to conformal covariance). As U(γ) operates inside the common domain ∩_k=1^∞𝒟((L_0+1)^k) called smooth vectors and denoted by 𝒱^∞ or B^∞ for module B, the vector U(γ)1 would be inside 𝒱^∞ and generally not inside 𝒱. Even with this condition one needs to prove that the expectation value for those insertion points exist which brings us to the second issue.The insertion points have the same norm and they are not distinct. To see this, by taking the adjoint of the fields on the left side and using conformal covariance it can be shown that computing the numerator is the same as computing|(U(γ)^†1,∏_i=1^n 𝒴_i(η_A_i(a_i),z_i)∏_i=n^1𝒴_i(a_i,z_i)1)|,where η_A_i is the anti-linear involution corresponding to the unitary structure of A_i (in minimal models η_A_i(a_i)=a_i). It is known that correlation functions (u,∏𝒴_i(b_i,w_i)v) for u,v with finite energy, can be evaluated at distinct insertions points with the same norm by an analytic extension of the region |w_1|<…<|w_2n| to the configuration space of ℂ^2n; This is <cit.> for chiral and <cit.> for full CFT. We first need to prove similar theorems for u,v smooth vectors (this could be accomplished by proving that the corresponding correlation function satisfies the same ODE as in <cit.>). But another issue would remain, which is that the insertion point z_i is repeated. There is a singularity when insertion points are the same which makes it impossible to define the above quantity. But in C_chiral^p, we could see that the singularities cancel each other. Notice that for any correlation function of the form (u,∏𝒴_i(b_i,w_i)v), the order of the singularity w_i-w_j only depends on the fields b_i,b_j; see <cit.> for 𝒴_i=Y the vertex operator (this can be easily generalized to intertwiners). For the denominator, the first norm is: ||∏_i=n^1𝒴_i(a_i,z_i)1||=|(1,∏_i=1^n𝒴_i(η_A_i(a_i),z_i)∏_i=n^1𝒴_i(a_i,z_i)1)|^1/2.The above is not defined but if we consider the insertions generically at z_i',z_i, it should give us a meromorphic function ∏1/(z_i'-z_i)^s_ii'∏1/(z_i-z_j)^s_ij∏1/(z_i'-z_j')^s_i'j'F(z_1',…,z_n',z_1,…,z_n) where F is a polynomial in z_i,z_i'. For the second norm in the denominator, by conformal invariance, it can be evaluated at z_i instead of γ(z_i) and the same singularities will appear with the same order. Therefore, one can see that the order of (z_i-z_i') is the same in the numerator and denominator. In the end,C_chiral^p would be of the form F_γ/F for two functions in z_i and the dependence of the normalization on γ, as mentioned, can be avoided by conformal invariance of vacuum to vacuum correlation functions.Even if C_chiral^p is defined, it is not obvious that C_chiral^p ≤ 1, which is crucial for quantum computation. We believe that one should be able to obtain C_chiral^p as a limit of the smeared version of the problem called C_chiral^s, which will be at most one simply due to Cauchy inequality. Define the smeared correlation function as follows:C_chiral^s=|(∏_i=n^1𝒴_i(a_i,β_d_a(γ)(f_i))1,U(γ)∏_i=n^1𝒴_i(a_i,f_i)1)|/||∏_i=n^1𝒴_i(a_i,β_d_a(γ)(f_i))1||. ||∏_i=n^1𝒴_i(a_i,f_i)1||,where f_i are smooth functions on S^1 and β_d_a(f)=γ'(γ^-1(z))^wta-1f(γ^-1) (this function appears in the conformal covariance of primary fields; see e.g. <cit.>). It can be shown that C_chiral^s is defined due to energy-boundedness, see e.g. <cit.>. The next step is to take a limit by considering sequence of functions converging to the Dirac delta at z_i, i.e. δ_z_i. To do so rigorously, a similar theorem to <cit.> when the expectation value is taken for smooth vectors could be helpful. This would relate the smeared and the point-like version:0.95(u,∏_i=1^n𝒴_i(η_A_i(a_i),g_i)∏_i=n^1𝒴_i(a_i,f_i)v) 0.95=∫_-π^π⋯∫_-π^π(u,∏_i=1^n𝒴_i(η_A_i(a_i),e^iϕ_i)∏_i=n^1𝒴_i(a_i,e^iθ_i)v)∏_i=1^n g_i(e^iϕ_i) 0.95∏_i=n^1f_i(e^iθ_i)∏_i=1^n dϕ_i ∏_i=n^1 dθ_iwhere dθ=e^iθ/2π, and f_i,g_i can be thought to be distributions centered on z_i,z_i' with ∫_S^1 f_i=∫_S^1 g_i=1. These functions must have disjoint support (this is required to apply <cit.>). Then, one has to analyze the limit when g_i, f_i →δ_z_i'=δ_z_i. This limit should be taken on the fraction C_chiral^s as the terms in the numerator and denominator diverge individually. It is worthy to note that on distinct points the limit works well. Assuming continuity of(u,∏_i=1^n𝒴_i(η_A_i(a_i),e^iϕ_i)∏_i=n^1𝒴_i(a_i,e^iθ_i)v)with respect to ϕ_i,θ_i (this is already true if u,v have finite energy but we need it for smooth vectors), it is not hard to show that if g_i →δ_z_i',f_i→δ_z_i for distinct z_i,z_i' we have 0.87|(u,∏_i=1^n𝒴_i(η_A_i(a_i),g_i)∏_i=n^1𝒴_i(a_i,f_i)v)| → |(u,∏_i=1^n𝒴_i(η_A_i(a_i),z_i')∏_i=n^1𝒴_i(a_i,z_i)v)|. Recall that as mentioned at the beginning, the direct analog of the TQFT case is the full CFT point-like correlation function. Assuming that one can prove that C_chiral^p is well-defined, it is not hard to show that the point-like and smeared version of the problem for the full CFT case can also be defined. In fact, for the smeared version, just like the chiral case, there is nothing to prove and it is already well-defined. For the point-like case, the full CFT intertwiner is a finite sum of pair of chiral intertwiners, hence the correlation function will be a finite sum of chiral correlation functions. We have the following analogous quantities for the full CFT: 0.9999C_full^p= |(∏_i=n^1𝕐(a_i⊗ a_i',γ(z_i),γ(z_i))1,U_L(γ) U_R(j ∘γ∘ j)∏_i=n^1𝕐(a_i⊗ a_i',z_i,z_i)1)|/||∏_i=n^1𝕐(a_i⊗ a_i',γ(z_i),γ(z_i))1||.||∏_i=n^1𝕐(a_i⊗ a_i',z_i,z_i)1||, 0.98C_full^s= |(∏_i=n^1𝕐(a_i⊗ a_i',β_d_a(γ)(f_i),β_d_a(γ)(f_i)∘ j)1,U_L(γ) U_R(j ∘γ∘ j)∏_i=n^1𝕐(a_i⊗ a_i',f_i,f_i∘ j)1)|/||∏_i=n^1𝕐(a_i⊗ a_i',β_d_a(γ)(f_i),β_d_a(γ)(f_i)∘ j)1||.||∏_i=n^1𝕐(a_i⊗ a_i',f_i,f_i∘ j)1||,where a_i' ∈ A_i' are primary fields from the contragredient module of A_i, j is the conjugation map, and U_L,U_R are the unitary evolution for the left and right moving part, respectively.Finally, as we know how to take the unitary evolution as input (outlined in previous section), formulating the correlation function problem in all four versions is possible; the inputs are the primary fields from a nice fixed VOA, the insertion points z_i (or Fourier coefficients of the smearing functions) and the decomposition of γ as exp(f)s. It is not entirely clear how the AC approach would help solve this question for general minimal models as it is hard to realize finite version of fields on a lattice or a spin chain as mentioned in [5.2]section 5.2. Similar to unitary evolution, the correlation function problem is expected to be in BQP and typically BQP-complete.§ ACKNOWLEDGEMENTS We would like to thank James Tener for many helpful discussions, in particular pointing out the relevance of Kaplansky's theorem for the proof of [cor4.10]Corollary 4.10. We thank P. Fendley, V. Jones, R. Koenig, H. Saleur, and anonymous referees for helpful comments to improve our paper and their encouragements.The second author is partially supported by NSF grant DMS-1411212.apa 1Dept of Mathematics, University of California, Santa Barbara, CA 93106-6105, U.S.A. 2Microsoft Station Q and Dept of Mathematics, University of California, Santa Barbara, CA 93106-6105, U.S.A.§ APPENDIX: SCALING LIMIT OF ISING ANYONIC CHAINS§.§ 1. Obtaining Virasoro representations and their actions In each case, we start with some operator that is supposed to become the desired one converging to L_m, and it will undergo some changes (all being some scalings) before becoming the desired operator. As an example, for the Hamiltonian, we will always start with -∑ e_j but during the process, it will change by some scaling which can be easily obtained by following the procedure until it produces the actual Hamiltonian L_0^c that converges to L_0. These scalings are the scaling factors mentioned in [identities]L_m^c,s identities called α_n^c,α_n^s,β_n^m,c, and β_n^m,s. §.§.§ Case 1(a): (1/2,1/2) _0+_1/2The method and the notations used in <cit.> will be followed closely and we will apply it case by case on Ising ACs to obtain the Virasoro modes throughout this section. It is therefore necessary to review the general procedure described for the Hamiltonian diagonalization of 1(a) in <cit.>. Consider the operator -∑_j=1^2n-1 t_je_j which after a scaling due to the equalitiese_2j=1/√(2)(1+σ_j^zσ_j+1^z), e_2j-1=1/√(2)(1+σ_j^x),becomesH= -∑_j=1^n t_2j-1σ_j^x-∑_j=1^n-1 t_2jσ_j^z σ_j+1^z,where the coefficients t_j are fixed. With this expression of H, it is easy to see the famous ℤ_2 symmetry provided by the spin-flip operator, called(-1)^F:= ∏_j=1^2nσ_j^x.As detailed in <cit.>, in order to diagonalize this Hamiltonian, the Majorana operators should be defined asψ_2j-1=( ∏_k=1^j-1σ_k^x) σ_j^z, ψ_2j= i(∏_k=1^jσ_k^x) σ_j^zwhich satisfy the ACR (Anticommutative Canonical Relations):{ψ_a,ψ_b}=2δ_ab , ∀ a,b=1, …,2n.It is a well-known fact that these operators and their monomials are linearly independent and this representation of the Clifford algebra is faithful. By usinge_a=1/√(2)(1+iψ_aψ_a+1),we rewrite the Hamiltonian H=i ∑_a=1^2n-1 t_a ψ_a+1ψ_a.Next, raising (creation) and lowering (annihilation) operators are introduced, i.e. Dirac operators for which[H,Ψ]=2ϵΨ.Notice that for any operator linear in the Majorana operators, the commutator with H is also linear in the Majorana operators. Let us choose the following form for ΨΨ = ∑_b i^b μ_b ψ_b,where μ_b are numbers that will turn out to be real. The i^b's factor will ensure that the matrix in [eq15](15) is hermitian and not skew-hermitian, thus making the computations easier. Computing μ_a's, Ψ'=[H,Ψ]= ∑_a i^aμ_a' ψ_a,is same as the following matrix equation[μ_1';μ_2'; ⋮;; μ_2n' ]= 2[0t_10… ;t_10t_2;0t_20;⋮t_2n-1;t_2n-10 ][μ_1;μ_2;⋮; ; μ_2n ].This hermitian matrix has determinant (-1)^n∏_j=1^nt_2j-1^2. The eigenvectors of this matrix give the Dirac operators and each corresponding eigenvalue is the energy that is raised or lowered. Specializing the values of t_js will give the different boundary conditions. (1/2,1/2) can be seen to correspond to the case t_j=1 for all j. Therefore, we will work with the matrix [eq15][eq15](15) assuming t_j=1. Notation. for n ∈ℕ, set [n]:={1,…,n}. E.g. [2n]-[n]={n+1,…,2n}. Similarly define [-n]:={-1,…,-n}.The Dirac operators Ψ_k for k ∈ [2n], are given by the eigenvectors μ_a,k=sin(akπ/2n+1) with corresponding energy ϵ_k=4cos(kπ/2n+1), satisfying (<cit.>)[H,Ψ_± k]=2 ϵ_± kΨ_± k,{Ψ_± k, Ψ_± k'}=0,{Ψ_± k, Ψ_∓ k'} = N_k δ_k,k'1,where Ψ_-k:=Ψ_2n+1-k, and N_k=2 ∑_aμ_a,k^2. The relations are obtained using the identities{Ψ,}=∑_a,b i^a+bμ_a ν_b {ψ_a,ψ_b}=2∑_a (-1)^a μ_aν_a,for any two linear Majorana forms Ψ = ∑_b i^b μ_b ψ_b,=∑_b i^b ν_b ψ_b. As a hermitian matrix has orthogonal eigenvectors, and for any eigenvector (μ_a,k)_a giving eigenvalue ϵ_k, there is a corresponding eigenvector((-1)^a+1μ_a,k)_a giving eigenvalue ϵ_-k:= -ϵ_k, equations [eq16](16) follow including the fact that Ψ_k^†=Ψ_-k. We will always work with the normalization of Ψ_k by √(N_k), hence {Ψ_± k, Ψ_∓ k'} = δ_k,k'1.From now on, the Dirac operators Ψ_k for k ∈ [n] will be called the raising or creation operators and the Dirac operators Ψ_k for k ∈ [2n]-[n] will be called the lowering or annihilation operators. This terminology will similarly apply for future cases. Further, at the end of each case, there will be a renumbering of the operators indices which will make the creation operators have negative index while the annihilation operators will have positive index.Therefore, Ψ_ks satisfy the ACR while the dimension of 𝒲_n (the Hilbert space) is 2^n. This implies the existence of an orthonormal basis of 𝒲_n given by∏_i ∈ SΨ_i 1_n,∀ S ⊂ [n],all of which will turn out to be eigenvectors of H, where 1_n is the vacuum or ground state annihilated by the annihilation operators. As mentioned in <cit.>, the energy symmetry of H and well-known properties of the representations of the algebra generated by the Ψ_ks, can be used to prove this. Let us recall these general facts.Notation. Denote by ℱ_n the algebra generated by the Ψ_ks and ℱ_n^+ the sub-algebra generated by the creation operators. Similarly define ℱ_n^-. We will use S as any subset of the indices of creation operators. Let 𝒲 be a representation of ℱ_n which is a Hilbert space with 𝒲=2^swhere s ≥ n and Ψ_k^†=Ψ_-k with respect to the inner product of 𝒲. Consider the image 𝒲_0 of the product of all annihilation operators. For any vector v ∈𝒲_0, by definition of 𝒲_0 and ACR relations, in particular Ψ_k^2=0, we get ℱ_n^-(v)={0}. Further, the space 𝒲_v=ℱ_n^+(v) generated by the creation operators acting on v has dimension 2^n with, assuming v is a unit vector, an orthonormal basis {∏_i ∈ SΨ_i v|∀ S}. The fact that this is an orthonormal basis can also be checked directly by computing the inner products using ACR, Ψ_k^†=Ψ_-k and that ℱ_n^-(v)={0}. Finally, with the same direct calculations, for any two orthonormal vectors u,v ∈𝒲_0, we have 𝒲_v ⊥𝒲_u. This implies that for any chosen orthonormal basis for 𝒲_0, a direct sum of irreducible representations with dimension 2^n of ℱ_n is obtained. We claim that this decomposition exhausts 𝒲, or equivalently 𝒲_0=2^s-n. Assume ∃ v ∈𝒲 which is orthogonal to the decomposition. One needs to find a sequence of Ψ_ks acting on v such that v is sent to a vector in 𝒲_0 and we will reach a contradiction. This is done by noticing that for any non-zero vector u, if Ψ_k u=0 for Ψ_k ∈ℱ_n^+ then Ψ_kΨ_k^† u=u ≠ 0. Therefore, we can start by acting on v by the annihilation operators in increasing order of indices (n+1,…,2n) and whenever the result is zero when acted by Ψ_k, acting by the creation operator Ψ_k^† and then Ψ_k resolves this issue. By using this procedure and ACR, there is a reordering of the action by Ψ_ks such that the end result is Ψ_2n…Ψ_n+1(∏_i ∈ SΨ_iv)=v_0 ≠ 0 where S is some subset from the creation indices. v_0 is in the image of theproduct of all annihilation operators, i.e. v_0 ∈𝒲_0. But v was assumed to be orthogonal to the decomposition, implying that(v_0,v_0)=(v, (∏_i ∈ SΨ_i)^†Ψ_n…Ψ_1v_0)=0v_0=0,which is a contradiction.With the same settings of [fct1]Fact 1, consider a matrix D satisfying [D,Ψ_k]=0 for all Ψ_k ∈ℱ_n. It follows that D preserves 𝒲_0 and it is uniquely determined based on how it acts on 𝒲_0. In particular, if there is a decomposition of 𝒲 into 2^s-n irreducible representations where D preserves the corresponding vacuums, then D acts as a scalar on each one of them. This will be always the case in the proofs.In addition to the spin-flip symmetry (-1)^F, the matrix H has charge conjugation symmetry (which will also be called energy symmetry) provided by C=∏_i σ_i^z ∏_i (σ_i^x)^i which satisfies CH=-HC implying each energy has one corresponding opposite energy. This is a necessary property which helps us to show that some non-zero scalar from the previous fact for H can not happen as that would break the symmetry. From [fct1]Fact 1, (𝒲_n)_0 is one dimensional from which a unit vector 1_n is chosen. DefineH':= ∑_k∈ [n]ϵ_k(Ψ_+kΨ_-k-Ψ_-kΨ_+k).H's eigenvectors are {∏_i ∈ SΨ_i1_n| ∀ S}, each with the corresponding eigenvalue ∑_i ∈ Sϵ_i - ∑_j ∉Sϵ_j. So H' has C-symmetry. Further, one can easily see that [H',Ψ_k]=2ϵ_kΨ_k and so, forD=H-H', [D,Ψ_k]=0. As(𝒲_n)_0 is one dimensional, D=α1. But H' shifted by any α does not satisfy the energy symmetry. Therefore, α=0 and H'=H. Taking the shift H → H+ ∑ϵ_k and using {Ψ_+k,Ψ_-k}=1,H=∑_k ∈ [n] 2ϵ_kΨ_+kΨ_-k.The final change to H is H →2n+1/8πH and the desired Hamiltonian L_0^c is given by:L_0^c=2n+1/π∑_k ∈ [n]cos(kπ/2n+1)Ψ_+kΨ_-k.Defining the scaling limit requires defining the connecting maps. Before doing so, a renumbering k → k-1/2-n is performed to get the creation operators indices as {-1/2,…,-(n-1/2)}. Notation. [(n+1/2)]:={1/2,…,(n-1/2)} and [-(n+1/2)]:={-1/2,…,-(n-1/2)}.This will also change the coefficients from cos(kπ/2n+1)=-sin((k-1/2-n)π/2n+1) to sin(-kπ/2n+1) and we will haveL_0^c=2n+1/π∑_k ∈ [(n+1/2)]sin(kπ/2n+1)Ψ_-kΨ_k.Next, we define ϕ_n:𝒲_n ↪𝒲_n+1, where ∀ Swe have ϕ_n(∏_i ∈ SΨ_i1_n)=∏_i ∈ SΨ_i1_n+1.This is consistent with an embedding of ℱ_n ↪ℱ_n+1 where Ψ_i ↪Ψ_i giving us in the limit the algebra of Dirac fermion operator ℱ. We will prove that there is a strong scaling limit (see [dfn4]Definition 4), where the scaling limit space 𝒱 can be constructed as the algebraic colimit of the sequence coming with the natural embedding maps ρ_n: 𝒲_n ↪𝒱. The connecting maps will turn out to be the restriction of ϕ_n to energy M as it is required in [dfn4]Definition 4. The natural orthonormal spanning set is {∏_i ∈ SΨ_i1|∀ S ⊂ℤ_<0+1/2}for 𝒱 where 1=ρ_n(1_n) is the vacuum vector. We need to make sure that this is consistent with the definition of scaling limit obtained through the double colimit construction in [dfn3]Definition 3.Restricting to energy at most M, one has to check that ϕ_n gives isometries ϕ_n^M : 𝒲_n^M →𝒲_n+1^M for large enough n. It is not hard to see that any eigenvector ∏_-k ∈ SΨ_-k 1_n with energy (2n+1)/π(∑_-k ∈ Ssin(kπ/2n+1)) < M has the energy (2n+3)/π(∑_-k ∈ Ssin(kπ/2n+3)) given by H_n+1 also smaller than M for large enough n. Indeed, by using the Taylor expansion we obtain(2n+1)/π(∑_-k ∈ Ssin(kπ/2n+1))=∑_k ∈ Sk - ∑_-k ∈ Sk^3 π^2/6(2n+1)^2+… =∑_-k ∈ Sk+O(1/n).This also shows that at the scaling limit we have the energy ∑_-k ∈ S k for ∏_-k ∈ SΨ_-k 1. It is then easy to check that L_0^cL_0, whereL_0=∑_k ∈ℕ-1/2kΨ_-kΨ_k.This gives the character ∏_k=1^∞ (1+q^k-1/2)which agrees with the character of _0+_1/2 ([1.2]section 1.2). Now consider the natural action of L_0^c on 𝒱 obtained through the embedding of ℱ_n ↪ℱ. We could alternatively take the “less” natural action by extending L_0^c by zero on the orthogonal complement of 𝒲_n in 𝒱 and no result on the convergence rate will be lost. By this embedding, the restriction of both L_0^c,L_0 to subspace with energy at most √(n) denoted by L_0^c|_√(n),L_0|_√(n), can be compared. In order to finish the proof of 1(a), one needs to proveL_0^c|_√(n)=L_0|_√(n)+O(1/n)This is a stronger result than restriction to some finite energy M. This equation demands M to be changing according to n and yet have a convergence. If L_0 gives an energy smaller than √(n) to some eigenvector ∏_-k ∈ SΨ_-k 1, then it must be shown that vector is inside 𝒲_n. This means that for large enough n, we have S ⊂ [-(n-1/2)]; that needs to be checked due to the less natural embedding used in the proofs of theorems in [4.1]section 4.1. This is easy to show as if ∑ k < √(n), then obviously there is no k>n-1/2 for large enough n. Further, we should show that L_0^c gives the same energy up to an error of O(1/n). That would imply that the error has norm at most O(1/n) as L_0^c|_√(n) and L_0|_√(n) share the same eigenvectors in 𝒱. Let us therefore estimate the difference |(2n+1)/π(∑_-k ∈ Ssin(kπ/2n+1))-∑_-k ∈ S k|,assuming ∑_-k ∈ Sk < √(n). Using [eq18](18),=|-∑_-k ∈ Sk^3 π^2/6(2n+1)^2+h.o.t| ≤ |∑_-k ∈ Sk^3 π^2/6(2n+1)^2|+|h.o.t|.In general, if the sum ∑_k x_k=t of non-negative numbers x_k is a fixed value t, then ∑_k x_k^j ≤ t^j with equality if and only if one of the numbers is t and the others are zero. This implies that in the above, the maximum happens when S={√(n)}. The h.o.t is (as a rough estimate) at most O(1/n^2) and the first term is exactly O(1/n). This finishes the proof of case 1(a).Before moving to the next case, we need to investigate what “separates” the two irreducible modules _0 and _1/2 at the level of the finite spaces 𝒲_n. The answer to this question will give some interesting identities that will be of use elsewhere.(-1)^F commutes with L_0^c and therefore preserves the vacuum as the eigenspace of the vacuum is one dimensional with energy zero. Further, it is easy to see that {(-1)^F,ψ_k}=0 and so {(-1)^F,Ψ_a}=0. Therefore any product of even number of creation operators (giving a vector inside _0) commutes with (-1)^F and any product of odd number of creation operators (giving a vector inside _1/2), anti-commutes with (-1)^F. It remains to determinein which ± 1 sector of (-1)^F the vacuum is. Going back to the previous labelling of creation and annihilation operators by integers [2n], we have the following identities involving the annihilation operators(-1)^F=i^n∏_j=1^2nψ_j, i^n ∏_j=1^2nψ_j ∏_k=1^n Ψ_-k=∏_k=1^n Ψ_-k .The first identitydirectly from the definition of the Majorana operators in terms of the Pauli operators.For the second identity, as (-1)^F preserves the one dimensional image of the product (vacuum), we deduce (-1)^F ∏_k=1^n Ψ_-k= α∏_k=1^n Ψ_-k for some α∈{± 1}. According to the second identity, α must be 1. To prove it, expand both sides of i^n ∏_j=1^2nψ_j ∏_k=1^n Ψ_-k= α∏_k=1^n Ψ_-kin terms of ψ_is. The monomials in ψ_is are linearly independent. On the RHS, all monomials have at most n terms. Suppose that the coefficient of a term with less than n Majorana operators is nonzero. This gives a monomial with more than n terms on the left side because of the product ∏_j=1^2nψ_j, so in the expansion of ∏_k=1^n Ψ_-k, only n-monomials will appear. To find α, one needs to compare the coefficient of ψ_1 …ψ_n on both sides. This means the coefficients of ψ_1…ψ_n and ψ_n+1…ψ_2n in the RHS. The coefficient of any of the n-monomials is the determinant of some matrix. For the first one, it is the determinant of (notice Ψ_-k=Ψ_2n+1-k) [iμ_1,2n i^2 μ_2,2n i^3 μ_3,2n…i^nμ_n,2n;iμ_1,2n-1 i^2 μ_2,2n-1 i^3 μ_3,2n-1 i^nμ_n,2n-1;iμ_1,2n-2 i^2 μ_2,2n-2 i^3 μ_3,2n-2;⋮i^nμ_n,n+2;i^n-1μ_n-1,n+1 i^nμ_n,n+1 ]_n × n, which has to be compared to the coefficient of ψ_n+1…ψ_2n, the determinant of0.89[i^n+1μ_n+1,2ni^n+2μ_n+2,2ni^n+3μ_n+3,2n…i^2nμ_2n,2n;i^n+1μ_n+1,2n-1i^n+2μ_n+2,2n-1i^n+3μ_n+3,2n-1 i^2nμ_2n,2n-1;i^n+1μ_n+1,2n-2i^n+2μ_n+2,2n-2i^n+3μ_n+3,2n-2;⋮i^2nμ_2n,n+2;i^2n-1μ_2n-1,n+1 i^2nμ_2n,n+1 ]_n × n,and by using μ_2n+1-t,k=(-1)^k+1μ_t,k, we obtain α=1. Hence, the vacuum is in the +1 sector of (-1)^F. Therefore, for all n, _0 is in the +1 sector and _1/2 is in the -1 sector.The procedure of taking scaling limit after the diagonalization and estimating the rate of convergence of the Hamiltonian to L_0 in all future cases will be similar and we will refer to this case. The focus will be only on the parts that have a different idea/explanation.§.§.§ Case 1(b) & 1(c): (0,0) & (1,1) _0 and (1,0) & (0,1) _1/2Projection under (-1)^F is non local. We would like to have exactly _0 and _1/2 in the scaling limit by at most a local projection. The boundary conditions involving 0 and 1-spin will provide that.As in previous case, consider the “would-be” Hamiltonian H=-∑_j=1^2n-1 t_je_j acting on the same Hilbert space, with t_1=t_2n-1=0 and all other t_js being 1. The Hilbert space (1/2,1/2) is the sum of four subspaces given by anyonic chains starting and ending with the following spins: (0,0),(1,0),(0,1),(1,1). It is easy to see that H with t_1=t_2n-1=0 preserves each of the four subspaces and so, restricted to any of those subspaces, it gives the operator derived in [1.4]section 1.4 for the four boundary conditions given by 0 and 1. The matrix [eq15](15) corresponding to H is2[ 0 0 0 …; 0 0 1; 0 1 0; ⋮; 0 1 0; 1 0 0; 0 0 ]_2n × 2nfrom which n-1 creation operators and annihilation operators Ψ_ks are derived corresponding to the same matrix of the case 1(a) for n → n-1. These creation and annihilation operators change the boundary conditions as each operator is a linear combination of ψ_j for 2 ≤ j ≤ 2n-1 and, according to their definition, all have the Pauli operator σ_1^x and do not have σ_n^x. Therefore, they flip the left boundary condition and pair the condition (0,1) with (1,1) and (1,0) with (0,0). Take the image of the product of all annihilation operators. As there are n-1 operators acting on a Hilbert space with dimension 2^n, by [fct1]Fact 1, a two dimensional vacuum space exists. As all Dirac operators preserve the spaces (0,1) ⊕ (1,1) and (1,0) ⊕ (0,0), both 2^n-1 dimensional, one can pick a unit vector in each space being the vacuum. Therefore, diagonalizing the Hamiltonian is done in exactly the same way by replacing n with n-1 in case 1(a), defining H' and showing D=H-H'=0 (proving the rate of convergence is also done similarly). The only difference is where the vacuum space (𝒲_n)_0 has (𝒲_n)_0=2 and, as noted in [fct2]Fact 2, D has to be shown to preserve two irreducible representations. This is clear as both H and H' preserve the spaces (0,1) ⊕ (1,1) and (1,0) ⊕ (0,0). Assume D=α1 on the first and D=β1 one the second subspace. But H' has the same symmetric spectrum on each of these subspaces and any shift α should be paired with another shift β=-α to preserve the energy symmetry. Therefore D=-ασ_n^z=-α i^n-1∏_j=1^2n-1ψ_j. This is not possible, unless α=0, as the monomials in Majorana operators ψ_js are linearly independent and we only have binomials or identity in the expansion of either H or H'.The Hamiltonian preserves the boundary conditions. As the two boundary conditions (1,1) and (0,0) are supposed to give _0, it is reasonable to expect that the two vacuum vectors are inside those spaces. Similar to identity [eq19](19), to show that the image of the product of annihilation operators is in the kernel of σ_1^z-σ_n^z:(σ_1^z-σ_n^z)∏_annihilationΨ_k=0 ⇔σ_1^z∏_annihilationΨ_k = σ_n^z ∏_annihilationΨ_k.As σ_1^z=ψ_1 and i^n-1∏_j=1^2n-1ψ_j=σ_n^z,⇔∏_annihilationΨ_k = i^n-1∏_j=2^2n-1ψ_j ∏_annihilationΨ_k.The argument used for proving [eq19](19) applies in this case as well by simply replacing n with n-1. As a result, taking the vacuum in, e.g. (0,0), and acting on it with even number of creation operators gives a vector inside (0,0) and with odd operators gives a vector inside (1,0). This implies _0 is the scaling limit corresponding to (0,0) (and (1,1)), and _1/2 corresponds to (1,0) (and (0,1)). §.§.§ Case 1(d) (1/2,0) ⊕ (1/2,1)2_1/16 To get _1/16 in the scaling limit, we specialize to the case t_j=1 for all j except t_2n-1=0. This corresponds to the AC with boundary condition 𝒲_n=(1/2,0) ⊕ (1/2,1), which are denoted by 𝒲_n^0 and 𝒲_n^1 respectively, and each will give a copy of _1/16. In this case, the matrix equation [eq15](15) becomes[μ_1';μ_2'; ⋮;; μ_2n-1'; μ_2n' ] = 2[ 0 1 0 …; 1 0 1; 0 1 0; ⋮; 0 1 0; 1 0 0; 0 0 ]_2n × 2n[μ_1;μ_2;⋮; ; μ_2n-1; μ_2n ]. The matrix has 2n-1 eigenvalues ϵ_k=4cos(kπ/2n) for k =1,…,n,…,2n-1 with corresponding eigenvectors (μ_b,k)_b=(sin(bkπ/2n))_b. In addition to those, there is the eigenvalue 0 with eigenvector (δ_b,2n)_b.This gives a set of (normalized) raising and lowering operators Ψ_± k for k=1,…,n-1 (the creation operators) satisfying the usual relations. The ϵ_n will give Ψ_n which also anticommutes with all other operators except its conjugate, which is easily checked (by looking at its corresponding eigenvector), to be exactly -Ψ_n,{Ψ_n,Ψ_n^†}=1Ψ_n^2=-1/2.Finally, the operator i^2nψ_2n corresponding to (δ_b,2n)_b (and the zero eigenvalue) anticommutes with all other operators.The algebra ℱ_n-1 acts on a 2^n dimensional space. By [fct1]Fact 1, the vacuum space (𝒲_n)_0 created by the product of all annihilation operators is two dimensional. Further, 𝒲_n^i (i=0,1) are preserved by the Hamiltonian. In fact, all the 2n-1 Dirac operators Ψ_k also preserve 𝒲_n^i as the coefficient for the only term containing σ_n^x in their linear expansion in terms of the ψ_js, i.e. ψ_2n, is sin((2n)k π/2n)=0. This implies (𝒲_n)_0 splits into two one-dimensional subspaces of 𝒲_n^0 and 𝒲_n^1.Hence, by restricting H to 𝒲_n^i, one can apply an argument similar to the case 1(a). Let us define H' which also preserves 𝒲_n^is H'=∑_k=1^n-1ϵ_k(Ψ_+kΨ_-k-Ψ_-kΨ_+k). As [H',Ψ_k]=2ϵ_kΨ_k, similar to H, we conclude that D=H-H' satisfies [D,Ψ_k]=0. [fct2]Fact 2 implies that D must be a scalar restricted to each 𝒲_n^is as they are both generated by a vacuum vector and H,H' both preserve 𝒲_n^is. If the C-symmetry argument is applied as usual, as C=∏_i σ_i^z ∏_i (σ_i^x)^i, only for even n, C preserves 𝒲_n^i. Hence, for even n, H would have an energy symmetry and so H=H' on each 𝒲_n^i and therefore on the whole 𝒲_n. Even if n is odd, a more involved argument is possible but as similar circumstances appear in the periodic chain case, an argument based on the (-1)^F symmetry will be proposed.As (-1)^F can be easily seen to commute (or anti-commute based on the parity of n) with the product of all annihilation operators, (𝒲_n)_0 is preserved by (-1)^F. It is similarly preserved by Ψ_n. But (-1)^F and Ψ_n anticommutes. Therefore, any eigenvector of (-1)^F in (𝒲_n)_0, by the action of Ψ_n, will go to another nonzero eigenvector (since Ψ_n^2=-1/2) with the opposite eigenvalue.The two unit eigenvectors 1_±∈ (𝒲_n)_0 with corresponding eigenvalue ± 1 of (-1)^F are sent to a scalar multiple of each other by Ψ_n. Then, defining H' as before, and noticing that [H',(-1)^F]=0, D preserves the sectors. By [fct2]Fact 2, H restricted to any of the ±1 sector of (-1)^F is equal to H' after some shift β_± in each ±1 sector. Hence, D|_+1=β_+1 and D|_-1=β_-1.Also, it is important to note that H's spectrum in both sector is the same. Indeed, given any index subset S of the creation operators, starting with either 1_± based on the parity of |S| ensures that the product gives a vector in the desired sector with the energy ∑_k ∈ Sϵ_k-∑_k ∉Sϵ_k.Now suppose β_++β_-<0, then the lowest energy x+β_+ for H=H'+β_+1 does not have its opposite in H'+β_-1 in the other sector; if not, then ∃ E such that -x-β_+=E+β_--β_+-β_-=x+E ≤ 0 as x is the lowest energy of H'. Similarly, β_++β_->0 is ruled out. So β_++β_-=0 and H=H'+β_+(-1)^F. But then looking at the expansion of H' and H in terms of Majorana operators, both have at most bilinear terms while (-1)^F=i^n∏ψ_j has 2n terms. Due to the linear independence of the Majorana monomials, H=H' is the only possibility.As was mentioned before, H and the creation operators preserve 𝒲_n^i and therefore, one can pick the vacuum vectors 1_n^i ∈𝒲_n^i. Then, similar to 1(a), after a suitable shift and scaling, the Hamiltonian L_0^c is constructedL_0^c=2n/π∑_k=1^n-1cos(kπ/2n)Ψ_+kΨ_-k+1/161,the restriction of which to each 𝒲_n^i has eigenvectors {∏_k ∈ SΨ_k 1_n^i|∀ S ⊂ [n-1]}. The shift 1/161 is not the natural one, but for computational issues, it is better to have the exact shift. After the renumbering Ψ_k →Ψ_k-n for k≠ n and Ψ_n → iΨ_0, one defines the scaling limit vector space in each boundary condition spanned by the orthogonal vectors {∏_k ∈ SΨ_k 1_n|∀ S ⊂ℕ}. Also, similar to 1(a), the scaling limit of L_0^c is obtained using the Taylor series of the coefficients in L_0^c = 2n/π∑_k ∈ [(n-1)]sin(kπ/2n)Ψ_-kΨ_k+1/161,which leads toL_0=∑_k ∈ℕ kΨ_-kΨ_k+1/161with the desired rate of convergence. This gives the character∏_k=1^∞ (1+q^k)for the scaling limit which is that of _1/16. This finishes the proof for this case.§.§.§ Case 1(e) periodic and full CFT The periodic case, as mentioned in [1.6]section 1.6, will be diagonalized differently from <cit.>, which involves taking the usual Fourier transform of the Majorana operators to get the Dirac operators. Here, we will continue applying the method in <cit.> and have cos() and sin() transform of the Majorana operators.In the periodic chain with 2n TL operators acting on, there will be operators of the formH= -∑_j=1^nt_2j-1σ_j^x-∑_j=1^n t_2jσ_j^z σ_j+1^z,where the case 1(a) Pauli-TL relation are used. Rewriting this in the language of the Majorana operators givesH=i (∑_a=1^2n-1 t_a ψ_a+1ψ_a - t_2nψ_1ψ_2n(-1)^F ).H has the (-1)^F symmetry and divides the spectrum in two ± 1 sectors which we can analyze separately. The method pursued is to first restrict H to one of those sectors so that the sign of (-1)^F is determined, and then extend H in the obvious way to both sectors (as the Majorana operators can be extended). So effectively, two matrices each of which equal to H in one of the ±1 sector will be diagonalized. Then the spectrum at the scaling limit will be easy to find.We will show that if n is even, the scaling limit is the diagonal full CFT _0_0+_1/2_1/2+_1/16_1/16 and if n is odd, it is _0_1/2+_1/2_0+_1/16_1/16. In this case, similar to how [eq15](15) was derived, the matrix is[μ_1';μ_2'; ⋮;; μ_2n' ]= 2[0t_10… (-1)^F+(n+1)t_2n;t_10t_2;0t_20;⋮t_2n-1; (-1)^F+(n+1)t_2n t_2n-10 ][μ_1;μ_2;⋮; ; μ_2n ], where by (-1)^F in the entries, the sign of the operator (-1)^F when restricted to ± 1 sector is considered. From now on, we will specialize to t_j=1 for all j. There are two cases based on the parity of n. Even n.Restricting to +1 sector, the matrix [eq23](23) becomes2[010… -1;101;010;⋮ 1; -110 ]_2n× 2n,with the corresponding operator beingH^(+1)=i (∑_a=1^2n-1 t_a ψ_a+1ψ_a - t_2nψ_1ψ_2n).As mentioned before, we should think of H^(+1) as an operator acting on the 2^n dimensional Hilbert space (on both ± 1 sectors), and once the spectrum of H^(+1) is found, we will restrict to the +1 sector.The matrix [eq24](24) has eigenvalues ϵ_k=4cos((2k-1)π/2n) for k ∈ [2n] and there are repetitions. The corresponding eigenvectors are (μ_b,k)_b=(cos((2k-1)bπ/2n))_b for k=1,…,n and (μ'_b,k)_b=(sin((2k-1)bπ/2n))_b for k=n+1,…,2n, where (μ_b,k)_b and (μ'_b,2n+1-k)_b correspond to the same eigenvalueϵ_k=4cos((2k-1)π/2n)=4cos((2(2n+1-k)-1)π/2n)=ϵ_2n+1-k. These in turn will give orthogonal eigenvectors constructing an algebra ℱ_n of (normalized) creation operators ℱ_n^+ for k = 1,…,n/2,3n/2+1,…,2n and (normalized) annihilation operators ℱ_n^- for k = n/2+1,…,3n/2 whereΨ_-k:=Ψ_k^†=Ψ_n+1-k for1 ≤ k ≤ n,andΨ_-k:=Ψ_k^†=Ψ_2n+1-k forn+1 ≤ k ≤ 2n.One can see that the adjoint of operators corresponding to first quadrant (k ≤ n/2) are the ones in the second quadrant with the opposite eigenvalue and for those in the fourth quadrant (3n/2+1 ≤ k ≤ 2n), the adjoint is the one with the opposite eigenvalue in the third quadrant. Next, defineH'^(+1)= ∑_k=1^n/2ϵ_k(Ψ_+kΨ_-k-Ψ_-kΨ_+k) + ∑_k=3n/2+1^2nϵ_k(Ψ_+kΨ_-k-Ψ_-kΨ_+k),and as [D,Ψ_k]=0 for D=H'^(+1)-H^(+1) for all k, D=α1 is a scalar by [fct2]Fact 2 as there are n Dirac operators acting on 2^n dimensional Hilbert space, therefore the vacuum space is one dimensional. The charge conjugation symmetry C always satisfies C(-1)^F=(-1)^n(-1)^FC. As n is even, the charge conjugation symmetry applies on the +1 sector. Therefore D|_+1=0 α=0H^(+1)=H'^(+1) on both sectors. Applying a suitable scaling gives the Hamiltonian𝕃_0^c,(+1)=n/4π(∑_k=1^n/2ϵ_kΨ_+kΨ_-k+∑_k=3n/2+1^2nϵ_kΨ_+kΨ_-k).Let us restrict to the +1 sector. Again, (-1)^F preserves the vacuum but whether the vacuum itself is in the +1 sector or -1 is important. One has to prove a similar identity like [eq19](19) where the product of annihilation operators ∏_L/2+1^3L/2Ψ_k is one side of the equation:i^2n(-1)^F ∏_annihilationΨ_k=∏_annihilationΨ_k,We will have to compute similar determinant of matrices while using the equalities:(-1)^kμ_s,k=μ'_s+n,2n+1-kfor 1≤ s ≤ n, n/2+1 ≤ k ≤ n,(-1)^k+1μ_s,k=μ'_s-n,2n+1-kforn+1≤ s ≤ 2n, n/2+1 ≤ k ≤ n,which can be compactly presented as(-1)^kcos((2k-1)sπ/2n)=sin((n+s)(2(2n+1-k)-1)π/2n)= sin((n-s)(2(2n+1-k)-1)π/2n).This means that for the two matrices, i-th row from one matrix will be equal to the opposite (n+1-i)-row on the other matrix up to a (-1)^k factor. Since i^2n=(-1)^n, for n even, the vacuum is in the +1 sector. As the eigenvectors of 𝕃_0^c are {∏_k∈ SΨ_k1_n|∀ S ⊂indices of creation operators} with 1_n the vacuum, and (-1)^F anticommutes with all creation operators, the eigenvectors of interest are those with even number of creation operators. We shall call all operators with index 1≤ k≤ n the left-moving (LM) operators and n+1≤ k≤ 2n the right-moving (RM) operators. If one takes odd number of operators from the LM part (giving us some energy in ℕ-1/2), then odd number of operators from the RM part should be taken as well and the same for even. Therefore the character of the scaling limit will be _0_0+_1/2_1/2. Scaling limit can be derived using the Fourier series and the rate of convergence can be proved similar to 1(a). Clearly, the relabelling will be Ψ_k →Ψ_k-n+1/2 for the LM and Ψ_k →Ψ_3n+1/2-k for the RM part, giving us𝕃_0^c,(+1)=n/π(∑_k ∈ [(n/2-1/2)]sin(kπ/n)Ψ_-kΨ_k+∑_k∈ [(n/2-1/2)]sin(kπ/n)Ψ_-kΨ_k),with scaling limit𝕃_0|_+1=L_0|_+1+L_0|_+1=(∑_k ∈ℕ-1/2 kΨ_-kΨ_k+∑_k∈ℕ-1/2 kΨ_-kΨ_k) This finishes the proof for the sector +1 and even n.For the -1 sector, we have the matrix[ 0 1 0 … 1; 1 0 1; 0 1 0; ⋮ 1; 1 1 0 ],with the corresponding operator, extended to act on both sectors, given byH^(-1)=i (∑_a=1^2n-1 t_a ψ_a+1ψ_a + t_2nψ_1ψ_2n).Matrix [eq27](27) has eigenvalues ϵ_k=4cos(2kπ/2n) for k=1,…,2n or equivalently k=0,≤,2n-1, where the eigenvalues corresponding to k=1,…,n/2 are repeated twice and the one corresponding to k=n,0 are repeated once. The corresponding eigenvectors are (μ_b,k)_b=(cos(2kbπ/2n))_b for k=0, …, n and another set of eigenvectors (μ'_b,k)_b=(sin(2kbπ/2n))_b for k=n+1,…,2n-1. Note that (μ_b,k)_b and (μ'_b,2n-k)_b are eigenvectors for the same eigenvalue as long as k ≠ n,0. The corresponding (normalized) Dirac operators areΨ_-k:=Ψ_k^†=Ψ_n-k for0 ≤ k ≤ n& k≠n/2, Ψ_-k:=Ψ_k^†=Ψ_2n-k forn+1 ≤ k ≤ 2n-1& k≠3n/2.Similar to the previous case, LM creation operators are in the first quadrant (0 ≤ k ≤n/2-1), and the LM annihilation operators (adjoint to the first quadrant) in the second quadrant. The RM creation operators belong to the fourth quadrant (3n/2+1 ≤ k ≤ 2n-1) with their adjoint in the third quadrant. Also similar to the case of _1/16, we have the (“would-be” zero-mode) operators Ψ_n/2,Ψ_3n/2 corresponding to ϵ_n/2=ϵ_3n/2=0 with their adjoint being Ψ_n/2,-Ψ_3n/2 respectively. All operators anticommute except with their adjoint, with which they give the identity. Hence, Ψ_n/2^2=1/2=-Ψ_3n/2^2.Summing up, there are n/2+(n/2-1)=n-1 creation operators and the situation is same as _1/16. We have two vectors in the image of the product of all n-1 annihilation operators. As (-1)^F preserve that image (since it anti commutes with all Dirac operators) but also since it anticommutes with Ψ_n/2 (which also preserves the vacuum space for the same reason), there are eigenvectors 1_n^± in each ±1 sector in the vacuum space. Next, definingH'^(-1)=∑_k=1^n/2-1ϵ_k(Ψ_+kΨ_-k-Ψ_-kΨ_+k)+∑_k=3n/2+1^2nϵ_k(Ψ_+kΨ_-k-Ψ_-kΨ_+k),it can be shown that [H^(-1)-H'^(-1),Ψ_k]=0. Further, both operators commute with (-1)^F, so D=H^(-1)-H'^(-1) preserves both sectors and acts as a scalar on each. As n is even, we have charge conjugacy in each sector, therefore D=0. It then becomes clear that one must define 𝕃_0^c,(-1) as𝕃_0^c,(-1)=n/4π(∑_k=0^n/2-1ϵ_kΨ_+kΨ_-k+∑_k=3n/2+1^2n-1ϵ_kΨ_+kΨ_-k),where we note that a shift by some scalar (which will be 1/8 in the limit) is not included yet and will be discussed later. As we are interested in the -1 sector, all combinations of the form * (odd LM) (odd RM) 1_n^-* (even LM) (even RM) 1_n^-* (odd LM) (even RM) 1_n^+* (even LM) (odd RM) 1_n^+are in the subspace the scaling limit should be taken. To do so, one needs to first apply the renumbering Ψ_k→Ψ_k-n/2 for 0 ≤ k ≤ n and Ψ_k→Ψ_3n/2-k for n+1 ≤ k ≤ 2n-1 except for Ψ_3n/2→ iΨ_0, and accordingly𝕃_0^c,(-1)=n/π(∑_k ∈ [n/2]sin(kπ/n)Ψ_-kΨ_k+∑_k ∈ [n/2]sin(kπ/n)Ψ_-kΨ_k). In order to build the scaling limit vector space 𝒱, the four possibilities outlined above need to be considered. The embeddings ϕ_n : 𝒲_n^(-1)↪𝒲_n+2^(-1), done by mapping each vector to its obvious corresponding vector (also consistent with the embeddings of the Dirac operators algebra), gives already the character of 𝕃_0^c,(-1) in the scaling limit as that of _1/16_1/16 but we need to identify the scaling limit with the space_1/16_1/16. This is also clear as every configuration above can be identified with its counterpart in _1/16_1/16 * (odd LM) (odd RM) |1/16⟩⊗|1/16⟩* (even LM) (even RM) |1/16⟩⊗|1/16⟩* (odd LM) (even RM) |1/16⟩⊗|1/16⟩* (even LM) (odd RM) |1/16⟩⊗|1/16⟩This map is unitary and “character”-preserving. Notice that only 1^- (the scaling limit of 1_n^-) is identified with |1/16⟩⊗|1/16⟩. The vector 1^+ (the scaling limit of 1_n^+) is not identified with anything inside _1/16_1/16 as it is not even present in the finite spaces 𝒲_n^(-1). What we observe, is a merging of two copies of _1/16_1/16 (generated by 1^+ and 1^-)and selection of a subspace of both, which together form a copy of _1/16_1/16. Finally, the scaling limit Hamiltonian is𝕃_0|_-1=L_0|_-1+L_0|_-1=(∑_k ∈ℕ kΨ_-kΨ_k+∑_k∈ℕ kΨ_-kΨ_k).Let us discuss the issue of the scalings done to H in both sectors. They can be seen to be clearly different. In fact, after the restriction to each sector and proving that H=H', there was a scaling H →n/8π(H+∑_k=1^n/2ϵ_k + ∑_k=3n/2+1^2nϵ_k) for the +1 sector and the other being H →n/8π(H+∑_k=0^n/2-1ϵ_k + ∑_k=3n/2+1^2n-1ϵ_k). There can only be one scaling to H. So the different scalings should differ by some scalar which is 1/8 in the scaling limit as the ground state in the (-1)^F=-1 sector is |1/16⟩⊗|1/16⟩. Indeed, by some trigonometric calculation, the two scalings differ by n/8π(4tan(π/4n))=1/8+O(1/n^2) which in the limit n→∞ is 1/8. Hence, the scaling limit for even n is the diagonal full CFT with the Hamiltonian 𝕃_0.Odd n.This case will not be analyzed as it is not a full diagonal CFT and not used in any of the main results. Nevertheless, the proof can be seen to be similar to the n even case, with the difference that the +1 sector gives _1/16_1/16 and the -1 sector gives _0_1/2+_1/2_0. The reason the -1 sector is not diagonal is the fact that odd(even) number of LM operators have to act with even(odd) numbers of RM operators to take the vacuum from the +1 sector to -1 sector.§.§.§ Case 2; The higher Virasoro modes L_ms Changing the coefficients t_j to a cos() and sin() tranform of the e_js is necessary to obtain the higher Virasoro modes L_ms.We will prove the case 2(a) (_0+_1/2) in [thm3.1]Theorem 3.1 with the rate of convergence. All other cases, including the periodic case, have a similar proof although there will be some comments for the periodic chain. The cos() transform and L_m^c+L_-m^c.Let us fix m ∈ℕ. The operator L_m+L_-m is given by (see e.g. <cit.>)∑_k ≥m+1/2,k ∈ℤ+1/2 (k-m/2) Ψ_-k+mΨ_k+∑_k ≥-m+1/2,k ∈ℤ+1/2 (k+m/2) Ψ_-k-mΨ_k,which we want to obtain in the scaling limit. To understand what the observable O=i∑ t_j(m)ψ_j+1ψ_j will be in terms of Ψ_k's, one has to use the matrix equation [eq15](15). We need to build another observable O' using Ψ_k's which has scaling limit L_m+L_-m, and that also satisfies [O-O',Ψ_k]=0. Then, going through the usual arguments, after some suitable scaling, the sequence O_nL_m+L_-m will be constructed.Notice that [L_m+L_-m,Ψ_k] is the sum of exactly two Dirac operators with indices differing by m from k. t_j(m) should be such that the same result for [O,Ψ_k] happens, with coefficients going to k±m/2 in the scaling limit. Using the indices before the renumbering, i.e. k ∈ [2n], a natural candidate for the coefficients would be cos((k∓m/2)π/2n+1). Hence, computing [O,Ψ_k] using [eq15](15), the following must holdt_j(m)μ_k,j+1+t_j-1(m)μ_k,j-1= cos((k+m/2)π/2n+1)μ_k+m,j+cos((k-m/2)π/2n+1)μ_k-m,j.From simple trigonometric identities, the right side is equal tocos((k+m/2)π/2n+1)sin((k+m)jπ/2n+1)+cos((k-m/2)π/2n+1)sin((k-m)jπ/2n+1) =cos(m(j+1/2)π/2n+1)sin(k(j+1)π/2n+1)+cos(m(j-1/2)π/2n+1)sin(k(j-1)π/2n+1),which is in factcos(m(j+1/2)π/2n+1)μ_k,j+1+cos(m(j-1/2)π/2n+1)μ_k,j-1.So t_j(m) are forced to be cos(m(j+1/2)π/2n+1). However, the coefficients cos(mjπ/2n), as used in the conjecture <cit.>, do not satisfy the identity [eq30](30). The implications will be discussed more in the second subsection. As the actual identity for the matrix [eq15](15) involves a two factor, O is changed to O/2 to cancel this factor. This is important to compute the scaling factors until O becomes the desired operator.Although the identities above determine what O' should be, what happens at the boundaries when k+m>2n or k-m<1 must be examined. In these cases, one has to consider sin((k+m)jπ/2n+1)=-sin((2(2n+1)-k-n)jπ/2n+1)=-μ_2(2n+1)-(k+m),j if k+m>2n and sin((k-m)jπ/2n+1)=-sin((m-k)jπ/2n+1)=-μ_m-k,j if k-m<1. Therefore, O' is defined as0.93(∑_k+m ≤ 2ncos((k+m/2)π/2n+1) Ψ_k+mΨ_k^†-∑_k+m>2ncos((k+m/2)π/2n+1)Ψ_2(2n+1)-k-mΨ_k^†) 0.93+(∑_k-m ≥ 1cos((k-m/2)π/2n+1)Ψ_k-mΨ_k^†-∑_k-m<1cos((k-m/2)π/2n+1)Ψ_m-kΨ_k^†).For k close to the boundaries, i.e. 1 or 2n (which corresponds to high energy creation/annihilation), another operator of high energy is associated. Therefore, there are no low-high energy mix in the above formula. There are also no terms like Ψ_k^†Ψ_k as m ≠ 0. Further, similar to the case of L_0, where H' had terms like (Ψ_kΨ_k^†-Ψ_k^†Ψ_k) for each Ψ_k with 1 ≤ k ≤ n, there are also two terms in each parenthesis in the summation above which have Ψ_k^†. Indeed, one is Ψ_k+mΨ_k^† and the other is Ψ_2n+1-kΨ_2n+1-k-m^†=Ψ_k^†Ψ_k+m, and we have0.99cos((k+m/2)π/2n+1)Ψ_k+mΨ_k^†=cos(((2n+1-k-m)+m/2)π/2n+1)Ψ_2n+1-kΨ_2n+1-k-m^†.Let us prove O=O'. The equation for O' implies [O-O',Ψ_k]=0, meaning that O-O' is a scalar. To prove that the shift is zero, we use the fact that monomials in Majorana operators are linear independent. The left side in O=O'+α1 is bilinear, but the right side might have scalar terms coming from the multiplication of ψ_jψ_j=1 when for O' is expanded in terms of the ψ_js. Those terms have to cancel each other so that a non-zero shift α will not be needed. In fact, it can be shown that the scalar from each of Ψ_k+nΨ_k^† is zero. Indeed, calculating the contribution from Ψ_k+nΨ_k^†, ∑_j=1^2n i^jsin((k+m)jπ/2n+1).i^jsin((2n+1-k)jπ/2n+1)= ∑_j=1^2n i^2jsin((k+m)jπ/2n+1).(-1)^j+1sin(kjπ/2n+1)= -∑_j=1^2nsin((k+m)jπ/2n+1).sin(kjπ/2n+1)=0,the last equality holds as they are eigenvectors of different eigenvalues of matrix [eq15](15) for t_j=1. Hence, O=O'. To find the scaling limit, taking note of the fact that each term is repeated twice, we have the following operatorL_m^c+L_-m^c=2n+1/2πO'where O' is defined in [eq32](32) and L_m^c is the first, and L_-m^c corresponds to the second parenthesis. Further the factor 2 in the denominator is due to the repetition of each term. As the proof for the convergence rate is similar, only L_m^c will be done. [thm3.1]Theorem 3.1 states thatL_m^c|_√(n)=L_m|_√(n)+O(1/n),where m ≤√(n) can also depend on n. By applying the energy restriction up to √(n), for large enough n, it is not hard to observe that all the bilinear terms of Dirac operators in L_m^c|_√(n) and L_m|_√(n) which will be nonzero operators after the energy restriction are the same. Indeed, most of the bilinear Dirac terms in both of the operators, are composed of an annihilation and a creation operator. If the annihilation operator annihilates energy more than √(n) then its restriction to vectors with energy at most √(n) is clearly zero. Lastly, for large enough n, as 2√(n) < < n, there are no terms close to the boundaries (with index close to 1 or 2n) and so, all terms inside of L_m|_√(n) are also present inside L_m^c|_√(n) and vice-versa, although with a different coefficient which their difference shall be estimated. After the renumbering Ψ_k →Ψ_k-n-1/2, the term cos((k+m/2)π/2n+1)Ψ_k+mΨ_k^† becomes sin((k-m/2)π/2n+1)Ψ_-k+mΨ_k.By taking the difference in the coefficient, we have|2n+1/πsin((k-m/2)π/2n+1)-(k-m/2)|≤ O((√(n))^3/(2n+1)^2),as k,m are of order √(n) due to the discussion in the previous paragraph. Next, since there are O(√(n)) of these differences (as k can vary), and each Dirac bilinear term has norm at most one,||L_m^c|_√(n)-L_m|_√(n)|| ≤ O((√(n))^3.√(n)/(2n+1)^2)=O(1/n),finishing the proof of the convergence rate. This obviously implies L_m^cL_m. The very similar reasoning can be made for all the other ACs. But we would like to note that when changing the boundary condition, the denominators in the fractions inside the trigonometric functions must be changed (e.g. in the _1/16 case, one needs 2n instead of 2n+1). The changes happen the most in the periodic case and wewill have to apply the following * replace 2n+1 by 2n,* replace m in t_j(m) by 2m,* replace k by 2k (or 2k-1), where the change to even numbers 2k is for _1/16_1/16 and odd number for _0_0+_1/2_1/2. One also has to check that the trigonometric identities like [eq31](31) holds for all the eigenvectors of the two different matrices (they are of two types: μ_i and μ'_i corresponding to cos() and sin()). As an example, for the LM part (with (μ)_b=(cos())_b eigenvectors) of _0_0+_1/2_1/2, the corresponding identity is:0.97cos((2m)(j+1/2)π/2n)cos((2k-1)(j+1)π/2n)+cos((2m)(j-1/2)π/2n)cos((2k-1)(j-1)π/2n)= 0.97cos((2k-1+m)π/2n)cos((2k-1+2m)jπ/2n)+cos((2k-1-m)π/2n)cos((2k-1-2m)jπ/2n)and there is a similar identity when the second cos() terms in each product are replaced with sin() for the RM operators. This gives the operator 𝕃_m^c+𝕃_-m^c and the convergence to 𝕃_m+𝕃_-m can be proved in a similar way.The sin() transform and i(L_m^s-L_-m^s).Having found the sum of the higher Virasoro modes, their difference is required to recover an operator L_mL_m, as shown in the remark after [thm3.1]Theorem 3.1. Again, this case will be demonstrated for _0+_1/2. As the proof to the convergence and scaling limit is very similar to the previous cos() transform, we will refer to the previous arguments for that part of the problem. We need the sin() transform of the form O= -i∑_j=1^2n-2 t_j(n)[e_j,e_j+1] which is same as O=i∑_j t_j(n)ψ_jψ_j+2. The corresponding matrix for [O,Ψ] where Ψ=∑ i^bμ_b ψ_b can be found as follows:[O,Ψ]=Ψ'=i(∑_b i^bμ'_bψ_b)where the i factor is needed to obtain i(L_m-L_-m), and we haveμ'_b=2(t_b(m)μ_b+2-t_b-2(m)μ_b-2)except at the boundaries where the formula will be different. In the case of the non-periodic chain (1/2,1/2), this can be turned into the corresponding matrix[00 t_1(m)0;000 t_2(m);-t_1(m)00 ⋱ ; -t_2(m);;⋱00t_2n-2(m); 000;0 -t_2n-2(m)00 ].The matrix corresponding to the periodic case can be recovered similarly and as expected, it will have entries (-1)^n+Ft_2n-1(m),(-1)^n+Ft_2n(m) and their opposite in the diagonals in the corners of the matrix.First, an identity similar to [eq31](31) only with sin() functions is needed to force the values for t_j(m). Since the coefficients (k±m/2) needs to be recovered in the limit using sin() functions, the natural candidate for the coefficients of the bilinear Dirac terms are sin((2k+m)π/2n+1). This can be seen to only work when t_j(m)=sin(m(j+1)π/2n+1), satisfying the following identityt_j(m)μ_k,j+2-t_j-2(m)μ_k,j-2= 0.999sin(m(j+1)π/2n+1)sin(k(j+2)π/2n+1)-sin(m(j-1)π/2n+1)sin(k(j-2)π/2n+1)= 0.999sin((2k+m)π/2n+1)sin((k+m)jπ/2n+1)-sin((2k-m)π/2n+1)sin((k-m)jπ/2n+1).It can be checked that at the boundaries j=1,2,2n-1,2n the first equality above still holds since sin(0)=sin(kπ)=sin(mπ)=0. Repeating the argument in the previous case, we set -iO'=0.93(∑_k+m ≤ 2nsin((2k+m)π/2n+1) Ψ_k+mΨ_k^†-∑_k+m>2nsin((2k+m)π/2n+1)Ψ_2(2n+1)-k-mΨ_k^†) 0.93-(∑_k-m ≥ 1sin((2k-m)π/2n+1)Ψ_k-mΨ_k^†-∑_k-m<1sin((2k-m)π/2n+1)Ψ_m-kΨ_k^†),And the argument for O/2=O' is exactly the same as before (where the factor 2 is due to the matrix equation [eq15](15) as in the previous case). Therefore, let us seti(L_m^s-L_-m^s)=2n+1/2πO',where O' is defined in [eq36](36) and L_m^s is the first, and L_-m^s corresponds to the second parenthesis. For the coefficients, after the renumbering Ψ_k→Ψ_k-n-1/2, sin((k-m/2)π/n+1/2) is the coefficient of Ψ_-k+mΨ_k in L_m^s, which means that in the scaling limit, the coefficient for the Dirac terms is as desired. The proof of convergence and its rate, are similar to the cos() transform. As previously, the denominators need to change for a change in boundary condition (e.g. 2n instead of 2n+1 for _1/16). For the periodic chain, the following changes need to performed: * replace 2n+1 by 2n,* replace m in t_j(m) by 2m,* replace k by 2k (or 2k-1),where the change to even numbers 2k is for _1/16_1/16 and odd number for _0_0+_1/2_1/2. It can be checked that the new identity involving the eigenvectors of type cos() for the +1 sector _0_0+_1/2_1/2 is0.9sin((2m)(j+1)π/2n)cos((2k-1)(j+2)π/2n)-sin((2m)(j-1)π/2n)cos((2k-1)(j-2)π/2n)= 0.9sin((2(2k-1)+2m)π/2n)cos(((2k-1)+2m)jπ/2n)-sin((2(2k-1)-2m)π/2n)cos(((2k-1)-2m)jπ/2n),which will also hold if cos()s are replaced with sin(), giving the identity for the sin() type eigenvectors. This finishes the proof of [thm3.1]Theorem 3.1.§.§ 2.TEXT or TEXT It was mentioned in [1.6]section 1.6, and in the remark after [cnj5.5]Conjecture 5.5, that the operators involved in a similar conjecture in <cit.> have undesirable low-high energy mix. These claims and why the “reasonable” guess that the sin() transform of the e_j must give i(L_m-L_-m) is not true, will be demonstrated by direct calculations. Let us start with the latter.We start with the nonperiodic chains. As before, the example _0+_1/2 will be used to show the argument which can be applied similarly to other nonperiodic cases. In order to obtain -∑_j=1^2n-1sin(m(j+1/2)π/2n+1)e_jor in other words O_s=i∑_j=1^2n-1sin(m(j+1/2)π/2n+1)ψ_j+1ψ_j in terms of the creation and annihilation operators, ψ_j should be expressed in terms of the Ψ_ks (the inverse of the transformation given by the eigenvectors of the matrix [eq15](15)). In <cit.>, it is shown thatψ_2j=(-1)^j∑_k=1^n i/N_kQ_2j-1(ϵ_k^2)(Ψ_+k+Ψ_-k),ψ_2j+1=(-1)^j∑_k=1^n 1/N_kQ_2j(ϵ_k^2)(Ψ_+k-Ψ_-k),where un-normalized Dirac operators are considered, and Q_a-1(ϵ_± k^2)=sin(akπ/2n+1). Using the above, the coefficient of a term Ψ_xΨ_-y in O_s for any 1 ≤ x,y≤ n is0.9991/N_xN_y∑_j=1^2n-1sin(m(j+1/2)π/2n+1)( sin(jxπ/2n+1)sin((j+1)yπ/2n+1)+sin((j+1)xπ/2n+1)sin(jyπ/2n+1) )This can be computed by direct computation (substituting sin(θ)=e^iθ-e^-iθ/2i) and it can be shown that it is zero if and only if m+x+y ≡ 0 2. This implies that (even after any scaling of O_s), e.g. for m=4, there are terms like Ψ_1Ψ_-(n-1) for n odd and Ψ_1Ψ_-(n-2) for n even with nonzero coefficient. Therefore, this is not an energy local operator and further, it can not be a candidate for the sequence associated to L(f) to obtain a strong SL-algebra in [thm4.6]Theorem 4.6. As another example with chain size even and with scaling limit _1/16, we have the following0.9991/N_xN_y∑_j=1^2n-2sin(m(j+1/2)π/2n)( sin(jxπ/2n)sin((j+1)yπ/2n)+sin((j+1)xπ/2n)sin(jyπ/2n) )which can be computed and it is zero ⇔ m+x+y ≡ 0 2 and the same conclusions follow. There is a similar argument for the sin() transform of [e_j,e_j+1].The conjecture <cit.> asserts that l_m+l_-m L_m+L_-m wherel_m+l_-m is some scaling of the sum of the terms cos(mjπ/2n)e_j. The difference with [cnj5.5]Conjecture 5.5 is the factor j instead of m(j+1/2)π/2n+1 and the denominator. We recall that we interpret the point (j+1/2)π/2n+1 as the “center” of the action of e_j in the half-circle in [fig1]Figure 1. Also, we recall that the identity [eq30](30) forced the coefficient cos(m(j+1/2)π/2n+1), therefore any other coefficient (including cos(mjπ/2n)) should have some undesirable effect, as it does not satisfy the identity, although these effects could vanish in the scaling limit.We can compute the coefficient of a term Ψ_xΨ_-y in O_s'=i∑_j=1^2n-1cos(mjπ/2n)ψ_j+1ψ_j for any 1 ≤ x,y≤ n:0.9991/N_xN_y∑_j=1^2n-1cos(mjπ/2n)( sin(jxπ/2n+1)sin((j+1)yπ/2n+1)+sin((j+1)xπ/2n+1)sin(jyπ/2n+1) ),which can be observed to have the similar property of being zero if and only if m+x+y ≢0 2. As an example, for m=9,x=14,n=52,y=49=n-3, the above gives approximately -0.0256625≠ 0. This suggests that the conjecture <cit.> does not provide the right candidates if a strong SL-algebra is desired as there are low-high energy mix. These terms could make even the convergence of simple products such as the convergence of commutators to the commutators of scaling limit impossible. Of course, they should vanish at the scaling limit, along with all those terms with non zero coefficient and energy shift other than m and a numerical simulation shows that happening but with a slower rate (as in <cit.>). Hence, the rate of convergence could also be another reason to consider the operators in[cnj5.5]Conjecture 5.5 for obtaining the higher Virasoro modes. Interchiral observables 𝕃_m.The summations considered above for the chiral CFTs need to be also analyzed for the full CFT. This time, an interchiral observable in the limit is obtained which could be thought of as:∂_zψ(z) ψ(z) +∂_zψ(z) ψ(z) .Note that the stress energy tensor is ∂_zψ(z) ψ(z) +∂_zψ(z) ψ(z) .Therefore the LM and RM part will mix due to the identities0.98sin(2m(j+1/2)π/2n)sin((2k-1)(j+1)π/2n)+sin(2m(j-1/2)π/2n)sin((2k-1)(j-1)π/2n)= 0.98cos((2k-1-m)π/2n)cos((2k-1-2m)jπ/2n)-cos((2k-1+m)π/2n)cos((2k-1+2m)jπ/2n),for the summation -∑_j=1^2nsin(2m(j+1/2)π/2n)e_j in the +1 sector. The RM part (sin() eigenvectors) pairs up with the LM part (cos() eigenvectors). As expected, one needs a similar identity for the LM part to RM part0.98sin(2m(j+1/2)π/2n)cos((2k-1)(j+1)π/2n)+sin(2m(j-1/2)π/2n)cos((2k-1)(j-1)π/2n)= 0.98cos((2k-1+m)π/2n)sin((2k-1+2m)jπ/2n)-cos((2k-1-m)π/2n)sin((2k-1-2m)jπ/2n).Both identities above can be modified (replacing 2k-1 with 2k) to make them work for the (-1)^F=-1 sector.There are similar equations for -i∑_j=1^2ncos(2m(j+1)π/2n)[e_j,e_j+1]. Thus, as in the case of 𝕃_m+𝕃_-m, we get an operator 𝕃_m converging to 𝕃_m in the scaling limit which contains bilinear terms Ψ_-k+mΨ_k instead of Ψ_-k+mΨ_k. Finally, all results on the convergence (rate) for 𝕃_m applies to its interchiral counterpart 𝕃_m.The above illustrates why the similar conjecture <cit.> likely involves a different diagonalization (as noted in [1.6]section 1.6), in order to be true. | http://arxiv.org/abs/1706.08497v4 | {
"authors": [
"Modjtaba Shokrian Zini",
"Zhenghan Wang"
],
"categories": [
"math-ph",
"math.MP",
"math.QA",
"quant-ph"
],
"primary_category": "math-ph",
"published": "20170626174103",
"title": "Conformal Field Theories as Scaling Limit of Anyonic Chains"
} |
Triangle singularities in B^-→ D^*0π^-π^0η and B^-→ D^*0π^-π^+π^- E. Oset December 30, 2023 =================================================================We propose a novel linear minimum-mean-squared-error (MMSE) precoder design for a downlink (DL) massive multiple-input-multiple-output (MIMO) scenario. For economical and computational efficiency reasons low resolution 1-bit digital-to-analog (DAC) and analog-to-digital (ADC) converters are used. This comes at the cost of performance gain that can be recovered by the large number of antennas deployed at the base station (BS) and an appropiate precoder design to mitigate the distortions due to the coarse quantization. The proposed precoder takes the quantization non-linearities into account and is split into a digital precoder and an analog precoder. We formulate the two-stage precoding problem such that the MSE of the users is minimized under the 1-bit constraint. In the simulations, we compare the new optimized precoding scheme with previously proposed linear precoders in terms of uncoded bit error ratio (BER).Massive MIMO, Precoding, 1-bit quantization, Transmit signal processing § INTRODUCTIONThe massive MIMO system, or named large-scale antenna system has been seen as a promising technology for the next generation wireless communication systems <cit.>. The huge increase in the number of antennas at BS can improve spectral efficiency (SE), energy efficiency (EE) and reliability. The BS with large number of antennas, say 100 antennas or more, simultaneously serves a much smaller number of single-antenna users. With the knowledge of CSI at the BS (CSIT), this large spatial DoF of massive MIMO systems can be exploited to significantly increase the spatial multiplexing/diversity gain using MU-MIMO precoding <cit.>. The linear precoders, such as MF, ZF <cit.> and the regularized zero-forcing (RZF) scheme <cit.> are shown to be near-optimal. Thus, it is more practical to use low-complexity linear precoding techniques in massive MIMO systems. Therefore, we mainly focus on linear precoding techniques in this work. The price to pay for massive MIMO systems is increased complexity of the hardware (number of radio frequency (RF) and ADC/DAC chains) and the signal processing and resulting increased energy consumption at the transmitter <cit.>. Several approaches are considered in the literatureto decrease the power consumption such as spatial modulation <cit.>, load modulation <cit.>, the use of parasitic antennas <cit.> and the use of low-cost transceivers <cit.>.One attractive solutionto overcome the issues of high complexity and high energy consumption associated with massive MIMO,is the use of very low resolution ADCs and DACs. The power consumption of the ADC and the DAC, one of the most power-hungry devices, can be reduced exponentially by decreasing the resolution <cit.> and 1-bit quantization can drastically simplify other RF-components, e.g., amplifiers and mixers. Therefore, we design an MMSE linear precoder in a DL massive MIMO scenario where the resolution of the DACs and ADCs is restricted to 1 bit. This precoder design aims at mitigating the distortions due to the coarse quantization in addition to inter-user interference (IUI). A similar work has been presented in <cit.>, where the authors optimize first the quantizer's levels and then give a closed-form expression of an MMSE precoder that takes into account the quantizer non-linearities. However, only quantization at the transmitter was considered. In this contribution, we do not optimize the quantizer. The quantizer in our work has constant levels. But we introduce a second precoding stage in the analog domain after the quantizer to minimize the distortions due to 1-bit DAC/ADC in i.i.d. complex Gaussian channels. The proposed two-stage precoder is designed based on iterative methods. We assume perfect CSIT and study how the new precoder scheme is improving the BER compared to the precoder introduced in <cit.>.This paper is organized as follows: in Section <ref> the system model is presented. In Section <ref> some derivations related to the 1-bit quantization are introduced. In Section <ref> we formulate our optimization problem and show the derivations and the corresponding solution. In Sections <ref> and <ref> we interpret the simulation results and summarize this work.Notation: Bold letters indicate vectors and matrices, non-bold letters express scalars. The operators (.)^*, (.)^ T, (.)^ H and [.] stand for complex conjugation, the transposition, Hermitian transposition and the expectation, respectively. The n × n identity matrix is denoted by 𝐈_n while the zeros (ones) matrix with n rows and m columns is defined as 0_n,m (1_n,m). We define (∙)_R = {∙}, (∙)_I = {∙} and (x) = (x_R) + (x_I). Additionally, (A) denotes a diagonal matrix containing only the diagonal elements of A. σ_α and ρ_αβ denote the standard deviation of α and the correlation coefficient between α and β, respectively. For a circular distributed Gaussian complex-valued signal α we have σ_α_R = σ_α_I. § SYSTEM MODEL We consider a massive MIMO downlink scenario as depicted in Fig. <ref>. The BS with N antennas serves M single-antenna users, where N ≫ M. The signal vector s∈𝒪^M contains the data symbols for each of the M users, where 𝒪 represents the set of QPSK constellation. We assume that s∼𝒪𝒩 ( 0_M,σ_s^2I_M ). In this system 1-bit quantization at the transmitter 𝒬_t as well as at the receiver 𝒬_r is deployed. Therefore, in order to mitigate the IUI, we make use of a two-stage precoder consisting of the digital precoder P and the analog precoder D. The use of the 1-bit quantizer at the transmitter 𝒬_t delivers a signal y_Q that belongs to the set of {± 1 ±}. It means that the magnitude of the entry y_Q,n,n=1,...,N, is constant and its phase belongs to {π/4, 3π/4, 5π/4, 7π/4}. As a result, all the antennas end up getting the same power. To recover the information loss of the power allocation due to 𝒬_t, we employ an analog precoder D of real-valued diagonal structure. So, we end up with y_Q_D = D_t ( Ps ), where D ∈ℛ^N× N, P ∈𝒞^N× M and P^ =[ p_1 p_N ]. The received decoded signal vector ŝ∈𝒞^M × 1 of the M single-antenna users reads as ŝ = _r ( H y_Q_D + η), where H ∈𝒞^M× N is the channel matrix with i.i.d. complex-valued entries of zero mean and unit variance and η∼𝒞𝒩 ( 0_M,C_η = I_M ) is the AWG noise vector.§ STATISTICAL THEORY OF 1-BIT QUANTIZATIONTo design a precoder which takes into account the effects of 𝒬_t and 𝒬_r, we need to know some statistical properties of quantization especially the auto and cross-correlation properties for a Gaussian input signal. Since quantization is non-linear, it has strong effects on the statistical properties of the signal. The statistical properties of hard limiters dealing with real-valued Gaussian-distributed signals are derived in <cit.>. These derivations are applied to complex-valued Gaussian-distributed signals and introduced in this section. For a complex-valued signal vector x we get (x)= ( x_R+ x_I )= ( x_R )+(x_I ).The covariance matrix between an unquantized circular distributed complex-valued signal x of covariance matrix C_x and its 1-bit quantized signal x_= ( x ) is given by C_x_x=√(4/π)KC_x, whereK=diag ( C_x )^-1/2.The covariance matrix of the 1-bit quantized circular distributed complex-valued signals x_ is given by C_x_=4/π ( arcsin ( K{C_x}K )+jarcsin ( K{C_x}K ) ).Note that the diagonal entries of C_x_ are the squared norm of quantized signals, which lead to the following resultdiag ( C_x_ )=2 I.These four equations are the basis for solving the optimization problem presented in the next section. § OPTIMIZATION PROBLEMThe optimization problem is formulated as follows{P_MMSE-Q,D_MMSE-Q}=_P,D[ ŝ- s_2^2 ] s.t. [ y_Q_D_2^2 ]≤ E_tx andD ∈ℛ^N× Nis diagonal. §.§ Objective functionWe aim at minimizing the MSE between the desired signals s and the received signals ŝ given that the power of the transmitted signal y_Q_D is limited by the available transmit power E_tx. We end up with the following expression for MSE MSE=σ_s^2( I_M )+ ( [ŝŝ^H ]- [sŝ^H ]- [ŝs^H ]).We have to find the three expectation terms in the above for which we make use of the covariance and cross correlation matrices. We have already mentioned that the input signal covariance matrix C_s = σ_s^2I_M. The covariance matrix of the precoder's output y is given by C_ y={yy^H}=σ_s^2PP^H.To find a linear expression for the covariance matrix C_y_Q={y_Qy^H_Q}, we make use of (<ref>), (<ref>) and the approximation arcsin(x)≃ x,forx ≠ 1. So, we get C_y_Q =4/π [ K_2PP^HK_2 + cI_N ], with K_2= ( PP^H ) ^-1/2and c = ( π/2 -1 ). The covariance matrix C_y_Q_D of the transmitted signal y_Q_D is given byC_y_Q_D=DC_y_QD.The received signal covariance matrix reads asC_x=HDC_y_QDH^H+C_η.If we look at our MSE expression in (<ref>), one of the terms which we need to find is ( [ŝŝ^H ]). Since the structure of C_ŝ= [ŝŝ^H ] is very similar to C_y_Q, we end up with( [ŝŝ^H ])=2MWe still need to find two more terms in the MSE, i.e. ( [ŝ s^H ] ) and ([ sŝ^H ] ) before we can proceed to solve for P. Note that ( [ŝ s^H ] )= ([ sŝ^H ] ^H ). We use(<ref>) to calculate the above mentioned expectations, which can be expressed as follows[ŝ s^H ] =4σ_s/π K_1 H D K_2 P[ sŝ^H ] =4σ_s/π P^H K_2 D H^H K_1,withK_1= ( C_ x )^-1/2. Finally, putting the expressions of (<ref>), (<ref>) and (<ref>) in (<ref>), we end up with the following closed-form expression for MSEMSE =σ_s^2M+2M -4σ_s/π ( K_1 H D K_2 P + P^H K_2 D H^H K_1 ).The MSE expression in (<ref>) contains two unknown variables P and D. Intuitively, D should be a function of P, since it reallocates the power to the transmit signal originally intended by P which gets lost due to 𝒬_t. To this end, we define a new matrix P'= K_2 P. P' is a row-normalized version of P, such that each row of P' has unit norm. Note that the MSE expression contains the productK_2 P and P^H K_2. K_1 also contains these products. Thus, the MSE expression found so far is in P' rather than P. The purpose ofD is to remove this row-normalization of P introduced by K_2. Therefore, an obvious choice for D isD = K^-1_2=diag (P P^H ) ^1/2.In other words, the optimization with respect to D is reformulated as a one with respect to the norm of each row of P.§.§ ConstraintThe constraint can be simply expressed as[y_Q_D_2^2 ]= (DC_y_Q D) ≤ E_tx. Using (<ref>) and the fact that D is a diagonal matrix, we can simplify the constraint to 2 (D^2 ) ≤ E_tx.§.§ Final Optimization ProblemUsing (<ref>), (<ref>) and (<ref>), we can finally write our optimization problem asmin_ P σ_s^2M+2M-4σ_s/π ( K_1 H P + P^H H^H K_1 )s.t. (P P^H ) ≤E_tx/2 andD = diag (P P^H ) ^1/2. §.§ Solving the Optimization ProblemNote that our objective function in (<ref>) is non-linear in P because of K_1. Furthermore, the solution set has to satisfy the constraint in (<ref>). Thus, we resort to the gradient projection algorithm to solve our optimization problem <cit.>. The steps for this algorithm can be found in Table <ref>.The needed derivative of the MSE with respect to P is given by∂MSE (P )/∂ P=-4/πσ_s[ H^T K_1-2/π H^T K_1^3 diag (H^* P^* )H^* P^*. .-2 c/πdiag (H^Tdiag (H^* P^* K_1^3 ) H^* ) P^*. . -2/π H^T K_1^3 diag (P^T H^T )H^* P^*. -. 2 c/πdiag (H^Tdiag (K_1^3 P^T H^T ) H^* ) P^*]. § SIMULATION RESULTSIn this section, we compare our proposed precoder with different precoding schemes in terms of the uncoded BER. All the simulation results are averaged over 200 channel realizations. The used modulation scheme is QPSK, where σ^2_s =2. N=20 antennas at the BS serve M=4 users with N_b=1000 transmit symbols per channel use. The tolerable error ϵ and the iteration step μ of the gradient projection algorithm are set to 10^-6 and 0.05, respectively. In Fig. <ref> the uncoded BER is simulated as function of the available transmit power E_tx. "WF, no Quant." refers to the linear Wiener filter precoder while no quantization is applied in the system model. "WF, D=I" is the linear Wiener filter precoder that does not take the quantization into account and equal power allocation is performed <cit.>. The transmit power constraint is still satisfied by appropriate scaling. "QP-GP" denotes our proposed precoder design: Quantized Precoder with Gradient Projection method. "QP-GP, D=I" refers to the proposed precoder design when the power allocation is equal for all transmit antenns. So, no additional analog processing D is required. "QWP" designates the Quantized Wiener filter Precoder introduced in <cit.>. It can be seen from the results that ignoring the distortions due to the 1-bit quantization in WF leads to the worst case scenario. When taking them into account in QP-GP, D=I a significant improvement in the uncoded BER can be achieved. This performance improvement can be further increased when unequal power allocation at the transmit antennas is deployed, as shown in the case of QP-GP and QWF. The proposed precoder design QP-GP outperforms the other designs. This iterative design converges to the same solution for different initial values.In general, the analog processing exhibits higher complexity and lower accuracy as compared to the digital counterpart due to hardware implementations and imperfections (aging, temperature,...). The analog precoder D offers less complexity due to its positive real-valued diagonal structure, and since it has to be updated only every coherence time. The BER performance sensitivity to inaccuracy in D implementation is studied and plotted in Fig. <ref>. It can be seen that even with 10% error in D, the BER performance does not degrade much as compared to the ideal case.The analog real-valued diagonal precoder D can be built within the power amplifiers at each antenna. Fig. <ref> shows the distribution of the normalized diagonal coefficients of D. We observe that the deviation of these coefficients among the different antennas and the channel realizations with respect to the mean value (at max 6dB) is quite small. Therefore, the requirements in terms of the dynamic range of the power amplifier are still reasonable. § CONCLUSION We present a new MMSE precoder design to mitigate the IUI in a DL massive MIMO scenario assuming perfect CSIT. The proposed precoder design takes into account the signal distortions due to the 1-bit quantization at the transmitter and at the receiver. The precoder is split into a digital precoder that separates the users in the direction and a real-valued diagonal analog precoder for the power allocation at each antenna. Our precodingmethod shows better performance in terms of the uncoded BER compared to the precoder designed in <cit.>. The analog precoder involved in the proposed scheme is updated every coherence time thus reducing the implementation complexity. Furthermore, the BER performance is insensitive to imperfections in the analog precoder implementation.IEEEbib | http://arxiv.org/abs/1706.08717v1 | {
"authors": [
"Ovais Bin Usman",
"Hela Jedda",
"Amine Mezghani",
"Josef A. Nossek"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170627082536",
"title": "MMSE precoder for massive MIMO using 1-bit quantization"
} |
§ INTRODUCTIONIn most of the cases the outcome of the process of data analysis is a set of objects in the form of statistical models, charts or tables. Three requirements are often superimposed to ensure sufficient quality of such results: they should be reproducible, verifiable and accessible. Reproducibility means that there is a process that reproduces results. Verifiability means that it is possible to check whether the newly generated results are identical to previously obtained results, and it is possible to check the context of object's creation. Accessibility means that results can be easily accessed for future computer based processing. Reproducibility gets increasing attention in the academic literature across various disciplines, see for example <cit.> for bioinformatics or <cit.> for the econometric research or <cit.> for more general discussion about differences between replicability and reproducibility. The R ecosystem of packages is equipped with wonderful tools such as knitr <cit.> or Sweave <cit.> which allow to create reproducible reports or articles. They follow the literate programming principle, and the R code, its results and its explanations appear together in a single document. It is assumed that the same input and identical instructions executed on the same operating system with the same local settings and with identical versions of installed libraries will result in the same output. Under these assumptions knitr or Sweave reports are sufficient to recreate the previously obtained results. But there are cases in which it is not convenient to recreate results from scratch, from raw input. Consider the following situations:* the input data is large or with limited/restricted access (e.g., for genomic data the raw input may easily hit few TB); * computations take a lot of time or require specialized hardware (e.g., calculations tuned for Graphics Processing Unit cards); * calculations are based on a very specific version of software or require commercial versions of software or some functions aredeprecated or removed over time. It can be an issue even for open software, e.g., due to rapid development of R, even widely used packages experience significant changes, like ggplot2 or lme4 in the year 2015;* results are generated and processed periodically and you wish to restore and compare models across all reports.In such situations it is desirable to retrieve the results that were calculated in the past rather than reproducing them from scratch. Objects that are backed up can be reused even if they cannot be reproduced or the reproducibility will be too complex or time consuming. Alternatively, it may be desired to check whether the reproduced results are the same as those obtained previously.An interesting example of such a feature are StatLinks <cit.> commonly used in reports prepared by OECD (Organization for Economic Co-operation and Development). In addition to scripts that generate results, most tables and plots that are presented in the reports are equipped with their own DOIs (digital object identifier) and web hooks. Through these links readers may download selected tables and plots, in the Excel format.The xls and xlsx formats are not ideal as they are proprietary and difficult to read in an automated way. But for extensive studies it is convenient and faster to access final results in such formats instead of having scripts that reproduce them.If the only result from the data analysis is a single plot, a model or a table, it is easy to save it in the rda format and make it accessible for the others. But increasing amounts of heterogeneous data results in growing complexity of the process of data analysis. The complexity comes either from data volume, data heterogeneity, numerous steps required for data preparation, results validation etc. Moreover, working with data is often a highly iterative process that generates large amount of partial or final results. For all the above reasons the management of versions of results becomes a task in itself. Neglecting this process results in Reproducibility Debt and may consequently lead to huge additional workload when it comes to recreation of results. The Reproducibility Debt is a part of wider category called Technical Debt <cit.>.It should be noted that the concept of recording and exploring relations between objects is not new. Potential applications in auditable data analyses werediscussed almost 30 years ago <cit.>. What we present in this article may be perceived as implementation of some of these concepts. It is now easier due to lower costs of data storage.The archivist package helps in managing, sharing, storing, linking and searching for R objects in a platform agnostic way. Its core functionalities allow for many interesting applications - some of them are presented in the Section <ref>. The archivist package automatically retrieves the object's meta-data and creates a rich structure that allows for easy management of stored R objects. The meta-data covers object's properties such as: name, creation date, class, versions of attached packages, structure and relations between R objects (as for example, that an object A was used for creation of an object B). All examples presented here are related to R objects. In the Section <ref> we discuss how this approach can be extended to other languages.The rest of the article has the following structure. In the Section <ref> (Motivation) we introduce key motivations and use-cases behind archivist. In the Section <ref> (Functionality) we present all functions available in the package and point out some further directions how this functionality can be integrated with GitHub, knitr, or be extended on other languages / formats. In the Section <ref> (Conclusions) we gather some final thoughts related to recordable and restorable research. § MOTIVATIONIn this section we present key concepts and some use-cases behind the archivist package. In the Section <ref> we present all functions available in the archivist in a more formal way. First let us introduce some terminology. * Artifact - an R object that is saved to the repository. Artifacts are identified by their MD5 hashes.* Repository - a collection of artifacts stored as binary files outside of the R session. Repositories are either local (with a read-write access) or remote (with a read access only). The API for repositories allow for following actions: add, delete, read or search for an artifact with selected Tags. In the current version of the archivist local repositories are folders in the file system while remote repositories are Git or Mercurial based repositories. The same mechanism can be used to access repositories pointed as URL addresses or folders attached to R packages.* MD5 hash - a unique identifier of an artifact. It's a 32-character-long string, result of cryptographical hash function MD5 (Message Digest algorithm 5). Here, we are using implementation of hash function available in thedigest package <cit.>. In the archivist package MD5 hashes are used as object's hooks. * Tag- an attribute of an artifact. Tags are represented as character strings; they usually have the following structure: key:value. An artifact may have many tags, even with the same key. Some tags are automatically derived from artifacts, others may be added manually. Tags may be referred as meta-data of artifacts as they describe either properties of artifacts (e.g., class, name, date of creation) or relations between artifacts (e.g., being a part of, being a result of). The archivist package manages R objects outside the R session. It stores binary copies of R objects in rda files and provides easy access for seeking and restoring these objects based on timestamps, classes or other properties.But, why anybody would like to store copies of R objects? Let's imagine the following use-cases:* A data scientist creates a report or an article and would like to provide an access to results presented in the article. Typically, these results are presented as plots, tables or models. Apart from including these results in the report or article in a human-readable form, it may be beneficial to be able to restore a given result in a machine-readable form for further processing. Having a possibility to retrieve an Rplot or table, one can perform some further transformation of it. The opportunity to retrieve a regression model enables additional residuals' validation or applying model to the new data. The archivist creates a hook to a copy of R object which restores the object in a remote R session. Such hooks are short one-line instructions and can be embedded in figures' or tables' captions. An example report that illustrates this use-case is available at . A part of it is presented in Figure <ref>. The report is created with the use of knitr package. It contains both R code and it's results in the form of tables and plots created with ggplot2 package <cit.>. In addition, there are also hooks to selected results. These hooks allow to restore a given plot or table directly in the local R session. Hooks of such a form restore a gg object in an R session.archivist::aread("pbiecek/Eseje/arepo/ba7f58fafe7373420e3ddce039558140")* A team of data scientists is working for some time on a forecasting model. During a certain period of time a large set of competing models is created. The team needs a tool that stores all models with additional metadata, such as model performance, information which data was used for model training and testing. The archivist creates a shared repository which can be used for storing models along with their meta-data and provides API for searching objects with specific meta-data. The example below reads all objects of the class lm, calculates a BIC score for them and sorts objects with respect to these scores. R> library("archivist") R> models <- asearch("pbiecek/graphGallery", patterns = "class:lm") R> modelsBIC <- sapply(models, BIC) R> sort(modelsBIC)990861c7c27812ee959f10e5f76fe2c3 2a6e492cb6982f230e48cf46023e2e4f39.05577 67.527350a82efeb8250a47718cea9d7f64e5ae7 378237103bb60c58600fe69bed6c7f11 189.73593189.735937f11e03539d48d35f7e7fe7780527ba7 c1b1ef7bcddefb181f79176015bc3931 189.73593189.735930e213ac68a45b6cd454d06b91f991bc7 e58d2f9d50b67ce4d397bf015ec1259c 243.49450243.4945018a98048f0584469483afb65294ce3ed 396.16690* Results are generated in a remote R process, like for example with a Shiny application. The archivist saves created R artifacts in an URL repository. See for example Figure <ref> that presents a screenshot from the Shiny application . All plots generated by this application are stored in an archivist repository and may be accessed with hooks presented below plots. Following line downloads a single plot directly to the local R session.archivist::aread("https://cogito.shinyapps.io/archivistShiny/arepo/ca680b829abd8f0a4bd2347dcf9fe534").§ FUNCTIONALITYThe key functionality of the archivist package is to manage copies of R objects, called artifacts, stored as binary files. Artifacts are stored in collections called repositories. Properties of artifacts and relations between artifacts are described by their tags.Typical lifetime of the repository is presented in Figure <ref>. The local repository is created with the createLocalRepo function. It can be set as a default repository so that calls of the other archivist functions can be simplified.Once the repository is created, new R objects can be archived with the saveToLocalRepo function or can be removed with the rmFromLocalRepo function. Artifacts can be restored from the repository with loadFromLocalRepo function. One can also get all objects that match given criteria with the function named searchInLocalRepo. Both functions have wrappers called aread and asearch, respectively, with the simplified and shorter interface. To summarise what kind of artifacts are in the repository one can use summaryLocalRepo or showLocalRepo functions. The repository can be removed with the deleteLocalRepo function.Table <ref> presents all functions available in the archivist package. These functions are divided into four core groups:* Functions for repository management. In this group there are functions used to create a new empty repository, to create a repository as a copy of an existing local or GitHub repository, to backup an entire repository into a single zip file, to present summary statistics of objects stored in the repository and to delete existing repository.* Functions for saving artifacts to a repository, loading artifacts from a repository and removing artifacts from a repository. Functions that show relations between artifacts, present artifacts' history or context in which they were created.* Functions for searching for artifacts within a repository. Artifacts may be accessed through date of creation, a tag or a list of tags.* Other features that do not fit previous categories. In sections <ref>-<ref> each group of these functions is presented separately. §.§ Repository managementA repository is a collection of artifacts and their meta-data. In this section you will find a list of functions for repository management (used to create a new empty repository, create a copy, present summary statistics or delete existing repository).Technically, repository is a directory with the following structure (see Figure <ref>). * A backpack.db file which contains an SQLite database. The database contains two tables with a structure presented in Figure <ref>. The table named artifact contains artifacts’ MD5 hashes and basic information about the artifacts. The table called tag contains artifacts’ tags. Since both artifacts and tags may be added into the database an unspecified number of times, each tag and artifact has one or more time points - one for each attempt to artifact's or tag's archiving to the repository.* A subdirectory called gallery with artifacts' storage. Artifacts are stored as separate files.Names of files start with MD5 hashes of corresponding artifacts. Extensions correspond to formats in which artifacts are saved. The current implementation for R stores artifacts in the rda format, but it can be easily extended to handle other formats. Additionally, also an artifact’s miniature is saved.For plots the default format for miniatures is raster file withpng extension, for other objects it is a text file with txt extension (e.g., for data frames it contains first few rows).A repository may be accessed in two ways. * Local - in this case repository is identified by its path in the local file system. The repository is in the read-write mode. If the file system is shared (shared file system on HPC cluster, a Dropbox directory, a mounted folder on Network File System, Secure Shell Filesystem, etc.) multiple users may read and write into the repository at the same time. *Remote - Currently archivist supports GitHub and Bitbucket repositories, but it can be easily extended to support any git or mercurial repository, see Section 4. Repository is identified by it's type (github/bitbucket), a username and the repository's name. The repository is accessible in read-only mode. Multiple users can read from such repository at the same time.In order to write to a remote repository one should either synchronize a local directory with GitHub/Bitbucket account or use a archivist.github package, which is archivist's first extension <cit.>. The logic behind this is as follows.Depending on the user’s needs it is possible to create a single repository per project or per group of projects or keep all artifacts ever created in a single repository. Since (i) a local repository is accessible even without an Internet connection, (ii) the access is faster and (iii) there is both read and write access, it is easier to work with local repositories, which are just a directory identified by its path.If the user wants to share a repository with artifacts with a general public then he or she can publish the local repository on GitHub or Bitbucket or make it available as a subdirectory of an R package. §.§.§ Creation of a new empty repository The createLocalRepo function creates a new local repository. The repoDir argument points to a directory that will be used as a repository root. The directory will be created if it does not exist. The default=TRUE argument marks the newly created repository as a default one.The directory may be specified either by global path or local path. The example below will create a repository named arepo in the current working directory.R> repo <- "arepo" R> createLocalRepo(repoDir = repo, default = TRUE) §.§.§ Deletion of an existing repository The deleteLocalRepo function deletes all artifacts, miniatures, the database with meta-data and the directory identified by the repoDir argument. R> repo <- "arepo" R> deleteLocalRepo(repoDir = repo) §.§.§ Copying artifacts from other repositories Functions copyLocalRepo and copyRemoteRepo copy selected artifacts from either local or remote (GitHub or Bitbucket) repository into a local repository. Artifacts to be copied are identified by their MD5 hashes. In the example below the artifact identified by hash 7f3453331910e3f321ef97d87adb5bad is copied along with its meta-data from remote GitHub repositorypbiecek/graphGallery to the local repository arepo. R> repo <- "arepo" R> createLocalRepo(repoDir = repo, default = TRUE) R> copyRemoteRepo(repoTo = repo,+md5hashes = "7f3453331910e3f321ef97d87adb5bad", +user = "pbiecek", repo = "graphGallery", repoType = "github") Functions zipLocalRepo and zipRemoteRepo download all artifacts and create a single zip archive.§.§.§ Showing repository's statistics A repository is a collection of artifacts and their meta-data. Functions summaryLocalRepo and summaryRemoteRepo summarize basic statistics about artifacts in the repository. Functions showLocalRepo and showRemoteRepo list all MD5 hashes and artifact’s meta-data. Functions show*Repo take argument method which may be either "tags" (the result is a data frame with artifact’s tags) or "md5hashes" (default, result is a data frame with artifact’s MD5 hashes). In the previous example we copied a single artifact from GitHub repository to the local one. The artifact is copied with its tags. In the example below we list all the tags within this single-artifact repository. R> showLocalRepo(repoDir = repo, method = "tags") artifacttag createdDate 1 7f3453331910e3f321ef97d87adb5b format:rda 2016-02-09 14:37:06 2 7f3453331910e3f321ef97d87adb5b class:gg 2016-02-09 14:37:06 3 7f3453331910e3f321ef97d87adb5b class:ggplot 2016-02-09 14:37:06 4 7f3453331910e3f321ef97d87adb5blabelx:Sepal.Length 2016-02-09 14:37:06 5 7f3453331910e3f321ef97d87adb5blabely:Petal.Length 2016-02-09 14:37:06 6 7f3453331910e3f321ef97d87adb5b date:2016-02-09 14:37:06 2016-02-09 14:37:06 7 7f3453331910e3f321ef97d87adb5b format:png 2016-02-09 14:37:06 8 ff575c261c949d073b2895b05d1097 relationWith:2166d...... 2015-06-22 17:17:14 8 ff575c261c949d073b2895b05d1097 sessionInfo:3b8c60...... 2015-06-22 17:17:14 In the example below the function summaryLocalRepo is used to list summaries of artifacts in the repository called graphGallery which is attached to the archivist package. One can find information about dates on which artifacts were added, classes of artifacts and the total number of artifacts in the repository. R> summaryLocalRepo(repoDir =+system.file("graphGallery", package = "archivist"))Number of archived artifacts in Repository:7Number of archived datasets in Repository:3Number of various classes archived in Repository:Number gg2 ggplot2 lm3 data.frame2 summary.lm1 Saves per day in Repository:Saves 2016-02-07 6 2016-02-0815 §.§.§ Setting a default repository In most of the cases we work with one repository per project. In such cases it is convenient to set a default local or remote repository. It can be done with setLocalRepo or setRemoteRepo functions. Look at the example below. R> setRemoteRepo(user = "pbiecek", repo = "graphGallery", repoType = "github") R> setLocalRepo(repoDir =+system.file("graphGallery", package = "archivist")) After setting a default repository, one can use the following functions * saveToLocalRepo,* loadFromLocalRepo, loadFromRemoteRepo, * rmFromLocalRepo, * searchInLocalRepo, searchInRemoteRepo, without specification of repoDir or user/repo/branch/subdir/repoType arguments. For example, the instruction below will add iris data frame to the default local repository. R> setLocalRepo(repoDir = repo) R> data("iris") R> saveToRepo(iris) Another option for setting a default value for an argument is the function aoptions(). It sets the default value for any argument that is used by archivist. For example the instruction below sets the default value for repoType to "github".R> aoptions("repoType", "github")§.§ Artifact managementAn artifact is an R object with its meta-data. Artifacts are stored in repositories. Key functions for artifact's management are functions for saving, loading and removing artifacts from a repository. §.§.§ Saving an R object into a repository The saveToLocalRepo function saves any R object into the selected repository. It stores in the repository both the object and its tags.Some tags and some meta-data are extracted in an automated way. The saveToLocalRepo function recognizes the class of the artifact and extracts tags typical for that class. It is possible to add support for a new class of objects or change list of tags extracted for selected classes, just extend the generic function extractTags(). Table <ref> lists classes that are recognized in the current version of the package and lists tags that are derived automatically from objects of a given class. For other classes the following attributes are extracted: name, creation time and MD5 hash.The saveToLocalRepo function takes at least two arguments: artifact - an R object which is about to be saved and repoDir which is a path to the local repository. The process of adding an R object to the repository triggers a chain of actions listed below. By setting some arguments of saveToLocalRepo to FALSE some of these actions may be skipped. * The name of the object is derived and stored as the object's tag name:xxx. It may be useful when searching for an object. One can search for all objects that had a specific name with asearch(pattern="name:iris"). * An MD5 hash is calculated for the object with the use of digest package. Then the object is saved as a binary file named md5hash.rda with the use of save function. * If there is any dependent object, it is saved separately to the repository (e.g., for object of class gg or lm the data slot is extracted from the object and saved separately. Additionally a tag relationWith:xxx is added, where xxx is the MD5 hash of the dataset).* The current session info, with the list of versions of attached packages, is saved to the repository.The session info is linked to the artifact. The link is a tag of the form sessionInfo:xxx, where xxx stands for MD5 hash of the object with session info. * A set of tags is extracted automatically and these tags are saved to the repository. See Table <ref> for the list of tags that are automatically derived. Tags extracted for a given class are defined by the generic extractTags function.* Additional tags specified by a user (with the userTags argument) are saved to the repository as well.* A miniature for the object is created – for plots it is a png file while for data frames or models it is a text description of the object. The following example creates a plot of the class gg and saves the object into the repository. Plots created with the use of ggplot2 package are objects and can be serialized in the same way as any other R objects <cit.>. A hash of the recorded object is returned. In the example below it is 11127cc6ce69a89d11d0e30865a33c13. By default, the related data object is also saved. In this case the dependent object is a dataset iris which is saved with the hash ff575c261c949d073b2895b05d1097c3.R> library("ggplot2") R> repo <- "arepo" R> pl <- qplot(Sepal.Length, Petal.Length, data = iris) R> saveToLocalRepo(pl, repoDir = repo) [1] "11127cc6ce69a89d11d0e30865a33c13" attr(,"data") [1] "ff575c261c949d073b2895b05d1097c3" The function saveToLocalRepo extracts additional tags such as the name of the original object (here: name:pl), its class (class:gg), labels on OX and OY axes (labelx:Sepal.Length) and MD5 hash of the data object. These tags are listed if we use showLocalRepo function on the repository. R> showLocalRepo(repoDir = repo, "tags")artifact tag createdDate 111127cc6ce69a89d11d0e30865a33cformat:rda 2016-02-09 16:42:59 211127cc6ce69a89d11d0e30865a33c name:pl 2016-02-09 16:42:59 311127cc6ce69a89d11d0e30865a33cclass:gg 2016-02-09 16:42:59 411127cc6ce69a89d11d0e30865a33cclass:ggplot 2016-02-09 16:42:59 511127cc6ce69a89d11d0e30865a33c labelx:Sepal.Length 2016-02-09 16:42:59 611127cc6ce69a89d11d0e30865a33c labely:Petal.Length 2016-02-09 16:42:59 711127cc6ce69a89d11d0e30865a33cdate:20160209 16:42:59 2016-02-09 16:42:59 811127cc6ce69a89d11d0e30865a33c session_info:e0373..... 2016-02-09 16:42:59 911127cc6ce69a89d11d0e30865a33cformat:png 2016-02-09 16:42:59 10 e037375a5f757efcc28561c0a1a2efformat:rda 2016-02-09 16:42:59 11 ff575c261c949d073b2895b05d1097format:rda 2016-02-09 16:42:59 12 ff575c261c949d073b2895b05d1097 session_info:e0373..... 2016-02-09 16:42:59 13 ff575c261c949d073b2895b05d1097format:txt 2016-02-09 16:42:59 14 ff575c261c949d073b2895b05d1097 relationWith:11127..... 2016-02-09 16:42:59By default, for each artifact also it's context, i.e., session info, is saved. It can be accessed with the function asession(). See the example below. Such additional information may be very useful if we cannot replicate previous results and we are in the need of recovering the exact versions of important packages, which can be done with restoreLibs function. R> asession("11127cc6ce69a89d11d0e30865a33c13") Session info ——————————————————settingvalue versionR version 3.2.2 (2015-08-14)system x86_64, darwin13.4.0ui RStudio (0.99.441) Packages ———————————————————-package* versiondate source acepack1.3-3.32013-05-03 CRAN (R 3.1.0) archivist* 1.9.7.32016-02-09 CRAN (R 3.2.2) ggplot2* 2.0.02015-12-16 Github (hadley/ggplot2@11679cd)gridExtra* 2.0.02015-07-14 CRAN (R 3.2.0) ...§.§.§ Serialization of an object creation event into repository The archivist provides a new operator %a% that works as the extended pipe operator %>% from the magrittr package <cit.>. In addition, it saves the resulting object to the default archivist repository together with the function call and its parameters. The default repository should be set first, see the setLocalRepo function for instructions how to do this. With this functionality it is possible to trace function calls and extract pedigree for some artifacts.R> library("archivist") R> createLocalRepo("arepo", default = TRUE) R> library("dplyr") R> iris+dpyr::filter(Sepal.Length < 6)+lm(Petal.Length Species, data=.)+summary() -> tmp How to recreate an object's history? The function ahistory extracts the chain of calls that leads to the selected object. As an argument one can specify either an object's value or its MD5 hash. The value of ahistory function is a data.frame with two columns – first contains function calls while second contains MD5 hashes of partial results.In the example above, a chain of three operations converts input iris data into the tmp object. The dplyr package <cit.> has to be loaded first since the function filter is used in this example. Following lines present the chain of consecutive transformations that are recorded in the repository. R> ahistory(tmp) R> ahistory(md5hash = "050e41ec3bc40b3004bc6bdd356acae7")iris[ff575c261c949d073b2895b05d1097c3] -> filter(Sepal.Length < 6)[d3696e13d15223c7d0bbccb33cc20a11] -> lm(Petal.Length Species, data = .)[990861c7c27812ee959f10e5f76fe2c3] -> summary() [050e41ec3bc40b3004bc6bdd356acae7] In order to restore an object's pedigree all partial results must be saved in a repository. So this option will work only for objects created by a chain of calls that use the %a% operator. §.§.§ Loading an object from a repository To read an object from repository we may consider the following four scenarios. * We know the object's MD5 hash and the object is in a local directory.* We know the object's MD5 hash and the object is in a remote repository, i.e., on GitHub or BitBucket.* We do not know the hash but we know some properties of the object so we need to find it first by its tags. The object is in a local repository.* As above, but the object is in a remote repository. If we know the MD5 hash of the requested artifact, we can directly load the object from the repository and in this section we are going to show how this can be done. If we do not know the MD5 hash, then we need to use one of search* functions presented in Section <ref>.Functions loadFromLocalRepo and loadFromRemoteRepo read artifacts from either local or remote repositories. The local repository is defined by a path to it's root; remote repository is defined by it's type (currently "github" (default) or "bitbucket"), the username, repository's name and a subdirectory within the repository. In both functions the argument value specifies whether the function should return the object by value (value=TRUE) or it should load the object into the namespace with its original name (value=FALSE).For the purpose of this example we have created a repository graphGallery, with two objects: a plot and a regression model.The repository is available both on GitHub (see https://github.com/pbiecek/graphGallery) and within the archivist package (see the graphGallery directory). Two archived objects have 7f3453331910e3f321ef97d87adb5bad and 2a6e492cb6982f230e48cf46023e2e4f hashes respectively.The full MD5 hash of an artifact is a 32-characters-long string but it is enough to set only the first few characters. In the example below it is enough to use "7f34533" prefix to load an artifact with the "7f3453331910e3f321ef97d87adb5bad" hash. There is only one artifact with prefix "7f34533" in its MD5 hash. If there is more, all that match the prefix are returned. Note that one should not use this feature unless is sure that new objects with colliding hashes will not be added. For small repositories conflicts are unlikely even for first five characters, but be careful when using this feature.Both following instructions retrieve an R object from GitHub, load it intoR session and make it accessible for further processing. In this case it is a ggplot2 object so after being loaded the print function is triggered and a plot is generated(see Figure <ref>). Note that by default the GitHub is assumed, but this may be changed with the parameter repoType. R> loadFromRemoteRepo("7f3453331910e3f321ef97d87adb5bad", +repo = "graphGallery", user = "pbiecek", value = TRUE) R> loadFromLocalRepo("7f34533",+system.file("graphGallery", package = "archivist"), value = TRUE)The aread function is a wrapper over loadFromRemoteRepo with more compact form. Shorter instructions and shorter code snippets might be placed in a figure or table caption. The single line below reads an object with the 7f34533... hash from graphGallery GitHub repositorythat is owned by the pbiecek user. R> archivist::aread("pbiecek/graphGallery/7f3453331910e3f321ef97d87adb5bad")The following instructions retrieve the same R object but this time from the graphGallery repository attached to the archivist package. Note that the default repository is set first with the setLocalRepo function. R> library("archivist") R> setLocalRepo(system.file("graphGallery", package = "archivist")) R> aread("7f3453331910e3f321ef97d87adb5bad")The use of MD5 hashes as objects identifiers has some advantages.In some use cases we may be restricted to use only models approved by some authority. For example, due to some hypothetical regulatory issues in production it might be advisable to use only a specific version of a model (such as credit scoring model or some forecasting model).In the archivist package all objects have their cryptographical hash calculated with the MD5 algorithm. One can use the digest function to validate the object's MD5 hash at any moment. One can also call an object the from repository by its MD5 hash. Having a list of MD5 hashes of allowed objects one can validate their identity.In the example below the downloaded regression model is digested to confirm its identity. R> setLocalRepo(system.file("graphGallery", package = "archivist")) R> model <- aread("2a6e492cb6982f230e48cf46023e2e4f") R> digest::digest(model) "2a6e492cb6982f230e48cf46023e2e4f" §.§.§ Removal of an object from a repository To remove an artifact from a repository one can use the rmFromLocalRepo function.In the example below the artifact 92ada1e052d4d963e5787bfc9c4b506c and all its tags are removed from the repository called repo. R> rmFromLocalRepo("7f3453331910e3f321ef97d87adb5bad", repoDir = repo) A list of artifact's hashes that should be removed may be obtained with the search* function. The example below searches for all artifacts older than 30 days and removes them from the repo repository. R> obj2rm <- searchInLocalRepo(list(dateFrom = "2010-01-01",+dateTo = Sys.Date()-30), repoDir = repo) R> rmFromLocalRepo(obj2rm, repoDir = repo, many = TRUE) It is also possible to remove many artifacts with one call. Broader examples of this function are explained in the package manual page accessed from R with ?rmFromLocalRepo. §.§ Search for an artifact and explore the repositoryOne of the advantages of the archivist package is the automated derivation of artifact’s tags and meta-data. It is useful when one wants to find previously calculated results in a large collection of R objects. Relations between artifacts are useful when we want to process the structure dependencies between artifacts. Below we present a list of functions for searching for artifacts on the basis of their properties.§.§.§ Search in a local or remote repository If we do not know the MD5 hashes of artifacts that are of our interest, we can find them with the use of search* functions.Searching within a local repository and a remote repository is very similar. Functions searchInLocalRepo or searchInRemoteRepo differ only in the way in which the repository is specified.In both functions the pattern argument may be either a tag (name, class, varname or other) or a date period in which given artifact was created. Hashes of all artifacts that meet all criteria (i.e., were created within a given time interval or have a given tag attached) are returned.For example, the following command retrieves MD5 hashes of all objects of the class gg from the pbiecek/graphGallery repository. R> searchInLocalRepo(pattern = "class:gg",+repoDir = system.file("graphGallery", package = "archivist")) [1] "7f3453331910e3f321ef97d87adb5bad" "369227e67f9164dcbe934dadf2b53cc2" To get a list of artifacts created within a given date range one can use following instruction. R> searchInLocalRepo(pattern = list(dateFrom = "2016-01-01", +dateTo = "2016-02-07"),+repoDir = system.file("graphGallery", package = "archivist")) [1] "d9313a0de3e2980201a8971e3384ff26" "ff575c261c949d073b2895b05d1097c3" [3] "2a6e492cb6982f230e48cf46023e2e4f" "93ecfdf1436932e2860c6dbdf2abc2ad" [5] "afb2550d0f886f0cf3b050f04c5cd4f8" The searchInLocalRepo and searchInRemoteRepo functions allow to use more than one searching criteria. Additional argument intersect specifies if the resulting objects have to met all or any of the search criteria. R> searchInLocalRepo(pattern=c("class:gg", "labelx:Sepal.Length"), +repoDir = system.file("graphGallery", package = "archivist")) [1] "369227e67f9164dcbe934dadf2b53cc2" "7f3453331910e3f321ef97d87adb5bad" These two functions return MD5 hashes of artifacts. In order to load these artifacts from repository one needs to use either loadFrom*Repo or aread functions. Since both operations are usually performed together (search for MD5 hashes of artifacts by their tag / load artifacts with given MD5 hashes), one can use the asearch function which retrieves MD5 hashes and returns a list with values of artifacts that meet all selected criteria. §.§.§ Retrieval of a list of R objects with given tags When working in a team or for a longer period of time, one produces a lot of partial results and it becomes harder and harder to trace what kind of analyses were conducted in the past and where are the results. The archivist extracts meta-data from R objects in the very same moment they are archived in a repository. For many researchers objects are so valuable, due to their pedigree and meta-data, that they can be regarded as artifacts. Having such additional meta-data it is easier to search for previously generated partial results, e.g., by specifying what kind of model with which variables we are looking for.For example, the code below retrieves all objects of the lm class with the Sepal.Length variable from within a list of dependent variables. In this repository only two artifacts (here lm models) match both conditions.The following instruction searches within the default local repository. R> setLocalRepo(system.file("graphGallery", package = "archivist")) R> models <- asearch(patterns = c("class:lm", "coefname:Sepal.Length")) Below is the code that searches within the GitHub repository. R> models <- asearch("pbiecek/graphGallery", +patterns = c("class:lm", "coefname:Sepal.Length")) R> lapply(models, coef) `18a98048f0584469483afb65294ce3ed`(Intercept) Sepal.Length -7.101443 1.858433`2a6e492cb6982f230e48cf46023e2e4f` (Intercept)Sepal.Length SpeciesversicolorSpeciesvirginica -1.7023422 0.6321099 2.2101378 3.0900021The following instruction retrieves all artifacts of the gg class (created with the package ggplot2) with label Sepal.Length on the X axis. Two objects are returned as a result. They are plotted together by the grid.arrange function from gridExtra package <cit.>. R> plots <- asearch(patterns = c("class:gg", "labelx:Sepal.Length")) R> length(plots) [1] 2 R> library("gridExtra") R> do.call(grid.arrange, plots) Result of these instructions is presented in Figure <ref>. §.§.§ Interactive search in a local repositoryFor local repositories, it is also possible to explore the repository interactively with the shinySearchInLocalRepo function. This function launches a Shiny application <cit.> which is dynamically created and whichallows for interactive specification of tags and sorting criteria. See Figure <ref> with an example screenshot of this application.In the text box area one can specify tags that filter out objects presented on the right panel. Only miniatures of objects that meet all these criteria are presented. Additionally, the instruction sort:key sorts the artifacts along the key. For example, use "sort:createdDate" to sort miniatures along the date of creation of the object.R> arepo <- system.file("graphGallery", package = "archivist") R> shinySearchInLocalRepo(arepo)§.§ ExtensionsThe archivist package is designed as a multi-purpose manager of objects. In this section we present some specific extensions.§.§.§ Archiving all results of a specific function The trace() function from the base package allows to insert a specific instruction to the body of a selected function. It can be used for example to call saveToLocalRepo() function at the end of a selected function.In the example below we modify the lm() function so that after it's each execution the created lm model is automatically added to the default local repository allModels. R> library("archivist") R> createLocalRepo("allModels", default = TRUE) R> atrace("lm", "z") Tracing function "lm" in package "stats" R> lm(Sepal.Length Sepal.Width, data=iris) Tracing lm(Sepal.Length Sepal.Width, data = iris) on exit Call: lm(formula = Sepal.Length Sepal.Width, data = iris)Coefficients: (Intercept)Sepal.Width6.5262-0.2234R> sapply(asearch("class:lm"), BIC) 42fcf77af2c40f70c445cbba513aeabd381.0236§.§.§ Integration with the knitr package The knitr package is a tool that transforms a mixture of R code and descriptions in natural language into a md, html or pdf report. Moreover the produced report contains results generated by the included R code. On one hand reader knows that presented results are generated by presented code. On the second hand the author does not waste time on coping the results, since they are automatically included in the output. Results included in a report are usually plots or tables. In such form they cannot be loaded from the pdf/html file directly to R. The archivist package records objects and makes them easier to access through local, GitHub or BitBucket repositories.The function addHooksToPrint combines these two tools. A call to this function should be included on the beginning of a knitr report. It creates a new generic print functions for classes specified by the class argument. These functions save objects to the repository and add corresponding hooks to the report after every attempt to print the object. Hooks are short instructions on how the recorded objects can be accessed. An example is presented in the report <http://bit.ly/1nW9Cvz>. Part of this report is presented in Figure <ref>. On the beginning there is a snippet presented below. It automatically adds hooks to the html report for all objects of classes ggplot or data.frame.R> addHooksToPrint(class=c("ggplot", "data.frame"), +repoDir = "arepo",+repo = "Eseje", user = "pbiecek", subdir = "arepo") As a result, just before each plot, there are automatically created hooks to corresponding objects e.g., archivist::aread("pbiecek/Eseje/arepo/24ea7c04b861083d4bf56eee1c5a17b7"). These hooks serve also as links to the corresponding R objects.The biggest advantage of this integration is that a single call to addHooksToPrint is needed to enrich the knitr report in archivist hooks for all interesting objects.§.§.§ Gallery of artifacts in the repositoryInformation about artifacts is stored in an SQLite database in the backpack.db file. The createMDGallery function creates a single markdown file with gallery of all artifacts in the repository. Such gallery, if saved as file named readme.md, will automatically list all artifacts with miniatures and tags in the GitHub web portal user interface. See an example gallery at <http://bit.ly/1Q62Tpz>. This gallery was created with the following instruction. A part of the result is presented in Figure <ref>. R> createMDGallery("arepo/readme.md",+ repo = "Eseje", user = "pbiecek", subdir = "arepo",+ addMiniature = TRUE, addTags = TRUE) §.§.§ Support for other repositories, other languages and other formatsThe current implementation of archivist supports local, GitHub and BitBucket repositories. The package is implemented in R and saves artifacts in the rda format. In order to support other repositories one can extend the function getRemoteHook. It is used internally by other archivist functions to generate URL addresses to files in remote repositories. In order to support other repositories it's enough to extend this function.All metadata related to artifacts is sorted in an SQLite database in backpack.db file. This database can be accessed from other languages. Objects are stored as files and can be added in different formats. Each artifact has an additional tag format:xxx that specifies in which format the artifact is saved, one artifact can be saved in more than one format. Currently artifacts are stored as rda files. In order to save objects in other formats, like json or csv, it is enough to extend the saveToLocalRepo function. In order to load objects from other formats it is enough to overload loadFromLocalRepo and loadFromRemoteRepo functions. §.§.§ Restoring older versions of packagesIn some cases, in order to use an artifact it is not enough to restore it. A good example of this problem are objects of the gg class created with ggplot2 package. The structure of gg objects is different in package ggplot2 in the version 1.0, different in the version 2.0 and different in the version 2.1. It means that even if we have restored an object that was created with package in version 2.0 we will not be able to use the plot function for this object if one uses ggplot2 package in the version 2.1 nor 1.0.To use the object we need to downgrade ggplot2 package to the version 2.0. This is possible with the restoreLibs function. For a given hash of an artifact the restoreLibs function restores it's session_info and reinstalls required packages with versions attached during the artifact's archiving. Packages can be reinstalled in the new directory, not to affect the default R libraries.For example, the 600bda83cb840947976bd1ce3a11879d object was created with ggplot2 version 2.0. The asession() function checks versions of packages that were then attached. R> asession("pbiecek/graphGallery/arepo/600bda83cb840947976bd1ce3a11879d") ...Formula1.2-12015-04-07 CRAN (R 3.1.3) ggplot22.0.02015-12-16 Github (hadley/ggplot2@11679cd)gridExtra* 2.0.02015-07-14 CRAN (R 3.2.0) ... Here the ggplot2 was in the version 2.0 and was installed from GitHub. The restoreLibs() function reinstalls all libraries from proper repositories (here GitHub) to proper versions (here commit 11679cd). R> restoreLibs("pbiecek/graphGallery/arepo/600bda83cb840947976bd1ce3a11879d") After that one can load and plot the ggplot object since the structure of gg object is compatible with installed libraries. R> aread("pbiecek/graphGallery/arepo/600bda83cb840947976bd1ce3a11879d") § CONCLUSIONSThe goal of a data analysis is not only to answer a research question based on data but also to collect findings that support that answer. These findings usually take the form of a table, plot or regression/classification model and are usually presented in articles or reports. Such objects are mostly well presented graphically, but they are hard to recreate back in a computer.In this paper we have presented the R package called archivist, which implements the logic of recordable research. The archivist stores R objects in repositories. The data scientist may share obtained results with other users, create hooks to models and then embed these hooks in articles, reports or web applications. One may also search within a repository and look for artifacts with given properties or relations with other artifacts. One may also validate the object’s identity or derive its pedigree.Repositories may be shared among team members or between different computers or systems. Statistical models or plots may be stored in a single repository which simplifies the object management.In this article we have also presented some use-cases for the archivist package, such as: hooks for R objects that can be embedded in reports or articles, interactive searching within repository or retrieving object’s pedigree.§ ACKNOWLEDGMENTS Thanks go to Ross Ihaka, Łukasz Bartnik, Cezary Chudzian and two anonymous reviewers for valuable discussions and comments on the idea of recordable research and early versions of this paper. We would like to thank Witold Chodor for his great contributions to the development of this package. The package archivist was initiated as an open project in the company iQor Polska sp. z o.o.. | http://arxiv.org/abs/1706.08822v1 | {
"authors": [
"Przemyslaw Biecek",
"Marcin Kosinski"
],
"categories": [
"stat.CO",
"stat.ME"
],
"primary_category": "stat.CO",
"published": "20170627124439",
"title": "archivist: An R Package for Managing, Recording and Restoring Data Analysis Results"
} |
^1UK Astronomy Technology Centre, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK^2Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK^3East Asian Observatory, 660 N. A`ohōkū Place, University Park, Hilo, HI 96720, USA Observational astronomyIan [email protected] The James Clerk Maxwell Telescope (JCMT) has been the world's most successful single dish telescope at submillimetre wavelengths since it began operationsin 1987. From the pioneering days of single-element photometers and mixers, through the first modest imaging arrays, leading to the state-of-the-artwide-field camera SCUBA-2 and the spectrometer array HARP, the JCMT has been associated with a number of major scientific discoveries. Famous for thediscovery of “SCUBA” galaxies, which are responsible for a large fraction of the far-infrared background, to the first images of huge discs of cooldebris around nearby stars, possibly giving us clues to the evolution of planetary systems, the JCMT has pushed the sensitivity limits more than any otherfacility in this most difficult of wavebands in which to observe. Now approaching the 30th anniversary of the first observations the telescope continues tocarry out unique and innovative science. In this review article we look back on just some of the scientific highlights from the past 30 years. Celebrating 30 Years of Science from the James Clerk Maxwell TelescopeIan Robson^1,2, Wayne S. Holland^1,2 and Per Friberg^3 December 30, 2023 ====================================================================== § INTRODUCTION AND OVERVIEWThe James Clerk Maxwell Telescope (JCMT) has been the world's premier single-dish submillimetre telescope since its opening in 1987. At 13,425 ft abovesea-level on Mauna Kea in Hawaii it has the benefit of being above much of the water vapour that restricts ground-based, submillimetre astronomy to a fewnarrow “windows” through which observations are possible. It was a purpose-built facility with a 15 m diameter, high-surface-accuracy primary mirrorthat feeds the incoming radiation to a receiver cabin at the Cassegrain focus behind the main dish and also to two Nasmyth platforms at each side of theelevation bearings.1mmThe scientific output of any astronomical facility is a combination of many factors: the intrinsic efficiency of the design, location and weather,instrumentation capability and reliability, operability and the effectiveness of the data-reduction software. Throughout the life of the JCMT, theobservatory management strove to provide the world's best instrumentation, improve the reliability of the facility and to maximise the scientific outputfor the funding agencies and astronomical community through science-ranked, weather-dependant, queue-based flexible scheduling of observations. Indeed, thethree funding agencies of the UK (55%), Canada (25%) and the Netherlands (20%) deserve credit for their continued support of the developments of thefacility and instrumentation even in times of severe financial pressure on home budgets.1mmThis article discusses the scientific output of the JCMT, doing so through the eyes of the instrumentation suite as this provides one of the most coherentways of seeing how technology has driven the scientific capability and new discoveries emanating from the facility. We have subdivided this treatment intofour sections: the three continuum instrument epochs of UKT14, SCUBA, SCUBA-2, and the heterodyne spectrometers, culminating in HARP. Within thesesections the science output is naturally grouped into astronomical themes. Whilst it has not been possible to include all the science from the 30 years ofoperations, the science highlights readily stand-out.1mm1mmIn terms of a timeline, the story can be broken down into some very clear regimes. In the continuum, we begin with the single-pixel, common-userbolometer, UKT14 <cit.> which reigned supreme in the world from 1988 until the arrival of SCUBA in 1997. SCUBA, the Submillimetre Common-UserBolometer Array <cit.>, was the world's first submillimetre “camera” with 128 pixels in two arrays, one operating at a primary wavelengthof 850 m with 37 pixels and one at 450 m with 91 pixels. It is not an overstatement to say that SCUBA brought about the “submillimetrerevolution” in astronomy and the science highlights from this instrument are described in Section 3. The outstanding success of SCUBA immediately showedthe need for a next-generation camera, one with many more pixels to provide a larger field-of-view and with improved sensitivity. This resulted inSCUBA-2 <cit.>, a revolutionary instrument that was fraught with technological challenges. Nevertheless, it became operational on thetelescope in 2011, and with over 5000 pixels at each of the main SCUBA wavelengths it heralded the onset of large-scale mapping of the submillimetre sky.SCUBA-2 continues in operation to this day and Section 4 describes the major inroads of science this instrument has brought, moving from the snapshots ofdiscrete objects or mapping tiny areas of sky, to large-scale imaging, resulting in statistically significant samples of objects and addressing evolutionacross many scale-sizes and cosmological timescales.1mmIn terms of heterodyne spectroscopy the JCMT had a slower start and went through a number of iterations of single- and dual-pixel instruments operatingin most of the submillimetre and near-millimetre atmospheric windows. The most successful of these was the 350 GHz (850 m) receiver RxB3<cit.>, a dual-channel instrument receiving orthogonal linear polarisations from the same position on the sky, and which operated onthe telescope between 1997 and 2006. The Digital Autocorrelating Spectrometer (DAS) <cit.> was used in conjunction with many of the earlyheterodyne mixers. The arrival of the 350 GHz 16-element Heterodyne ARray Program (HARP) receiver <cit.> heralded the ability to carry outhigh resolution spectroscopy over large areas of sky. This instrument was introduced in 2007 and came with a new digital spectral correlator, the AutoCorrelation Spectrometer Imaging System (ACSIS) <cit.>. The heterodyne suite of instruments has proven to be very successful over the yearsand some of the major science results are described in Section 5.1mmThe eventual move to flexible scheduling of observations, based on the science priority of the proposal and the weather at the telescope, led to anenormous increase in productivity and in subsequent scientific impact of the facility <cit.>. Although flexible scheduling was somewhatresisted by the users at the start, the eventual implementation meant that no longer were top-ranked proposals at the risk of being blighted by poorweather that happened to coincide with the fixed schedule of their observing run, but they would be undertaken throughout the semester when the weatherwas best suited to the scientific requirements. The JCMT was one of the first observatories to bring about this innovation and was a major operationaladvantage for SCUBA and the instruments that followed. Finally, the importance of an easily-accessible and user-friendly archive was duly recognised,particularly when the volume of data being generated increased significantly (e.g. with the introduction of SCUBA). Ths led to the creation of the JCMTScience Archive (JSA) <cit.>, hosted at the Canadian Data Archive Centre <cit.>. The JSA is designed to increase the productivityof the telescope by making not only the raw data, but also science-quality reduced images available to the JCMT and wider astronomy communities. Thiswill allow the astronomers of the future to interrogate the data to explore, for example, time-dependent phenomena over the lifetime of the telescope.§ SCIENTIFIC RESULTS FROM UKT14 Although only a single-pixel device, UKT14 came with many improvements over its predecessors. It was constructed by the Royal Observatory Edinburgh andfrom the start it was designed as a “common-user” instrument and crucially came with a user-friendly data reduction software suite. It was originallydesigned for and operated on the United Kingdom Infrared Telescope (UKIRT) but when moved to the JCMT brought more than a 100 times increase in sensitivityover UKIRT along with an increase of a factor of 4 in angular resolution, with beam sizes (full-width at half-maximum) of 14 arcseconds at850 m and 6 – 7 arcseconds at 350/450 m. The instrument had very carefully designed optics to minimise stray radiation and a range offilters to select the atmospheric “windows” allowing photometric observations to be made from 2 mm to 350 m. This was very important for studiesof spectral energy distributions, albeit the most often used filters tended to be in the most stable windows in the submillimetre at 800 and 450 m.It was a very sensitive photometer, the composite germanium bolometer being cooled to 0.35 K by liquid He^3, and capable of detecting point-sources downto a level of ∼8 mJy at 800 m in one hour of observing time. UKT14 turned out to be critically important: it was state-of-the-art in theearly days of the JCMT; it opened up a range of new science ventures for submillimetre study; but perhaps most importantly, it introduced a whole newgeneration of astronomers to the field, many of whom were not necessarily submillimetre astronomy experts. Indeed, these were the same astronomers whowould go on to make revolutionary discoveries with the introduction of SCUBA.1mmThe science output from UKT14 was indeed huge, both in extent and depth. According to the JCMT Annual Reports, over 180 papers in refereed journalscontained data from UKT14 before it was superseded by SCUBA in 1996. This is a staggering 44% of all the JCMT refereed papers over the same period,showing the dominance of UKT14 and continuum science in the first decade of the JCMT. The science topics ranged from observations of comets in the SolarSystem to high-redshift galaxies, mostly detecting the thermal emission from cold dust grains. There was a long-standing and very successfulprogram of monitoring the flaring emission from blazars, however, which originated from non-thermal emission from relativistic electrons. Because ground-basedsubmillimetre astronomy was still in its infancy, UKT14 was also used extensively for assessing observing techniques, studying the atmospheric extinctionand identifying calibration sources, all of which would provide the sound basis for visiting astronomers to build upon <cit.>. In compilingexamples of UKT14 science, we have tried to show the breadth of the different astrophysical topics opened up by this remarkably versatile instrument, andso present a wide but relatively shallow selection of topics, albeit selected mainly through citation indexes. §.§ Solar System studies One of the early observations with UKT14 was the first detection of a comet at submillimetre wavelengths in 1989 by Jewitt & Luu <cit.>.They found that the emission of comet P/Brorsen-Metcalf could be modelled by a population of transient, large grains with a total mass of∼10^9 kg, which could have been produced by some form of breakdown of part of the refractory mantle of the comet. Later observations of the cometHyakutake in 1996 by Jewitt & Matthews <cit.> found that from 1.1 mm to 350 m the emission can be described as thermalemission from large (∼1 mm) dust grains in the coma and a resulting total mass of around 2 × 10^9 kg. The spectral index indicates thatthe opacity factor is similar to that found in the circumstellar discs of young stars. Remarkably, a small map was made at 800 m, which showedthat the emission was consistent with the steady emission of solid particles from the cometary nucleus on timescales less than 1 day. A major study byRedman and co-workers of the asteroid 4-Vesta in 1989 <cit.> showed that the submillimetre emission might originate from a form of dusty,porous regolith. Furthermore, unlike the single-peaked rotational light-curve in the optical, the millimetre light-curve was seen to be double-peaked,indicating that it was most likely dominated by the triaxial shape of the asteroid. A major investigation was undertaken by Griffin & Orton<cit.> who measured the emission from Uranus and Neptune from 2 mm to 350 m. These precise data allowed the brightnesstemperatures of the planetary atmospheres to be calculated with greater accuracy (with uncertainties of <2 K) based on data from Mars, the primarycalibration source at submillimetre wavelengths. These new values enabled Uranus to become a valuable calibrator for submillimetre observations, both forground- and space-based facilities. Addressing both calibration purposes and intrinsic properties of asteroids, Müller and Lagerros <cit.>used JSA data on 1-Ceres, 2-Pallas, 4-Vesta, 532-Herculina, 10-Hygiea, 106-Dione and 313-Chaldaea to determine thermal models for the asteroids as wellas defining new far-infrared (far-IR) photometric standards to be used by the ISOPHOT instrument on the ISO satellite. This demonstrated the valueand accessibility of the JCMT science archive. The work also built on a programme of observations that produced a major publication by Redman, Feldman &Matthews <cit.> in which the spectral energy distribution was obtained for seven asteroids, five of which were non-metallic and two weremetallic. The data showed that there was a notable range of physical properties of the surfaces, even for the non-metallic bodies.1mmTo conclude this section Stern, Weintraub & Festou <cit.> succeeded where many had previously failed and detected Pluto at 1.3 mm and800 m, deducing a surface temperature of 30 – 44 K with a most probable range of 35 – 37 K. This range is significantly lower than had beenpredicted from radiative equilibrium models and from other observations and showed that the methane features in Pluto's spectrum were from solid, ratherthan gas-phase, absorptions, demonstrating that Pluto's atmosphere is dominated by nitrogen or carbon monoxide rather than methane. §.§ Star formation The very early stages of star formation, detecting the emission from cold dust, turned out to be one of the key areas for UKT14 study and was one of themost popular series of targets resulting in many publications over the period. One of the most spectacular set of observations and indeed, the most citedUKT14 result, came from maps by André, Ward-Thompson & Barsony <cit.> of the ρ Ophiuchus molecular cloud in which the protostellarsource VLA 1623 was proposed as a new category, “Class 0”, as the earliest phase in the star formation sequence. The observations of the core of thecloud at 800 m and 450 m took 18 hours of integration over three nights of excellent and stable weather. The results are shown inFig. <ref>. The emission shows four compact clumps with masses ∼1 solar masses (M_⊙) embedded in a ridge of about15 M_⊙. VLA 1623 is the coldest clump with a temperature estimated to be 15 – 20 K, appearing to have no central heat source and not detected bythe IRAS satellite. The mass was estimated to be 0.6 M_⊙ with a luminosity of around the same as the Sun. These observations showed thepotential of the submillimetre for uniquely being able to study the earliest phases of star formation and this paper was a landmark in the field.1mm 1mmThe observations of ρ Oph were rapidly followed up with the next most cited UKT14 paper, in which Ward-Thompson and co-workers <cit.>made observations of 21 cold molecular cores in dark clouds with no infrared source; the so-called “starless cores”. These clumps have insufficientbolometric luminosity to be typical of a “Class 1” protostar and a crucial discovery was that these cores differed from those that had an IRASfar-IR detection in that they are all more diffuse and less centrally peaked. The clumps had densities 10^5 to 10^6 cm^-3 but the densityprofile was inconsistent with the r^-2 or r^-3/2 profiles predicted by standard theory and instead were more consistent with magnetic support. Theauthors concluded that these submillimetre bright, dark cores are indeed pre-protostellar and are in the very earliest stages prior to protostellarcollapse.1mmSaraceno et al. <cit.> conducted a series of observations of a sample of 45 Class I and Class 0 young stellar objects (YSOs) at 1.3 mmusing UKT14 and the Swedish-ESO Submillimetre Telescope (SEST) telescope and made a number of important conclusions regarding their evolutionarysequencing. These included that the evolution of a protostar was mainly controlled by the mass of both the central object and circumstellar material andthat the Class 0 sources were indeed the earliest stages of star formation yet observed. Also, the Class I sources showing outflow had dynamicaltimescales exceeding ten thousand years and that they were probably in the deuterium burning phase where they spend most of their lifetime accretingmaterial. On the other hand, the Class I sources with no outflow behave like Class II sources with no outflow, and the authors suggest that these aremost probably Class II sources suffering high extinction from foreground emission. Further examples of early phases of star formation can be found in thesearch for protostellar cores in Bok globules <cit.> and in IRAS sources <cit.>.1mmAlthough the Orion complex of clouds and star formation were a notable source of study, perhaps surprisingly they hardly feature in the highly-ratedcitations of UKT14. Chini and co-workers <cit.> produced probably the most definitive study of submillimetre emission from OMC-2 and OMC-3detecting six probable Class 0 protostars as well as describing the general dust morphology and temperature of these complexes. To conclude this section onprotostars, the value of using UKT14 with its associated polarimeter <cit.> to detect the polarisation properties of protostars wasdemonstrated by Holland et al. <cit.> who observed the polarised 800 μm emission from aligned dust grains in the prototypical Class 0source VLA 1623 and in Sharples 106-IR, a high-mass, YSO along with its associated protostar S 106-FIR. For VLA 1623, the magneticfield was found to be almost exactly perpendicular to the highly collimated CO outflow, suggesting that the outflow is not collimated by the magneticfield. However, for the S 106 region, the situation was more complex, and it was clear that more extensive imaging would be the way forward, and thatwould have to wait for SCUBA. §.§ Stars and discs The study of nearby stars and their associated circumstellar discs attracted a lesser degree of attention, probably because most of the observations turnedout to be very difficult and at the limit of the capability of UKT14. On the other hand, the observations provided unique insights into a number ofphenomena. Mannings & Emerson <cit.> observed six T Tauri stars to investigate the dusty, circumstellar discs surrounding these starsin the early stages of stellar evolution using UKT14's full filter set of 2 mm to 350 m. For the optically thin sources, the spectral indexesindicated that the dust grains were larger than found in the interstellar medium, suggesting that grain growth in the protoplanetary discs had already occurredand was ongoing. The authors estimated that the rate of growth was of order 10^6 M_⊙ per year.1mmThe Vega phenomenon (excess thermal emission above that expected from the the stellar photosphere at far-IR wavelengths) was first discovered by theIRAS satellite and subsequently investigated by several UKT14 observational campaigns. The breakthrough was made by Zuckerman & Becklin<cit.> who made the first detections of excess 800 m emission from the stars Vega, β Pictoris and Fomalhaut. Sylvesterand co-workers <cit.> followed up with observations from 2 mm to 450 m in a major paper describing the far-infrared emission from alarge sample of stars including Vega. Nine stars were detected at millimetre/submillimetre wavelengths and the data suggested that the surrounding dust wasgenerally more likely to be in a ring rather than a spherical cloud and that they were composed of larger grains than found in the interstellar medium.Although there remained some uncertainty about the precise evolutionary state of all of the stars (some may have been younger than main sequence),nevertheless, these important papers pointed the way forward to some of the earliest and most important observations that SCUBA would make (see Section 3).1mmThe extended dusty envelope of five highly evolved stars (including one planetary nebula) were observed by Knapp, Sandell & Robson <cit.> whofound that the spectral index was just less than unity for all the sources, irrespective of whether the envelope of the star was carbon or oxygen rich. Thegas-to-dust ratio was calculated to be around 100. Observations of one of the sources (CRL618) suggested that it might be slowly variable, which wasimportant as one of the primary aims of this programme was to investigate whether these sources might be suitable as calibrators in themillimetre/submillimetre region. Finally, addressing the latest stages of stellar evolution Williams and co-workers <cit.> used UKT14 as partof a multifrequency study of the Wolf-Rayet system WR 147 and showed that the presence of non-thermal emission between the two stars was most probably duesto colliding winds in the system. §.§ The Galactic Centre The Galactic Centre region was a difficult target for UKT14 due to the complex emission over an extended region and the southerly declination. Dent andcollaborators <cit.> made maps of the 10 – 20 pc region at 1100 m and 800 m, as well as the inner region at 450 m. The2 pc inner ring was clearly detected as was the dust emission from three giant molecular clouds, which seemed to be connected by a ridge of thermalemission. The 2 pc ring revealed a two-component structure in the submillimetre: northern and southern emission, which were bounded by the radio continuumspiral arms. The initial results were followed up by Zylka et al. <cit.> who made maps with UKT14 at 800 m, 600 m and450 m of the 2 arcminute region surrounding the strong radio source Sgr A*. This was one of the rare examples of the difficult-to-calibrate600 m data being used from UKT14, especially in a map. A number of important conclusions were derived from these observations: that warm dustemission was definitely responsible for most of the far-IR emission from the region and that the heating of the dust was not from the centralsupermassive black hole but from a cluster of hot and luminous stars in the central parsec region. §.§ Dust emission in external galaxies Although the IRAS satellite had detected strong dust emission from many relatively nearby galaxies, the low surface brightness and the small size ofthe JCMT beam in comparison made UKT14 detections relatively difficult. Chini et al. <cit.> carried out observations at 800 m and400 m of seven of the 32 spiral galaxies that were previously mapped at 1.3 mm by the IRAM 30 m telescope. It was found that the sample split intotwo, with one half being dominated by cold interstellar dust with a temperature of ∼20 K, while for the rest, much colder dust (∼10 K)dominated. The ratio of the infrared luminosity to the gas mass turned out to be equivalent to the star formation rate in the Milky Way. Fich & Hodge<cit.> observed a sample of 22 early-type galaxies detected by IRAS and managed to detect six of them with tight upper limits beingobtained on a further eight. These allowed upper limits to be determined for the dust temperatures, the upper limit being principally because of the veryextended size of the IRAS beam compared to that of UKT14 on the JCMT. Depending on the value of the emissivity, these temperatures lay between 20 Kand 40 K.1mmMapping of nearby galaxies was a difficult proposition, requiring the most stable and driest conditions and so the results were relatively sparse given thefickle nature of matching observing conditions to requirements with the fixed-date scheduling at the time. However, two programmes stand out. Hughes, Gear& Robson <cit.> followed up their earlier maps of the nearby starburst galaxy M82 at 1100 m and 800 m by succeeding in making adiffraction-limited map of the 1.5 kpc diameter nuclear regions at 450 m <cit.>. The 9 arcsecond resolution of the map showed that thethermal emission from the central dust cloud was double in structure and the dust temperature was around 48 K. Both the earlier 800 m map and450 m image are shown in Fig. <ref>. Hawarden and co-workers <cit.> made an 800 m map of the central region of thenearby radio galaxy NGC 5128 (Centaurus A); a difficult observation due to the very southerly declination of the source. The map was combined withphotometry at all the UKT14 filters apart from 600 m and the results showed that the non-thermal central emission was surrounded by a circumnucleartorus of dust. Farther out from the centre extended dust emission was observed and even farther out, the dark optical dust lanes in the galaxy were alsodetected.1mm1mmMany programmes sought to detect submillimetre thermal emission from dust in radio quiet quasars and active galactic nuclei (AGN) and in 1992, Barvainis,Antonucci & Coleman <cit.> made the breakthrough by detecting the Seyfert I galaxy PG1434+590 and the gravitationally lensed quasar 1413+117(the “Cloverleaf”) at 450 m and 350 m, the latter was also detected at 800 m. The measured spectral indexes favoured thermalemission from dust but non-thermal, synchrotron emission could not be completely ruled out. At a redshift of 2.546 the strong suggestion was that this wasthe emission from typical far-IR galaxies red-shifted into the submillimetre. This was a milestone observation. At the same time, Clements et al.<cit.> detected the high-redshift (z = 2.286) IRAS galaxy 10214+4724 at 800 m and 450 m. The authors concluded that thedust-enshrouded Seyfert model and the primeval galaxy model were both excluded by their observations but the submillimetre emission was consistent with amassive starburst of around 100 M_⊙ per year. However, the observations were unable to determine whether this particular starburst was responsiblefor the formation of a significant fraction of the stars in the galaxy and indeed, much of the dust probably existed from previous bursts of starformation. This work was immediately followed up by Hughes, Robson, Dunlop & Gear <cit.> who detected three out of a sample of tenIRAS-selected radio-quiet quasars at 800 m and 450 m. In this case the very steep spectral index confirmed that the emission wasindeed thermal radiation from warm dust and conclusively ruled-out any significant non-thermal synchrotron contribution. The dust grain temperature wasestimated to be between 45 – 60 K.1mmIRAS detections provided the sources for many samples of submillimetre observations and a number of multi-wavelength investigations of ultraluminousIRAS galaxies (ULIRGs) were undertaken using UKT14. An example of one of these was by Rigopoulou, Lawrence & Rowan-Robinson <cit.>who detected 9 out of the ten brightest ULIRGs at 350 m, 450 m, 800 m and 1.1 mm. One of the key aims of this programme was todetermine whether the far-IR emission was powered by thermal emission from dust in a starburst (as in the above), or by accretion onto a centralsupermassive black hole. The observations showed that for these galaxies, the submillimetre emission was consistent with thermal emission. Although two ofthe objects might have housed an AGN, nevertheless, their very weak X-ray emission implied a very large optical depth and so again argued strongly againstaccretion being responsible for the far infrared luminosity.1mmMoving on to radio galaxies, in 1994 Dunlop and collaborators <cit.> investigated the concept that the stars in elliptical galaxies werebelieved to have been formed in a rapid bust of star formation in the early Universe. They used UKT14 observations to search for evidence in the form ofthermal emission from dust from the anticipated starburst phase in these distant galaxies; this same dust making them very faint for optical studies. Froma sample of six high redshift radio galaxies they were successful in detecting the redshift 3.8 galaxy, 4C41.17, at 800 m with a strong upper limitat 450 m. These two points were sufficient to establish thermal emission from dust at a temperature around 40 K and that the dust mass was >10times that found in corresponding low-redshift radio galaxies. The dust luminosity was ∼5 × 10^13 solar luminosities (L_⊙),corresponding to a starburst of a few thousand M_⊙ per year. The authors concluded that the observations were consistent with around 10% of theeventual mass of the galaxy still to be converted into stars and that it was possible that we were witnessing the end-product of star formation that hadbegun at much higher redshifts. However, it was not possible to say more about the formation epoch of elliptical galaxies per se. This was probably theearliest UKT14 observation relating to cosmology and the epoch of galaxy formation and early evolution.1mmThese exciting high-redshift studies were continued and the context of unambiguously identifying dust emission from relatively low signal-to-noiseobservations were discussed in detail by Hughes, Dunlop and Rawlings <cit.>, reviewing the observations to date, including some recent and veryfaint detections at 800 μm from their on-going programme. This was an important overview, especially in the context of the upcoming SCUBA instrument. §.§ Flat spectrum radio sources and blazars Extensive programmes of observations of flat-spectrum radio sources, in particular blazars, were a feature of UKT14 on the JCMT, following on from earlierwork on UKIRT. These observations sought to answer specific questions for these relativistic-jet dominated sources: to determine the “snapshot” spectralenergy distribution from radio to gamma-ray regions; to monitor and determine the variability behaviour; to test theoretical models such as the“shock-in-jet” model of Marscher & Gear <cit.>. Being bright, these sources were readily observable in many weather conditions andalso required only short integration times. They were also a feature of the Discretionary Time available to the Director of the telescope. They featuredprominently in many coordinated multi-wavelength campaigns from the radio to gamma-rays.1mmStevens et al. <cit.> published the last in the series of multi-frequency observations of blazars in 1994 which covered 17 blazars atwavelengths from 800 m to 13 mm. Good agreement was found between the variability and the shock-in-jet models and it was found that the flares inthe BL Lac objects tended to reach a maximum at a longer wavelength than those of the optically violently variable (OVV) quasars. This might indicate astronger shock in the former objects. The data also showed that the flaring behaviour was complex, with multiple maxima and flickering being present andthat “clean” flares were relatively rare, all of which indicated greater temporal sampling being needed.1mmGear and collaborators <cit.> further investigated the differences between the BL Lacs and OVVs through a large sample of 22 of the former and 24of the latter. Quasi-simultaneous data were obtained across a wide wavelength range, which showed that the overall spectral shape was relatively consistentacross all the sources. While this indicated that the same basic mechanism was at work in both classes, however, as noted above, a clear difference wasagain found in the millimetre-region spectra indicating a subtle difference in the jet properties between the two classes of flat-spectrum radio source. Theauthors suggested that the parent sources for the two classes might be the Fanaroff-Riley Class I and Class II sources, in which case submillimetrepolarimetry would be an acid test for the future.1mmThis was eventually undertaken using the UKT14 polarimeter to observe 26 flat-spectrum radio sources at 1.1 mm and 800 m <cit.>.Although a significant level of linear polarisation was detected in most sources (of order 10 – 15%) the magnetic field seemed less well-ordered onsub-parsec scales than on parsec scales and in the most highly ordered cases it was perpendicular to the jet axis. No significant difference was foundbetween the BL Lac and the flat-spectrum quasars and whilst the emission from many of the most highly-polarised sources could be well-fitted byshock-in-jet models, for most sources this was not the case. Conical shock models seemed to be the best descriptor for the diverse emission from the jetsin the sample.1mmExtensive multi-frequency observing campaigns from radio to gamma-rays were carried out on particular sources, for example, results from the quasar 3C279were reported by Hartman and co-workers between 1996 and 2001 <cit.>. These extensive observations showed that the variability wasvery complex, with different correlations being seen for different flares. The spectra could be best modelled with a relativistic electron dominated jetwith gamma-ray production arising through a combination of synchrotron self-Compton and external Compton processes. Interestingly, when 3C279 was in itshigh state, the gamma-ray luminosity dominated everything else by at least a factor of ten.1mmAnother very popular quasar for multi-frequency observations was 3C273, typified by the work by Robson et al. <cit.> who reported on a four-yearobserving campaign from infrared through centimetre wavelengths during which a number of flares were seen. A period of relative inactivity allowed thequiescent spectrum to be obtained, from which flaring behaviour could be subtracted to determine the flare emission itself. Caution was noted in that thebehaviour of a flare was critically dependent on the temporal overlap of the observations at differing wavelengths but where these were simultaneous theinfrared emission preceded that at longer wavelengths and there was distinct evidence for the evolution of the turnover of the flare to propagate to longerwavelengths. The emission between the infrared and 2 mm wavelengths was commensurate with a single synchrotron component associated with the innermostpart of the relativistic jet or the injection zone itself. A major paper by Turler et al. in 1999 <cit.> reported on thirty years ofmulti-wavelength monitoring of 3C273 showing the complex behaviour of this source and providing a database for emission modelling purposes.1mmWith the demise of UKT14 and the introduction of SCUBA, these programmes tended to fall in popularity due to the high impact of imaging science compared tosingle-pixel photometry. However, the work exploring dust emission from radio quiet galaxies and AGNs at cosmological redshifts was ideally suited toSCUBA, as will be seen in the next section. § THE FIRST CAMERA ARRAYS: THE SCUBA ERA SCUBA, the Submillimetre Common-User Bolometer Array, built by the Royal Observatory Edinburgh, was in the late 1990's the most versatile and powerful of anew generation of submillimetre cameras <cit.>. It uniquely combined a sensitive dual-waveband imaging array with a three-band photometer, andhad a sensitivity background-limited by the emission from the Mauna Kea atmosphere and telescope at all observing wavelengths from 350 m to 2 mm.The increased sensitivity and array size mean that SCUBA mapped ∼10,000 times faster than UKT14 to the same signal-to-noise. Most importantly, SCUBAwas a facility instrument, open to the world community of users, and was provided with an unprecedented high level of user support.1mmThe dual-camera system consisted of a short-wavelength (SW) array of 91 pixels and a long-wavelength (LW) array of 37 pixels. Each array had approximatelythe same field-of-view on the sky (2.3 arcmin in diameter) and could be used simultaneously by means of a dichroic beamsplitter. The SW array was optimisedfor operation at 450 m (but could also be used at 350 m), whilst the LW array was optimised for 850 m (with observations at750 m and 600 m also possible). The array pixels were arranged in a close-packed hexagon, with the photometric pixels positioned around theoutside of the LW array. The detectors were cooled to ∼100 mK using a dilution refrigerator, ensuring close to a factor of 10 increase in sensitivityper pixel over UKT14. SCUBA was delivered to the JCMT in April 1996, and first light on the telescope was obtained in July. After extensive commissioning,the first astronomical data for the community were taken in May 1997 using two modes of operation: photometry and “jiggle-mapping”, the latter usingnovel movement of the secondary mirror to create a fully-sampled image. The final major mode of data acquisition, “scan-mapping”, was released inFebruary 1998. Although issues with the filter drum meant that SCUBA became a fixed 450/850 m imager by the early 2000's, thepopularity of the instrument remained very high during its entire lifetime. §.§ New perspectives on galaxy formation and evolution In late 1997 SCUBA made several monumental discoveries, particularly in the area of galaxy formation and evolution. Capitalising on a spectacular period ofgood weather on Mauna Kea (the El Niño event of late 1997/early 1998) observations revealed a population of galaxies responsible for at least part ofthe far-IR background, detected a number of high redshift galaxies and provided new insights into galaxy evolution. In the next section we summarisejust a few of these high profile discoveries.§.§.§ “SCUBA galaxies”The most vigorously star-forming galaxies in the nearby Universe are also those in which dust obscuration is the most significant. It was long suspected,therefore, that the early evolution of galaxies would take place inside shrouds of dust. The first deep SCUBA maps outside of the Galactic Planeimmediately confirmed this suspicion, revealing a large population of hitherto unknown, star-forming galaxies. This discovery was reported by Smail, Ivison& Blain <cit.> in a series of targeted observations towards lensed galaxy clusters, exploiting the amplification of all background sources bythe clusters. The authors concluded that the observed source counts needed a significant increase in the number density of star-forming galaxies in thehigh redshift Universe and suggest that previous optical surveys may have underestimated the star formation density by a large factor. This workwas the first peer-reviewed paper to emerge from SCUBA observations.1mmSubsequent unbiased (“blank-field”) surveys by groups led by Barger <cit.>, Hughes <cit.> and Eales <cit.> confirmed thatthe surface density of submillimetre sources was several orders of magnitude above that expected for a non-evolving galaxy population. The conclusion wasthat strongly star-forming galaxies must have been substantially more common in the early Universe than they are today. Having an instrument with hithertounprecedented imaging capability and sensitivity meant that SCUBA could maximise the use of good-weather periods for statistically significant wide anddeep surveys. For example, 14 nights of the some of the best weather seen on Mauna Kea were used to produce the deepest ever submillimetre image to date.The 850 m image of the Hubble Deep Field (HDF) by Hughes and co-workers <cit.> reached a 1σ noise limit of 0.7 mJy/beam at850 m over an area of around 5arcmin^2. It was concluded that the radiation from the five most significant detections in this iconic image, asshown in Fig. <ref> (left), accounted for 30 – 50% of the previously unresolved background emission in the HDF area. The star formationrate implied from these redshift 2 – 4 galaxies was a factor of five higher than that inferred from optical observations (right panel ofFig. <ref>). The paper describing this seminal discovery has at the time of writing (June 2017) reached over 1000 citations.1mm1mmBlain, Smail, Ivison & Kneib <cit.> went on to conclude that these first deep submillimetre surveys confirmed a large population of dustygalaxies was missing from optical inventories of star formation activity. Further support for this was obtained with the submillimetre detection of anextremely red galaxy, HR10, at z = 1.4 by Cimatti and collaborators <cit.> and the radio galaxy, 8C1435+635 at z = 4.25 by Ivison et al.<cit.>. The former is a relatively common class of galaxy previously thought to consist of very old, quiescent ellipticals, but which SCUBArevealed to comprise young, star-forming systems similar to local ultraluminous IRAS galaxies <cit.>. The distant,submillimetre-selected galaxies discovered by Smail, Ivison & Blain <cit.> were also shown to resemble ULIRGs, at least in the rest-frameultraviolet/optical, with a similar proportion of mergers <cit.>. The diversity of SCUBA-selected galaxies was first shown fromobservations of the distant, galaxies detected in the field of the massive cluster lens Abell 1835 by Ivison et al. <cit.>. One galaxy showedalmost pure starburst characteristics, whilst the others had varying degrees of AGN activity. The study showed that although almost identical spectralenergy distributions are seen for many galaxies, they often exhibit strikingly different optical/UV spectral characteristics. It was concluded thatoptical/UV spectral classifications can hence be misleading when applied to distant, highly-obscured galaxies, and that other means of determining thevarious contributions to the overall energy budget of submillimetre galaxies (and hence to the far-IR extragalactic background) are needed.§.§.§ Extragalactic surveys go deeper and wider By the early 2000's deep extragalactic surveys had become more ambitious and included the 3 hr and 10 hr fields of the Canada-UK Deep Submillimetre Survey<cit.>, the 8 mJy survey of the ELAIS N2 and Lockman-Hole E fields <cit.> and wider map of the HDF North region<cit.>. In this latter work Borys and co-workers mapped 165 arcmin^2 of the region surrounding the HDF detecting 19 sources at >4σsignificance, and concluded that the number of galaxies detected accounted for approximately 40% of the 850 m submillimetre background. Moreover,the nature of the galaxies uncovered in these surveys was becoming clearer, with critical measurements such as the determination of a median redshift of2.4 from radio measurements reported by Chapman et al. <cit.>.1mmTowards the end of 2002, the first data were taken in what was to be the most ambitious extragalactic survey undertaken to date at the JCMT. This major,collaborative survey was called SHADES (the SCUBA HAlf Degree Extra-galactic Survey) and aimed to cover 0.5 degree^2 to a 4σ detection limit of8 mJy/beam at 850 m. SHADES was motivated by many science drivers, particularly the desire to clarify the number density, redshift distribution, andclustering properties of the bright submillimetre-selected galaxy population. To make further progress in this field required a large and complete sampleof 850 m sources (analogous to the 3C radio source sample, which ultimately revolutionised extragalactic radio astronomy). The main issue(particularly for the non-cosmologists that used the JCMT!) was that the survey would require approximately one third of the usable UK time on thetelescope over the subsequent 3 years. The resulting 850 m maps of the Lockman Hole and SXDF/UDS fields (the latter is shown in the left panel ofFig. <ref>) formed the largest submillimetre imaging survey of meaningful depth ever undertaken to date, and provided a uniquely powerfulresource for the study of the bright submillimetre galaxy population. The results from this survey, reported by Coppin and collaborators <cit.>,included a new sample of 120 sources and a definitive measurement of the source number counts in the 1 – 10 mJy range (as shown in the right panel ofFig. <ref>), resolving some of 20 – 30% of the far-IR background.1mm1mm§.§.§ New insights on radio galaxies and AGN The study of galaxies with AGN was also revolutionised by SCUBA. Early use of the jiggle-mapping mode led to the discovery of SMM02399-0136 by Ivison etal. <cit.>, a hyper-luminous galaxy at z ∼ 2.8 hosting an AGN. Such galaxies could not be easily detected in conventional AGN/QSOsurveys, so the presence of SMM02399-0136 in the very first submillimetre image of the distant Universe suggested that estimates of the prevalence of AGNmay require substantial revision. The unprecedented sensitivity of SCUBA's photometry mode allowed the study of radio selected and optically selected AGNto move from the pioneering world of bare detections to the reliable extraction of physical parameters. For the high-redshift radio galaxy 8C 1435+635,Ivison et al. <cit.> presented 450 m and 850 m detections of sufficient quality to infer that the formative starbursts of suchmassive ellipticals may still be in progress at z ∼ 4. Observations of a sample of radio galaxies by Archibald and co-workers <cit.>,spanning a range of redshifts between 1 and 5, showed that the submillimetre luminosity of radio galaxies is primarily a function of redshift, andfurthermore may be representative of massive ellipitcals in general. The authors concluded that the observed increase in submillimetre detection rate andcharacteristic luminosity with redshift is due to the increasing youthfulness of the stellar population of radio galaxies in their sample.1mmThe steep-spectrum, narrow-line radio galaxy 53W002 was especially interesting as it had been shown to contain an over-density of compact, Ly-αemission-line galaxies at z ∼ 2.4. SCUBA observations of the 53W002 field by Smail et al. <cit.> uncovered four luminous submillimetregalaxies. By matching the submillimetre source position using an astrometrically-precise 1.4 GHz map one of these sources was shown to be coincident witha Lyman-α-selected galaxy at z = 2.39, 330 kpc away from the radio galaxy in projection. This confirmed the presence of ultraluminous, dustygalaxies in the over dense structure around 53W002 at a look-back time of ∼11 Gyrs. SCUBA galaxies, as the progenitors of massive ellipticalgalaxies, should therefore be strongly clustered in the highest density regions of the distant universe.§.§.§ Gamma-ray bursts: galaxy evolution at high redshifts An alternative method for studying the characteristics and evolution of galaxies at high redshift is to use gamma-ray bursts (GRBs). SCUBA pioneered earlyobservations both of the host galaxy and of the afterglow from GRBs, speculating that the hosts were early starburst galaxies in contrast to thepreviously SCUBA-selected galaxies, which tended to host populations of more evolved stars. The implication was that the submillimetre surveys, which havecertain selection biases, miss a fraction of the cosmic star formation, which can be possibly recovered by observations of GRB hosts. Although theobservations were somewhat of a struggle (due to low flux levels), and only a few host galaxies were detected in the submillimetre <cit.>, theresults paved the way for future studies, particularly from space missions such as SWIFT, Spitzer and Herschel. For some bursts theearly afterglow (hours to weeks following the burst itself) peaks in emission in the submillimetre. By tracking the evolving afterglow emission across theentire spectrum, Smith and co-workers <cit.> showed that it was possible to study aspects such as the types of shocks involved, whether theoutflow has a jet or spherical geometry, and to also investigate the geometry of the surrounding medium (uniform versus prior stellar wind).§.§.§ Probing large-scale structure: The Sunyaev-Zel'dovich effect One of the most versatile probes of large-scale structure of the Universe is the Sunyaev-Zel'dovich (SZ) effect – the distortion of the cosmic microwavebackground (CMB) radiation through inverse Compton scattering by high energy electrons in galaxy clusters. This distortion produces a characteristic“increment” in the CMB temperature above frequencies of around 200 GHz, an effect that had only been measured in eight clusters before SCUBA madeobservations of a further two compact galaxy clusters, thereby providing a significant addition to this field of study. Constraining the full spectralshape of a cluster's SZ distortion allows separation of the thermal SZ effect, which is caused by the random motions of the cluster's electrons, from thekinetic effect, caused by the cluster's motion relative to the CMB rest frame. For the observations a large (180 arcsecond) chop throw of the secondarymirror had to be employed to ensure that no significant SZ flux (a small amplitude signal) appeared in the reference beams. In addition, 450 m datawere used to remove the effects of atmospheric emission from the 850 m data since standard, in-band atmospheric corrections would cancel the SZintensity. The JCMT's high angular resolution also allowed rejection of possible point-source contaminants which plague SZ measurements with smaller apertureinstruments. The results of this work by Zemcov and collaborators <cit.> provided robust, high S/N measurements of the SZ increment towards theclusters Cl 0016+16 and MS 1054.4–0321. §.§ The nearby universe: cold dust and giant magnetic bubbles Another key area of research focused on utilising SCUBA's sensitivity and mapping capabilities to make the deepest images to date of the location of colddust reservoirs in nearby spiral galaxies. The bulk of star formation activity in nearby spirals is often missed by IR studies, since most of the dust massresides in cold, extended, low-surface brightness discs, often far from the galactic nucleus. Studies of nearby galaxies such as NGC891 (see the leftpanel of Fig. <ref>) and NGC7331 by Alton and co-workers <cit.> revealed that up to 90% of the total dustmass can be located within galactic discs at large radii. The images also detected spectacular dust “chimneys” escaping from the main absorption layer upto z-heights of nearly 2 kpc. Further observations of cold dust emission in the “Whirlpool Galaxy” (M51) by Meijerink et al. <cit.>showed that the 850 m originated in an underlying exponential disc with a scale length of 5.5 kpc. This reinforced the view that the submillimetreemission from spiral galaxy discs traces the total hydrogen column density (i.e. the sum of H_2 and HI).1mm1mmDunne and collaborators <cit.> observed 104 galaxies from the IRAS Bright Galaxies sample to provide the first statistical survey of thesubmillimetre properties of the Local Universe. They made the first direct measurements of the submillimetre luminosity function, concluding that the slopeof the function must flatten at luminosities values lower than in the survey. They postulated the existence of a population of “cold” galaxies (<25 K)emitting strongly in the submillimetre that would have been missed from far-IR selected samples. By comparing the global galaxy properties with theirsubmillimetre/far-IR properties, average gas-to-dust ratios of close to 600 were found, compared to the Galactic value of only 160. The conclusion was thatmost galaxies in the sample must contain a “cold dust” component with a temperature of <20 K.1mmThe sensitivity of SCUBA was also used in earnest to make deep submillimetre images of the central 8 × 2 kpc region of Centaurus A, the nearestgiant elliptical galaxy. The remarkable images at 450 m and 850 m by Leeuw et al. <cit.> revealed an unresolved central AGN core,an inner jet interacting with gas in a dust lane, an S-shaped inner disc of active star formation, and a colder outer disk. These images are shown in theright-hand panel of Fig. <ref>. The results supported the theory that the inner disc material is consistent with a warped-discmodel of tilted rings. Finally, with magnetic fields believed to play an important role in the star formation process in the central region of starburstgalaxies, Greaves et al. <cit.> used the SCUBA polarimeter <cit.> to map the magnetic field morphology surrounding the innerregions of M82. The polarised dust emission showed that the major magnetic features found in M82 are ordered fields over scales of hundreds of parsecswithin the torus, and an outer “bubble-like” field associated with the dusty halo, with a diameter of at least 1 kpc. §.§ Large-scale mapping of the Galactic Centre One of the most ambitious, large-scale projects undertaken with SCUBA was to map the Central Molecular Zone (CMZ) of the Galactic Centre over an extent of3 degrees in galactic longitude. This stunning data set by Pierce-Price and co-workers <cit.>, shown in Fig. <ref>, containsdetailed information on both the warm cloud population near SgrA* (the non-thermal radio source at the centre of the Galaxy), and the circumnuclear disc, with asensitivity RMS limit of approximately 30 mJy/beam at 850 m, equivalent to just a few M_⊙. There is clearly an extraordinary amount ofstructure, and such data are vital for understanding cloud evolution in a dynamic (sheared and rapidly-rotating) environment. The images also provide inputto models of the starburst phenomenon in other galaxies as well as the periodic star-formation inferred in our own Galactic Centre. The SCUBA datarepresented the first optically thin map to trace essentially all the mass in the CMZ at high spatial resolution.1mm1mmAitken and collaborators <cit.> reported the first detections of linear polarisation from SgrA* at 750 m, 850 m, 1.35 mm and2 mm, confirming the contribution of synchrotron radiation. Large changes in the position angle between the submillimetre and millimetre measurementswere observed and the best model to explain these changes was one in which the synchrotron radiation from the excess flux is self-absorbed in themillimetre but becomes optically thin in the submillimetre. The authors conclude that this suggests the flux originates from an extremely compact source of∼2 Schwarzschild radii. §.§ Debris discs: the fallout of planetesimal collisions around stars Observations with UKT14 had already shown an intriguing glimpse (via point-by-point photometry) of what was possible in terms of imaging the faint discsthat surround many main sequence stars <cit.>. The fact that such material exists suggests the presence of larger unobservablebodies in these systems, such as planets. SCUBA was well-suited to measure the low-level thermal emission from the dust grains in such discs. The resultsfrom the work of Holland and co-workers <cit.> were spectacular, and included the first images of the debris discs around the well-knownstars Fomalhaut and Vega. For example, around Fomalhaut, the peak flux in the map (see the left panel of Fig. <ref>) was seen tooccur in two distinct regions, offset from the stellar position. The image is consistent with an edge-on torus (“doughnut-like”) structure of a sizesimilar to our own Edgeworth-Kuiper Belt (EKB), and with a central cavity containing significantly less dust emission. The cavity is about the diameterof Neptune's orbit, and a possible explanation is that the region has been cleared of gas and dust by the formation of planetesimals <cit.>.More recent observations by the Hubble Space Telescope (HST) <cit.>, Herschel <cit.> and the Atacama Millimeter Array(ALMA) <cit.>, at higher angular resolution, have shown that the disc is actually a thin ring, possibly shepherded by one or more planets.1mmHow typical is our Solar System architecture around other stars is one of the most fundamental questions in astronomy. Fomalhaut and Vega (see right panel ofFig. <ref>) are extremely luminous, relatively short-lived A-stars, and hence any planetary system that may exist around them wouldlikely be very different from our Solar System. Further work with SCUBA therefore targeted G and K stars in an attempt to address the uniqueness of ourSolar System architecture. The image of the debris disc around the young, nearby (only 3 pc) ϵ Eridani by Greaves et al. <cit.>,shown in the centre panel of Fig. <ref>, revealed a dust ring peaking at 60 AU from the star with a void of emission in the inner3 AU radius <cit.>. Substructure, observed as asymmetries within the ring, was interpreted as possibly being due to perturbations by planets.Moreover, observations of the Sun-like G8 star τ Ceti by Greaves and colleagues <cit.> revealed a vast EKB-like disc. Modellingshowed that the mass in colliding bodies up to 10 km in size is around 1.2 Earth masses, compared with 0.1 Earth masses in the EKB, and hence theevolution around the two stars has been very different. One possibility is that τ Ceti has lost fewer comets from the outskirts of the system,compared with the Sun.1mm1mm §.§ Large-scale mapping of star-forming regions One of the key goals for SCUBA was to provide the capability to carry out large-scale (several degrees), high dynamic range imaging of star-forming regionsin the Milky Way. One of the first regions to be imaged was the central region of the Orion A molecular cloud. Fig. <ref> from Johnstoneand Bally <cit.> shows that the SCUBA images trace the morphology and spectral index of the optically thin emission from interstellardust. The famous Orion “bright bar” is clearly seen in the image together with a chain of compact sources embedded in a narrow, high column-densityfilament that extends over the entire length of the map. The region is also believed to be a site of progressive star formation (from the south to thenorth), and so offers an opportunity to compare dust core properties (such as the spectral index - see right panel in Fig. <ref>) over arange of evolutionary stages.1mm1mmFurther large-scale mapping of regions such as the ρ Ophiuchus molecular cloud by Johnstone et al. <cit.> used clump-finding algorithmsto identify, and compare the properties of individual objects. Thus it became possible to determine the mass distribution of clumps based on submillimetrefluxes for the first time. In the case of ρ Ophiuchus, the clumps spanned a mass range of 0.02 to 6.3 M_⊙ and the distribution was characterisedby a broken power-law, N(M) ∝ M^-α, where α is typically 0.5 – 1.5. As with other studies it was concluded that the observed clumpsmay represent a evolutionary stage, being fragments produced during the collapse of a larger and gravitationally unstable core within the cloud. Theobservations of the ρ Oph A region of the cloud complex also highlighted the vast improvement in performance of SCUBA over UKT14. To achieve the sameS/N as SCUBA over the 4 × 3 arcminute region shown in Fig. <ref> would have taken UKT14 over 10,000 hours! §.§ The Holy Grail: protostars One of the long-standing challenges facing infrared and submillimetre astronomy is the understanding of the earliest stages of star formation. SCUBA readilydemonstrated the power of deep imaging to discover new candidate protostars, as well as obtaining reliable statistics on the early stages of stellarevolution, including the protostellar Class 0 phase. Unbiased surveys of extended dark clouds, for example by Visser and co-workers <cit.>, werealso carried out to identify complete samples of protostellar condensations, allowing the measurement of star formation efficiencies, mass accretion ratesand evolutionary lifetimes.1mmFurthermore, as a result of SCUBA observations interest began to focus on the starless (or “pre-stellar”) cores, which are significant in that theyconstrain the initial conditions of protostellar collapse. Over 40 such cores in the Orion molecular cloud were studied by Nutter and Ward-Thompson<cit.>, who concluded that the high-mass, core mass function (CMF) follows a roughly Salpeter-like slope, just like the initial mass function (IMF)seen in earlier studies. The deep SCUBA maps showed that the CMF turns over at ∼ 1.3 M_⊙, about a factor of 4 higher than the completenesslimit. This turnover, never previously observed and only revealed by the much deeper SCUBA maps, mimics the turnover seen in the stellar IMF at∼0.1 M_⊙. The low-mass side of the CMF is a power-law with an exponent of, 0.35 – 0.2, which is consistent with the low-mass slope of theyoung cluster IMF of 0.3 – 0.1. This shows that the CMF continues to mimic the shape of the IMF all the way down to the lower completeness limit of thesedata at ∼0.3 M_⊙.1mmOne of the the most spectacular images of the earliest stages of star formation came from SCUBA imaging by White and co-workers <cit.> of thefamous Eagle nebula (M16). As shown in the 450 m SCUBA-2 image presented Fig. <ref> some differences are immediately evident fromthe HST optical image, particularly in terms of the dominant thermal emission from the tips of the “fingers” seen in the SCUBA map. Thecontinuum spectra of these cores show that they are much cooler (∼20 K) than the surrounding molecular gas in each of the fingers. The results of athermal and chemical model of the environment concluded that the fingers appear to have been formed after the primordial dense clumps in the originalcloud were irradiated by light from its own OB stars. During the subsequent photoevaporative dispersal of the cloud the clumps shielded material lyingbehind it, facilitating the formation of the fingers. The absence of embedded IR sources or molecular outflows suggest that the cores at the tips of thefingers have the characteristics of the earliest stages of protostellar formation.1mm1mmOne of the most ambitious projects to be undertaken with SCUBA and it's polarimeter was an attempt to make the first observations of the magnetic fieldgeometry in pre-stellar cores. By doing so such results would test the theoretical ideas about the way in which the field geometry affects the starformation process. The first published maps by Ward-Thompson et al. <cit.> for three cores revealed smooth, uniform polarisation vectorsin the plane of the sky, showing no evidence of the type of geometry that might be expected in magnetically dominated stage of evolution. It was concludedthat no current model of magnetically-regulated star formation could explain the existing observations. §.§ Solar System Science: Comet Hale-Bopp and the subsurface of Pluto It has been long suspected that large particles may be present in comets sufficient to dominate the total mass of the coma. Believed to be products ofagglomerative growth in the proto-solar nebula these particles are the local analogues of the dust observed in the disks around many young stars. Some ofthe first evidence for this was published for the Comet Hale-Bopp by Jewitt and Matthews <cit.> based on SCUBA 850 mobservations. The dust coma surface brightness is well described by a steady-state outflow model, in which the dust density varies with the inverse squaredistance from the nucleus. Submillimetre observations have proved vital in studying the properties of these large particles; the data provide an estimateof the total mass, the dust mass production rate as a function of heliocentric distance, and the size of the particles in comparison with those incircumstellar discs.1mmThe subsurface of Pluto is known contain a reservoir of frozen volatiles but very little is known about it. Greaves, Whitelaw and Bendo <cit.>used archival light curves of the brightness of Pluto to probe just below the skin depth of the thermal changes over Pluto's day. With the light curve inthe submillimetre differing significantly from those measured in the mid- and far-IR, in a region that is optically dark on the planet's surface, thesuggestion is that the layers a few centimetres below the surface have not undergone any major temperature change. One possibility is that these regionscould have a different emissivity, perhaps with a subsurface layer richer in nitrigen or methance ices than the surface. Results from the NASA NewHorizons probe concluded that the surface composition is suprisingly complex, with the nitrogen, methane and water-rich areas creating a puzzle forunderstanding Pluto's climate and geologic history.§ SCUBA-2: WIDE-FIELD IMAGING IN THE SUBMILLIMETRE BECOMES A REALITY Although SCUBA had made so many pioneering discoveries it was obvious by the turn of the century that an even more sensitive camera was required,specifically to allow wide-field surveys to be undertaken, in line with the planned work by satellites such as Herschel in the far-IR. The projectbecame known as SCUBA-2, and involved an international partnership between institutes in the UK, USA and Canada. As was the case with SCUBA, SCUBA-2 hadtwo imaging arrays working simultaneously in the atmospheric windows at 450 and 850 m, the vast increase in pixel count to over 10,000 meant thatSCUBA-2 would map the sky 100 – 150 times faster than SCUBA to the same signal-to-noise. SCUBA-2 was a major step forward in technology. It was the firstastronomical camera to use superconducting transition-edge sensors in a time-domain multiplexed readout scheme. The instrument itself was also a majorchallenge having a liquid cryogen-free dilution refrigerator to cool the detector to <100 mK and over 600 kg of optics cooled to less than 4 K.SCUBA-2 was delivered to the JCMT in April 2008 with two engineering sub-arrays (one quarter of the field-of-view at each wavelength), and eventually beganscience operations in late 2011 with fully populated, science-grade focal planes. In February 2012, SCUBA-2 began a series of unique legacy surveys for theJCMT community. These surveys took almost 3 years and the results provided complementary data to the shorter wavelength, shallower, larger area surveysfrom Herschel. The SCUBA-2 surveys have also provided a wealth of information for further study with new facilities such as ALMA, and futurepossible telescopes such as SPICA and ground-based large, single-aperture dishes. §.§ The first generation surveys with SCUBA-2 The key scientific driver for SCUBA-2 was the ability to carry out large-scale surveys of the submillimetre sky to unprecedented depth. Six firstgeneration “legacy-style” survey programmes were approved covering a very broad base, ranging from the studies of debris discs around nearby stars togalaxy populations and evolution in the early Universe. The SCUBA-2 element of these surveys was initially approved to run from February 2012 to September2014, with several benefiting from an extension to February 2015. In the next sections we briefly describe these surveys and summarise some of the keyfindings so far.§.§.§ Galactic plane survey The JCMT Plane Survey (JPS) sought to achieve a full census of star-formation activity in the plane of the Galaxy observable from JCMT to a detected masslimit of around 40 M_⊙ at the far edge of the Galaxy. The aims included examining triggered and large-scale star formation and to study the evolutionof massive YSOs, infrared dark clouds and filaments, along with dust evolution and molecular cloud structure. Surveys of the Galactic Plane in themillimetre/submillimetre are currently the only approach to determine the relative importance of the physical processes that are likely to affect the starformation efficiency on Galactic scales (>1 kpc, e.g., spiral density waves) and within individual molecular clouds (e.g., temperature and pressure). Toachieve this, the JPS observed six fields along the Galactic Plane at longitudes of 10, 20, 30, 40, 50 and 60 degrees (see left panel ofFig. <ref> with each field just over 5 × 1.7 degrees in area as described by Moore et al. <cit.>. The argument was thatlarge fractions of the Plane needed to be surveyed in order to account for the statistical distribution of cloud masses and YSO luminosities, as well aslocal variance.1mmThe final survey covers an area of approximately 50 degree^2 and achieved an average noise level of 7.2 mJy/beam at 850 m, when smoothed over abeam diameter. An example of one of the fields, highlighting the W43 star-forming region, is shown in the right panels of Fig. <ref>. Thesurvey is approximately 10× more sensitive than the complementary ATLASGAL survey carried out on the APEX telescope, which studied the innerGalactic Plane at 870 m, covering galactic longitudes between 60 and 270 degrees. A catalogue of ∼7800 compact sources was generated from theJPS, and it was shown that these sources are responsible for 42% of the total emission from the maps with the remaining flux lying in filamentarystructures. One of the key outcomes of the survey, which also included large-scale observations with HARP (see section 5), is that thedominant scale of variations in star formation efficiency in the Galactic disc is that of individual molecular clouds as described in one of the firstpapers by Eden and collaborators <cit.>, with spiral arms having only a relatively minor influence. Using the Herschel 70 m data,the survey team showed that 38% of the sources detected show evidence of ongoing star formation <cit.>. The JPS images and associated sourcecatalogue represent a valuable resource for studying the role of environment and spiral arm structure on star formation in the Galaxy.§.§.§ Gould Belt survey The Gould Belt is a large (∼1 kpc diameter) ring of molecular clouds and OB star associations that is inclined at ∼20^∘ to the GalacticPlane. It is important for star formation studies as it contains most of the nearby low- and intermediate-mass star formation regions such as the Orion andTaurus-Auriga molecular clouds. The JCMT Gould Belt Survey (GBS) aimed to address several of the major unsolved questions in star formation: the evolutionof pre- and protostellar cores, the origin of the IMF, and the link between star formation and molecular cloud properties <cit.>. Thetargets were molecular clouds within 500 pc of the Sun where the angular resolution is high enough to separate individual pre/protostellar cores(0.1 pc). The survey was awarded 612 hours of observing time, which included both SCUBA-2 and HARP observations of 14 nearby clouds, covering a totalarea of almost 700 degree^2. The improved resolution of the JCMT also allows for more detailed study of large-scale structures such as filaments,protostellar envelopes, extended cloud structure and morphology down to the Jeans length.1mm1mmThe resulting maps of the Gould Belt molecular clouds are amongst the deepest ever undertaken with typical noise levels of 3–4 mJy/beam at 850 mover tens of square degrees. The uniqueness of the SCUBA-2 observations is that they predominantly trace cold, dense cores that are more likely to bepre-stellar than the more evolved clumps seen by Herschel. For example, using observations from both SCUBA-2 and HARP, Pattle et al.<cit.> identified significant fractions of pressure-supported starless cores in Ophiuchus that are unlikely to ever become gravitationallybound. Furthermore, by combining SCUBA-2 data with shorter wavelength data from Spitzer and Herschel it is also possible to measure thevariation in temperature along the line-of-sight, as shown in Fig. <ref>, for the Ophiuchus L1688 cloud complex. Mairs and collaborators<cit.> presented a catalogue of sources from observations of the Southern Orion A cloud, showing that the larger-scale regions of emission withinthe cloud are often subdivided into smaller dense fragments that are usually invisible in shorter wavelength surveys. One of the key aims of the survey wasto investigate the prestellar mass function in the various molecular clouds. Salji and collaborators <cit.> determined that it peaked at 1.39M_⊙ in Orion A, revealing a star-forming efficiency of 14% when compared the Orion nebula cluster IMF. Furthermore, the prestellar mass function wasfound to decay with a high-mass power-law exponent of 2.53, similar to the Salpeter IMF value of 2.35 for stars in the Solar neighbourhood. The extensiveGBS data continues to be analysed and has already produced a number of new source catalogues, characterising thousands of cores and clumps in terms oftheir properties and evolutionary status.§.§.§ Nearby galaxy survey SCUBA pioneered some of the earliest observations of the extent of cold dust in nearby spiral galaxies (see Section 3b). As described by Wilson andcollaborators <cit.> the JCMT Nearby Galaxies legacy survey (NGLS) aimed to use both SCUBA-2 and HARP to investigate both the physicalproperties of gas and dust in galaxies, along with the effect that galaxy morphology and unusual environments (such as metallicity) has on the properties of thedense ISM. The SCUBA-2 part of the survey was allocated 100 hours of observing time, with a goal of reaching a sensitivity level of 1.8 mJy/beam at850 m. A total of 48 spiral galaxies were observed, the majority of which came from the Spitzer SINGS survey. One of the major findings wasthe presence of significant levels (up to 25%) of CO in the centres of galaxies. Much of the analysis of the data is still underway; the emission from themajority of the galaxies is very weak, and recovering the flux on scales of several arcminutes has been a challenge for the data reduction. Combining thedata with shorter wavelength data from Herschel will also allow the dust temperature variation in these galaxies to be mapped across large galacticscales for the first time.§.§.§ Cosmology legacy survey The JCMT Cosmology Legacy Survey (S2CLS) sought to capitalise on the pioneering high redshift galaxy work undertaken by SCUBA and other early submillimetrecameras. Submillimetre galaxies (SMGs) are amongst the most luminous dusty galaxies in the Universe but their true nature remained unclear. It could bethat they are the progenitors of the massive elliptical galaxies seen in the local Universe, or a short-lived phase of a more typical star-forming galaxy.As described by Geach et al. <cit.> the key driver of the 850 m survey was to deliver a sufficient number of galaxies to address thisquestion by reliably measuring the clustering of the submillimetre population (providing valuable constraints on galaxy formation models) and to detect andstudy the (rare) progenitors of rich clusters. The 850 m survey would also establish, unambiguously, the faint end of the counts of SMGs in thisband. The goal of the deep 450 m component of the survey was to resolve a significant fraction of the extragalactic background light at450 m into individual galaxies by getting as close as possible to the confusion limit at this shorter wavelength (similar to that achieved withSCUBA for the HDF at 850 m). The survey plan was to map an area of 10 deg^2 at 850 m to a depth of 1σ =1.5 mJy/beam and 0.25 deg^2 at 450 m to a depth of 1σ = 1.2 mJy/beam. To achieve this, several extragalactic survey fields (includingthe original SHADES fields from SCUBA) were to be mapped for which a wide range of ancillary data is available from other wavelengths. The wide-field850 m included a number of well-studied fields, such as the UKISS-UDS, COSMOS, Akari-Northern ecliptic pole, and Lockman Hole north regions,whilst the ultra-deep 450 m maps were centred in the COSMOS and UDS fields. The S2CLS was the by far the largest of the first generation legacysurveys, and was awarded close to 1800 hours or 51% of the total survey time over the 3 year period.1mm1mmThe results from the S2CLS increased the size sample of 850 m-selected SMGs by an order of magnitude. Fig. <ref>shows the first 850 m maps by Geach and collaborators <cit.> from this extensive survey, covering a total area of ∼5 degree^2 anddetecting approximately 3000 sources. The average RMS noise in the maps is 1.2 mJy/beam, close to the expected confusion limit of 0.8 mJy/beam. Such alarge survey also allows a comprehensive measurement of the number counts of submillimetre sources and the results show both a distinctive upturn in thecounts caused by strong gravitational lensing of high redshift galaxies, and a contribution from local sources of submillimetre emission. For the firstunbiased, blank-field assessment of the number counts of galaxies at 450 m Geach et al. <cit.> showed that 16% of the cosmic infraredbackground was resolved into individual galaxies (see Fig. <ref>), whilst a further ∼40% was recovered in the SCUBA-2 map bycomparing to Spitzer-detected 24 m emitters. Koprowski et al. <cit.> utilised multi-frequency data to determine the redshiftdistribution of the 106 galaxies detected in the deepest, central area of COSMOS field and found a median redshift of 2.38 ± 0.09. Roseboom andcollaborators <cit.> explored the physical properties of these galaxies from their spectral energy distributions, revealing correlations, forexample, between the dust temperature and infrared luminosity. Some 24% of the 450 m sources were found to be starbursts, i.e. displaying ananomalously high star formation rate.1mmThe nature of SMGs was further explored by Wilkinson and co-workers <cit.> who investigated the clustering of galaxies in the S2CLS fields. Across correlation analysis carried out on a sample of ∼600 counterparts from the UKIDSS Ultra Deep survey led to an estimation of the halo massesof these SMGs and a comparison with passive and star-forming galaxies selected in the same field. It was found that, on average, the SMGs occupy high-massdark matter halos (M_halo > 10^13 M_⊙) at redshifts z > 2.5, consistent with being the progenitors of massive elliptical galaxiesfound in present-day galaxy clusters. It was also found the that SMG clustering strength was consistent with star-forming population and that this appearsto be the case across all redshifts. Recent work by Bourne and co-workers <cit.> explored the evolution of cosmic star formation in the S2CLSdata sample. They concluded that the star formation history appears to undergo a transition at z ∼ 3 – 4, as unobscured structure growth in the earlyUniverse is surpassed by obscured star formation, driven by the gradual build-up of the most massive galaxies in the Universe during the peak of cosmicassembly. The S2CLS catalogue and images have also presented an opportunity for follow-up work, both in terms of the properties of individual sources (e.g.with ALMA) and also in the statistical analysis of the entire sample.1mm1mm§.§.§ SASSy: The SCUBA-2 all sky survey Prior to the start of the legacy survey observing campaign the original “all-sky” survey (SASSy) was reborn as the more modest “SCUBA-2 Ambitious SkySurvey”, with the aim of making the largest submillimetre map of the Outer Galaxy to identify the coldest and earliest regions of star formation. Asdescribed by Thompson et al. <cit.> the survey would cover of order 700 degree^2 between longitudes 120^∘ and 240^∘,extending to ±2^∘ from the Plane. The primary goal was to detect all the compact sources within the survey bounds above a few times the∼40 mJy noise level at 850 m. The survey was allocated 480 hours of mainly poor weather (“band 5”) observing time, and was often used afallback project in poor weather conditions. The sources identified are in the process of being compared to the IRAS and Planck catalogues todetermine if any new objects have been detected. For example, in the 120 degree^2 region of the Galactic Plane covering longitude 120^∘ < l< 140^∘ and latitude |b| < 2.9^∘, Nettke et al. <cit.> produced a catalogue of ∼300 sources, of which 19 were newdetections in comparison to IRAS, 41 new detections compared to Planck and 13 that were not found in either catalogue. Analysis continues ofthis very extensive dataset.§.§.§ SCUBA-2 observations of nearby stars survey (SONS) Although the main strength of SCUBA-2 is in wide-field mapping, the camera can also image compact sources very quickly and with high image fidelity. TheSCUBA-2 Observations of Nearby Stars survey (SONS) targeted 100 nearby stars looking for evidence of debris discs – the extrasolar analogues of the EKB inour Solar System. As described by Matthews et al. <cit.> the survey aimed to characterise these discs by: (1)providing direct dust masses that could not be obtained from shorter wavelengths; (2) adding to the far-IR/submillimetre spectrum to constrain the dust sizedistribution; (3) using the power of a 15 m telescope to resolve disc structures around the nearest systems; and (4) looking for evidence of resonantclumps and other features in resolved structures that could be indicative of unseen perturbers, such as planets. Of particular importance was toinvestigate the diversity of exo-planetary system architectures, as this represents a key piece of information that will help link the formation andevolution of planetary systems with the evolution of planetary building blocks (planetesimals). The survey used 325 hours of observing time and for the 100targets reached an average RMS noise of ∼1.2 mJy/beam. A total of 49 discs were detected, many for the first time, and 16 of the nearest discs werealso spatially resolved by the JCMT.1mmThe results from SONS, presented by Holland and collaborators <cit.>, more than doubled the number of imaged discs from submillimetreobservations. The discs are characterised in terms of their flux density, size (radial distribution of the dust) and derived dust properties from theirspectral energy distributions. The mass of a disc, for particles up to a few millimetres in size, is uniquely obtainable from submillimetre observations,and shows a slow decline with age over hundreds of millions of years of stellar evolution (see the left panel of Fig. <ref>). Manyindividual objects from SONS have also been studied with some surprising results. For example, observations of the nearby Sun-Like star HD 38858 revealed alarge, extended structure, clearly with a flux peak offset from the star position (right panel of Fig. <ref>). Kennedy et al.<cit.> used multiple wavelength data, including from Herschel, to determine that although the disc is clearly resolved by the SONSobservations the peak to the south is most likely a background object. The offset nature of the peak emission is still puzzling, but the emission mightindicate a perturbed disc that could have detectable volatiles. Similarly, observations of the nearby main sequence star 61 Vir, a system which has atleast two known inner planets, reveal a resolved disc with a diameter of at least 80 AU, i.e. very similar to the EKB in our Solar System. Marino andco-workers <cit.> combined the SCUBA-2 data with ALMA observations to conclude that the disc is very likely extended from 60 to over 100 AU andso represents a very broad parent planetesimal belt. The observations of 61 Vir have already shown the legacy of the SONS survey in that it is providingcomprehensive target list for high angular resolution follow-up observations with submillimetre interferometers, such as SMA an ALMA.1mm §.§ The EAO era: A new call for large programsOn the 1st March 2015 the East Asian Observatory officially took over responsibility for the operations of the JCMT. Shortly after this (1st July) therewas also a second call for “Large Programs” to be undertaken with the JCMT, covering the period from late 2015 until late 2018. A total of 7 programswere approved (6 would use SCUBA-2) and observations began in November 2015. Some 50% of the total telescope time, amounting to at least 2400 hours, wasto be dedicated to these programs. A second large program call for either extension to the existing ones or new initiatives was issued in February 2017.§.§.§ STUDIES - the SCUBA-2 Ultra Deep Imaging EAO Survey The objective of STUDIES is to obtain the first confusion-limited 450 m map, centred on the COSMOS field at the northern edge of the CANDELS region<cit.>. The single pointing (“Daisy map”) of around 10 arcminutes in diameter will eventually use 330 hours of the best weather available onMauna Kea. This will take advantage of the high angular resolution offered by the JCMT/SCUBA-2 at 450 m, compared to Herschel at 350 and500 m, and allow detections of faint galaxies with a significant higher surface density. The goal is to reach an RMS noise of 0.6 mJy/beam at thecentre of the field and to detect the dominant members in the dusty galaxy population that give rise to the bulk of the far-IR extragalactic background.Such a deep map will enable the detection of nearly all L_IR > 10^12 L_⊙ galaxies at z < 4, and the majority of L_IR > 10^11L_⊙ galaxies at z < 2. The observations will also allow, for the first time, a substantial overlap in the star formation rate range with galaxiesdetected by deep optical surveys. This will provide a more complete census of the cosmic star formation that is both obscured and unobscured by dust. Injust over a year (Nov 2015 – Feb 2017) the observations were 40% complete, and have reached an RMS of 1 mJy/beam in the centre of the image. A total of 98sources have been detected so far at a significance of > 4σ, with this number expected to dramatically increase as the map goes deeper<cit.>.§.§.§ SC2-COSMOS COSMOS is a survey of ∼1000 submillimetre galaxies in the 2 degree^2 COSMOS field <cit.>. The region is the pre-eminent ALMA-visible,degree-scale, extragalactic survey field, and has been studied extensively from the X-ray to the radio. The goal is to first complete the 850 m mapof the full COSMOS field (partly covered by SCUBA-2 Cosmology Legacy Survey) to a depth of 1.5 mJy/beam, and to then increase the depth of this map to1.2 mJy. This map will have twice the area of similar surveys in a single contiguous field, allowing unique tests of the clustering of the submillimetregalaxy population on scales up to ∼60 Mpc. By March 2017 the observations were 79% complete against an allocation of 223 hours, and an RMS noise of<1.5 mJy/beam has been achieved across the entire field (as shown in Fig. <ref>). Already, some 1400 submillimetre sources have beendetected in the field and an analysis of the multi-wavelength properties of these galaxies is underway. Follow-up observations with ALMA of the 150brightest sources are also planned.1mm1mm§.§.§ JINGLE - the JCMT dust and gas in Nearby Galaxies Legacy Exploration JINGLE is a survey designed to systematically study the cold ISM of galaxies in the local Universe <cit.>. The survey will provide 850 mimages with SCUBA-2 for a sample of 192 Herschel-selected galaxies, as well as integrated CO(2-1) line fluxes with the heterodyne Receiver A (RxA)for a subset of 62 of these galaxies. A total of 780 hours has been allocated to the survey over the 3 year period. The sample builds on multiple surveysincluding Herschel/H-ATLAS and the MaNGA optical integral-field spectroscopy surveys. By combining the results from the JCMT observations with allthese ancillary data, JINGLE will allow for a detailed characterisation of the gas and dust properties of galaxies in the local universe. Scientificobjectives include studying the dust-to-gas ratio and how it varies across the galaxy population, correlating the molecular gas content withspatially-resolved galaxy properties, and investigating the correlation between ISM properties and the dynamics of galaxies. The scaling relations betweendust, gas, and global physical properties will also provide critical benchmarks for high-redshift studies with JCMT and ALMA. As of early 2017, the surveyis 36% complete with 106 galaxies already observed with SCUBA-2.§.§.§ BISTRO - B-fields in star-forming region observations Without accurate knowledge of the collapse process of molecular clouds, it is not possible to understand how a star forms. The exact role of magneticfields (B-fields) in this process is still open to considerable debate, and so the BISTRO survey will address this by tracing the direction and strength ofthe magnetic field on scales of ∼1000 – 2000 AU in the central regions of several nearby molecular clouds (e.g. Orion, Ophiuchus, Taurus L1495)already observed with the previous Gould Belt legacy survey <cit.>. The scientific objectives are to assess the relative importance of magneticfield and turbulence in the star formation process, to test models of magnetic “funnelling” of materials onto filaments and to investigate the role ofB-fields in shaping protostellar evolution (including bipolar outflows from young protostars). The survey uses a rotating half-waveplate polarimeter thatwas developed specifically for use with SCUBA-2 <cit.> and was awarded a total observing time of 224 hours of good weather (“band 2”). As ofearly 2017, some 38% of the programme has been completed, with regions such as Orion A, Ophiuchus, and Serpens Main already observed. An example of one ofthe early observations is shown in Fig. <ref> from Ward-Thompson et al. <cit.> for the central region of the Orionmolecular cloud. The image shows that magnetic field lies perpendicular to the famous “integral-shaped filament” and may beresponsible for “funnelling” matter onto filaments to aid the formation of dense cores that eventually become protostars.1mm1mm§.§.§ SCOPE - SCUBA-2 Continuum Observations of Pre-protostellar Evolution It is known that stars form in the densest regions within molecular clouds, the earliest phase being linked with the so-called pre-stellar cores. However,the formation and early evolution of these cores in different environments is not well known. The SCOPE survey is carrying out an “all-sky” survey at850 m of a sample of 2000 cold clumps identified by the Planck surveyor <cit.>. The JCMT/SCUBA-2 is more sensitive to cold dustthan Herschel, and also has high angular resolution to resolve the substructure of Planck cold clumps. The aims of the surveyinclude the study of how dense cores form and how star formation varies as a function of environment, the universality of filaments in the cold ISM andtheir roles in generating dense cores, how dust properties change in different environments, and how dust properties affect the chemical evolution of densecores. The survey is being supplemented with observations from other millimetre and radio telescopes (such as Purple Mountain observatory and the NobeyamaRadio Observatory 45 m telescope), and will also form a legacy database for such studies with other instruments, especially ALMA. The survey was awarded300 hours of band 3 and 4 time, and as at March 2017 was 69% complete.§.§.§ TRANSIENT - A transient search for variable protostars: How do stars gain their mass? The TRANSIENT program is using SCUBA-2 to measure accretion variability in protostars in eight fields within nearby star-forming regions<cit.>. It has been found that an outburst in accretion luminosity heats the dust in the envelope, which is then seen as brighter emission at850 m <cit.>. Far-IR and submillimetre observations provide a snapshot of accretion rate, averaged over the few-weeks heatingtimescale for the luminosity burst to propagate through the envelope. Given the difficulty of carrying out such long-term observations in the far-IR (theneed for a space-based observatory) submillimetre monitoring may be the only way to probe the earliest stages of stellar growth, since these stars are soheavily embedded they are not visible at optical/near-IR wavelengths. The monitoring programme includes a total of 182 embedded protostars (Class 0/I YSOswith envelopes) and 800 disc/flat-spectrum objects. Each region will be observed once a month for a total period of 3 years to search for signs ofvariability across epochs. The program was awarded 150 hours of observing time, split equally in weather bands 1, 2 and 3. As of March 2017 the program is27% complete, and the first variable candidate has already been identified. §.§ Other science highlights from SCUBA-2 §.§.§ Observations of Comet ISON In late 2013 the JCMT launched a campaign to study the chemistry of the sun-grazing comet C/2012 S1 (ISON). Several groups of observers concentrated onmeasuring the production rate of HCN and water as the comet approached perihelion. The comet was also observed multiple times with SCUBA-2, catching thefinal hours before it disintegrated. Keane led the team on these observations <cit.> which showed that as ISON approached perihelion thecontinuum emission from the nucleus became an elongated dust column spread out over 60 arcseconds (∼10^5 km) in a direction away from the Sun. Oneof the final images reveals distinct clumps, consistent with the catastrophic disruption of the comet, producing ∼5 × 10^10 kg ofmillimetre-sized dust.§.§.§ New insights on planet formation SCUBA-2 has also been used to explore and place new constraints on the dust and gas mass of protoplanetary discs during the giant planet building phase.Williams and collaborators <cit.> surveyed a half-degree field towards the σ Orionis cluster, which contains almost 300 young stellarobjects with estimated ages of 3 Myr. Only nine stars were detected from the observations at 850 m with these having estimated disc masses ofbetween 5 and 17 Jupiter masses. Using a stacking analysis the mean mass for 83 infrared-detected objects that were not detected by SCUBA-2 was determinedto be 0.5 Jupiter masses, effectively ruling them out of ongoing planet formation. The lack of emission illustrates how little raw material must remain inthe environs of the vast majority of these young objects. This suggests that planet forming must start very early on, and that the growth of planetary coresmust be largely complete within a couple of Myr after the host star becomes optically visible.§.§.§ Dust surrounding a pulsar The nearby Geminga pulsar is believed to have crossed the Galactic Plane in the last 100,000 years. Greaves & Holland <cit.> reportthe detection of a shell of material surrounding Geminga that could have formed from compression of the local interstellar medium. A compact source isdetected from 450 m observations which may evidence for the existence of a circum-pulsar disc, the first time any such structure has been detectedin the submillimetre. The inferred mass of dust is expected to exceed 6 Earth masses, and so has the potential to form low-mass planets such as thearchetypes around PSR B1257+12 <cit.>. Further imaging at high angular resolution is planned for this object.§.§.§ Embedded binaries and their dense cores The relationship between young, embedded binary stars and their parent cores is not well understood. Sadavoy & Stahler <cit.> used VLAand SCUBA-2 observations of a number of young stars and cores in the Perseus molecular cloud to explore the origin of binary stars. It was revealed thatmost embedded binaries are found towards the centres of their parent cores. Wide (>500 AU separation) binaries tend to be aligned with the long axes ofthe core, whilst tight systems show no preferred orientation. The authors tested a number of evolutionary models in an attempt to account for thepopulations of both single and binary Class 0 and I sources. The model that best fits the observations suggests that all stars form initially as widebinaries, and then either break up into separate stars or shrink into tighter orbits. Future observations will explore whether the high mass fraction ofdense cores that become stars in Perseus is similar in other star-forming regions.§.§.§ SUPER GOODS: ultra-deep imaging of the GOOD-N field In addition to the extensive surveys that formed the Cosmology Legacy survey, Cowie and co-workers <cit.> carried out ultra-deep imaging withSCUBA-2 of the GOODS-N field. The maps, covering 450 arcmin^2, detected 31 and 186 sources at 450 m and 850 m, respectively, and reachedRMS noise levels well below the confusion limit at 850 m. Using extensive VLA and SMA observations to pinpoint exact galaxy locations, and Keckspectra to determine resdhifts, it was shown that the star formation rate of these galaxies reaches a peak at z = 2 – 3, before dropping at higherredshifts. It was also suggested that the shape in the number density of galaxies per unit volume as a function of star formation rate is invariant overthis particular redshift range.§.§.§ The space density of galaxies at z > 4 Until the advent of Herschel only a handful of dusty star forming galaxies were known to exist at redshifts greater than 4 and most of thesewere amplified by gravitational lensing. Ivison and co-workers <cit.> selected 109 galaxies for SCUBA-2 imaging based on their extremely redfar-infrared colours and faint 350 m and 500 m fluxes from the Herschel-ATLAS imaging survey. The addition of the submillimetre dataallowed the peak of the spectral energy distribution to be identified and so led to better constraints on the redshifts of these objects. The galaxies weredetermined to be in the redshift range 3.3 to 4.3 (median value of 3.66), with a third lying at z > 4 suggesting a space density of ∼6 ×10^-7 Mpc^-3. The sample contains some of the most luminous star-forming galaxies and the most overdense cluster of early starburst ellipticalsknown to date. § SCIENCE WITH THE JCMT HETERODYNE INSTRUMENTATION At the time of the dedication of the JCMT in March 1987, two heterodyne receivers were in operation on the telescope; namely a polarisation splittingdual-channel 230 GHz band receiver (RxA) <cit.> and a single channel 345 GHz band receiver (RxB) <cit.>. Both receiverswere equipped with Schottky diode mixers. The 230 GHz receiver, RxA, covered the range 220 – 280 GHz using two sets of mixers – one set for the lowerpart and one for the upper part of the band. RxA was operated for a short-time as a dual-channel receiver but for most of the time operated in a hybridmode with mixers in the opposite polarisation covering different frequency ranges. The 345 GHz receiver covered the range 320 – 370 GHz and used acarcinotron as the local oscillator (LO) source.An early goal was to equip the telescope with state-of-the-art single- or dual-pixel SIS receivers andsubsequently array receivers with a priority for the 345 GHz band. This was made more feasible with Canada joining the UK/Netherlands project in thespring of 1987, injecting additional resources into the JCMT instrument development fund. A number of single-feed SIS receivers were deployed in the earlypart of the 1990s followed by polarisation-splitting, dual-channel receivers.1mmIn the early years heterodyne science observations were constrained to single-point spectra (as shown in Fig. <ref>) or making small maps upto a few arcminutes in size. Observing larger areas was too time consuming for the single pixel instruments with their limited sensitivity and spectrometerdump-time. The first change occurred in 1992 with the introduction of the DAS spectrometer <cit.> allowing dump-times to be reduced to a second andthe associated development of “On-The-Fly” software for heterodyne mapping. This, combined with more sensitive instruments, caused a noticeable increasein the data rate and generated a need for more storage space. In 2007 with the Heterodyne Array Receiver Program (HARP) <cit.> the JCMT becamethe premier observatory for mapping lines in the 350 GHz band. HARP has a array of 16 mixers, each spaced by 30 arcseconds on the sky, and was supportedby a 16-channel auto-correlation spectrometer (ACSIS) <cit.> capable of observing a 2 GHz bandwidth in each pixel. HARP is now used extensivelyfor large programmes and is still, after 10 years of operation, a very competitive instrument.In parallel with the development of receivers by the partners, agreements were also made with external groups to bring receivers to the telescope. A345 GHz SIS receiver from Sutton's group was brought in for two short periods in 1989 and 1990. A 600 and 800 GHz receiver from Genzel's group wasused in each year from 1988 to 1996 <cit.>, whilst the South Pole Imaging Fabry-Pérot Instrument (SPIFI) <cit.> visited the JCMT for short periods between 1999 – 2000. Although a few upgrades were carried out, no new heterodyne projects were started after 2000, anda high-frequency receiver project (RxE) was cancelled due to budget constraints. The receiver and spectrometer deployments are summarised inTable <ref> and Table <ref>, respectively. §.§ Chemistry Chemistry is an integral part of heterodyne line observing: without an understanding of the chemistry the data cannot be exploited to the fullest extent.An example is the much studied conversion from observed CO intensity to the total molecular mass. Thus, many papers study chemistry in different regionssuch as galaxies, hot cores, protostellar envelopes and discs, shocks and evolved stellar envelopes. This not only gives a better understanding of thechemistry but also an improved knowledge of the source morphology and physics. The information from spectral scans aims to catalogue molecules, lines andabundances, typically in well-known objects, and often without extensive modelling or analysis of the physical conditions. There have been ten spectralline surveys published using JCMT data covering a large fraction of the 230, 345, 460 and 650 GHz windows. Most of the surveys were conducted in the 1990susing single pixel receivers, and groups, for example, led by Greaves <cit.>, Sutton <cit.> and MacDonald<cit.> studied sources including OMC1, Sgr B2, W3, G34.4, IRAS 16293–2422 and IRC+10216. The citation count is still steadily increasing forthese papers, demonstrating their great legacy value. The JCMT Spectral Line Legacy Survey (SLS) by Plume and co-workers <cit.> was designed tostudy and catalogue the lines in some typical regions – a low-mass core (NGC 1333 IRAS 4), three high-mass cores spanning a range of star-formingenvironments and evolutionary states (W49, AFGL 2591, and IRAS 20126), and the Orion Bar photo-dissociation region. The SLS used HARP and in contrast tomost spectral surveys, a region around the central source was also observed, thereby giving additional information about the morphology and chemicalvariations in the local environs. In their SLS paper on the Orion bar, van der Wiel et al. <cit.> find that the molecular abundances, ingeneral, followed the layered structure as predicted by models of photo-dissociation regions, but there were also discrepancies between the models and theobservations.1mmChemistry and physics in particular regions and/or molecular species have been the topic of a large number of papers. Two such papers by van Dishoeck et al.<cit.> and van der Tak et al. <cit.> have well over 200 citations each. The paper by van Dishoeck is a spectral scan of IRAS16293–2422 and uses the observations to isolate the physical regions around the source with different physical and chemical properties. Results forsulphur and silicon species in the same source were reported by Blake et al. <cit.>, and a similar study of IRAS 16293–2422 using moreextensive JCMT and IRAM observations was published by Caux and collaborators <cit.>. The paper by van der Tak studied thestructure around high-mass YSOs using a number of spectral lines as well as continuum and mid-IR data. The data were used to delineate the physics andchemistry in the different parts of the envelope, where freeze-out of CO was demonstrated as well as grain evaporation in the inner region. Thechemistry and photo-ionisation of the Orion bar was studied by Hogerheijde et al. <cit.> and van der Werf et al. <cit.>, andthese papers resolved not only some of the stratification observed in the earlier mentioned SLS paper, but also concluded that the gas in the bar must beclumpy.1mmSulphur chemistry in hot cores was studied by Hatchell et el. <cit.> who found that abundance ratios of the major sulphur species did notvary between different hot cores, and, with the exception of carbonyl sulphide (OCS), were in agreement with models. Hence, no evolutionary sequence wasfound for hot cores. Van der Tak and van Dishoeck <cit.> used the H^13CO^+ abundance to constrain the cosmic ray ionisation rate in theenvelope of YSOs: cosmic ray ionisation forms H_3^+, which is then destroyed by reaction with CO in molecular regions forming HCO^+. Models ofgas-grain chemistry were tested by van der Tak and collaborators <cit.> by observing H_2CO and CH_3OH towards massive YSOs; a largenumber of lines allowed excitation temperatures and abundances to be determined. The CH_3OH/H_2 abundance shows a jump from 10^-9 to 10^-7that could be attributed to grain evaporation due to radiation based on a corresponding jump in excitation temperature and correlated (IR measuredexcitation) temperature of C_2H_2. Schöier et al. <cit.> observed IRAS 16293–2422 and derived a detailed temperature and densitystructure and a detailed comparison of the observations with models strengthened the evidence for infall in the envelope. Furthermore, the molecularspecies are divided into those that have constant abundance in the envelope and those that have increased abundance close to the core. The latermolecules, like CH_3OH, SO and SO_2, increase in abundance where the temperature can thermally evaporate molecules from grains. Low mass YSOs, suchas IRAS 16293–2422, have hot cores but the chemical timescales are much shorter than in the hot cores of massive YSOs.1mmOrganic molecules were observed towards T Tauri and Herbig Ae stars by Thi et al. <cit.>. The detections showed that the emission was from densegas at moderate temperature with some species, such as CN, enhanced by photo-dissociation. This is consistent with accretion disc models with a coldmid-plane having chemistry affected by freeze-out onto grains, whilst molecules formed by photo-dissociation are enhanced on the disc surface byradiation from the central object. The ortho/para ratio in H_2 was been studied by Pagani and co-workers <cit.> using N_2D^+, N_2H^+and ortho H_2D^+ lines. The ortho/para ratio in H_2 is important for understanding the deuteration amplification in the clouds. Under someconditions the abundance of HDCO and CH_2DOH has been found to be higher than their un-deuterated analogue species, showing a deuterium enhancement of10^6 times the D/H ratio. This only occurs if molecules like CO are heavily frozen-out onto grains. Hence, deuterium enrichment is important forunderstanding and studying the freezing-out of molecules onto grains. Frozen-out deuterium-enriched molecules serve as a marker of pristinematerial in comets. §.§ Extragalactic sources About 150 papers studying galaxies using JCMT heterodyne data have been published, and not unexpectedly, many of the these papers utilise the 346 GHz CO J= 3 – 2 line, which is easily observable during typical Mauna Kea weather conditions. An early example is the paper by Devereux and co-workers<cit.> that reported on observations of CO(3–2) in the centre regions of starburst galaxies. They found that the ratio between the CO(3–2)and CO(1–0) lines was higher in the centre of starburst galaxies than in Galactic molecular clouds, whilst the gas mass was typically 10% of the totaldynamical mass. To explain the difference to Galactic molecular clouds from earlier studies, the larger ratio required a more complicated model than justone in which the gas was hotter. Yao et al. <cit.> extended these investigations and observed 60 IR-luminous sources selected from the SCUBA LocalUniverse Galaxy Survey (SLUG). The authors reported an almost identical average CO(3–2) to CO(1–0) line ratio but with a much larger spread in values,indicating a large variation in excitation of the gas in IR-luminous galaxies. In parallel, higher level CO lines as well as lines from HCN, HCO^+,HNC^+ and CS were used to study the gas excitation in starburst and IR-bright galaxies. The CO(6–5) transition, observed by Harris et al. in 1991<cit.>, clearly showed that the gas in nearby starburst galaxies such as M 82 and NGC 253 was hotter and denser than in typical Galacticclouds. Such studies were extended to high redshift starbursts by Padadopoulous and collaborators <cit.> with the detection of CO(4–3) ingalaxies at redshifts of 3.79 and 3.53. A number of possible sources for the excitation were discussed in these papers, such as violent turbulence, thepresence of OB stars, or cosmic rays due to AGNs. Bradford et al. <cit.> concluded that in the case of NGC 253, the CO(7–6) emission wasexcited by cosmic rays due to the high supernova activity in the region.1mmAalto et al. <cit.> discovered an unexpectedly high HNC/HCN line ratio in star-forming galaxies. In Galactic warm dense gas this ratio is lower,even in photo-dissociation regions. The observed high line ratio was explained by IR excitation of HNC, which has a much lower energy bending mode than HCN,or, alternatively, by X-ray dominated chemistry due to the presence of AGNs. The paper by Greve et al. <cit.> studied the starburst galaxies Arp220and NGC 6240 in several molecular species and transitions. The authors showed that the emission from these molecules traced different densities and thereis a size-density relationship for the gas, similar to, but steeper than that observed in Galactic clouds. The bulk of the gas mass ~(1–2)×10^10 M_⊙ resides in a dense n = 10^5 – 10^6 cm^-3 warm phase. Papadopoulos et al. <cit.> presentedspectral line energy distributions for 70 LIRGs, with the galaxies covering a range of infrared luminosities and morphologies showing a broad range of ISMconditions. On the high excitation side the ISM is dominated by hot (>100 K) and dense (N>10^4 cm^-3) molecular gas with gas mass reservoirs of~(few) 10^9 M_⊙. The authors conclude that the gas excitation in merger driven ULRIGs is dominated by turbulence and cosmic raysrather than UV/optical photons and supernova shocks. This new understanding of the gas phase in massive star-forming galaxies was used to guide laterobservations with the Herschel satellite and ALMA. Another noteworthy and highly-cited galaxy paper was presented by Edge <cit.> who showedthat hot gas in galaxy clusters, cooled by X-ray emission, generates a cooling flow of gas onto the galaxies in the clusters. Searches for CO in thecentral galaxies of clusters with cooling flows had only provided one detection before Edge reported 16 more detections of CO in galaxies at the centre ofclusters with cooling flows.1mmThe NGLS (see Section 4a(iii)) observed an HI-selected sample of 155 galaxies spanning all morphological types with distances less than 25 Mpc. Thesurvey has so far produced 10 papers e.g. Wilson et al. <cit.>. The objective of the heterodyne component of the survey was to study the gasproperties, gas–to–dust ratio and to compare radial profiles of the dust, HI and CO emission. The authors find a wide range of molecular gas massfractions in the galaxies in the sample. By comparing the NGLS data with merging galaxies at low and high redshift, which have also been observed in theCO J = 3–2 line, they show that the correlation of far-IR and CO luminosity shows a significant trend with luminosity. This trend is consistent with amolecular gas depletion time that is more than an order of magnitude faster in the merger galaxies than in nearby normal galaxies.There is also astrong correlation of the L_farIR/L_CO(3-2) ratio with the atomic-to-molecular gas mass ratio. This correlation suggests that some ofthe far-infrared emission originates from dust associated with atomic gas and that its contribution is particularly important in galaxies where most ofthe gas is in the atomic phase.§.§ Clouds, Cores and Galactic structure Observing and mapping molecular clouds and cores in CO or other lines is a common topic for JCMT heterodyne papers, with around 150 papers having “cloud”and/or “core” in their title. Some well-cited examples include the paper by Davis et al. <cit.> who mapped the Serpens molecular cloud in theCO 2 – 1 line and continuum, identifying cores and outflows and estimating their ages. The paper by Kirk and collaborators <cit.> studied thekinematics of dense cores in the Perseus molecular cloud with the N_2H^+ 1 – 0 and CO 2 – 1 lines, and found that the internal motion measured by theN_2H^+ line-width in the SCUBA-selected dense cores was more than sufficient to support against gravitational collapse. Whilst many cloud regions weremapped early on with the JCMT heterodyne instruments, large-scale mapping did not start until 2007. The change was triggered by the decision to allocatelarge amounts of time to large surveys and the arrival of HARP/ACSIS made this feasible on the heterodyne side. This shift in policy was instigated to keepthe JCMT competitive in the era of large submillimetre interferometers as the superior angular resolution from the latter would make them muchbetter-suited to observe compact objects, such as protostellar accretion discs.1mmThe first generation JCMT Legacy surveys began in 2007, concentrating on heterodyne observations (SCUBA-2 would join the surveys, but not until 2012, asdiscussed in Section 3). Of these surveys, SLS and NGLS have already been described, with a third major survey concentrating on mapping the extent of^13CO and C^18O(3 – 2) in a number of molecular clouds in the Gould belt. The results from these surveys have been published in a series ofpapers, including the Orion B region by Buckle et al. <cit.>, the Perseus molecular cloud by Curtis and co-workers <cit.>, Taurus byDavis et al. <cit.>, Serpens from Graves and co-workers <cit.>, Orion A by Buckle et al. <cit.> and the Ophiuchus region byWhite and collaborators <cit.>. As an example of the results, the Buckle et al. paper <cit.> presents temperature, opacity, mass andenergy content and location of outflow regions in Orion B. Several follow-up papers, including those from Curtis et al. <cit.> andDrabek-Maunder and co-workers <cit.>, went on to analyse the data further in terms of detailing the extent and properties of outflows in theregions. In addition, the Perseus region was mapped in the dense gas tracers HCO^+ and HCN by Walker-Smith et al. <cit.>. Outside of theformal legacy surveys, additional surveys included the CO High Resolution Survey (COHRS) by Dempsey et al. <cit.> that mapped the Galacticplane in CO(3 – 2) within the area 10 < l < 65 and |b| < 0.5. The 13CO/C18O(3 – 2) Heterodyne Inner Milky Way Plane Survey (CHIMPS) published byRigby et al. <cit.> covers the Galactic plane area 28 < l < 46 for |b| < 0.5 (see Fig. <ref>. All of the surveys were obtained witha spatial resolution of 14 arcseconds and are publicly available: part of the JCMT's legacy for the future. §.§ Star formation: outflows and discs Spectroscopy of specific star formation regions (as opposed to entire molecular clouds described above) has also been an extensive area of research for theJCMT. Around 200 papers based on JCMT heterodyne data have “star formation”, “outflow” or “disc” in the title. The papers cover many aspects fromcompact accretion discs and cores to envelopes and large-scale outflows. Some the most cited papers in the area of star formation with the JCMT arediscussed below.1mmIndividual outflows have been observed to determine their morphology and physical characteristics. For example, Richer et al. <cit.> mapped theoutflow in Orion B and modelled the outflow as driven by a neutral highly collimated jet, the collimation increasing with velocity in the outflow. Lada andFich 1996 <cit.> observed the outflow in NGC 2264G and again found collimation increasing with outflow velocity. The outflow obeyed a “Hubblelaw” with the gas velocity increasing further away from the central source. The NGC 1333/IRAS4 outflow source was studied by Blake et al. (1995)<cit.> with the authors deducing a depletion of CO and other molecules in the flow, as well as observing the additional presence of SiO in theoutflow. Indeed, SiO was later identified as an outflow indicator by Nasini et al. <cit.>. Large parts of the NGC1333 cloud complex weresurveyed by Knee and co-workers <cit.> identifying 10 protostellar sources, each of which was found to be driving an outflow. A number of surveypapers addressed the issue of whether all protostellar and YSOs have outflows. The paper by Parker et al. <cit.> surveyed IRAS sourcesrepresentative of low-mass YSOs embedded in dark molecular clouds and found outflows in 70% of the targets. Other surveys also detected outflows in largefractions of the targets, suggesting that many, if not all, such objects have outflows. The survey of the Perseus Cloud complex by Hatchell et al.<cit.> found outflows in 65% of the 51 SCUBA-identified cores, and all but four of the outflows were also identified by Spitzer asYSOs. Indeed, only one of the Spitzer sources did not have a detected outflow, again showing the almost complete correlation between YSO andoutflows, and that a large fraction of the cores also have embedded YSOs. Outflows deposit kinetic energy into the circumstellar envelope and cloud, whichhas the potential to stop accretion, disrupt the envelope, and generate turbulence that supports the cloud (or can even disrupt the cloud). Hence, althoughoutflows have a clear impact on the star formation process, there is still no clear consensus of just how significant a factor this is.1mmProtostellar accretion discs were directly studied by using short-baseline interferometry involving the nearby Caltech Submillimeter Observatory (CSO) andthe Smithsonian Millimeter Array (SMA), as discussed in section 5(f). Without the sub-arcsecond resolution afforded by an interferometer, accretiondiscs are unresolved by the JCMT, and the disc emission needs to be disentangled from that of the ambient cloud. This can be achieved in a number of ways;for example, using the chemical or physical characteristics of the region e.g. line line width, or, select sources that have separated from the parent cloudeither spatially or in velocity space (it is not uncommon that T Tauri stars have left their parent cloud or the cloud has been disrupted). The paper byThi et al. <cit.> studied discs around T Tauri and Herbig Ae stars using ISO H_2 and JCMT CO(3 – 2) and CO(6 –5) data, selecting sourcesthat are spatially separated from their ambient clouds. The H_2 emission arises from hot (100 – 200 K) gas whilst the CO emission from cooler (20 –80 K) gas. The lower level CO emission profile was shown to be double peaked, characteristic of a disc in Keplerian rotation. By comparing mass estimatesfrom the CO line and continuum, the CO abundance was found to be lower than predicted, which was attributed to freeze-out in the disc centre andphoto-dissociation on the disc surface. Zadelhoff and co-workers <cit.> observed the sources LkCa 15 and TW Hya in a number of highexcitation lines, showing that the line emission mainly originated from an intermediate disc layer with high densities of 10^6 – 10^8 cm^-3 andmoderately warm temperatures. The authors found evidence for significant freeze-out of CO and HCO^+ at low temperatures, but the abundance in the warmerupper layer was also low and attributed to photo-dissociation. The first detection of DCO^+ in the disc of TW Hya was reported by van Dishoeck et al.<cit.>. The DCO^+/HCO^+ ratio was found to be 0.035 ± 0.015, similar to values in pre-stellar cores. Organic molecules inprotoplanetary discs surrounding T Tauri and Herbig Ae stars were studied by Thi and collaborators <cit.>, using the JCMT for high excitation linesand the IRAM 30 m telescope for low excitation lines. The main conclusions were that abundances were lower compared to the envelopes around protostars.The importance of photo-dissociation is shown by the CN/HCN ratio that is found to be higher than in Galactic photo-dominated regions, which has enhancedCN/HCN abundance ratio due to photo-dissociation. §.§ Solar System: planetary atmospheres and comet chemistry The study of Solar System objects using the JCMT heterodyne instruments started with observations of the Sun during the total Solar eclipse over Hawaii on11th July 1991. As well as continuum observations of sunspots, spectroscopy of the chromosphere during limb occultation was carried out by Lindsey et al.<cit.> resulting in a measurement of the chromospheric temperature profile in the near-millimetre region. The limited bandwidth of theheterodyne instrument reduced the problem with saturation – sensitivity was not an issue. Other Solar System observations included the first detection ofCO and HCN in Neptune by Martin et al. <cit.>, and of CO in the atmosphere of Pluto at millimetre wavelengths by Bockelée-Morvan et al.<cit.> and Greaves et al. <cit.>. The detection of the important catalyst H_2O_2 in the Martian atmosphere by Clancy andco-workers <cit.> was the first such detection in a planetary atmosphere outside that of the Earth. Venus is the planetary atmospheremost studied with the JCMT and a dozen papers, including those by Clancy et al. <cit.> and Sandor and collaborators <cit.>, havereported observations of temperature structure, wind patterns through the Doppler shifts and atmospheric chemistry and its variability.1mmComets are by far the most studied Solar System objects with the JCMT heterodyne instruments, having been observed and monitored from the early 1990s. Anearly influential paper by Senay et al. <cit.> highlighted that enough CO gas was observed to explain coma outburst in the most distant comets. For comets close to the Sun, the sublimation of water ice is a dominant driver, but the temperature in distant comets is too cold to allow sublimation togenerate the outbursts. The JCMT has also monitored the gas abundance in a number of comets often in conjunction with other telescopes. The cometsHale-Bopp and Hyakutake were studied in a number of highly-cited papers by Biver and collaborators <cit.> <cit.> <cit.>. Thedetection of HNC from Comet Hyakutake by Irvine et al. <cit.> was first seen as evidence for the existence of interstellar ices in comets. TheHNC/HCN ratio was found to be similar to the interstellar gas phase value and higher than the equilibrium ratio expected in the outermost Solar nebula,where comets are thought to be formed. The HNC ratio was later explained by the same authors <cit.> as being due to photo-chemistry in the cometcoma and not an indication of interstellar origin, whereas isotopic studies supported the view that comets contained pristine unprocessed interstellarices. Meier et al. <cit.> reported the third detection of HDO in a long-period, Oort Cloud comet; the data all giving HDO/H_2O abundancesratios about twice the terrestrial ocean values. Such detections did not support the view that comets supplied the majority of the water for Earth'soceans. The detection of DCN in Comet Hale-Bopp by Meier at al <cit.> revealed an even higher deuterium enrichment in HCN of about 6–8 timesthe ratio in H_2O. Different deuterium enrichment is a hallmark of interstellar ion-neutral and grain-phase chemistry, whilst it is not expected inmaterial processed in denser and warmer part of the Solar nebula. These observations were strong evidence for the presence of interstellar ices in comets. §.§ Interferometry: accretion discs and supermassive black holes There was early interest in using the JCMT as part of an interferometer, with the first experiment taking place in January 1992. This was a millimetre-waveVLBI including the JCMT, Nobeyama, SEST and OVRO. No fringes, however, were found. About the same time the first tests were carried with the Short BaselineInterferometer (SBI) between the JCMT and the CSO <cit.> <cit.>. It operated at the 230, 345 and 460 GHz bands and was one of thefirst, if not the first interferometer operating at submillimetre wavelengths. The two-element interferometer became a forerunner of the high-frequencyinterferometers operating today, such as the Plateau de Bure, SMA and ALMA. The main scientific contribution of SBI was the detection of accretion discsaround a number of YSOs. With a resolution of 1 arcsecond or better it could resolve the accretion disc down to a size of ∼70 AU for nearby objects.Lay et al. <cit.> observed HL Tau and L1551 IRS5, partly resolving the sources with disc major axes of 60 AU and 80 AU respectively, whilst theminor axis was constrained to be <50 AU. The elongation of the disc was perpendicular to the outflow, as expected. In total, 16 protostellar sources wereobserved including class 0, I and II YSOs, with work by Brown et al. <cit.> confirming accretion discs with masses of at least 10^-2 M_⊙.Wiedner and collaborators <cit.> also carried out the first submillimetre interferometric observations of Arp 220 in both line and continuum.The continuum emission at 342 GHz clearly was binary in the east–west direction with a separation of 1 arcsecond. The CO(3–2) line showed a binarysource at some velocities but a more extended structure at other velocities.1mmThe construction of the SMA on Mauna Kea made it possible for both the JCMT and CSO to join the array of eight antennas, with the project becoming known asthe extended SMA (eSMA). Such an extension would contribute significantly longer baselines whilst affording more sensitivity by doubling the collectingarea, thus allowing the observation of weaker sources with even higher angular resolution than was possible with the SMA alone. The first science campaignoccurred in April 2008 with Bottinelli et al. <cit.> reporting the first detection of CI absorption towards the lensed system PKS1830–211at z = 0.866. The results also showed that it was possible to resolve regions with different CI/CO ratios in the image. Other projects includedobservations of the inner envelope of IRC+10216 by Shinnaga and collaborators <cit.> where the HCN maser emission was resolved andvibrationally excited KCl masers were detected for the first time. Two regions were observed in the inner envelope; the acceleration zone R < 5 R_* and ashell zone with a velocity close to the terminal expansion velocity. The shell zone extends to 30 R_* and has a clumpy structure in HCN(3 – 2) emissionin the v = (0, 1^1e, 0) state. Other papers reported on the formation of circumstellar discs around protostellar objects. One example was thedynamical velocity field of IRAS 16293-2422, investigated by Favre et al. <cit.> to within 50 AU from the central object, with the rotationdeviating from Keplerian due to the disc mass being dynamically significant. Technical development at the SMA together with the pending arrival of ALMA ledto a gradual decline in the use of eSMA.1mmIn April 2007 three telescopes, namely the JCMT, one antenna of the CARMA array in California and the SMT telescope in Arizona, observed Sgr A* at 230 GHzin the Galactic Centre. Doeleman and collaborators <cit.> reported, for the first time, on resolved structures of the size of the eventhorizon around the super-massive black hole (SMBH) at the centre of the Milky Way. The detected source size of ∼40 micro-arcseconds is slightlysmaller than the expected size of the event horizon of the (presumed) black hole, suggesting that the bulk of the SgrA^* emission may not be centred onthe black hole, but instead arises in the surrounding accretion flow. The project to image the shadow of the black hole at the Galactic Centre was laternamed the Event Horizon Telescope (EHT). Up until 2013 only the three original telescopes participated in the EHT, but subsequently a phased SMA joined in,and currently eight telescope are involved. Observations are now dual-polarisation, whilst other SMHB candidate sources, such as the centre of the Virgocluster M87, have also been observed <cit.>. Current observations are being interpreted by a range of different models including the geometriccrescent model proposed by Kamruddin & Dexter <cit.> (see, for example, Fig. <ref>), which qualitatively provides an excellentstatistical description of the existing data. The first real resolved image of the shadow of the black hole at the centre of our Galaxy is expected in thenext few years.1mm§ CONCLUDING REMARKS Looking back on the past history of the telescope it is clear that the scientific impact relied on a number of timely technological innovations in newinstrumentation. Fig. <ref> shows the number of (peer-reviewed) papers over the past 30 years split between the three continuum instruments, thesingle/dual pixel heterodyne receivers and the HARP array. The plot is dominated by SCUBA with just over 50% of the total number of papers. In terms of anoverall legacy, SCUBA had, without a doubt, one of the biggest impacts of any instrument built for an astronomical telescope. In the period 1997 – 2005 itrevolutionised our knowledge in a number of areas of astronomy. In particular, it led to a major advance in the understanding of the astronomical originquestions: how planets, stars, and galaxies form. SCUBA revealed discs of cold dust around nearby stars that are evidence that planet formation is ongoingor has already occurred. It has detected large numbers of young protostars and “pre-stellar cores” – objects on the brink of becoming stars, makingpossible the first statistical studies of the earliest stage of star formation. Finally, and perhaps most significantly, very shortly after the detectionof a strong submillimetre background, SCUBA showed that this background is composed of high-redshift, ultraluminous, dusty galaxies. These galaxies haveall the properties expected for elliptical galaxies in their formation stage, objects which have been looked for in vain for over a decade with opticaltelescopes. Indeed, the paper describing this seminal discovery now has over 1000 citations, making it by far the most cited scientific paper in thehistory of the JCMT. Further evidence of the scientific impact of the JCMT came from an analysis of the productivity/impact of 36radio/millimetre/submillimetre telescopes carried out by Trimble & Zaich <cit.> for the year 2001, which showed that the JCMT was the mostsuccessful facility with 21.1 citations per paper.1mmNow in its 30th year of operation, the JCMT continues to produce world-leading science. As of mid-2017 a number of new large scientific programmes haverecently been awarded time on the telescope, including an extension to the BISTRO magnetic field survey of Gould belt clouds, a new survey to resolve starformation in the Galactic plane with HARP, a dust and gas survey of nearby evolved stars, and an extensive study of the Andromeda galaxy. The East AsianObservatory is now looking for opportunities to expand the capabilities of the telescope with a series of instrument upgrades over the next five years<cit.>. Of particular importance in this regard is to capitalise on the key strengths of single-aperture telescopes in an era that is becomingincreasingly dominated by multi-element interferometers (such as the SMA and ALMA). There are initial design plans for a much larger (of order 100 pixel)850 m heterodyne array to replace the current HARP system. Despite the relatively large field-of-view, particularly compared to the predecessorinstruments on the JCMT, SCUBA-2 has still only covered some 5.3% of the total sky visible from Mauna Kea (as shown in Fig. <ref>). Newtechnologies are also emerging that could see SCUBA-2 upgraded with new detectors or indeed replaced by an even larger format (100,000+ pixel) imagingcamera <cit.>. Finally, a full replacement is planned for the Receiver A (operating at 1.3 mm) to allow science to continue even when theweather is not suitable for submillimetre observations. It is clear from these ambitious plans that the JCMT will have a bright and relevant future. § ACKNOWLEDGMENT For the period 1987 until February 2015 the JCMT was operated by the Joint Astronomy Centre on behalf of the UK Science and Technologies FacilitiesCouncil (STFC), the Netherlands Organisation for Pure Research, and the National Research Council of Canada. From March 2015 the telescope has beenoperated by the East Asian Observatory on behalf of the National Astronomical Observatory of Japan, the Academia Sinica Institute of Astronomy andAstrophysics of Taiwan, the Korea Astronomy and Space Science Institute, the National Astronomica Observatories of China and the Chinese Academy ofSciences. Additional operational support funding is also provided by STFC and participating universities in the UK and Canada. This paper has made use ofNASA's Astrophysics Data System. 300Duncan1990 Duncan WD, Robson EI, Ade PAR, Griffin MJ, Sandell G. 1990, A millimetre/submillimetre common user photometerfor the James Clerk Maxwell Telescope, Mon. Not. R. Astron. Soc. 243, 126–132. (doi:10.1093/mnras/243.1.126) Holland1999 Holland WS, Robson EI, Gear WK, Cunningham CR, Lightfoot JF, Jenness T, Ivison RJ, Stevens JA, Ade PAR, Griffin MJ, et al.1999, SCUBA: A common-user submillimetre camera operating on the James Clerk Maxwell Telescope, Mon. Not. R. Astron. 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"authors": [
"Ian Robson",
"Wayne S. Holland",
"Per Friberg"
],
"categories": [
"astro-ph.GA",
"astro-ph.CO",
"astro-ph.EP",
"astro-ph.IM",
"astro-ph.SR"
],
"primary_category": "astro-ph.GA",
"published": "20170626075823",
"title": "Celebrating 30 Years of Science from the James Clerk Maxwell Telescope"
} |
The optimal particle-mesh interpolation basis Xingyu Gao December 30, 2023 =============================================There exist many ways to build an orthonormal basis of ^̊N, consisting of the eigenvectors of the discrete Fourier transform (DFT). In this paper we show that there is only one such orthonormal eigenbasis of the DFT that is optimal in the sense of an appropriate uncertainty principle. Moreover, we show that these optimal eigenvectors of the DFT are direct analogues of the Hermite functions, that they also satisfy a three-term recurrence relation and that they converge to Hermite functions as N increases to infinity.0.15cmKeywords:eigenvectors, discrete Fourier transform, orthogonal basis,Hermite functions, uncertainty principle 2010 Mathematics Subject Classification : Primary 42A38, Secondary 65T50 § INTRODUCTION AND THE MAIN RESULTSThe following question about the discrete Fourier transform (DFT) has received a lot of attention in the research literature: How can we construct an orthonormal basis of ^̊N consisting of the eigenvectors of the DFT, which would be analogous to Hermite functions and would have some other desirable properties? In terms of “other desirable properties", we might ask forthis eigenbasis to be explicit or, at least, easily computable numerically; it would also be beneficial if the eigenbasis was in some sense unique. The reader can find some examples of such constructions in the papers byDickinson and Steiglitz <cit.>, Grunbaum <cit.>, Mehta<cit.> and Pei and Chang <cit.>. Let us state the main properties of Hermite functions that will be used later. Hermite functions are defined as followsψ_n(x):=(-1)^n (√(π) 2^n n!)^-1/2e^x^2/2^̣n/x̣^n e^-x^2n≥ 0,One can also write ψ_n(x)=(√(π) 2^n n!)^-1/2e^-x^2/2 H_n(x),where H_n are Hermite polynomials. It is well-known that Hermite functions form a complete orthonormal basis of L_2()̊ and that they are the eigenfunctionsof the continuous Fourier transform operator ℱ, defined as(ℱ g)(x)=1/√(2π)∫_ e^-ı x y g(y) ỵ,so that (ℱψ_n)(x)=(-ı)^n ψ_n(x).Many constructions of the eigenbasis of the DFT start either with the fact that the Hermite functions are theeigenfunctions of the operator ℒ:=^̣2/x̣^2-x^2, or that they satisfy therecursion relationsψ_n'(x)+x ψ_n(x)=√(2n)ψ_n-1(x),ψ_n'(x)-x ψ_n(x)=-√(2(n+1))ψ_n+1(x).The goal, typically, is to find a discrete counterpart of the operator ℒ or or the operators/̣x̣+x and /̣d x-x and to use them to construct discrete analogues of Hermite functions. A major problemwith this approach is that there are many ways to approximate a differential operator by a matrix, and this leads to many different constructions and results in lack of uniqueness of the “canonical eigenbasis" of the DFT. However, we do not need to be restricted to using differential operators if we want to characterize Hermite functions. It turns out that a much more fruitful approach, in terms of finding discrete analogues of Hermite functions, is to use Hardy's uncertainty principle. This result was derived back in 1933 and it states that if both functions f and ℱf are O(|x|^m e^-x^2/2) for large x and some m, then f and ℱf are finite linear combinations of Hermite functions (see <cit.>). The following theorem follows easily from Hardy's result. Assume that the functions f_n : ↦̊$̊ satisfythe following three conditions:(i)f_n are the eigenfunctions of the Fourier transform: (ℱ f_n)(x)=(-ı)^n f_n(x) forn≥ 0 and x∈$̊;(ii){f_n}_n≥ 0form an orthonormal set inL_2()̊, that is∫_ f_n(x)f_m(x)x̣=δ_n,m;(iii) for everyn≥ 0we have x^-n-1 e^x^2/2 f_n(x) → 0as|x|→∞. Then for alln≥ 0we havef_n ≡ψ_norf_n ≡ -ψ_n.The main goal of our paper is to find a set of vectors{ T_n}_0≤ n < N-1that wouldbe characterized uniquely by the three conditions similar to items (i), (ii) and (iii) in Theorem<ref>. It is clear what should be the analogues of conditions (i) and (ii) in the discrete setting.It turns out that condition (iii), which essentially expresses Hardy's uncertainty principle, has a counterpart in the formofa discrete uncertainty principle (see <cit.>), which states that the number of non-zero elements of a vector and of its discrete Fourier transform cannot be too small (the signal and its Fourier transform cannot be too localized). In order to state our results, first we need to present necessary definitions and notation. We denote by⌊ x ⌋the floor function and by⌈ x ⌉the ceiling function.LetNbe a positive integer and defineI_N:={ k ∈ℤ: - + 1≤ k ≤}.We consider the elements of a vector a ∈^̧Nas being labelled by the setI_N.Given a vector a, we define anN-periodic function a : ↦$̧ by specifyinga(k)= a(k) for k∈ I_N and extending it by periodicity to all of . This correspondence between vectors and N-periodic functions is clearly a bijection, and we will often view vectors as N-periodic functions and N-periodic functions as vectors, depending on situation; we will use the same notation a to denote both of these objects.The dot product between vectors a and b is denoted by a,b and the norm a is defined by a=√( a, a̅). The (centered) discrete Fourier transform is defined as a linear map that sends a vectora ∈^̧N to b= a∈^̧N according tothe ruleb(l) = 1/√(N)∑_k ∈ I_N e^-ıω k l a(k), l ∈ I_N,where we denoted ω:=2π/N. With the above normalisation,is a unitary operator on ^̧N. The next definition will play a crucial role in our paper.For a vector a ∈^̧N we define ( a) to be the integer n ∈{0,1,…,} such that a(n)0 or a(-n)0, but a(k) = 0 when n < k≤.To illustrate the concept of the width of a vector, we provide the following examples:if a=[0, 0, 1, 2, 3, 0, 0] then ( a)=1; if a=[0, 0, 0, 1, 2, 3, 0] then ( a)=2; if a=[0, 1, 2, 3, 0, 0] then ( a)=1; if a=[1, 2, 3, 0, 0, 0] then ( a)=2; if a=[0, 0, 0, 0, 0, 1] then ( a)=3.For clarity, we have underlined the term a(0). The next theorem is our first main result. For every N≥ 2 there exist unique (up to change of sign) vectors { T_n}_0≤ n<N in ^̊N that satisfy the following three conditions:(i)T_n are the eigenvectors of the DFT: T_n=λ_nT_n for 0≤ n < N, where λ_n=(-ı)^n for 0≤ n < N-1; (ii){ T_n}_0≤ n <N form an orthonormal basis in ^̊N;(iii)( T_n)≤⌊ (N+n+2)/4 ⌋ for 0≤ n<N. Note that Theorem <ref> is a counterpart of Theorem <ref>, as both identify unique orthonormal bases consisting of the eigenvectors of the Fourier transform that are optimal in the sense of corresponding uncertainty principles. Therefore, the following name is appropriate for the eigenbasis constructed in Theorem <ref>. The basis { T_n}_0≤ n<N, which is identified in Theorem<ref>, will be calledthe minimalHermite-type basis of ^̊N.The vectors { T_n}_0≤ n<N were introduced in <cit.>, building upon earlier work of Kong <cit.>. However, the optimality of this basis was not established in <cit.>, and the construction of the basis was done differently depending on the residue of N modulo 4. In the present paper we give a simpler construction of the basis { T_n}_0≤ n<N, which also has the advantage that it does not distinguish between residue classes of N modulo 4. Moreover, from this construction we are able to deduce that( T_n)=⌊ (N+n+2)/4 ⌋ for 0≤ n <N. Our construction provides new information about eigenspaces of the DFT. Let E_m be the eigenspace of the DFT with the eigenvalue (-ı)^m and let S_n be the linear subspace of vectors having width not greater than n. More formally, E_m:={ a ∈^̊N : a = (-ı)^ma}, S_n:={ a ∈^̊N :( a)≤ n},wherem ∈{0,1,2,3} and 0≤ n ≤. It is clear that for 0≤ n < we have S_n ⊂ S_n+1 and (S_n)=2n+1, while S_=^̊N. The dimensions of the eigenspaces E_m are also known: (E_m)=-K_m+1 for m ∈{0,2},(E_m)=-K_m for m ∈{1,3}, where K_m:=(N + 2 +m)/4.The result (<ref>) is usually presented in the form of a table, by considering different residue classes of N modulo 4, see Table <ref>.The dimensions of the eigenspaces of the DFT were first found by Schur in 1921, however they can also be easily obtained from a much earlier result of Gauss on the law of quadratic reciprocity. By using this result and the fact that the vectors{ T_n}_0≤ n <N satisfy conditions (i) and (ii) of Theorem <ref>, we can establish the eigenvalue corresponding to the eigenvector T_N-1: T_N-1=(-ı)^N-1 T_N-1 (respectively, T_N-1=(-ı)^NT_N-1) if N is odd (respectively, if N is even). For this reason we will find it convenient to introduce a “ghost" vector T_N= T_N-1, so that we can restore the symmetry and have T_N=(-ı)^NT_N when N is even. The following result, which follows from our construction of the minimal Hermite-type basis, provides more detailed information about the eigenspaces of the DFT. Let { T_n}_0≤ n<N be the minimal Hermite-type basis of ^̊N and denote T_N:= T_N-1 and K_m:=(N + 2 +m)/4.(i) If 0≤ n < K_0 then (E_0 ∩ S_n )=0. If K_0 ≤ n ≤ then(E_0 ∩ S_n )=n-K_0+1 and the vectors { T_4l}_0≤ l ≤ n-K_0 form an orthonormal basis of E_0 ∩ S_n. (ii) If 0≤ n < K_1 then (E_1 ∩ S_n )=0. If K_1 ≤ n < then(E_1 ∩ S_n )=n-K_1+1 and the vectors { T_4l+1}_0≤ l ≤ n-K_1 form an orthonormal basis of E_1 ∩ S_n. (iii) If 0≤ n < K_2 then (E_2 ∩ S_n )=0. If K_2 ≤ n ≤ then(E_2 ∩ S_n )=n-K_2+1 and the vectors { T_4l+2}_0≤ l ≤ n-K_2 form an orthonormal basis of E_2 ∩ S_n. (iv) If 0≤ n < K_3 then (E_3 ∩ S_n )=0. If K_3 ≤ n < then(E_3 ∩ S_n )=n-K_3+1 and the vectors { T_4l+3}_0≤ l ≤ n-K_3 form an orthonormal basis of E_3 ∩ S_n.We have argued above that the minimal Hermite-type eigenvectors { T_n}_0≤ n<N are analogues of Hermite functions {ψ_n}_n≥ 0, since both are characterized in a very similar way via uncertainty principles, as presented in Theorem<ref> and Theorem <ref>. There is another similarity between the vectors { T_n}_0≤ n<N and Hermite functions: they both satisfy a three term recurrence relation. In order to state this result, let us introduce an operatoracting on vectors in ^̊N in the following way:a(k)=a(k + 1) +a(k - 1) + 2 cos(ω k)a(k).Note that here we interpret vectors a and a as N-periodic functions on ℤ, see the discussion onpage page_N_periodic. It is clear thatis a self-adjoint operator and it is also known that it commutes with the DFT (see <cit.>).Let { T_n}_0≤ n<N be the minimal Hermite-type basis of ^̊N and denote T_N:= T_N-1. Then for 0≤ n <N-5 we haveT_n+4=( T_n-a_nT_n-b_n-4 T_n-4)/b_n,where the coefficients a_n and b_n satisfya_n= T_n,T_nandb_n^2= T_n-a_nT_n-b_n-4 T_n-4^2= T_n^2-a_n^2-b_n-4^2.In the above formulas we interpret b_n=0 and T_n=0 for n<0.When N is odd (respectively, even) formula (<ref>) holds also for n=N-5(respectively, n=N-4).In Section <ref> we will give explicit formulas for the vectors T_n for n=0,1,2,3. These explicit formulas, combined with the three-term recurrence (<ref>), lead to a simple and efficient algorithm for computing the remaining vectors { T_n}_4≤ n <N. We will discuss this numerical algorithm in Section <ref>. Given the many similarities between the vectors T_n and Hermite functions,it is natural to askwhether T_n converge to the corresponding Hermitefunctions ψ_n. This result was established in <cit.> for n≤ 7 when N≡ 1 (mod 4), and it was conjectured that this convergence holds true for all n (and all residue classes of N modulo 4). The following theorem confirms this conjecture and provides precise information about the rate of convergence. Let N≥ 2, ω=2π /N and { T_n}_0≤ n<N be the minimal Hermite-type basis of ^̊N. Definethe sequence of vectors {_n}_n≥ 0as _n(k)=√(ω)×ψ_n(√(ω) k),k ∈ I_N.Then it is possible to choose the signs of vectors T_n in such a way that for every n≥ 0 and any ϵ>0 we haveT_n - _n =O(N^-1+ϵ) as N→ +∞. The rest of the paper is organized as follows. In Section <ref> we give an explicit construction of the minimal Hermite-type basis and we prove Theorem <ref>, Proposition <ref> and Theorem<ref>. In Section <ref> we prove Theorem <ref>. Finally, in Section <ref> we discuss an algorithm for numerical computation of the minimal Hermite-type basis. § CONSTRUCTING THE MINIMAL HERMITE-TYPE BASISOur goal in this section is to construct the basis { T_n}_0≤ n <N and to investigate its properties, and ultimately to prove Theorem <ref>. While the construction is essentially the same as in <cit.>, the present version is simpler, both in notation and proofs. Another advantage of the present construction is that it does not distinguish between different residue classes of N modulo 4.We recall that ω=2π /N. We define S(0)=1 andS(k)= ∏_j = 1^k (2 sin(ω j / 2)), k≥ 1.In the next Lemma we list some properties of the sequence {S(k)}_k≥ 0.(i)S(k) S(N - 1 - k) = S(N - 1)=N when 0 ≤ k < N. (ii) If N is odd, thenS( - k) S( + k)=S( - k) S( + k)= N, |k|≤.(iii)If N is even, then[eq:ss2]S( - k) S( + k) = 2 N cos(ω k / 2) ,|k|≤, S( - k) S( + k) = N/2 cos(ω k / 2) |k|≤. The fact S(k) S(N - 1 - k) = S(N - 1) follows from (<ref>) by using formula sin(ω j/2)= sin(ω (N-j)/2). Next, evaluating the identity| ∑_j = 0^N - 1 z^j | = ∏_j = 1^N - 1z - e^ω j for z = 1, we obtain N = ∏_j = 1^N - 11 - e^ω j = S(N - 1),which ends the proof of item (i). Items (ii) and (iii) follow easily from (i).If N is even (respectively, odd) we set α_0:=1/2 (respectively, α_0:=1).For 1 ≤ n ≤ we defineα_n:=S(n)^-2 (S(2n))^1/2,if N is odd, S(n)^-2 (S(2n-1) sin(ω n/2))^1/2, if N is even.Similarly, for 0 < n < we defineβ_n:=S(n)^-2 (S(2n-1))^1/2, if N is odd,S(n)^-2 (S(2n-1) cos(ω n/2))^1/2, if N is even.We also denotet_k:=2/√(ω)sin( ω k/2).When 0 ≤ n ≤, we define the Gaussian-type vector u_n byu_n(k)=α_n ∏_j=n+1^( 1 - (t_k/t_j)^2 ), k∈ I_N.If n= the empty product is interpreted as one, thus u_(k)=α_ for k ∈ I_N. When 0 < n <, we define the modified Gaussian-type vector v_n byv_n(k)=β_n sin(ω k) ∏_j=n+1^( 1 - (t_k/t_j)^2 ),k∈ I_N.If n= the empty product is interpreted as one, thus v_(k)=β_sin(ω k) for k ∈ I_N. The vectors u_n and v_n were introduced in <cit.> and <cit.>, following the earlier work of Kong<cit.>. Note, however, that these vectors had different normalization constants in <cit.> and were labelled with different index in <cit.>.To simplify the statement of results, whenever we write an identity involving u_n, v_n or any of a number of objects introduced below, we implicitly assume that n is in the admissible range; for example, 0 ≤ n ≤ when speaking about u_n.We call a vector a ∈^̊N even (respectively, odd) if the corresponding N-periodic function is even (respectively, odd).Note that the vectors u_n are even and vectors v_n are odd.Assume that N≥ 2 and k∈ I_N. (i) The following identities are true:u_n(k) =α_n S(n)^2/S()^2∏_j=n+1^ (2cos(ω k)-2cos(ω j)), v_n(k) =β_n S(n)^2/S()^2sin(ω k) ∏_j=n+1^ (2cos(ω k)-2cos(ω j)).(ii)If N is odd then u_n(k)=α_nS(n)^2/ N^2S(N-n-1-k)S(N-n-1+k),and if N is even and n<N/2 thenu_n(k)=α_n S(n)^2/N^2cos(ω k/2)S(N-n-1-k)S(N-n-1+k),whereas if N is even and n=N/2 we have u_n(k)=1/(2√(N)). (iii) If N is odd thenv_n(k)=β_n S(n)^2/N^2sin(ω k) S(N-n-1-k)S(N-n-1+k),and if N is even thenv_n(k)=2β_n S(n)^2/N^2sin(ω k/2) S(N-n-1-k)S(N-n-1+k).Formulas (<ref>) and (<ref>) follow from (<ref>) and (<ref>) by noting thatω t_k^2=4 sin^2(ω k/2)=2-2cos(ω k)and using the identity ∏_j=n+1^m (ω t_j^2)=∏_j=n+1^m (2 sin(ω j/2))^2=S(m)^2/S(n)^2, with m= or m=. In order to prove formula (<ref>) we write for |k|≤ n u_n(k) =α_n S(n)^2/S()^2∏_j=n+1^ (2cos(ω k)-2cos(ω j))=α_n S(n)^2/S()^2∏_j=n+1^ (2 sin(ω (j-k)/2) 2 sin(ω (j+k)/2))=α_n S(n)^2 S(-k)S(+k)/S()^2S(n-k) S(n+k).After simplifying the above expression using Lemma <ref>, we see that formula (<ref>) is valid for all |k|≤ n. We extend its validity to all k∈ I_N, since both the left-hand side and the right-hand side are zero when n<|k|. The proof of formulas (<ref>), (<ref>) and (<ref>) follows exactly the same steps. We leave the details to the reader. The following result is the key to constructing the basis { T_n}_0≤ n <N. This result was first established in <cit.> via q-binomial Theorem; here we give a simpler proof based on Lemma <ref>.For all admissible n we have u_n=u_ - n andv_n = - v_ - n. The proof is based on the following observations:(i)if a ∈^̊N is even, then a ∈^̊N and is also even,and both a and a are polynomials of cos(ω k) (of degree ( a) and ( a), respectively);(ii)if a ∈^̊N is odd, then a ∈^̊N and is also odd,and both a and a are polynomials of cos(ω k) (of degree ( a) - 1 and ( a) - 1, respectively) multiplied by sin(ω k). Formula (<ref>) implies that u_n(k)=P_n(2cos(ω k)) for a polynomial P_n(z) of degree - n, which has- n zeroes located at points z=cos(ω j) with n < j ≤. Note, furthermore, that these properties describe the polynomial P_n(z) uniquely, up to multiplication by a constant.Next, we write 2cos(ω k)=e^ıω k+e^-ıω k, and we note that P_n(2cos(ω k)) can be written as a sum ofterms e^ıω k l with |l|≤ -n. It follows that u_n(l) = 0 when -n < |l| ≤. On the other hand, according to property (i) above, u_n(k)=Q_n(2 cos(ω k)) for some polynomialQ_n(z) of degree n. Since we already know that Q_n has n zeroes located at points z=cos(ω j) with - n < j ≤, it follows that Q_n = C_n P_ - n for some constant C_n. We conclude that u_n = C_nu_ - n.Now our goal is to identify the constant C_n. We observe that formula (<ref>) tells us that P_n(z)=α_n S(n)^2/S()^2 z^-n+ a polynomial of lower degree. Since u_n(k)=P_n(2cos(ω k))=P_n(e^ıω k+e^-ıω k) we conclude that the coefficient ate^ω k ( - n) in the expansion of u_n(k) in powers of e^ω k is equal to α_n S(n)^2/S()^2. Since u_n=^-1 ( u_n)=C_n ^-1 u_ - n, we see that the coefficient ate^ıω k ( - n) is also equal to C_n u_ - n( - n)/√(N). Thus we obtain an identityα_n S(n)^2/S()^2=C_n/√(N) u_ - n( - n),and we need to verify that this identity implies C_n=1.When N is odd, we use formulas (<ref>), (<ref>) and Lemma <ref> to find that α_n S(n)^2=S(2n)^1/2, S()^2=N and u_ - n( - n)=1/N S(N-1-2n)^1/2 S(2n).Substituting the above results into (<ref>) and using the identity S(N-1-2n)S(2n)=N we conclude that C_n=1. When N is even, we use formulas (<ref>), (<ref>) and Lemma <ref> to find thatα_n S(n)^2=(S(2n-1) sin(ω n/2))^1/2,S()^2=2N and u_ - n( - n)=1/N (S(N-1-2n)cos(ω n/2))^1/2 S(2n-1)Substituting the above results into (<ref>) and using the identityS(N-1-2n)S(2n-1)cos(ω n/2) sin(ω n/2)=N/4 we again conclude that C_n=1.In a similar way, one shows that v_n(k) = 2 sin(ω k) P̃_n(2 cos(ω k)) for a polynomial P̃_n of degree - n - 1, with zeroes at 2 cos(ω j), n < j <; and that v_n(k) = -2 sin(ω k) Q̃_n(2 cos(ω j)) for a polynomial Q̃_n of degree n - 1, with zeroes at 2 cos(ω j), - n < j <. It follows that v_n = -C̃_nv_ - n for some constant C̃_n.In order to show that C̃_n = 1, we follow the same argument as in the first part of the proof and we conclude that β_n S(n)^2/2 S()^2=C̃_n/√(N) v_ - n( - n). When N is odd, we use formulas (<ref>), (<ref>) and Lemma <ref>to find that β_n S(n)^2=S(2n-2)^1/2, S()^2=N and u_ - n( - n)=1/2N S(N-2n)^1/2 S(2n-1).Substituting the above results into (<ref>) and using the identity S(N-2n)S(2n-1)=N we conclude that C̃_n=1.When N is even, we use formulas (<ref>), (<ref>) and Lemma <ref>to find that β_n S(n)^2=(S(2n-1)cos(ω n/2))^1/2, S()^2=N/2 and u_ - n( - n)=2/N (S(N-1-2n)sin(ω n/2))^1/2cos(ω n/2) S(2n-1).Substituting the above results into (<ref>) and using the identityS(N-1-2n)S(2n-1)cos(ω n/2) sin(ω n/2)=N/4we conclude that C̃_n=1. We define K_m:=(N + 2 +m)/4 andw_n :=u_n +u_ - nwhen K_0 ≤ n ≤;x_n :=v_n +v_ - nwhen K_1 ≤ n <;y_n :=u_n -u_ - nwhen K_2 ≤ n ≤;z_n :=v_n -v_ - nwhen K_3 ≤ n <.Let us explain the motivation behind this definition.First of all, one can check that K_0= /2, K_1=/2, K_2=/2 + 1, K_3=/2 + 1.Next, given any even vector a, the vector a ± a is an eigenvector of the DFT with corresponding eigenvalue ± 1. Thus, we haveeigenvectors u_n+u_-n of the DFT with eigenvalue 1, but some of these will be repeated twice. It is easy to check that there are exactly -K_0+1 distinct eigenvectors of the formu_n+ u_-n.Similar considerations apply to vectors u_n- u_-n, the difference with the previous case is that one of these vectors may be zero: this happens if 2n= for some n. Thus, one can check that there exist precisely -K_2+1 distinct eigenvectors of the form u_n-u_-n.The same considerations apply in the case of odd eigenvectors v_n ± v_ - n: here we would use the fact that if a is an odd vector, then a ± a is an eigenvector of the DFT with the corresponding eigenvalue ∓ i.We recall that the subspaces E_m and S_n were defined in (<ref>). In the following proposition we collect some important properties of vectors w_n, x_n, y_n and z_n.(i) For K_0 ≤ n ≤ the vectors { w_l}_K_0≤ l ≤ n form the basis of the subspace E_0 ∩ S_n. In particular,(E_0 ∩ S_n )=max(n-K_0+1,0) for 0≤ n ≤.(ii) For K_1 ≤ n < the vectors { x_l}_K_1≤ l ≤ n form the basis of the subspace E_1 ∩ S_n. In particular,(E_1 ∩ S_n )=max(n-K_1+1,0) for 0≤ n <. (iii) For K_2 ≤ n ≤ the vectors { y_l}_K_2≤ l ≤ n form the basis of the subspace E_2 ∩ S_n. In particular,(E_2 ∩ S_n )=max(n-K_2+1,0) for 0≤ n ≤.(iv) For K_3 ≤ n < the vectors { z_l}_K_3≤ l ≤ n form the basis of the subspace E_3 ∩ S_n. In particular,(E_3 ∩ S_n )=max(n-K_3+1,0) for 0≤ n <. Note that for all admissible n we have(a)( w_n)=( x_n)=( y_n)=( z_n)=n,(b)w_n= w_n, x_n=-ı x_n, y_n=- y_n, z_n=ı z_n. Let us denote d^(m)_n=(E_m ∩ S_n).Note that a vector of width lcannot be obtained as a linear combination of vectors of strictly smaller width, thus the vectors{ w_l}_K_0≤ l ≤ are linearly independent. In particular, for K_0 ≤ n ≤ the vectors{ w_l}_K_0≤ l ≤ n are linearly independent and they lie in E_0 ∩ S_n due to items (a) and (b) above. This gives us the following inequalities:d^(0)_n<d^(0)_n+1 forK_0≤ n < andn-K_0+1≤ d^(0)_n forK_0≤ n ≤. Similarly,n-K_1+1≤ d^(1)_n forK_1≤ n < ,n-K_2+1≤ d^(2)_n forK_2≤ n ≤,n-K_3+1≤ d^(3)_n forK_3≤ n < .Using the fact that E_0 ∩ S_=E_0, E_1 ∩ S_=E_1, E_2 ∩ S_=E_2, E_3 ∩ S_=E_3,we conclude that d^(0)_+d^(1)_-1+d^(2)_+d^(3)_-1=∑_m=0^3 (E_m)=N.In deriving the above identity we have also used the fact that the linear subspaces E_m are orthogonaland E_0+E_1+E_2+E_3=^̊N. Next, one can check that (-K_0+1)+(-K_1)+(-K_2+1)+(-K_3)=N. The above result, combined with (<ref>) and the inequalities (<ref>)–(<ref>) proves that d^(m)_=(E_m)=-K_m+1 for n even andd^(m)_=(E_m)=-K_m for m odd.Note that this is equivalent to Schur's result as presented in Table <ref>. Now, considering the case m=0, we summarize what we have proved so far. We know that n-K_0+1 ≤ d^(0)_n for K_0≤ n ≤,d^(0)_n<d^(0)_n+1for K_0 ≤ n < ,d^(0)_=-K_0+1. These results imply that d^(0)_n=n-K_0+1 forK_0≤ n ≤. Since the n-K_0+1 vectors { w_l}_K_0≤ l ≤ n are linearly independent and they lie in the linear subspace E_0 ∩ S_n of dimension d^(0)_n=n-K_0+1, these vectors form the basis for E_0 ∩ S_n. This ends the proof of item (i). The proof of remaining items (ii), (iii) and (iv) follows exactly the same steps. We define { W_n}_K_0 ≤ n ≤, { X_n}_K_1 ≤ n <, { Y_n}_K_2 ≤ n ≤ and{ Z_n}_K_3 ≤ n < to be the sequences of unit vectors,obtained by applying Gram–Schmidt ortogonalisation to the corresponding sequences { w_n}_K_0 ≤ n ≤, { x_n}_K_1 ≤ n <, { y_n}_K_2 ≤ n ≤ and{ z_n}_K_3 ≤ n <. Furthermore, we defineT_n, 0 ≤ n < N, to be the rearrangement of the vectors W_n, X_n, Y_n and Z_n, obtained by enumerating the rows of the tableT_0 =W_K_0 , T_1 =X_K_1 , T_2=Y_K_2 , T_3=Z_K_3 , T_4 =W_K_0 + 1 , T_5 =X_K_1 + 1 , T_6=Y_K_2 + 1 , T_7=Z_K_3 + 1 , T_8 =W_K_0 + 2 , T_9 =X_K_1 + 2 , T_10=Y_K_2 + 2 , T_11=Z_K_3 + 2 , …………Note that the columns of the above table have unequal length.In the caseN≡ 0 (mod 4) we have =, K_3=K_2=K_1+1=K_0+1 and the last three rows of the table (<ref>) are …………T_N-8=W_-2 , T_N-7=X_-2 , T_N-6=Y_-1, T_N-5=Z_, T_N-4=W_-1 , T_N-3=X_ , T_N-2=Y_,T_N-1=W_.Similarly, when N≡ 1 (mod 4) we have =, K_3=K_2=K_1=K_0+1 and the last three rows of the table (<ref>) are …………T_N-9=W_-2 , T_N-8=X_-2 , T_N-7=Y_-1, T_N-6=Z_-2, T_N-5=W_-1 , T_N-4=X_ , T_N-3=Y_, T_N-2=Z_, T_N-1=W_;when N≡ 2 (mod 4) we have =, K_3=K_2=K_1=K_0and the last three rows of the table (<ref>) are …………T_N-10=W_-2 , T_N-9=X_-2 , T_N-8=Y_-2, T_N-7=Z_-2, T_N-6=W_-1 , T_N-5=X_ , T_N-4=Y_-1, T_N-3=Z_, T_N-2=W_,T_N-1=Y_;and when N≡ 3 (mod 4) we have =, K_3-1=K_2=K_1=K_0 and the last three rows of the table (<ref>) are …………T_N-11=W_-2 , T_N-10=X_-3 , T_N-9=Y_-2, T_N-8=Z_-2, T_N-7=W_-1 , T_N-6=X_-2 , T_N-5=Y_-1, T_N-4=Z_, T_N-3=W_,T_N-2=X_, T_N-1=Y_.The above examples show that when N is odd, the vectors { T_n}_0≤ n <N are a straightforward rearrangement of the vectorsW_n, X_n, Y_n and Z_n as shown in (<ref>). When N is even, then the vectors{ T_n}_0≤ n <N-1 are also a straightforward rearrangement of the vectorsW_n, X_n, Y_n and Z_n as shown in (<ref>), but the last vector T_N-1“skips" one spot in this table so that it corresponds to a W vector or aY vector. Thus, the last vector T_N-1 always has a real eigenvalue, irrespective of the residue class of N modulo 4. Also, note that the index of vectors T increases by four along each column in(<ref>)-(<ref>), except that if N is even then the column containing the vectorsT_N-8 and T_N-4 ends in T_N-1 and not in T_N. This discrepancy explains why in the statements of our results we introduce a “ghost" vector T_N= T_N-1: this simple trick allows us to restore the pattern of indices increasing by four along each column and makes the statements of our results more precise.Proof of Theorem <ref>: First let us establish “existence" part of Theorem <ref>: we will check that the vectors T_n constructed above satisfy conditions (i)-(iii) of Theorem <ref>. We note that( W_n)=( X_n)=( Y_n)=( Z_n)=nandW_n= W_n, X_n=-ı X_n, Y_n = -Y_n, Z_n=ı Z_nfor all admissible n. Thus, by construction, it follows that { T_n}_0≤ n <N is an orthonormal basis of ^̊N, and thatT_n=(-ı)^nT_n and ( T_n)=⌊ (N+n+2)/4 ⌋ for 0≤ n < N-4.Considering the case N≡ 0 (mod 4) (see (<ref>)), we see thatT_n=(-ı)^nT_n for N-4 ≤ n <N-1 and ( T_n)=⌊ (N+n+2)/4 ⌋ for N-4 ≤ n < N.Thus in the case N≡ 0 (mod 4) the vectors { T_m}_0≤ m ≤ N-1 satisfy conditions (i)-(iii) of Theorem <ref>. The remaining cases N≡ 1,2,3 (mod 4) can be considered in exactly the same way using(<ref>), (<ref>) and (<ref>). We leave all the details to the reader.Now we need to establish the the “uniqueness” part of Theorem <ref>. Assume that {T̃_n}_0≤ n <N is a set of vectors satisfying conditions (i)-(iii) of Theorem <ref>. We denote T̃_N:=T̃_N-1 and define W̃_n=T̃_4(n-K_0) for K_0 ≤ n ≤, X̃_n=T̃_4(n-K_1)+1 for K_1 ≤ n < , Ỹ_n=T̃_4(n-K_2)+2 for K_2 ≤ n ≤, Z̃_n=T̃_4(n-K_3)+3 for K_3 ≤ n < .As we argued on page T_N-1_discussion, conditions (i) and (ii) of Theorem <ref> and Schur's result (<ref>) imply thatT̃_N-1=(-ı)^N-1T̃_N-1 (respectively, T̃_N-1=(-ı)^N T̃_N-1) if N is odd (respectively, if N is even). Thus we can arrange {T̃_n}_0≤ n <N as in the table(<ref>), and in each case N≡ 0,1,2,3 (mod 4) the table would have the same form of last rows, as shown in (<ref>), (<ref>), (<ref>) and (<ref>).One can check that an equivalent way to definethe vectors{W̃_n}_K_0 ≤ n ≤, {X̃_n}_K_1 ≤ n <, {Ỹ_n}_K_2 ≤ n ≤ and{Z̃_n}_K_3 ≤ n < is through the table (<ref>) with T_n replaced by T̃_n. Let us consider the sequence of vectors {W̃_n}_K_0≤ n ≤. From the definition it is clear that these vectors are orthonormal,they satisfy W̃_n=W̃_n and (W̃_n)=(T̃_4(n-K_0))≤⌊ (N+4n-4K_0+2)/4 ⌋=⌊ (N+2)/4 ⌋+n-K_0=n. In the case N≡ 0 (mod 4) and n=, the last computation should be replaced by the inequality (W̃_)≤, which is trivial, since the width of any vector in ^̊N is not greater than .These conditions imply that for K_0 ≤ n ≤ the vectors {W̃_l}_K_0≤ l ≤ n give an orthonormal basis of the space E_0 ∩ S_n.Consider the case n=K_0.As we established in Proposition <ref>(i), we have (E_0 ∩ S_K_0)=1 and W_K_0 is a unit vector lying inE_0 ∩ S_K_0. Thus W̃_K_0= W_K_0 orW̃_K_0=- W_K_0. Next, consider n=K_0+1. Again,according to Proposition <ref>(i), we have (E_0 ∩ S_K_0+1)=2 and { W_K_0,W_K_0+1} is an orthonormal basis of E_0 ∩ S_K_0+1. Since{W̃_K_0, W̃_K_0+1} is also an orthonormal basis of E_0 ∩ S_K_0+1 and we have already proved thatW̃_K_0= W_K_0 or W̃_K_0=- W_K_0, we conclude thatW̃_K_0+1= W_K_0+1 orW̃_K_0+1=- W_K_0+1. Proceeding in this way, we show thatfor K_0≤ n ≤ we haveW̃_n=W_n or W̃_n=- W_n.In exactly the same way we show that for all admissible n we have X̃_n= ± X_n,Ỹ_n= ± Y_n and Z̃_n= ± Z_n, and this implies that T̃_n=± T_n for 0≤ n <N.Proof of Proposition <ref>: The proof follows from Proposition <ref> andtables (<ref>)–(<ref>). Proof of Theorem <ref>: The proof will be based on the following key observation: if a ∈ E_m, thena ∈E_m and ( a)≤( a)+1. The first statement is true sincecommutes with the DFT, and the second statement follows from (<ref>). Let us denote ν(n)=⌊ (N+n+2)/4 ⌋, so that ( T_n)=ν(n). Consider 0≤ n < N-5. Then T_n ∈ S_ν(n)∩ E_mfor some m ∈{0,1,2,3}, which impliesT_n ∈ S_ν(n)+1∩ E_m. Thus we can expandT_n=γ_n+4 T_n+4 + γ_n T_n + γ_n-4 T_n-4 + γ_n-8 T_n-8+…,where we interpret γ_j=0 and T_j=0 for j<0. Using the orthonormality of T_n, formula (<ref>) and the fact thatis self-adjoint imply that for any m,n ∈{0,1,…, N-5} such that|m-n|>4 we have0= T_n,T_m.From (<ref>) and (<ref>) we findγ_n-8= T_n,T_n-8=0 and similarly for γ_n-12 and all other coefficients γ_j with j<n-4. Thus we have proved that there exist sequences a_n, b_n and c_n such that for all 0≤ n <N-4 we haveT_n=c_nT_n+4 + a_nT_n + b_n-4 T_n-4.From (<ref>) we find a_n= T_n,T_n and c_n= T_n,T_n+4= T_n,T_n+4 = T_n, c_n+4 T_n+8 + a_n+4 T_n+4 + b_n T_n=b_n. Thus we can write T_n=b_n T_n+4 + a_nT_n + b_n-4 T_n-4.Finally, we compute b_n^2 =b_n T_n+4^2= T_n-a_nT_n - b_n-4 T_n-4^2 = a_nT_n^2+ T_n^2+b_n-4 T_n-4^2- 2a_nT_n, T_n -2 T_n,b_n-4 T_n-4 +2a_nT_n, b_n-4 T_n-4=a_n^2+ T_n^2+b_n-4^2-2a_n^2-2b_n-4^2+0=T_n^2-a_n^2-b_n-4^2.This ends the proof of Theorem <ref> in the case n<N-5. The remaining cases n=N-5 if N is odd andn=N-4 if N is even are left to the reader.§ CONVERGENCE OF VECTORS T_N TO HERMITE FUNCTIONS In the previous section we considered N to be fixed. Now our goal is to study the behaviour of T_n as N →∞. Note that dependenceof T_n (as well as many other objects in this section) on N is not visible in our notation.The proof of Theorem <ref> will be preceded by several lemmas. Recall that we denoted ω = 2 π / N. (i) Assume that P, Q are real polynomials (that do not depend on N) and a and b are vectors in ^̊N, having elements a(k) = P(√(ω) k) exp(-ω k^2 / 2), b(k) = Q(√(ω k)) exp(-ω k^2 / 2) for k∈ I_N. Then for any ϵ>0 √(ω) a,b = ∫_ P(x) Q(x) exp(-x^2) x̣ + O(e^-(π/2-ϵ)N),as N→ +∞. (ii)Assume that P, Q are real polynomials (that do not depend on N) such that Q(x) exp(-x^2/2) is the continuous Fourier transform ofP(x) exp(-x^2/2). Let a and b be vectors in ^̊N, having elements a(k) = P(√(ω) k) exp(-ω k^2 / 2) and b(k) = Q(√(ω) k) exp(-ω k^2 / 2) for k ∈ I_N. Then for any ϵ>0 a -b= O(e^-(π/4-ϵ)N),as N→ +∞. The proof is based on the Poisson summation formula: for a function f in Schwartz class and any a>0, y∈$̊ we have a ∑_k ∈ f(a k)e^-ı a k y=√(2 π)∑_k∈ (ℱf)(y+2π k/a).We will also need the following estimates: for anyα>0,β>0,ϵ>0and for any polynomialP(which does not depend onN) we have √(ω)∑_k≥α N P(√(ω)k) e^-βω k^2=O(e^-(2πα^2 β-ϵ) N),∑_k=1^∞ P(k/√(ω)) e^-β k^2/ω=O(e^- (β/(2π)-ϵ)N),asN→ +∞. We leave it to the reader to verify these estimates. Let us prove item (i). Let(P)=nand(Q)=mand letR(x)e^-x^2/4be the continuousFourier transform ofP(x)Q(x) e^-x^2, for some polynomialRof degreen+m. We write √(ω) a,b =√(ω)∑_k∈ I_N P(√(ω)k)Q(√(ω)k)e^-ω k^2=√(ω)∑_k∈ P(√(ω)k)Q(√(ω)k)e^-ω k^2+O(e^-(π/2-ϵ)N)=∫_ P(x)Q(x)e^-x^2x̣+√(2π)∑_k∈, k≠ 0R(2π k/√(ω))e^-(2 π k/√(ω))^2/4+O(e^-(π/2-ϵ)N)=∫_ P(x)Q(x)e^-x^2x̣+O(e^-(π/2-ϵ)N),where in the second step we used estimate (<ref>), in the third step we applied the Poisson summation formula(<ref>) and in the last step we used estimate (<ref>). The above result gives us (<ref>).Let us now prove item (ii). Assume that(P)=n. We fixl∈ I_Nand computea(l) =1/√(2π)×√(ω)∑_k∈ I_N P(√(ω) k)e^-ω k^2/2-ı√(ω) k ×√(ω) l= 1/√(2π)×√(ω)∑_k∈ P(√(ω) k)e^-ω k^2/2-ı√(ω) k ×√(ω) l+O(e^-(π/4-ϵ)N)=∑_k∈ Q(√(ω) l+2π k/√(ω))e^-(√(ω) l+2π k/√(ω))^2/2+O(e^-(π/4-ϵ)N).Note that sincel ∈ I_N, we have|l|≤ N/2, thus for allk≠ 0we have| √(ω) l+2π k/√(ω)|≥π/√(w) (2|k|-1)≥π/√(w) |k|.Thus the sum in the right-hand side of (<ref>) can be estimated as follows∑_k∈ Q(√(ω) l+2π k/√(ω))e^-(√(ω) l+2π k/√(ω))^2/2=b(l)+O(∑_k≥ 1 (k/√(ω))^n e^-π^2 k^2/(2ω))=b(l)+O(e^-(π/4-ϵ)N).The above result combined with (<ref>) imply (<ref>). Recall that we denotedt_k = 2 ω^-1/2sin(ω k / 2). Note the following two properties:(i)t_k/(√(ω)k) → 1 as N→∞;(ii)2 √(ω)k / π≤t_k≤√(ω)k for k∈ I_N. Let us define a vector G ∈^̊Nas follows:G(k)=√(ω/π)e^-ω k^2/2, k∈ I_N.Note that G=1+O(e^-N), due to Lemma <ref>(i).Suppose that P is a real polynomial (that does not depend on N) and a ∈^̊N is defined bya(k)=(P(t_k)-P(√(ω)k)) G(k) for k∈ I_N. Then for any > 0we have a=O(N^-1 + ) as N→∞.Let us define δ =min(1/2,ϵ/4). Then for|k|≤ N^1/2+δ we havet_k - √(ω) k= √(ω)k×(ω k / 2)^-1sin(ω k / 2) - 1= √(ω)k× O((ω k)^2) = O(N^-1+3δ).Since the functionP'(t) exp(-t^2/2) is bounded on $̊ and G(k)=O(N^-1/4),we have | a(k)| =|P(t_k)-P(√(ω)k)| G(k) ≤max_t∈[ |P'(t)| e^-t^2/2]×t_k - √(ω) k× O(N^-1/4)= O(N^-5/4+3δ),for|k|≤ N^1/2+δ. It follows that∑_k≤ N^1/2 + δ a(k)^2= O(N^1/2 + δ) × O(N^-5/2 + 6 δ) = O(N^-2 + 7δ) .At the same time, for anyp > 0we haveP(t_k)G(k) = O(N^-p)andP(√(ω) k) g(k) =O(N^-p)uniformly inksuch thatk > N^1/2 + δ. Thus,∑_k∈ I_Nk>N^1/2+δ a(k)^2 = O(N^1 - 2 p) .Estimates (<ref>) and (<ref>) imply that a=O(N^-1+7δ/2)=O(N^-1+ϵ).We recall that the vectors u_nand v_nwere introduced in Definition <ref> and we definethe normalised vectors U_n =u_n^-1 u_nand V_n =v_n^-1 v_n. The next result is crucial in the proof of Theorem <ref>: it shows that the vectors U_n(respectively, V_n) are analogues of Gaussian functione^-x^2/2(respectively,x e^-x^2/2) ifn=N/4+O(1)asN→ +∞. This result was first established in <cit.> using the Euler-Maclaurin summation formula, here we give a simpler and shorter proof.Assume that P is a real polynomial that does not depend on N and define vectorsa and b in ^̊N by a(k)=P(t_k)( U_n(k)- G(k)) andb(k)=P(t_k)( V_n(k)-√(2) t_kG(k)) for k∈ I_N.Then for anyϵ>0 we have a=O(N^-1+ϵ)and b=O(N^-1+ϵ) as N→∞ and n=N/4+O(1). Let us first prove that a=O(N^-1+ϵ).We define the vector Ũ_n∈^̊N via Ũ_n(k)=√(ω/π)×∏_j=n+1^( 1 - (t_k/t_j)^2 ), k∈ I_N.Denote δ = min(1/16, ϵ/4). Formula (<ref>) gives ust_k-√(ω) k=O(N^-1+3δ), in particular t_k=O(N^δ) for |k|≤ N^1/2+δ.Next, for n+1≤ j ≤ we have π/4+O(N^-1) ≤π j/N≤π/2, thus sin(π j/N)>1/2 and|t_j|>C N^1/2 for C=1/√(2π)when N is large enough. We conclude that (t_k/t_j)^2=O(N^-1+2δ)for |k|≤ N^1/2+δ and n+1≤ j ≤. Using the approximation ln(1-x)=-x+O(x^2), we obtain for |k|≤ N^1/2+δ ln[ √(π/ω) Ũ_n(k)] =- t_k^2 ∑_j=n+1^ t_j^-2 + O(N)× O(N^-2+4δ) = -t_k^2/2×π/N∑_j=n+1^1/sin(π k/N)^2+O(N^-1+4δ).Next, we use the Riemann sum approximationπ/N∑_j=n+1^1/sin(π k/N)^2= ∫_π /4^π/2x̣/sin(x)^2+O(N^-1)= (π /4)+O(N^-1)=1+O(N^-1). Combining the above two computations and the facts t_k-√(ω) k=O(N^-1+3δ) and t_k=O(N^δ) we obtainln[ √(π/ω) Ũ_n(k)]=- ω k^2/2+O(N^-1+4δ), |k|≤ N^1/2+δ which is equivalent toŨ_n(k)=G(k)(1+O(N^-1+4δ)), |k|≤ N^1/2+δdue to (<ref>).Now, let us consider a vector ã∈^̊N having elements ã(k)=P(t_k) (Ũ_n(k)- G(k)) for k∈ I_N.Note that we have P(t_k) G(k)=O(N^-2) for |k|>N^1/2+δ. Using this result, the fact that|Ũ_n(k+1)|≤ |Ũ_n(k)| and formula (<ref>), we conclude that we also have P(t_k)Ũ_n(k)=O(N^-2) for |k|>N^1/2+δ, thus ã(k)=O(N^-2) for |k|>N^1/2+δ. We apply the above results and use Lemmas<ref> and <ref> and obtainã^2 = ∑_k∈ I_Nk≤ N^1/2+δP(t_k)^2(Ũ_n(k) -G(k))^2 +∑_k∈ I_Nk>N^1/2+δP(t_k)^2(Ũ_n(k) -G(k))^2 =O(N^-2+8δ)∑_k∈ I_Nk≤ N^1/2+δP(t_k)^2G(k)^2 +O(N)× O(N^-4) =O(N^-2+8δ) √(ω/π)∑_k∈ I_NP(t_k)^2 e^-ω k^2 +O(N^-3) =O(N^-2+8ϵ)×1/√(π)∫_ℝ P(x)^2e^-x^2x̣+O(N^-3)=O(N^-2+8δ)and we conclude that ã=O(N^-1+4δ)=O(N^-1+ϵ). Considering a constant polynomial P≡ 1 we conclude thatŨ_n= G+O(N^-1+ ϵ)=1+O(N^-1+ ϵ). Since U_n/U_n = Ũ_n / Ũ_n the above two facts give us the result: the norm of the vector ais O(N^-1+ϵ).Now we will prove that that b=O(N^-1+ϵ).In this case we define Ṽ_n(k)=√(4ω/π)× t_k √(1-ω t_k^2/2)×∏_j=n+1^( 1 - (t_k/t_j)^2 ), k∈ I_N.Again, we set δ = min(1/16, ϵ/4). Then for |k|≤ N^1/2+δ we have√(1-ω t_k^2/2)=1+O(N^-1+2δ),since t_k=O(N^δ) and ω=2π /N. Repeating the calculation in (<ref>) we conclude that Ṽ_n(k)= √(2) t_kG(k)(1+O(N^-1+4δ)), |k|≤ N^1/2+δ.The rest of the proof proceeds as above. Note that Lemmas <ref> and <ref>tells us that the norm of the vectorH∈^̊N having elements H(k)=√(2) t_kG(k) is 1+O(N^-1+ϵ) and that V_n/ V_n=Ṽ_n/Ṽ_n.The details are left to the reader. Using the methods from the proof of Lemma (<ref>) one could prove the following, more general result. Take any real polynomial P that does not depend on N and any c∈ (0,1/2) and define a vector a ∈^̊N via a(k)=P(t_k)( U_n(k)- ( ω(π c)/π)^1/4e^-(π c) ω k^2/2), k∈ I_N. Then a=O(N^-1+ϵ) as N→ +∞ and n=cN + O(1).The above result tells us that the vectors u_n, defined in (<ref>), are discrete analogues of Gaussian functions f_c(x)=(π c)^1/4 e^-(π c)x^2/2,provided that n=CN+O(1) and N is large. Thus the discrete Fourier transform identity u_n= u_-n that we established in Theorem <ref> is the counterpart of the continuous Fourier transform identity ℱ f_c=f_1/2-c.As an immediate consequence of the above lemmas we have the following result. (i) If n = N/4 + O(1), P, Q are polynomials (that do not depend on N) and A(k) = P(t_k)U_n(k), B(k) = Q(t_k)U_n(k) for k∈ I_N, then for any ϵ>0 A,B= 1/√(π)∫_ P(x) Q(x) exp(-x^2) x̣ + O(N^-1+ϵ),as N→ +∞. (ii) Suppose that n = N/4 + O(1) and that P, Q are polynomials (that do not depend on N) such that Q(x) exp(-x^2/2) is the continuous Fourier transform of P(x) exp(-x^2/2). If A(k) = P(t_k)U_n(k) and B(k) = Q(t_k)G(k) for k∈ I_N, then for any ϵ>0 A -B =O(N^-1+ϵ),as N→ +∞. Proof of Theorem <ref>: The proof of Theorem <ref> will proceed by induction with respect ton, and first we will consider the case of evenn. Let us first present several observations. We denoteν(n)= ( T_n) = (N + n + 2)/4 .From the definition of vectors T_ngiven on page page_def_T it is clear that ifnis even (and admissible) then T_nis a linear combination of U_ - ν(n),U_ - ν(n) + 1, …,U_ν(n).Formula (<ref>) implies that for0≤ m < n ≤we haveU_m(k)= C ∏_j=m+1^n ( 1-(t_k/t_j)^2) × U_n(k), k∈ I_N,for some constantC=C(m,n,N).From the above two facts we can conclude that for every even and admissiblenwe haveT_n(k)= P_n(t_k)U_ν(n)(k)for some even polynomialP_nof degree(P_n) = 2(ν(n) - ( - ν(n))) = 4 ν(n) - 2.It is easy to see that the sequence(P_n)forn = 0, 2, 4, 6, …is equal to either0, 4, 4, 8, 8, 12, 12, …(ifN= 0orN = 1modulo 4) or2, 2, 6, 6, 10, 10, 14, …(otherwise). In particular,(P_n)is always equal to eithernorn + 2.Forγ∈$̊ and a vector a={ a(k)}_k∈ I_N we will write a=(N^γ) if a=O(N^γ)as N→∞. Thus our goal is to prove that for every even m we have T_m=_m+(N^-1+ϵ) as N →∞. Note this is true for m=0 as was established in Lemma <ref>.Now, fix an even m≥ 2. Suppose that we have already chosen the signs of { T_n}_n=0,2,…,m-2 in a proper way and have proved that T_n=_n+(N^-1+ϵ) for n =0,2,…,m-2. Our goal is to prove that T_m=_m+(N^-1+ϵ). Let us define the normalized Hermite polynomials h_n(x):= (2^n n!)^-1/2 H_n(x), where H_n are the classical Hermite polynomials (see (<ref>)).Note that with this normalization we have _n(k)=h_n(√(ω)k)G(k) (see (<ref>)) and (<ref>)).Next, let γ_j be the coefficients in the expansion of the polynomial P_m(x) in the basis h_j(x), that is P_m(x)=γ_0 h_0(x)+γ_2 h_2(x)+…+γ_m h_m(x)+γ_m+2 h_m+2(x). Note that here we have used the fact that P_m is an even polynomial of (P_n)≤ m+2.Let us define the vectors { A_j}_j=0,2,…,m+2 via A_j(k) = h_j(t_k)U_ν(m)(k), k∈ I_N,so that we have T_m=∑_j=0,2,…,m+2γ_j A_jAccording to Lemmas <ref> and <ref>, we have A_j=_j+(N^-1+ϵ) for all j ∈{0,2,…,m+2}.Lemma <ref> implies _i,_j=δ_i,j+O(e^-N),for i,j ∈{0,2,…,m+2}, thus A_i, A_j=δ_i,j+O(N^-1+ϵ) for i,j ∈{0,2,…,m+2}. Therefore, the Gramian matrixof vectors { A_j}_j=0,2,…,m+2 converges to the identity matrix as N→ +∞, and the same must be true for the inverse of this Gramian matrix. This proves that the norm of a linear combination of vectors { A_j}_j=0,2,…,m+2 is comparable (uniformly as N →∞) with the norm of the coefficients. Since T_m=1, we conclude that the coefficients {γ_j}_j=0,2,…,m+2 are uniformly bounded as N→ +∞.Using the above result combined with formula(<ref>) and the estimate A_j=_j+(N^-1+ϵ) we conclude thatT_m=∑_j=0,2,…,m+2γ_j_j + (N^-1+ϵ). Next, by induction hypothesis, we have T_n=_n+ (N^-1+ϵ), for n ∈{0, 2, …, m - 2}.Using this result and orthogonality of the vectors { T_n}_0≤ n <Nwe conclude that 0= T_n, T_m =∑_j=0,2,…,m+2γ_j T_n, _j + O(N^-1+ϵ)= ∑_j=0,2,…,m+2γ_j_n, _j + O(N^-1+ϵ)=γ_n+O(N^-1+ϵ),for n ∈{0, 2, …, m - 2}. In other words, we have proved that γ_n=O(N^-1+ϵ) for all n ∈{0, 2, …, m - 2}; combining this result with the fact Ψ_j=1+O(e^N) we obtain T_m=γ_m_m +γ_m+2 _m+2+ (N^-1+ϵ).Our next goal is to show that γ_m+2=O(N^-1+ϵ). First, we use the fact that T_m is an eigenvector of theDFT to calculateT_m=(-ı)^m T_m=(-ı)^m γ_m_m + (-ı)^m γ_m+2 _m+2+ (N^-1+ϵ).At the same time, we can evaluate the same expression via Lemma <ref>: this gives us T_m= γ_m_m +γ_m+2 _m+2+ (N^-1+ϵ)= (-ı)^m γ_m_m + (-ı)^m+2γ_m+2 _m+2+ (N^-1+ϵ).Comparing the above two formulas we conclude that γ_m+2=O(N^-1+ϵ). ThusT_m=γ_m_m + (N^-1+ϵ), and since T_m=1 we conclude that|γ_m|=1+O(N^-1+ϵ). This implies that T_m= _m + (N^-1+ϵ) or - T_m= _m + (N^-1+ϵ) and this ends the induction step.The proof of the identity T_n=_n+(N^-1+ϵ) when n is odd follows the same steps.We provide only a sketch of the proof and leave all the details to the reader. First of all, we note that if n is odd then T_n is a linear combination of V_ - ν(n),V_ - ν(n) + 1, …,V_ν(n), and thereforeT_n(k)= Q_n(t_k)V_ν(n)(k)for an even polynomial Q_n of degree (Q_n) = 2(ν(n) - ( - ν(n))) = 4 ν(n) - 2.Again, it is easy to see that the sequence (Q_n) for n = 1, 3, 5, 7, … is equal to either 0, 4, 4, 8, 8, 12, 12, … (ifN= 0 or N = 3 modulo 4) or 2, 2, 6, 6, 10, 10, 14, … (otherwise). In particular, (Q_n) is always equal to either n-1 or n + 1.Thus our goal is to prove that for every odd m we have T_m=_m+(N^-1+ϵ) as N →∞. Note this is true for m=1 as was established in Lemma <ref>. Now, fix an odd m≥ 3.Suppose that we have already chosen the signs of { T_n}_n=1,3,…,m-2 in a proper way and have proved that T_n=_n+(N^-1+ϵ) for n =1,3,…,m-2. Our goal is to prove that T_m=_m+(N^-1+ϵ). We define γ_j to be the coefficients in the expansion of the odd polynomial √(2) xQ_m(x) in the basis h_j(x). This is equivalent to writingQ_m(x)=γ_1 h_1(x)/(√(2)x)+γ_3 h_3(x)/(√(2)x)+…+γ_m h_m(x)/(√(2)x)+γ_m+2 h_m+2(x)/(√(2)x). Note that for odd j the function h_j(x)/x is an even polynomial (since h_j is an odd polynomial in this case).Let us define the vectors { B_j}_j=1,3,…,m+2 via B_j(k) = h_j(t_k)/(√(2)t_k) × V_ν(m)(k),k∈ I_N, so that we have T_m=∑_j=1,3,…,m+2γ_j B_jAccording to Lemmas <ref> and <ref>, we have B_j=_j+(N^-1+ϵ) for all j ∈{1,3,…,m+2}.The rest of the proof proceeds exactly in the same way as in the case of even m:first we use orthogonality of T_n to show that γ_j=O(N^-1+ϵ) for j=1,2,…,m-2 and then we use the fact thatT_m is an eigenvector of the DFT to prove that γ_m+2=O(N^-1+ϵ).The details are left to the reader.§ NUMERICAL COMPUTATION OF THE MINIMAL HERMITE-TYPE BASIS The eigenvectors T_n can be efficiently evaluated numerically using the three-term recurrence relation stated in Theorem <ref>. The algorithm is quite straightforward. First we compute the vectors T_0,T_1,T_2,T_3 viaT_0 = c_0 ( u_K_0 +u_ - K_0), T_1 =c_1 ( v_K_1 +v_ - K_1), T_2 =c_2 ( u_K_2 -u_ - K_2), T_3 =c_3 ( v_K_3 -v_ - K_3),where K_m = (N + 2 + m) / 4, c_i are the normalisation constants that make T_i unit vectors and the Gaussian-type vectors u_n and the modified Gaussian-type vectors v_n are computed by expressions given in Definition <ref> or in Lemma <ref>. Then we compute the remaining vectors { T_n}_4≤ n ≤ N-1 using the three-term recursion described in Theorem <ref>. While doing this, we need to remember to set T_N-1 to the “ghost" vector T_N when N is even.Let us discuss the computational complexity of the above algorithm. It is easy to see that the number of arithmetic operations needed to evaluate the initial vectors T_0,T_1,T_2,T_3 is linear in N, and the same is true for each recursive step.Thus the vectorT_n can be evaluated using O(N n) arithmetic operations, and the entire basis requires only O(N^2) operations. This bound is clearly optimal: the complete basis consists of N^2 numbers, so it cannot be evaluated using fewer than O(N^2) operations. Considering the memory requirement, we note that we need O(N^2) memory to compute the entire basis (since we need to store the entire basis in memory) and we need only O(N) memory if our goal is to compute vector T_n for one fixed value of n: this last statement is true since the above recursive algorithm, based on Theorem <ref>, requires us to store only eight vectors { T_l}_n-8≤ l ≤ n-1 to compute vectors T_m with m≥ n. Clearly, these bounds for memory requirements are also optimal. However, the time complexity of the above algorithm is worse than O(N^2) due to rapid loss of precision. It is easy to see where this loss of precision comes from. Note that for N largewe have T_n = 4T_n +o(1), thus a_n=4+o(1) and b_n=o(1). Thus, when we calculate T_n+4via (<ref>), first we calculate the difference T_n-a_nT_n and we have subtract numbers of similar magnitude, then we normalize the resulting vector T_n-a_nT_n-b_n-4 T_n-4 (which is o(1))by multiplying it by a large number 1/b_n.Subtracting numbers of similar magnitude and multiplying the result by a large number inevitably results in loss of precision. Empirically, we have found that evaluation of T_N-1 for N = 256 leads to loss of approximately 110 digits of precision. This increases to over 440 digits when N = 1024. For this reason, high-precision arithmetic is necessary even for relatively small values of N.To facilitate applications, we have pre-computed the minimal Hermite-type basis for all N less than or equal to 1024, as well as a few larger values of N, and made theseresults publicly available on the Internet at https://drive.google.com/open?id=0B1hpG-8rGMJcQnhXbE8tR3NZVXM. We used a Wolfram Mathematica script to generate the vectors T_n; the source code is given in Listing <ref> (the output has been generated using the code given in Listing <ref>). For N = 1024, and using interval arithmetic with 1000 digits of precision, the script takes about 100 seconds on a modern computer. To test our results we have also computed the minimal Hermite-type basis using a Fortran90 program and David Bailey's MPFUN90 multiple precision package, this code can be found at www.math.yorku.ca/ akuznets/math.html. abbrv[p] Mathematica numer = (N[Interval[#], prec] ); (* function to be used for numerical evaluation *) k0 = -Ceiling[nn/2] + 1; (* auxiliary: evaluate vectors a[k0],a[k0+1],...,a[k0+nn-1] *) omega = 2 Pi/nn; S[k_?IntegerQ] := S[k] =If[2 k < nn, If[k > 0, Product[numer[2 Sin[omega j/2]], j, 1, k], numer[1]], nn/S[nn - 1 - k]]; alpha[n_?IntegerQ] := alpha[n] = (* multiplied by S[n]^2 *)If[OddQ[nn], If[n > 0, Sqrt[S[2 n]], numer[1]], If[n > 0, Sqrt[S[2 n - 1] Sin[omega n/2]], numer[1/2]]]; beta[n_?IntegerQ] := beta[n] = (* multiplied by S[n]^2 *)If[OddQ[nn], Sqrt[S[2 n - 1]], Sqrt[S[2 n - 1] Cos[omega n/2]]]; u[n_?IntegerQ] := u[n] = Table[If[Abs[k] > n, 0, If[OddQ[nn],alpha[n] S[nn - n - 1 - k] S[nn - n - 1 + k] / nn^2,If[n < nn/2, alpha[n] S[nn - n - 1 - k] S[nn - n - 1 + k] Cos[omega k/2] / nn^2, numer[1/Sqrt[4 nn]]] ]],k, k0, k0 + nn - 1]; v[n_?IntegerQ] := v[n] = Table[If[Abs[k] > n, 0, If[OddQ[nn],beta[n] S[nn - n - 1 - k] S[nn - n - 1 + k] Sin[omega k] / nn^2,beta[n] S[nn - n - 1 - k] S[nn - n - 1 + k] 2 Sin[omega k/2] / nn^2 ]],k, k0, k0 + nn - 1]; T[0] := T[0] = Normalize[u[Floor[(nn + 2)/4]] + u[Floor[nn/4]]]; T[1] := T[1] = Normalize[v[Floor[(nn + 3)/4]] + v[Floor[(nn + 1)/4]]]; T[2] := T[2] = Normalize[u[Floor[(nn + 4)/4]] - u[Floor[(nn - 2)/4]]]; T[3] := T[3] = Normalize[v[Floor[(nn + 5)/4]] - v[Floor[(nn - 1)/4]]]; Lm := Lm = Table[numer[2 Cos[omega k]], k, k0, k0 + nn - 1]; (* multiplier for L *) L[a_] := RotateLeft[a, 1] + RotateRight[a, 1] + Lm a; If[EvenQ[nn], T[nn - 1] := T[nn - 1] = T[nn]]; (* set T[nn-1] to the ghost vector T[nn] *) T[n_?IntegerQ] := T[n] = Block[LT, Ta, t,LT = L[T[n - 4]];Ta = T[n - 4].LT;t = If[n < 8, LT - Ta T[n - 4], LT - Ta T[n - 4] - Tb[n - 8] T[n - 8]];Tb[n - 4] = Norm[t];t / Tb[n - 4] ]; Mathematica script that was used to evaluate minimal Hermite-type eigenvectors T_n, represented as T[n] in the code. Input parameters are nn, equal to the dimension N, and precision, the number of digits to be used in interval arithmetic calculations. Lazy evaluation with memoization is used. Interval arithmetic allows us to keep track of rounding errors.[p] Mathematica digits = 400; maxoutput = 100; print[Interval[a_, b_]] := ToString[If[b - a > 10^(-digits - 2), Throw["Insufficient precision"], (* check precision *) If[Abs[a + b]/2 < 10^(-maxoutput), "0", (* treat zero in a separate way *)ScientificForm[ (a + b)/2, (* number to be printed *) Min[maxoutput, digits + MantissaExponent[(a + b)/2][[2]]], (*number of significant digits to be printed*) NumberFormat -> (SequenceForm[#1, "e", #3])] ]] ]; Export["/path/to/file.txt", Table[print /@ T[n], n, 0, nn - 1], "Table"] Mathematica script used for creating pre-evaluated tables of minimal Hermite-type eigenvectors. Only faithfully evaluated digits are printed: in case of excessive loss of precision execution is aborted. | http://arxiv.org/abs/1706.08740v1 | {
"authors": [
"Alexey Kuznetsov",
"Mateusz Kwaśnicki"
],
"categories": [
"math.CA",
"42A38 (Primary), 65T50 (Secondary)"
],
"primary_category": "math.CA",
"published": "20170627091430",
"title": "Minimal Hermite-type eigenbasis of the discrete Fourier transform"
} |
Non-Orthogonal Multiple Access combined with Random Linear Network Coded Cooperation Amjad Saeed Khan, Student Member, IEEE, and Ioannis Chatzigeorgiou, Senior Member, IEEE A. S. Khan and I. Chatzigeorgiou are with the School of Computing and Communications, Lancaster University, Lancaster, United Kingdom (e-mail: {a.khan9, i.chatzigeorgiou}@lancaster.ac.uk).June 23, 2017 ========================================================================================================================================================================================================================================================================================== This letter considers two groups of source nodes. Each group transmits packets to its own designated destination node over single-hop links and via a cluster of relay nodes shared by both groups. In an effort to boost reliability without sacrificing throughput, a scheme is proposed, whereby packets at the relay nodes are combined using two methods; packets delivered by different groups are mixed using non-orthogonal multiple access principles, while packets originating from the same group are mixed using random linear network coding. An analytical framework that characterizes the performance of the proposed scheme is developed, compared to simulation results and benchmarked against a counterpart scheme that is based on orthogonal multiple access.Network coding, non-orthogonal multiple access, sparse random matrices, decoding probability, throughput.§ INTRODUCTION Random Linear Network Coding (RLNC) is a scheme that allows an intermediate node to combine and forward the data of multiple users in a single transmission, and can effectively improvenetwork capacity <cit.>. RLNC has the inherent capability to achieve spatial diversity. For example, it has been shown in <cit.> that network coding can improve the diversity gain of networks that either contain distributed antenna systems or support cooperative relaying. Furthermore, RLNC can improve both the throughput <cit.> and the latency in a network <cit.> by reducing the number of distinct transmissions. The benefits of network coding have made it an attractive solution for challenges encountered in existing and future communication systems. For instance, it has been shown in <cit.> that by modifying the IEEE 802.11g frame structure, network coding combined with Orthogonal Frequency Division Multiplexing (OFDM) can significantly improve throughput. The importance of network-coded cooperation has been demonstrated in <cit.> and implemented in <cit.> with Orthogonal Frequency Multiple Access (OFDMA). Recently, Non-Orthogonal Multiple Access (NOMA) has been recognised as a promising multiple access technique for 5G mobile networks <cit.>. It has been shown in <cit.>, <cit.> that combining NOMA with OFDM can improve the spectral efficiency and accommodate more users than the conventional OFDMA-based systems. Moreover, the usefulness of RLNC for downlink NOMA-based transmissions has been studied in <cit.>.This letter considers network-coded cooperation in a NOMA-based scenario with two groups of source nodes. Each group communicates with a different destination node via multiple relay nodes. To the best of our knowledge, this work represents the first attempt to characterise the performance of NOMA-based RLNC cooperation. The main contributions of our work can be summarized as follows:(i) we propose a framework which integrates the benefits of NOMA-based multiplexing and RLNC-based cooperative relaying; (ii) using the fundamentals of RLNC and uplink/downlink NOMA, we derive closed-form expressions for the network performance, in terms of the decoding probability at each node, and the system throughput; (iii) we validate the accuracy of the derived expressions through simulations and we investigate the impact of the system parameters on the network performance and throughput. § SYSTEM MODELConsider a network with two source groups, two destination nodes and N commonly shared relay nodes r_1,r_2,, r_N. Each source group G_k contains K source nodes s_1^(k),s_2^(k),, s_K^(k) for k=1,2. The packets transmitted by source nodes in G_k are meant to be received by destination d_k, either directly or via relay nodes. The acceptable transmission rate for G_1 is R_1^* and for G_2 is R_2^*. Without loss of generality, we assume that all source nodes in G_1 require a comparatively high quality of service with R_1^*<R_2^*. In practice, G_1 could be a group of devices (e.g., sensors) associated to high risk applications that need to be connected quickly with low data rate, and G_2 could be a group of devices related to low risk applications that can afford opportunistic connectivity. All nodes operate in half duplex mode. The links connecting the nodes are modeled as quasi-static Rayleigh fading channels. The channel gain between nodes i and j is represented by h_ij, which is a zero-mean circularly symmetric complex Gaussian random variable with variance σ_ij^2. Before the communication process is initiated, source nodes from the two groups are paired according to their indices, such that s_i^(1) in group G_1 is paired with s_i^(2) in G_2. Only paired nodes are allowed to transmit simultaneously over the same frequency band. The simultaneous transmission of two nodes exploits the principle of superposition coding, which is a key component of NOMA. Node pairing in NOMA has been recently proposed for 3GPP Long Term Evolution Advanced (LTE-A) <cit.>. Source nodes in different pairs transmit over orthogonal frequency bands, and therefore can be recovered independently. This approach is also known as OFDM-NOMA <cit.> but, for the sake of brevity, we shall simply refer it to as NOMA. We consider the worst case scenario, in which both source groups contain an equal (i.e., K) number of source nodes, such that relay nodes always receive superimposed signals. The proposed communication process is divided into the broadcast phase andthe relay phase. During the broadcast phase, each source node broadcasts a packet in the form of an information-bearing signal to the relay and destination nodes. The signals transmitted by the i^th source pair (s_i^(1),s_i^(2)), and received by a relay node r_j and destination nodes d_1 and d_2, are respectively given byz_r_j^i =√(α_1P_s)h_s_i^(1)r_jx̃_i+√(α_2P_s)h_s_i^(2)r_jỹ_i+w_r_j^i,z_d_1^i =√(α_1P_s)h_s_i^(1)d_1x̃_i+w_d_1^i,z_d_2^i =√(α_2P_s)h_s_i^(2)d_2ỹ_i+w_d_2^i,where P_s is the total transmission power by the source pair, α_1 and α_2 are the fractions of P_s transmitted by s_i^(1) and s_i^(2), respectively, with α_1+α_2=1, and {x̃_i, ỹ_i} represent the modulated signals of data packets {x_i, y_i}.The additive white Gaussian noise components at the relay and destination nodes are represented by w_r_j^i and w_d_k^i, respectively. All relay nodes employ Successive Interference Cancellation (SIC) to recover the transmitted signals, and then disjointly demodulate and store the correctly received packets. During the relay phase, a relay node r_j employs RLNC on the successfully received and stored data packets of groups G_1 and G_2, and generates coded packets m_j^(1) and m_j^(2), respectively, given by m_j^(1)=∑_i=1^Kc_i,j^(1)x_i and m_j^(2)=∑_i=1^Kc_i,j^(2)y_i, where, c_i,j^(k) represents the coding coefficients over a finite field F_q of size q. The value of a coefficient is zero if a received packet contains irrecoverable errors; otherwise, the value of that coefficient is selected uniformly at random from the remaining q-1 elements of F_q. The probability mass function of c_i,j^(k) is given as Pr(c_i,j^(k)=t) ={ [][c]l'sϵ_s_i^(k)r_j, for t=0,1-ϵ_s_i^(k)r_j/q-1, for t∈ F_q∖{0}, .where 0≤ϵ_s_i^(k)r_j≤ 1 is the outage probability of the link connecting the source node s_i^(k)with the relay node r_j. The closed form expression of ϵ_s_i^(k)r_jwill be presented in Section <ref>. This type of RLNC at the relay nodes is known as sparse RLNC, where the sparsity level is determined by the outage probability ϵ_s_i^(k)r_j <cit.>, <cit.>.Each relay node, instead of transmitting two separate network-coded signals (one for each destination), generates a signal that is the superposition of the two network-coded signals and broadcasts it to both destinations. Relay transmissions are orthogonal, either in time or in frequency. The superimposed signal transmitted by relay r_j can be expressed as (√(P_rβ_1)m̃_j^(1)+√(P_rβ_2)m̃_j^(2)), where P_r is the total transmitted power, andβ_1, β_2 denote the power allocation coefficients, such that β_1+β_2=1 with β_1>β_2 in order to satisfy the quality of service requirement <cit.>. Thus, the received signal at destination d_k is given asẑ_d_k^j=h_r_jd_k(√(P_rβ_1)m̃_j^(1)+√(P_rβ_2)m̃_j^(2))+ŵ_d_k^jwhere ŵ_d_k^j is the Gaussian noise component. Each destination node employs SIC in order to separate the superimposed signals and retrieve the relevant coded packets. Destination d_k will recover the data packets of source group G_k if it collects K linearly independent packets directly from that source group and via the relay nodes.§ ACHIEVABLE RATE AND LINK OUTAGE PROBABILITY This section describes the achievable transmission rate of source-to-destination, source-to-relay and relay-to-destination links. An outage occurs when the achievable rate is less than the target rate of transmission. Therefore, the outage probability of each link can be expressed in terms of the corresponding achievable rate and the target rate. Let us first consider the broadcast phase, during which signals arrive at each destination node directly from the respective source group. The achievable rate of the s_i^(k)d_k link, which originates from group G_k, can be obtained asR_s_i^(k)d_k=B_slog(1+ P_sα_k|h_s_i^(k)d_k|^2/B_sN_0)where k∈{1,2}, i∈{1,2,,K}, N_0 represents the noise power and B_s denotes the bandwidth of the frequency band allocated to each source pair for simultaneous transmissions, as discussed in Section <ref>. The outage probability of the s_i^(k)d_k link can be derived if we combine expression (<ref>) with the cumulative distribution function of Rayleigh fading <cit.>, which givesϵ_s_i^(k)d_k=Pr(R_s_i^(k)d_k≤ R_k^*)=1-exp(-τ_k/ρ_sα_kσ_s_i^(k)d_k^2)where ρ_s=P_s/B_s N_0 and τ_k=2^R_k^*/B_s-1. The achievable rate of the link between one of the nodes of a source pair and a relay node r_j depends on the channel conditions of both links that connect the nodes of the source pair with r_j. For example, assume that α_1|h_s_i^(1)r_j|>α_2|h_s_i^(2)r_j|. In that case, SIC at the relay node r_j will first recover the signal of the node from G_1 and treat the other signal as interference. Thus, the achievable rate of a link between s_i^(k) and r_j can be expressed as <cit.>R_s_i^(1)r_j=B_slog(1+α_1|h_s_i^(1)r_j|^2/α_2|h_s_i^(2)r_j|^2+1/ρ_s) R_s_i^(2)r_j=B_slog(1+ρ_sα_2|h_s_i^(2)r_j|^2).The outage probability of a link between s_i^(k) and r_j can be obtained as ϵ_s_i^(1)r_j=Pr(R_s_i^(k)r_j<R_k^*), thusϵ_s_i^(1)r_j=1-α_1σ_s_i^(1)r_j^2/τ_1α_2σ_s_i^(2)r_j^2+α_1σ_s_i^(1)r_j^2exp(-τ_1/ρ_sα_1σ_s_i^(1)r_j^2) 0.4866!ϵ_s_i^(2)r_j=1-Pr[(R_s_i^(1)r_j>R_1^*)∩(R_s_i^(2)r_j>R_2^*)]=1-α_1σ_s_i^(1)r_j^2/τ_1α_2σ_s_i^(2)r_j^2+α_1σ_s_i^(1)r_j^2exp(-τ_1(τ_2+1)/ρ_sα_1σ_s_i^(1)r_j^2- τ_2/ρ_sα_2σ_s_i^(2)r_j^2). During the relay phase, the destination node d_2 can only successfully recover the coded signals corresponding to source group G_2, when R_r_jd_2>R_2^* provided thatR_r_jd_1>R_1^*. On the other hand, the destination d_1 can recover the coded signals of G_1, when R_r_jd_1>R_1^*. The achievable rates are given asR_r_jd_1=B_slog(1+β_1|h_r_jd_1|^2/β_2|h_r_jd_1|^2+1/ρ_r) R_r_jd_2=B_slog(1+ρ_rβ_2|h_r_jd_2|^2)where B_s is the bandwidth allocated to each pair of relays, and ρ_r=P_r/B_sN_0. It is assumed that β_1 ≥τ_1 β_2, otherwise the outage probability is always one <cit.>. The outage probability of links r_jd_1 and r_jd_2 can be respectively obtained asϵ_r_jd_1 =Pr(β_1|h_r_jd_1|^2/β_2|h_r_jd_1|^2+1/ρ_r≤τ_1)=1-exp(-τ_1/(ρ_rβ_1-τ_1ρ_rβ_2)σ_r_jd_1^2), ϵ_r_jd_2 =1-Pr(β_1|h_r_jd_2|^2/β_2|h_r_jd_2|^2+1/ρ_r> τ_1,ρ_rβ_2|h_r_jd_2|^2>τ_2)=1-exp(-1/ρ_rσ_r_jd_2^2max(τ_1/β_1-τ_1β_2,τ_2/β_2)). §.§ OMA-based Benchmark schemeIn this letter, we consider conventional OFDMA as the benchmark Orthogonal Multiple Access (OMA) scheme. According to this scheme, all nodes s_i^(k) and r_j transmit over orthogonal frequency bands. As a result, likewise (<ref>), the achievable rates of source-to-relay and source-to-destination links during the broadcast phase, and the relay-to-destination links during the relay phase can be respectively obtained asR_s_i^(k)u =B_s/2log(1+ P_sα_k|h_s_i^(k)u|^2/0.5B_sN_0),R_r_j d_k =B_s/2log(1+ P_r β_k|h_r_jd_k|^2/0.5B_sN_0)where u∈{r_j,d_k}. The factor 1/2 is due to the fact that, unlike NOMA, each sub-band is now further split between two transmitting nodes. Note that, using the achievable rates, we can derive the outage probabilities. These results can be further extended to RLNC-based analysis, which will be presented in the next section, and can be used as benchmarks against the proposed NOMA-based scheme.In the remainder of the letter, we assume that links connecting co-located transmitting nodes with receiving nodes are statistically similar, hence ϵ_r_jd_k=ϵ_rd_k, ϵ_s_i^(k)r_j=ϵ_s^(k)r and ϵ_s_i^(k)d_k=ϵ_s^(k)d_k for all valid values of i, j and k. § DECODING PROBABILITY AND ANALYSIS This section analyses the system performance in terms of the probability of a destination node successfully recovering the packets of all nodes in the corresponding source group. Furthermore, the system throughput is derived as a function of the number of packet transmissions.The destination node d_k can recover the packets of all source nodes in group G_k if and only if it collects packets that yield K degrees of freedoms (dofs). Note that dofs at a destination node represent successfully received linearly independent packets, which can be either source packets delivered during the broadcast phase, or coded packets transmitted during the relay phase. According to <cit.> and <cit.>, the probability that the N≥ K coded packets, which have been transmitted by the N relay nodes, will yield K dofs can be bounded as follows: lCr P^'(K, N,ϵ_s^(k)r,q)≥max{∏_i=1^K(1-Γ_max^N-i+1),1-∑_w=1^KKw××(q-1)^w-1[q^-1+(1-q^-1)(1-1-ϵ_s^(k)r/1-q^-1)^w]^N}where Γ_max=max{ϵ_s^(k)r,1-ϵ_s^(k)r/q-1}. In order to formulate the decoding probability at each destination node, let us assume that the destination d_k successfully received m packets, given that K+N packets were transmitted, i.e., K source packets during the broadcast phase and N coded packets during the relay phase. If we denote by f_ℓ(N_T,ϵ) the probability mass function of the binomial distribution, that is,f_ℓ(N_T,ϵ)=N_Tℓϵ^N_T-ℓ(1-ϵ)^ℓthen the probability that h of the m packets are source packets and the remaining m-h are coded packets is given byP_h/m(ϵ_s^(k)d_k,ϵ_rd_k)=f_h(K,ϵ_s^(k)d_k)f_m-h(N,ϵ_rd_k). The contribution of the h recovered source packets to the m-h coded packets can be removed, so that the m-h coded packets become linear combinations of the remaining K-h source packets only. Thus, at this point of the decoding process, the destination node d_k can successfully recover the remaining data packets if and only if the modified m-h coded packets yield K-h dofs. By employing (<ref>), (<ref>) and the law of total probability, the overall decoding probability at the destination d_k can be expressed as0.4866!P_d_k(K,N)=∑_m=K^N+K∑_h=h_min^KP_h/m(ϵ_s^(k)d_k,ϵ_rd_k)P^'(K-h,m-h,ϵ_s^(k)r,q)where h_min=max(0,m-N). Note that retransmissions are not allowed in case of packet failures during the broadcast phase or the relay phase. Therefore, by modifying the expression of the end-to-end throughput in <cit.>, the average system throughput can be defined asη=K/K+max{E_d_1(N),E_d_2(N)}where E_d_k(N) is the average number of relay nodes needed by each destination node d_k to recover the entire source group G_k, and can be calculated using <cit.>E_d_k(N)=N-∑_v=0^N-1P_d_k(K,v).Moreover, by following (<ref>), the average number of relays required for both destinations to decode the packets of the respective source groups can be represented as E_T(N)=N-∑_v=0^N-1P_joint(K,v), where .§ NUMERICAL RESULTS In this section, the accuracy of the derived analytical bound in (<ref>), when used in combination with the decoding probability in (<ref>), is verified through simulations. In the considered system setup, the bandwidth of each sub-band is normalized to 1, i.e., . The source nodes and relay nodes have been positioned such that σ_s^(1)d_1^2=0.1458, σ_s^(2)d_2^2=0.1458, σ_s^(1)r^2=2.9155, σ_s^(2)r^2=1, σ_rd_1^2=1.3717 and σ_rd_2^2 =1.9531. We set α_1=0.6 and α_2=0.4, while exhaustive search has been used to identify the values of β_1 and β_2 that maximize the joint decoding probability mentioned in Section <ref>. The average system SNR is set equal to ρ_s=ρ_r=ρ̅ and, unless otherwise stated, we consider R_1^*=1, R_2^*=1.5.Fig. 1 shows the decoding probabilities P_d_1 and P_d_2 at the two destination nodes in terms of the system SNR. The figure clearly demonstrates the tightness of the analytical curve to the simulation results. The decoding probability P_d_1 is greater than P_d_2 because node d_1 supports a lower target rate than node d_2, and d_1 is allocated more power than d_2 to ensure that the quality of service requirements are met. As expected, NOMA-RLNC outperforms OMA-RLNC because each source node in NOMA-RLNC benefits from being allocated twice the bandwidth that is allocated in OMA-RLNC.Fig. 2 shows the joint decoding probability, for different values of field size q, as a function of the number of relays. The analytical bound is close to the simulation results for q=2 and becomes tighter for greater values of q. A significant gain in performance can be observed when the field size increases from q=2 to q=4. However, the increase in gain is markedly smaller when q further increases from 4 to 64. This is because the certainty of linear independence between coded packets increases with the field size and approaches the highest possible degree even for relatively small values of q. We stress that the computational complexity of the decoder at the destination nodes also depends on the value of q. Thus, the choice of the field size over which RLNC is performed results in a trade-off between complexity and performance gain. Fig. 3 illustrates the relationship between the system SNR and the average number of relays required for the decoding of the source packets of both source groups by the respective destination nodes. The curves establish the diversity advantage offered by the combination of NOMA with RLNC as opposed to OMA with RLNC. For a fixed value of SNR, OMA-RLNC clearly needs more relays for cooperation than NOMA-RLNC. Alternatively, OMA-RLNC can achieve the same performance as NOMA-RLNC at the expense of a higher SNR.Fig. 4 presents the system throughput as a function of the system SNR, for different target rates. The performance gap between NOMA-RLNC and OMA-RLNC is evident. We observe that, for a fixed SNR value, when the target rate increases from R^*_2=1.5 to R^*_2=2, the outage probability increases and, therefore, the system throughput reduces. Interestingly, an increase in the target rate also increases the performance gap between NOMA-RLNC and OMA-RNC, that is, the throughput degradation of NOMA-RLNC is less severe than that of OMA-RLNC. An intuitive reason for this observation is that the 1/2 spectral loss in OMA dominates the system throughput. § CONCLUSIONS This letter investigated the benefits of NOMA-based multiplexing and RLNC-based cooperative relaying in terms of decoding probability and system throughput. Simulation results established the tightness of the derived expressions. Comparisons emphasized the importance of network-coded cooperation and demonstrated the impact of the filed size on network performance. This work showed that the combination of NOMA with RLNC can clearly provide a superior performance, in terms of diversity gain and system throughput, than the combination of conventional OMA with RLNC. We note that network performance can be further improved if modern techniques of node pairing <cit.> are employed.IEEEtran | http://arxiv.org/abs/1706.08487v1 | {
"authors": [
"Amjad Saeed Khan",
"Ioannis Chatzigeorgiou"
],
"categories": [
"cs.IT",
"math.IT"
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"primary_category": "cs.IT",
"published": "20170626171818",
"title": "Non-Orthogonal Multiple Access combined with Random Linear Network Coded Cooperation"
} |
uc3m,imdea]Arturo Azcorra [email protected]]Iñaki Ucarcorrauth [corrauth]Corresponding author [email protected]]Francesco Gringoli [email protected],imdea]Albert Banchs [email protected]]Pablo Serrano [email protected][uc3m]Universidad Carlos III de Madrid, 28911 Leganés, Spain [imdea]IMDEA Networks Institute, 28918 Leganés, Spain [unibs]Università degli Studi di Brescia, 25123 Brescia, Italy[copy]©2017. This manuscript version is made available under the http://creativecommons.org/licenses/by-nc-nd/4.0/CC-BY-NC-ND 4.0 license. DOI: http://doi.org/10.1016/j.comcom.2017.06.00810.1016/j.comcom.2017.06.008In this paper, we revisit the idea of putting interfaces to sleep during packet overhearing (i.e., when there are ongoing transmissions addressed to other stations) from a practical standpoint. To this aim, we perform a robust experimental characterisation of the timing and consumption behaviour of a commercial 802.11 card. We design , a local standard-compliant energy-saving mechanism that leverages micro-sleep opportunities inherent to the CSMA operation of 802.11 WLANs. This mechanism is backwards compatible and incrementally deployable, and takes into account the timing limitations of existing hardware, as well as practical CSMA-related issues (e.g., capture effect). According to the performance assessment carried out through trace-based simulation, the use of our scheme would result in a 57% reduction in the time spent in overhearing, thus leading to an energy saving of 15.8% of the activity time. energy efficiency energy measurement CSMA wireless networks § INTRODUCTION IEEE 802.11 is the standard de facto for broadband Internet access. The recent 802.11ac amendment opens up new opportunities by bringing Gigabit to wireless local area networks (WLANs). Since the seminal work <cit.>, energy efficiency stands as a major issue due to the intrinsic CSMA mechanism, which forces the network card to stay active performing idle listening.The 802.11 standard developers are fully aware of the energy issues that WiFi poses on battery-powered devices and have designed mechanisms to reduce energy consumption. One of such mechanisms is the Power Save (PS) mode, which is widely deployed among commercial wireless cards, although unevenly supported in software drivers. With this mechanism, a station (STA) may enter a doze state during long periods of time, subject to prior notification, if it has nothing to transmit. Meanwhile, packets addressed to this dozing STA are buffered and signalled in the Traffic Indication Map (TIM) attached to each beacon frame.The PS mechanism dramatically reduces the power consumption of a wireless card. However, the counterpart is that, since the card is put to sleep for hundreds of milliseconds, the user experiences a serious performance degradation because of the delays incurred. The automatic power save delivery (APSD) introduced by the 802.11e amendment (<cit.> gives a nice overview) is based on this mechanism, and aims to improve the downlink delivery by taking advantage of QoS mechanisms, but has not been widely adopted.More recently, the 802.11ac amendment improves the PS capabilities with the VHT TXOP power save mechanism. Basically, an 11ac STA can doze during a TXOP (transmission opportunity) in which it is not involved. This capability is announced within the new VHT (Very High Throughput) framing format, so that the AP knows that it cannot send traffic to those STAs until the TXOP's natural end, even if it is interrupted earlier. Still, the potential dozing is in the range of milliseconds and may lead to channel underuse if these TXOPs are not fully exploited.Considering shorter timescales, packet overhearing (i.e., listening to the wireless while there is an ongoing transmission addressed to other station) has been identified as a potential source of energy wastage <cit.>. Despite this, we have performed an extensive measurement campaign and have not found any attempt from manufacturers[Using our setup described in Section <ref>, we have tested cards from different manufacturers with the latest available firmwares and drivers: Broadcom BCM43224, Realtek RTL8191SEvB, Atheros AR9280, Intel Wireless-AC 7260 and Qualcomm Atheros QCA9880, which is a very recent state-of-the-art 11ac card.] to implement solutions to lessen its impact.In this work, we revisit this idea of packet overhearing as a trigger for sleep opportunities, and we take it one step further to the range of microseconds. To this end, we experimentally explore the timing limitations of 802.11 cards and, building on this knowledge, we design , a local standard-compliant energy-saving mechanism for 802.11 WLANs. With , a STA is capable of saving energy during packet overhearing autonomously, with full independence from the 802.11 capabilities supported or other power saving mechanisms in use, which makes it backwards compatible and incrementally deployable.In summary, the main contributions of this paper are the following: * A robust experimental characterisation of the timing and consumption behaviour of a COTS (commercial off-the-shelf) wireless card. * Design of , a local standard-compliant energy-saving mechanism that takes into account these timing limitations, as well as practical CSMA-related issues (e.g., capture effect, hidden nodes) not considered in prior work. * Performance evaluation ofbased on our measurements and real wireless traces, and performance degradation analysis due to channel errors. * Discussion of the impact and applicability of our mechanism. We draw attention to the need for standardising the hardware capabilities in terms of energy in 802.11. The remainder of this paper is organised as follows. Section <ref> reviews related work. Section <ref> experimentally explores the timing limitations of 802.11 COTS devices. Section <ref> analyses micro-sleep opportunities in CSMA as well as several practical issues of 802.11 networks, and proposes . Section <ref> presents the performance evaluation and Section <ref> summarises the conclusions of this paper.§ RELATED WORK It has been empirically proved that, in any network, the so-called power-law distribution, also known as Pareto distribution, holds for the traffic generated by its nodes. In other words, typically a few heavy hitters-generate most of the traffic while the majority of the nodes are only responsible for a small fraction, and this is true regardless of the network load. This means that the majority of nodes within any WLAN would spend most of the time in idle state.There are two main strategies to save energy in this idle time: the first one targets idle listening (the wireless channel is empty), and the second one targets packet overhearing (there are other nodes communicating). To support these savings, COTS devices have two main operational states as a function of the reference clock used: the active state and the sleep state. The more a card stays in sleep state, the less power it consumes.Since its conception, 802.11 has attempted to minimise idle listening with the introduction of the PS mode, and some previous work followed this path. For instance, Liu and Zhong <cit.> proposed μPM to exploit short idle intervals (<100 ms) without buffering or cooperation. μPM predicts the arrival time of the next frame and puts the interface in PS mode while no arrivals are expected. This mechanism demonstrated poor granularity (tens of ms) on existing hardware and leads to performance degradation due to frame loss. Therefore, it is only suitable for low-traffic scenarios.Others propose a PS-like operation. Jang et al. <cit.> described Snooze, an access point (AP)-directed micro-sleep scheduling and antenna configuration management method for 11n WLANs. As a consequence of its centralised design, the granularity of the so-called micro-sleeps in this approach is poor (few milliseconds), which poses doubts on its performance under heavy loads.Zhang and Shin <cit.> addressed the issue from a different standpoint with their Energy-Minimizing Idle Listening (E-MiLi). E-MiLi adaptively downclocks the card during idle periods, and reverts to full rate when an incoming frame is detected. To achieve this purpose, they need to change the physical layer (PHY) all the way down to enable downclocked detection, which severely limits the potential gains. For instance, the E-MiLi downclocking factor of 16 would yield a high power consumption in a modern card compared to its sleep state (see <ref>).On the other hand, all indicators show that we should expect an exponential grow in the number of wireless devices connected. Thus, there is a rough consensus about that densification will become one of the main aspects of next-generation wireless networks, which brings us back to the problem of packet overhearing. In this way, the recent 11ac amendment adds the ability to save energy during TXOPs, but this mechanism is restricted to QoS traffic, and the potential sleeps are coarse, in the range of milliseconds. Any sub-millisecond approach must take into account the timing parameters of the hardware. In fact, some early studies realise the importance of this issue when WiFi technology began to take off commercially <cit.>.Baiamonte and Chiasserini <cit.> were the first to chase fine-grained micro-sleep opportunities during packet overhearing. They define the Energy-efficient Distributed Access (EDA) scheme, which uses the 802.11 virtual carrier-sensing mechanism for power-saving purposes. Basically, a STA dozes when the Network Allocation Vector (NAV) or the backoff counter are non-zero. Unfortunately, this work lacks an empirical characterisation of the timing constraints needed to design a practical mechanism. Moreover, dozing during the backoff window is not 802.11-fair: in 802.11, STAs must sense the channel every single time slot during the contention period and, if another STA seizes the channel first, the backoff timer must be stopped in order to receive the incoming frame and set the NAV to the proper value. The EDA scheme allows STAs to doze during the contention period and, therefore, breaks the CSMA operation.Balaji et al. <cit.> revisited the problem of packet overhearing with a scheme called Sleep during Neighbor-Addressed Frame (SNAF). With SNAF, a wireless card checks the destination MAC address and switches to sleep state during the payload duration if it was addressed to other host. They assume, without any experimental validation though, an instantaneous switch-off and that the time required to wake up is equivalent to a Short Interframe Space (SIFS). In order to prevent the risk resulting from errors in the frame header that would lead to an incorrect NAV counter, the authors propose to introduce a new framing format with a new FCS devoted to the MAC header only. This solution lacks compatibility and introduces more overhead based on no evidence.Building on the same idea, Sudarshan et al. <cit.> proposed Übersleep. This time, the authors do not consider it necessary to add any extra FCS, as they claim (without any specific basis) that such errors are very unlikely.More recently, Palacios-Trujillo et al. modified DCF <cit.> and PCF <cit.> to exploit per-packet sleeps. They also applied these ideas to network coding <cit.> and to a polling-based version of 11ac's TXOP PS mode <cit.>. Unfortunately, all these papers rely on these early studies mentioned before <cit.>, which analysed old wireless cards unable to perform sub-millisecond transitions between states.§ STATE TRANSITION TIMES ON 802.11 CARDS From the hardware point of view, the standard PS mechanism requires supporting two states of operation: the awake state and the sleep state. The latter is implemented using a secondary low-frequency clock. Indeed, it is well-known that the power consumption of digital devices is proportional to the clock rate <cit.>. In fact, other types of devices, such as microcontroller-based devices or modern general-purpose CPUs, implement sleep states in the same way.For any microcontroller-based device with at least an idle state and a sleep state, one would expect the following behaviour for an ideal sleep. The device was in idle state, consuming P_idle, when, at an instant t_off, the sleep state is triggered and the consumption falls to P_sleep. A secondary low-power clock decrements a timer of duration Δ t_sleep = t_on - t_off, and then the expiration of this timer triggers the wake-up at t_on. The switching between states would be instantaneous and the energy saving would beE_save = (P_idle - P_sleep) ·Δ t_sleep This estimate could be considered valid for a time scale in the range of tens of milliseconds at least, but this is no longer true for micro-sleeps. Instead, Fig. <ref> presents a conceptual breakdown of a generic micro-sleep. After the sleep state is triggered at t_off, it takes Δ t_off before the power consumption actually reaches P_sleep. Similarly, after the wake-up is triggered at t_on, it takes some time, Δ t_on, to reach P_idle. Finally, the circuitry might need some additional time Δ t_ready to stabilise and operate normally. Thus, the most general expression for the energy saved in a micro-sleep is the following:E'_save= E_save - E_waste=(P_idle - P_sleep) · (Δ t_sleep -Δ t_ready)- ∫_Δ t_off∪Δ t_on (P - P_sleep) · dt where we have considered a general waveform P(t) for the transients Δ t_off and Δ t_on. E_waste represents an energy toll per sleep when compared to the ideal case.Our next objective is to quantify these limiting parameters, which can be defined as follows:Δ t_off is the time required to switch from idle power and to sleep power consumption.Δ t_on is the time required to switch from sleep power to idle power consumption.Δ t_ready is the time required for the electronics to stabilise and become ready to transmit/receive. The sum of this set of parameters defines the minimum sleep time, Δ t_sleep,min, for a given device:Δ t_sleep,min = Δ t_off + Δ t_on + Δ t_ready Performing this experimental characterisation requires the ability to timely trigger the sleep mode on demand. Most COTS cards are not suitable for this task, because they implement all the low-level operations in an internal proprietary binary firmware. After an extensive study, we found that cards based on the open-source driverare well suited for our needs, as they do not load a firmware to operate, and the driver has access to very low level functionality (e.g., supporting triggering the sleep mode by just writing into a register). Leveraging on these properties, we conducted our experimental characterisation using an Atheros AR9280 Half-height Mini PCI Express card.This card is attached to a PC through a flexible x1 PCI Express to Mini PCI Express adapter from Amfeltec, as the right part of Fig. <ref> depicts. This adapter connects the PCI bus' data channels to the host and provides an ATX port so that the wireless card can be supplied by an external power source. The same PC holds a NI PCI-6289, a high-accuracy multifunction data acquisition (DAQ) device, optimised for 18-bit input accuracy. Its timing resolution is 50 ns with an accuracy of 50 ppm of sample rate. In this way, the operations sent to the wireless card and the energy measurements can be correlated using the same timebase. A small command-line tool was developed[<https://github.com/Enchufa2/daq-acquire>] to perform measurements on the DAQ card using the open-source Comedi[<http://comedi.org/>] drivers and libraries.The power supply is a Keithley 2304A DC Power Supply, which is optimised for testing battery-operated wireless communication devices. It powers the wireless card through a measurement circuit that extracts the voltage and converts the current with a high-precision sensing resistor and amplifier. Considering that the DAQ card has a certain settling time, it can be modelled as a small capacitor which acts as a low-pass filter. Thus, two buffers (voltage followers) are placed before the DAQ card to decrease the output impedance of the measurement circuit <cit.>.The card under test is associated to an AP in 11a mode to avoid any interfering traffic from neighbouring networks. This AP is placed very close to the node to obtain the best possible signal quality, as we are simply interested in not losing the connectivity for this experiment. With this setup, the idea is to trigger the sleep state, then bring the interface back to idle and finally trigger the transmission of a buffered packet as fast as possible, in order to find the timing constraints imposed by the hardware in the power signature. From an initial stable power level, with the interface associated and in idle mode, we would expect a falling edge to a lower power level corresponding to the sleep state. Then the power level would raise again to the idle level and, finally, a big power peak would mark the transmission of the packet. By correlating the timestamps of our commands and the timestamps of the measured power signature, we are able to measure the limiting parameters Δ t_off, Δ t_on, Δ t_ready. The methodology to reproduce these steps required hacking thedriver to timely trigger write operations in the proper card registers, and to induce a transmission of a pre-buffered packet directly in the device without going through the entire network stack. The code for reproducing this experiment is available on GitHub[<https://github.com/Enchufa2/crap/tree/master/ath9k/downup>], and comprises the following steps:* Initially, the card is in idle state, connected to the AP. * A RAW socket (Linux AF_PACKET socket) is created and a socket buffer is prepared with a fake packet. * t_off is triggered by writing a register in the card, which has proved to be almost instantaneous in kernel space. * A micro-delay of 60 μs is introduced in order to give the card time to react. * t_on is triggered with another register write. * Another timer sets a programmable delay. * The fake frame is sent using a low-level interface, i.e., calling the functionfrom theoperations directly. By doing this, we try to spend very little time in kernel. The power signature recorded as a result of this experiment is shown in Fig. <ref>. As we can see, the card spends Δ t_off = 50 μs consuming P_idle and then it switches off to P_sleep in only 10 μs. Then, t_on is triggered. Similarly, the card spends Δ t_on = 50 μs consuming P_sleep and it wakes up almost instantaneously. Note that the transmission of the packet is triggered right after the t_on event and the frame spends very little time at the kernel (the time spent in kernel corresponds to the width of the rectangle labelled asin the graph). Nonetheless, the card sends the packet 200 μs after returning to idle, even though the frame was ready for transmission much earlier.To understand the reasons for the delay in the frame transmission observed above, we performed an experiment in which frame transmissions were triggered at different points in time by introducing different delays between the t_on andevents. Fig. <ref> shows that the card starts transmitting always in the same instant whenever the kernel triggers the transmission within the first 250 μs right after the t_on event (lines 0 and 200). Otherwise, the card starts transmitting almost instantaneously (line 350). This experiments demonstrate that the device needs Δ t_ready = 200 μs to get ready to transmit/receive after returning to idle.To sum up, our experiments show that, if we want to bring this card to sleep during a certain time Δ t_sleep, we should take into account that it requires a minimum sleep time Δ t_sleep,min=300 μs. Therefore, Δ t_sleep≥Δ t_sleep,min must be satisfied, and we must program the t_on interrupt to be triggered Δ t_on + Δ t_ready=250 μs before the end of the sleep. Note also that the card wastes a fixed time Δ t_waste consuming P_idle:Δ t_waste = Δ t_off + Δ t_ready which is equal to 250 μs also. Thus, the total time in sleep state is Δ t_sleep - Δ t_waste, and the energy toll from Equation (<ref>) can be simplified as follows:E_waste≈ (P_idle - P_sleep)·Δ t_waste §DESIGN The key idea ofis to put the interface to sleep during packet overhearing while meeting the constraint Δ t_sleep,min identified in the previous section. Additionally, the algorithm should be local in order to be incrementally deployable, standard-compliant, and should take into account real-world practical issues. For this purpose, Section <ref> analyses available micro-sleep opportunities in 802.11 and determines under which circumstances the NAV mechanism can be used to extend a micro-sleep while ensuring that no frames are lost within such time. Section <ref> explores well-known practical issues of WLAN networks that had not been addressed by previous energy-saving schemes. Finally, Section <ref> presents . §.§ Micro-sleep opportunities in 802.11 Due to the CSMA mechanism, 802.11 STAs receive every single frame from their SSID or from others in the same channel (even some frames from overlapping channels). Upon receiving a frame, a STA checks the Frame Check Sequence (FCS) for errors and then, and only after having received the entire frame, it discards the frame if it is not the recipient. In 802.11 terminology, this is called packet overhearing. Since packet overhearing consumes the power corresponding to a full packet reception that is not intended for the station, it represents a source of inefficiency. Thus, we could avoid this unnecessary power consumption by triggering micro-sleeps that bring the wireless card to a low-energy state.Indeed, the Physical Layer Convergence Procedure (PLCP) carries the necessary information (rate and length) to know the duration of the PLCP Service Data Unit (PSDU), which consists of a MAC frame or an aggregate of frames. And the first 10 bytes of a MAC frame indicate the intended receiver, so a frame could be discarded very early, and the station could be brought to sleep if the hardware allows for such a short sleeping time. Therefore, the most naive micro-sleep mechanism could determine, given the constraint Δ t_sleep,min, whether the interface could be switched off in a frame-by-frame basis. And additionally, this behaviour can be further improved by leveraging the 802.11 virtual carrier-sensing mechanism. Virtual carrier-sensing allows STAs not only to seize the channel for a single transmission, but also to signal a longer exchange with another STA. For instance, this exchange can include the acknowledgement sent by the receiver, or multiple frames from a station in a single transmission opportunity (TXOP). So MAC frames carry a duration value that updates the Network Allocation Vector (NAV), which is a counter indicating how much time the channel will be busy due to the exchange of frames triggered by the current frame. And this duration field is, for our benefit, enclosed in the first 10 bytes of the MAC header too. Therefore, the NAV could be exploited to obtain substantial gains in terms of energy. In order to unveil potential sleeping opportunities within the different states of operation in 802.11, first of all we review the setting of the NAV. 802.11 comprises two families of channel access methods. Within the legacy methods, the Distributed Coordination Function (DCF) is the basic mechanism with which all STAs contend employing CMSA/CA with binary exponential backoff. In this scheme, the duration value provides single protection: the setting of the NAV value is such that protects up to the end of one frame (data, management) plus any additional overhead (control frames). For instance, this could be the ACK following a data frame or the CTS + data + ACK following an RTS.When the Point Coordination Function (PCF) is used, time between beacons is rigidly divided into contention and contention-free periods (CP and CFP, respectively). The AP starts the CFP by setting the duration value in the beacon to its maximum value (which is 32 768; Table 8-3 of the IEEE Std 802.11-2012 <cit.> depicts the duration/ID field encoding). Then, it coordinates the communication by sending CF-Poll frames to each STA. As a consequence, a STA cannot use the NAV to sleep during the CFP, because it must remain CF-pollable, but it still can doze during each individual packet transmission. In the CP, DCF is used.802.11e introduces traffic categories (TC), the concept of TXOP, and a new family of access methods called Hybrid Coordination Function (HCF), which includes the Enhanced Distributed Channel Access (EDCA) and the HCF Controlled Channel Access (HCCA). These two methods are the QoS-aware versions of DCF and PCF respectively.Under EDCA, there are two classes of duration values: single protection, as in DCF, and multiple protection, where the NAV protects up to the end of a sequence of frames within the same TXOP. By setting the appropriate TC, any STA may start a TXOP, which is zero for background and best-effort traffic, and of several milliseconds for video and audio traffic as defined in the standard (Table 8-105 of the IEEE Std 802.11-2012 <cit.>). A non-zero TXOP may be used for dozing, as 11ac does, but these are long sleeps and the AP needs to support this feature, because a TXOP may be truncated at any moment with a CF-End frame, and it must keep buffering any frame directed to any 11ac dozing STA until the NAV set at the start of the TXOP has expired.HCCA works similarly to PCF, but under HCCA, the CFP can be started at almost any time. In the CFP, when the AP sends a CF-poll to a STA, it sets the NAV of other STAs for an amount equal to the TXOP. Nevertheless, the AP may reclaim the TXOP if it ends too early (e.g., the STA has nothing to transmit) by resetting the NAV of other STAs with another CF-Poll. Again, the NAV cannot be locally exploited to perform energy saving during a CFP.Finally, there is another special case in which the NAV cannot be exploited either. 802.11g was designed to bring the advantages of 11a to the 2.4 GHz band. In order to interoperate with older 11b deployments, it introduces CTS-to-self frames (also used by more recent amendments such as 11n and 11ac). These are standard CTS frames, transmitted at a legacy rate and not preceded by an RTS, that are sent by a certain STA to itself to seize the channel before sending a data frame. In this case, the other STAs cannot know which will be the destination of the next frame. Therefore, they should not use the duration field of a CTS for dozing. §.§ Practical issues §.§.§ Impact of capture effect It is well-known that a high-power transmission can totally blind another one with a lower SNR. Theoretically, two STAs seizing the channel at the same time yields a collision. However, in practice, if the power ratio is sufficiently high, a wireless card is able to decode the high-power frame without error, thus ignoring the other transmission. This is called capture effect, and although not described by the standard, it must be taken into account as it is present in real deployments.According to <cit.>, there are two types of capture effect depending on the order of the frames: if the high-power frame comes first, it is called first capture effect; otherwise, it is called second capture effect. The first one is equivalent to receiving a frame and some noise after it, and then it has no impact in our analysis. In the second capture effect, the receiving STA stops decoding the PLCP of the low-power frame and switches to another with higher power. If the latter arrives before a power-saving mechanism makes the decision to go to sleep, the mechanism introduces no misbehaviour.However, <cit.> suggests that a high-power transmission could blind a low-power one at any time, even when the actual data transmission has begun. This is called Message in Message (MIM) in the literature <cit.>, and it could negatively impact the performance of an interface implementing an energy-efficiency mechanism based on packet overhearing. In the following, we will provide new experimental evidence supporting that this issue still holds in modern wireless cards.We evaluated the properties of the MIM effect with an experimental setup consisting of a card under test, a brand new 802.11ac three-stream Qualcomm Atheros QCA988x card, and three additional helper nodes. These are equipped with Broadcom KBFG4318 802.11g cards, whose behaviour can be changed with the open-source firmware OpenFWWF <cit.>. We disable the carrier sensing and back-off mechanisms so that we can decide the departure time of every transmitted frame with 1 μs granularity with respect to the internal 1MHz clock.Fig. <ref> depicts the measurement setup, which consists of a node equipped with our Atheros card under test (ath), a synchronization (Sync) node, a high energy (HE) node and a low energy (LE) node. These two HE and LE nodes were manually carried around at different distances with respect to the ath node until we reached the desired power levels.The Sync node transmits 80-byte long beacon-like frames periodically at 48 Mbps, one beacon every 8192 μs: the time among consecutive beacons is divided in 8 schedules of 1024 μs. Inside each schedule, time is additionally divided into 64 micro-slots of 16μs. We then program the firmware of the HE and LE nodes to use the beacon-like frames for keeping their clocks synchronised and to transmit a single frame (138-μs long) per schedule starting at a specific micro-slot. This allows us to always start the transmission of the low energy frame from the LE node before the high energy frame from the HE node, and to configure the exact delay Δ t as a multiple of the micro-slot duration. For instance, we set up a Δ t = 32 μs by configuring LE node to transmit at slot 15, HE node at slot 17. By moving LE node away from the ath node while the HE node is always close, we are able to control the relative power difference Δ P received by the ath node between frames coming from the LE and HE nodes. With the configured timings, we are able to replicate the reception experiment at the ath node approximately 976 times per second, thus collecting meaningful statistics in seconds. We obtained the results shown in Table <ref>. When the energy gap is small (≤ 5 dB), the MIM effect never enters into play as we can see from the first part of Table <ref>. If the two frames are transmitted at the same time, then the QCA card receives the majority of the HE frames (92%) despite some of them are broken (17%); almost no LE frames are received. By increasing the delay to 16 μs, the QCA card stops working: the short delay means that the HE frame collide with the LE one at the PLCP level. The energy gap does not allow the QCA correlator to restart decoding a new PLCP and, in fact, only a few frames are sporadically received. Further increasing the delay allows the QCA card to correctly receive the PLCP preamble of the LE frame, but then the PDU decoding is affected by errors (e.g., delay set to 48 μs) because of collision. Finally, if the delay is high enough so that both frames fit into a schedule, the QCA card receives everything correctly (≥ 144 μs).When the energy gap exceeds a threshold (e.g., more than 35 dB), then the behaviour of the QCA card changes radically as we can see from the second part of Table <ref>: first, with no delay, all high energy frames are received (expected given that they overkill the others); second, when both frame types fit in the schedule, all of them are received, which confirms that the link between LE node and the QCA is still very good. But, unlike the previous case, HE frames are received regardless of the delay, which means that the correlator restarts decoding the PLCP of the second frame because of the higher energy, enough for distinguishing it from the first frame that simply turns into a negligible noise.Thus, our experiments confirm that the MIM effect actually affects modern wireless cards, and therefore it should be taken into account in any micro-sleep strategy. Let us consider, for instance, a common infrastructure-based scenario in which certain STA receives low-power frames from a distant network in the same channel. If the AP does not see them, we are facing the hidden node problem. It is clear that none of these frames will be addressed to our STA, but, if it goes to sleep during these transmissions, it may lose potential high-power frames from its BSSID. Therefore, if we perform micro-sleeps under hidden node conditions, in some cases we may lose frames that we would receive otherwise thanks to the capture effect. The same situation may happen within the local BSSID (the low-power frames belong to the same network), but this is far more rare, as such a hidden node will become disconnected sooner or later.In order to circumvent these issues, a STA should only exploit micro-sleep opportunities arising from its own network. To discard packets originating from other networks, the algorithm looks at the BSSID in the receiver address within frames addressed to an AP. If the frame was sent by an AP, it only needs to read 6 additional bytes (in the worst case), which are included in the transmitter address. Even so, these additional bytes do not necessarily involve consuming more time, depending on the modulation. For instance, for OFDM 11ag rates, this leads to a time increase of 8 μs at 6 and 9 Mbps, 4 μs at 12, 18 and 36 Mbps, and no time increase at 24, 48 and 54 Mbps.§.§.§ Impact of errors in the MAC header Taking decisions without checking the FCS (placed at the end of the frame) for errors or adding any protection mechanism may lead to performance degradation due to frame loss. This problem was firstly identified by <cit.> and <cit.> which, based on purely qualitative criteria, reached opposite conclusions. The first work advocates for the need for a new CRC to protect the header bits while the latter dismisses this need. This section is devoted to analyse quantitatively the impact of errors.At a first stage, we need to identify, field by field, which cases are capable of harming the performance of our algorithm due to frame loss. The duration/ID field (2 bytes) and the MAC addresses (6 bytes each) are an integral part of our algorithm. According to its encoding, the duration/ID field will be interpreted as an actual duration if and only if the bit 15 is equal to 0. Given that the bit 15 is the most significant one, this condition is equivalent to the value being smaller than 32 768. Therefore, we can distinguish the following cases in terms of the possible errors:* An error changes the bit 15 from 0 to 1. The field will not be interpreted as a duration and hence we will not go to sleep. We will be missing an opportunity to save energy, but there will be no frame loss and, therefore, the network performance will not be affected.* An error changes the bit 15 from 1 to 0. The field will be wrongly interpreted as a duration. The resulting sleep will be up to 33 ms longer than required, with the potential frame loss associated.* With the bit 15 equal to 0, an error affects the previous bits. The resulting sleep will be shorter or longer that the real one. In the first case, we will be missing an opportunity to save energy; in the second case, there is again a potential frame loss. Regarding the receiver address field, there exist the following potential issues:* A multicast address changes but remains multicast. The frame will be received and discarded, i.e., the behaviour will be the same as with no error. Hence, it does not affect.* A unicast address changes to multicast. The frame will be received and discarded after detecting the error. If the unicast frame was addressed to this host, it does not affect. If it was addressed to another host, we will be missing an opportunity to save energy.* A multicast address changes to unicast. If the unicast frame is addressed to this host, it does not affect. If it is addressed to another host, we will save energy with a frame which would be otherwise received and discarded.* Another host's unicast address changes to your own. This case is very unlikely. The frame will be received and discarded, so we will be missing an opportunity to save energy.* Your own unicast address changes to another's. We will save energy with a frame otherwise received and discarded. As for the transmission address field, this is checked as an additional protection against the undesirable effects of the already discussed intra-frame capture effect. If the local BSSID in a packet changes to another BSSID, we will be missing an opportunity to save energy. It is extremely unlikely that an error in this field could lead to frame loss: a frame from a foreign node (belonging to another BSSID and hidden to our AP) should contain an error that matches the local BSSID in the precise moment in which our AP tries to send us a frame (note that this frame might be received because of the MIM effect explained previously).Henceforth, we draw the following conclusions from the above analysis: * Errors at the MAC addresses do not produce frame loss, because under no circumstances they imply frame loss. The only impact is that there will be several new opportunities to save energy and several others will be wasted. * Errors at the duration/ID field, however, may produce frame loss due to frame loss in periods of time up to 33 ms. Also several energy-saving opportunities may be missed without yielding any frame loss. * An error burst affecting both the duration/ID field and the receiver address may potentially change the latter in a way that the frame would be received (multicast bit set to 1) and discarded, and thus preventing the frame loss. From the above, we have that the only case that may yield performance degradation in terms of frame loss is when we have errors in the duration/ID field. In the following, we are going to analytically study and quantify the probability of frame loss in this case. For our analysis, we first consider statistically independent single-bit errors. Each bit is considered the outcome of a Bernoulli trial with a success probability equal to the bit error probability p_b. Thus, the number of bit errors, X, in certain field is given by a Binomial distribution X∼B(N, p_b), where N is the length of that field. With these assumptions, we can compute the probability of having more than one erroneous bit, (X ≥ 2), which is three-four orders of magnitude smaller than p_b with realistic p_b values. Therefore, we assume that we never have more than one bit error in the frame header, so the probability of receiving an erroneous duration value with a single-bit error, p_e,b, is the following:p_e,b≈ 1 - (1 - p_b)^15 However, not all the errors imply a duration value greater than the original one, but only those which convert a zero into a one. Let us call Hw(i) the Hamming weight, i.e., the number of ones in the binary representation of the integer i. The probability of an erroneous duration value greater than the original, p_eg,b, is the following:p_eg,b(i) = p_e,b·15 -Hw(i)/15 which represents a fraction of the probability p_e,b and depends on the original duration i (before the error). In order to understand the implications of the above analysis into real networks, we have analysed the data set SIGCOMM'08 <cit.> and gathered which duration values are the most common. In the light of the results depicted in Table <ref>, it seems reasonable to approximate p_eg,b/p_e,b≈ 1, because it is very likely that the resulting duration will be greater than the original.Finally, we can approximate p_b by the BER and, based on the above data and considerations, the frame loss probability, p_loss, due to an excessive sleep interval using a single-bit error model is the following:p_loss = p_eg,b≈ p_e,b≈ 1 - (1 - BER)^15 The above analysis assumes that errors occur independently. However, it is well known that in reality errors typically occur in bursts. In order to understand the impact of error bursts in our scheme, we analyse a scenario with independent error bursts of length X bits, where X is a random variable. To this end, we use the Neyman-A contagious model <cit.>, which has been successfully applied in telecommunications to describe burst error distributions <cit.>. This model assumes that both the bursts and the burst length are Poisson-distributed. Although assuming independency between errors in the same burst may not be accurate, it has been shown that the Neyman-A model performs well for short intervals <cit.>, which is our case.The probability of having k errors in an interval of N bits, given the Neyman-A model, is the following:p_N(k) = λ_b^k/k!e^-λ_B∑_i=0^∞i^k/i!λ_B^i e^-iλ_b whereλ_b is the average number of bits in a burst.λ_B = Np_b/λ_b is the average number of bursts. This formula can be transformed into a recursive one with finite sums <cit.>:p_N(k)= λ_Bλ_b e^-λ_b/k∑_j=0^k-1λ_b^j/j!p_N(k-1-j)p_N(0)= e^-λ_B(1-e^-λ_b) Following the same reasoning as for the single-bit case, we can assume one burst at a time which will convert the duration value into a higher one. Then, the frame loss probability is the following: p_loss = ∑_k=1^15 p_15(k) with parameters λ_b and p_b ≈BER.Fig. <ref> evaluates both error models as a function of BER. As expected, the single-bit error model is an upper bound for the error burst model and represents a worst-case scenario. At most, the frame loss probability is one order of magnitude higher than BER. Therefore, we conclude that the frame loss is negligible for reasonable BERs and, consequently, the limited benefit of an additional CRC does not compensate the issues. §.§ Algorithm design In the following, we present , which builds upon the insights provided in previous sections and tries to save energy during the channel transmissions in which the STA is not involved. However, not all transmissions addressed to other stations are eligible for dozing, as the practical issues derived from the capture effect may incur in performance degradation. Therefore, the algorithm must check both the receiver as well as the transmitter address in the MAC header in order to determine whether the incoming frame is addressed to another station and it comes from within the same network.If these conditions are met, a basic micro-sleep will last the duration of the rest of the incoming frame plus an inter-frame space (SIFS). Unfortunately, the long times required to bring an interface back and forth from sleep, as discovered in Section <ref>, shows that this basic micro-sleep may not be long enough to be exploitable. Thus, the algorithm should take advantage of the NAV field whenever possible. Our previous analysis shows that this duration information stored in the NAV is not exploitable in every circumstance: the interface can leverage this additional time during CPs and it must avoid any NAV set by a CTS packet.Finally, after a micro-sleep, two possible situations arise: * The card wakes up at the end of a frame exchange. For instance, after a data + ACK exchange. In this case, all STAs should wait for a DIFS interval before contending again. * The card wakes up in the middle of a frame exchange. For instance, see Fig. <ref>, where an RTS/CTS-based fragmented transmission is depicted.In the latter example, an RTS sets the NAV to the end of a fragment, and our algorithm triggers the sleep. This first fragment sets the NAV to the end of the second fragment, but it is not seen by the dozing STA. When the latter wakes up, it sees a SIFS period of silence and then the second fragment, which sets its NAV again and may trigger another sleep. This implies that the STA can doze for an additional SIFS, as Fig. <ref> shows, and wait in idle state until a DIFS is completed before trying to contend again.Based on the above, Algorithm <ref> describes the main loop of a wireless card's microcontroller that would implement our mechanism. When the first 16 bytes of the incoming frame are received, all the information needed to take the decision is available: the duration value (Δ t_NAV), the receiver address (R_A) and the transmitter address (T_A). The ability to stop a frame's reception at any point has been demonstrated to be feasible <cit.>. Note that MAC addresses can be efficiently compared in a streamed way, so that the first differing byte (if the first byte of the R_A has the multicast bit set to zero, i.e., R_A is unicast) triggers our sleep procedure (Set_Sleep in Algorithm <ref>). In addition, the main loop should keep up to date a global variable (C) indicating whether the contention is currently allowed (CP) or not (CFP). This is straightforward, as every CFP starts and finishes with a beacon frame.The Set_Sleep procedure takes as input the remaining time until the end of the incoming frame (Δ t_DATA) and the duration value (Δ t_NAV). The latter is used only if it is a valid duration value and a CP is active. Then, the card may doze during Δ t_sleep (if this period is greater than Δ t_sleep,min), wait for a DIFS to complete and return to the main loop.Finally, it is worth noting that this algorithm is deterministic, as it is based on a set of conditions to trigger the sleep procedure. It works locally with the information already available in the protocol headers, without incurring in any additional control overhead and without impacting the normal operation of 802.11. Specifically, our analytical study of the impact of errors in the first 16 bytes of the MAC header shows that the probability of performance degradation is comparable to the BER under normal channel conditions. Therefore, the overall performance in terms of throughput and delay is completely equivalent to normal 802.11.t§ PERFORMANCE EVALUATION This section is devoted to evaluate the performance of . First, Section <ref>, through trace-driven simulation, shows thatsignificantly reduces the overhearing time and the energy consumption of a real network. Secondly, Section <ref> analyses the impact of the timing constraints imposed by the hardware, which are specially bad in the case of the AR9280, and discusses the applicability ofin terms of those parameters and the evolution trends in the 802.11 standard. §.§ Evaluation with real traces In the following, we conduct an evaluation to assess how much energy might be saved in a real network if all STAs implementusing the AR9280, the wireless card characterised in Section <ref>. The reasons for this are twofold. On the one hand, the timing properties of this interface are particularly bad if we think of typical frame durations in 802.11, which means that many micro-sleep opportunities will be lost due to hardware constraints. On the other hand, it does not support newer standards that could potentially lead to longer micro-sleep opportunities through mechanisms such as frame aggregation. Therefore, an evaluation based on an 11a/g network and the AR9280 chip represents a worst case scenario for our algorithm.For this purpose, we used 802.11a wireless traces with about 44 million packets, divided in 43 files, from the data set SIGCOMM'08 <cit.>. The methodology followed to parse each trace file is as follows. Firstly, we discover all the STAs and APs present. Each STA is mapped into its BSSID and a bit array is developed in order to hold the status at each point in time (online or offline). It is hard to say when a certain STA is offline from a capture, because they almost always disappear without sending a disassociation frame. Thus, we use the default rule in , the daemon that implements the AP functionality in Linux: a STA is considered online if it transmitted a frame within the last 5 min.Secondly, we measure the amount of time that each STA spends (without our algorithm) in the following states: transmission, reception, overhearing and idle. We consider that online STAs are always awake; i.e., even if a STA announces that it is going into PS mode, we ignore this announcement. We measure also the amount of time that each STA would spend (with our algorithm) in transmission, reception, overhearing, sleep and idle. Transmission and reception times match the previous case, as expected. As part of idle time, we account separately the wasted time in each micro-sleep as a consequence of hardware limitations (the fixed toll Δ t_waste). After this processing, there are a lot of duplicate unique identifiers (MAC addresses), i.e., STAs appearing in more than one trace file. Those entries are summarised by aggregating the time within each state.At this point, let us define the activity time as the sum of transmission, reception, overhearing, sleep and wasted time. We do not account for idle time since our goal is to understand how much power we can save in the periods of activity, which are the only ones that consume power in wireless transmissions (the scope of this paper). Using the definition above, we found that the majority of STAs reveals very little activity (they are connected for a few seconds and disappear). Therefore, we took the upper decile in terms of activity, thus obtaining the 42 more active STAs. The activity aggregation of all STAs is normalised and represented in Fig. <ref>. Transmission (tx) and reception (rx) times are labelled as common, because STAs spend the same time transmitting and receiving both with and without our algorithm. It is clear that our mechanism effectively reduces the total overhearing (ov) time from a median of 70% to a 30% approximately (a 57% reduction). The card spends consistently less time in overhearing because this overhearing time difference, along with some idle (id) time from inter-frame spaces, turns into micro-sleeps, that is, sleep (sl) and wasted (wa) time.This activity aggregation enables us to calculate the total energy consumption using the power values from the thorough characterisation presented in <ref>. Fig. <ref> depicts the energy consumption in units of mAh (assuming a typical 3.7-V battery). The energy savings overcome 1200 mAh even with the timing limitations of the AR9280 card, which (1) prevents the card from going to sleep when the overhearing time is not sufficiently long, and (2) wastes a long fixed time in idle during each successful micro-sleep. This reduction amounts to a 21.4% of the energy spent in overhearing and a 15.8% of the total energy during the activity time, when the transmission and reception contributions are also considered.Fig. <ref> provides a breakdown of the data by STA. The lower graph shows the activity breakdown per STA for our algorithm (transmission bars, in white, are very small). Overhearing time is reduced to a more or less constant fraction for all STAs (i.e., with the algorithm, the overhearing bars represent more or less a 30% of the total activity for all STAs), while less participative STAs (left part of the graph) spend more time sleeping. The upper graph shows the energy consumption per STA with our algorithm along with the energy-saving in dark gray, which is in the order of tens of mAh per STA. §.§ Impact of timing constraints The performance gains ofdepend on the behaviour of the circuitry. Its capabilities, in terms of timing, determine the maximum savings that can be achieved. Particularly, each micro-sleep has an efficiency (in comparison to an ideal scheme in which the card stays in sleep state over the entire duration of the micro-sleep) given byE'_save/E_save = E_save - E_waste/E_save≈ 1 - Δ t_waste/Δ t_sleep which results from the combination of Equations (<ref>), (<ref>) and (<ref>). Fig. <ref> represents this sleep efficiency for the AR9280 card (Δ t_waste=250) along with other values. It is clear that an improvement of Δ t_waste is fundamental to boost performance in short sleeps.Similarly, the constraint Δ t_sleep,min limits the applicability of , especially in those cases where the NAV cannot be used to extend the micro-sleep. For instance, let us consider the more common case in 11a/b/g networks: the transmission of a frame (up to 1500 bytes long) plus the corresponding ACK. Then,Δ t_sleep,min≤Δ t_DATA + Δ t_SIFS + Δ t_ACK + Δ t_SIFS and expanding the right side of the inequality,Δ t_sleep,min ≤8(14+l_min+4)/λ_DATA + Δ t_SIFS+ Δ t_PLCP + 8(14+2)/λ_ACK + Δ t_SIFS Here, we can find l_min, which is the minimum amount of data (in bytes, and apart from the MAC header and the FCS) that a frame must contain in order to last Δ t_sleep,min. Based on this l_min, Fig. <ref> defines the applicability in 802.11a DCF in terms of frame sizes (≤ 1500 bytes) that last Δ t_sleep,min at least. Again, an improvement in Δ t_waste would boost not only the energy saved per sleep, but also the general applicability defined in this way.The applicability ofmay also be affected by the evolution of the standard. Particularly, 802.11n introduced, and 802.11ac followed, a series of changes enabling high and very high throughput respectively, up to Gigabit in the latter case. This improvement is largely based on MIMO and channel binding: multiple spatial and frequency streams. Nevertheless, a single 20-MHz spatial stream is more or less equivalent to 11ag. Some enhancements (shorter guard interval and coding enhancements) may boost the throughput of a single stream from 54 to 72 Mbps under optimum conditions. Yet it is also the case that the PLCP is much longer to accommodate the complexity of the new modulation coding schemes (MCSs). This overhead not only extends each transmission, but also encourages the use of frame aggregation. Thus, the increasing bandwidth, in current amendments or future ones, does not necessarily imply a shorter airtime in practice, and our algorithm is still valid. Reducing PHY's timing requirements is essential to boost energy savings, but its feasibility should be further investigated. Nonetheless, there are some clues that suggest that there is plenty of room for improvement. In the first place, Δ t_off and Δ t_on should depend on the internal firmware implementation (i.e., the complexity of the saved/restored state). Secondly, Fig. <ref> indicates that a transmission is far more aggressive, in terms of a sudden power rise, than a return from sleep. From this standpoint, Δ t_ready = 200 μs would be a pessimistic estimate of the time required by the circuitry to stabilise. Last, but not least, the 802.3 standard goes beyond 802.11 and, albeit to a limited extent, it defines some timing parameters of the PHYs (e.g., Δ t_w_phy would be equivalent to our Δ t_on+Δ t_ready). These timing parameters are in the range of tens of μs in the worst case (see Table 78-4 of the IEEE Std 802.3-2008 <cit.>).Due to these reasons, WiFi card manufacturers should push for a better power consumption behaviour, which is necessary to boost performance with the power-saving mechanism presented in this paper. Furthermore, it is necessary for the standardisation committees and the manufacturers to collaborate to agree power consumption behaviour guidelines for the hardware (similarly to what has been done with 802.3). Indeed, strict timing parameters would allow researchers and developers to design more advanced power-saving schemes.§ CONCLUSIONS Based on a thorough characterisation of the timing constraints and energy consumption of 802.11 interfaces, we have exhaustively analysed the micro-sleep opportunities that are available in current WLANs. We have unveiled the practical challenges of these opportunities, previously unnoticed in the literature, and, building on this knowledge, we have proposedan energy-saving scheme that is orthogonal to the existing standard PS mechanisms. Unlike previous attempts, our scheme takes into account the non-zero time and energy required to move back and forth between the active and sleep states, and decides when to put the interface to sleep in order to make the most of these opportunities while avoiding frame losses.We have demonstrated the feasibility of our approach using a robust methodology and high-precision instrumentation, showing that, despite the limitations of COTS hardware, the use of our scheme would result in a 57% reduction in the time spent in overhearing, thus leading to an energy saving of 15.8% of the activity time according to our trace-based simulation. Finally, based on these results, we have made the case for the strict specification of energy-related parameters of 802.11 hardware, which would enable the design of platform-agnostic energy-saving strategies.§ ENERGY CONSUMPTION CHARACTERISATION§.§ State consumption parametrisation In order to gain insight into the energy savings of , we performed a complete state parametrisation (power consumption in transmission, reception, overhearing, idle and sleep) of the AR9280 card (the active state in the traces used for the evaluation, Section <ref>) using the same scenario as in Section <ref> (see Fig. <ref>). As in Section <ref>, all measurements (except for the sleep state) were taken with the wireless card associated to the AP in 11a mode to avoid any interfering traffic, and it was placed very close to the node to obtain the best possible signal quality. The reception of beacons is accounted in the baseline consumption (idle). The card under test performed transmissions/receptions to/from the AP at a constant rate and with fixed packet length. In order to avoid artifacts from the reception/transission of ACKs, UDP was used and the NoACK policy was enabled. Packet overhearing was tested by generating traffic of the same characteristics from a secondary STA placed in the same close range (∼cm). Under these conditions, several values of airtime percentage were swept. For each experiment, current and voltage signals were sampled at 100 kHz and the mean power consumption was measured with a basic precision of 1 mW over intervals of 3 s.Regarding the sleep state, the driverinternally defines three states of operation: awake, network sleep and full sleep. A closer analysis reveals that the card is awake, or in active state, when it is operational (i.e., transmitting, receiving or in idle state, whether as part of an SSID or in monitor mode), and it is in full sleep state when it is not operational at all (i.e., interface down or up but not connected to any SSID). The network sleep state is used by the 802.11 PS mechanism, but essentially works in the same way as full sleep, that is, it turns off the main reference clock and switches to a secondary 32 kHz one. Therefore, we saw that full sleep and network sleep are the same state in terms of energy: they consume exactly the same power. The only difference is that network sleep sets up a tasklet to wake the interface periodically (to receive TIMs), as required by the PS mode.Fig. <ref> shows our results for transmission, reception and overhearing. Idle and sleep consumptions were measured independently, are depicted with gray horizontal lines for reference. As expected, power consumptions in transmission/reception/overhearing state are proportional to airtime, thus the power consumption of such operations can be easily estimated by extrapolating the regression line to the 100% of airtime (gray vertical line).These mean values are shown in Table <ref>. First of all, reception and overhearing consumptions are the same within the error, and they are close to idle consumption. Transmission power is more than two times larger than reception. Finally, the sleep state saves almost the 70% of the energy compared to idle/reception. §.§ Downclocking consumption characterisation As the AR9280's documentation states, its reference clock runs at 44 MHz for 20 MHz channels and at 88 MHz for 40 MHz channels in the 2.4 GHz band, and at 40 MHz for 20 MHz channels and at 80 MHz for 40 MHz channels in the 5 GHz band. Thus, as Table <ref> shows, we measured two more results to gain additional insight into the behaviour of the main reference clock, which is known to be linear <cit.>.Using an 11n-capable AP, we measured the idle power in the 2.4 GHz band with two channel widths, 20 and 40 MHz. Note that the idle power in 11a mode (5 GHz band), with a 40 MHz clock, is higher than the idle power with a 44 MHz clock. This is because both bands are not directly comparable, as the 5 GHz one requires more amplification (the effect of the RF amplifier is out of the scope of this paper).With these two points, we can assume a higher error (of about 10 mW) and try to estimate a maximum and a minimum slope for the power consumed by the main clock as a function of the frequency f. The resulting averaged regression formula is the following:P(f) = 0.91(3) + 0.0051(5)f This result, although coarse, enables us to estimate how a downclocking approach should perform in COTS devices. It shows that the main consumption of the clock goes to the baseline power (the power needed to simply turn it on), and that the increment per MHz is low: 5.1(5) mW/MHz. As a consequence, power-saving mechanisms based on idle downclocking, such as <cit.>, will not save too much energy compared to the sleep state of COTS devices. For instance, the x16 downclock of <cit.> applied to this Atheros card throws an idle power consumption of 1.10(2) W in 11a mode, i.e., about a 15% of saving according to Table <ref>, which is low compared to the 70% of its sleep state. This questions the effectiveness of complex schemes based on downclocking compared to simpler ones based on the already existing sleep state. elsarticle-num | http://arxiv.org/abs/1706.08312v2 | {
"authors": [
"Arturo Azcorra",
"Iñaki Ucar",
"Francesco Gringoli",
"Albert Banchs",
"Pablo Serrano"
],
"categories": [
"cs.NI",
"cs.PF",
"C.2.2; C.4"
],
"primary_category": "cs.NI",
"published": "20170626103621",
"title": "$μ$Nap: Practical Micro-Sleeps for 802.11 WLANs"
} |
=1 | http://arxiv.org/abs/1706.08543v3 | {
"authors": [
"Xiaoyong Chu",
"Suchita Kulkarni",
"Pierre Salati"
],
"categories": [
"hep-ph",
"astro-ph.HE"
],
"primary_category": "hep-ph",
"published": "20170626180114",
"title": "Dark matter indirect signals with long-lived mediators"
} |
^1INRIM, Strada delle Cacce 91, I-10135 Torino, Italy ^2Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy ^3Politecnico di Milano, Dipartimento di Elettronica, Informazione e Bioingegneria, Piazza Leonardo da Vinci 32, 20133 Milano, Italy ^4H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, U.K ^5Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 6997801, Israel ^∗To whom correspondence should be addressed; E-mail: [email protected] 24ptDetermining the Quantum Expectation Value by Measuring a Single PhotonF. Piacentini,^1∗A. Avella,^1 E. Rebufello,^1,2 R. Lussana,^3 F. Villa,^3 A. Tosi,^3 M. Gramegna,^1G. Brida,^1 E. Cohen,^4 L. Vaidman,^5 I. P. Degiovanni,^1 M. Genovese^1 December 30, 2023 =================================================================================================================================================================================== Quantum mechanics, one of the keystones of modern physics, exhibits several peculiar properties, differentiating it from classical mechanics. One of the most intriguing is that variables might not have definite values. A complete quantum description provides only probabilities for obtaining various eigenvalues of a quantum variable. These and corresponding probabilities specify the expectation value of a physical observable, which is known to be a statistical property of an ensemble of quantum systems. In contrast to this paradigm, we demonstrate a unique method allowing to measure the expectation value of a physical variable on a single particle, namely, the polarisation of a single protected photon. This is the first realisation of quantum protective measurements.Quantum theory has led to an unprecedented success in predicting a vast amount of experimental results, with a perfect agreement between its predictions and every realised connected experiment. However, until this day there is no consensus about the foundational concepts of quantum mechanics. The reality of the wavefunction is still under hot debate <cit.>. Probably the most puzzling feature of quantum theory, standing in stark contrast with classical physics, is the fact that physical observables lack definite values. A complete description of a quantum system only predicts the spectrum and probabilities for the measurement outcomes of a physical observable. Given the quantum state of the system |Ψ⟩, which, according to standard quantum mechanics, comprises its complete description, to each observable Awe can associate a definite number: ⟨Ψ|A|Ψ⟩ =∑ p_i a_i (p_i being the probability to obtain the (eigen)value a_i as the result of the measurement of A). The meaning of this number is statistical: for finding the expectation value of A it is necessary to have an ensemble of identically prepared systems and to perform numerous measurements.Single measurement yielding the expectation value of a physical variable seems to be against the spirit of quantum mechanics. However, it has been suggested that, in certain special situations, one can find the expectation value of an observable performing only a single measurement. This is the method of protective measurement (PM), originally proposed as an argument supporting the reality of the quantum wavefunction <cit.>.The conceptual strength of the PM argument for the reality of the quantum state is a highly controversial issue <cit.>, namely, does the procedure allow observing the state or only the protection mechanism? Nevertheless, the idea of PM triggered and helped studying various foundational topics beyond exploring the meaning of the wavefunction, such as the analysis of Bohmian trajectories <cit.>, determination of the stationary basis <cit.> and analysis of measurement optimisation for minimising the state disturbance <cit.>. The concept of protection was also extended to measurement of a two-state vector <cit.>. In spite of the rich and diverse analysis of the theory behind PM, to this date PM have not been realised experimentally.Protection can be realised <cit.> both actively or passively: in this paper we employ an active protection technique based on the Zeno effect <cit.>. It is worth mentioning that, although weak measurements (WMs) <cit.> and Zeno effect <cit.> have been largely considered in experiments for several physical systems, up to now no experiment joining them in a PM has been realized yet.Our main result is the extraction of the expectation value of the photon polarisation by means of a measurement performed on a single protected photon (see Fig. <ref>), that survived the Zeno-type protection scheme. The polarisation operator is defined byP = |H⟩⟨ H|-|V⟩⟨ V|,where H and V are the horizontal and vertical polarisations, respectively. We note that, because of the presence of the active protection in our experiment, the single click of a multi-pixel camera tells us that the expectation value of the polarisation operator of the single protected photon is ⟨ P ⟩= -0.3± 0.3 (see Fig. <ref>), in agreement with the theoretical predictions (⟨ P ⟩= -0.208).Our experiment (see Fig. <ref>) is analogous to a Stern-Gerlach experiment.Heralded single photons, prepared in the polarisation state | ψ_θ⟩ = cosθ | H ⟩ + sinθ | V ⟩, pass through a birefringent material shifting them in the transverse direction x (according to their polarisation). The spatial mode is close to a Gaussian one with a 4.1 pixels σ (being σ the source of uncertainty associated with the estimation of ⟨ P ⟩ presented in Fig. <ref>). The WM interaction is obtained exploiting K=7 birefringent units, while the state protection is realised via the quantum Zeno scheme (see Methods for experimental details). At the end of the optical path, the photons are detected by a spatial-resolving single-photon detector prototype <cit.>. Without protection, the photons end up in one of the two regions corresponding to the vertical and horizontal polarisations, centered around x=± a (see Fig. <ref>a).Then, the expectation value can be statistically found by counting the ratio of counts:⟨ P ⟩= N_H - N_V/N.In contrast, when employing PM, the photons end up in a region whose center is located atx=a ⟨ P ⟩ (see Fig. <ref>b). A large ensemble of measurements allows finding the center with arbitrarily good precision, but even a single photon detection provides information about ⟨ P ⟩, albeit with a finite precision defined by the width of the distribution.In Fig. <ref>(a-d) we show the results obtained collecting heralded single photons for a measurement time of 1200 s. In panels (a) and (c) we see, respectively, a histogram and a contour plot of the photon counts distribution observed in the unprotected case for the input state |ψ_17π/60⟩ =0.629|H⟩+0.777|V⟩. As in a standard Stern-Gerlach experiment, we observed photons only in two regions corresponding to the eigenvalues of P. The expectation value of thepolarisation ⟨ P ⟩ evaluated using (<ref>) from this distribution (after dark count subtraction) is ⟨ P_17π/60⟩ = -0.21(4), inagreement with theoretical expectations, ⟨ P_17π/60⟩ = -0.208. Panels (b) and (d) show histogram and contour plot of the photon counts distribution obtained in the protected case for the same polarisation state. Instead of two distributions around x=± a, here we find a single distribution of photon detections centered very close to the expected value x = ⟨ P ⟩ a. The measured expectation value is ⟨ P_17π/60⟩ = -0.19(2) (dark counts subtracted). This result demonstrates that we have been able to realise and exploit the PM concept, providing the estimation of the polarisation operator, ⟨ P ⟩, by the detection of a single photon.This is further confirmed in Fig. <ref> (e) and (f), presenting typical photon detection maps for the input state | ψ_17π/60⟩ obtained from a small number of detected photons. Specifically, Fig. <ref>(e) and (f) correspond respectively to the unprotected case (N=14 detection events) and to the protected one (N=17 detection events); the circles drawn in the two figures represent the width of the distributions reported in Fig. <ref>(a-d). As expected, there is a clear concentration of the counts inside the circles that - despite the non-ideality of our SPAD array, presenting a non-negligible level of dark counts likely responsible for the detection events outside the circles- demonstrates the validity of our technique even when just few detections are considered. The first detected photons in the runs are signified with white pixels. We see that while the white pixel of Fig. <ref>(f) provides a good estimate of the expectation value, we cannot learn much from the white pixel of Fig. <ref>(e).Indeed, using a single photon is what makes PM special. Nevertheless, one could argue that our experiment concerns a single post-selected photon (i.e. that survived all protection stages) and allowing post-selection, one can perform a measurement yielding the expectation value in the case of both weak and strong interaction.It is obvious that the photon survival probability decreases when the interaction strength grows, while increasing the number of interaction-protection stages decreases the uncertainty, but also the survival probability. To discuss quantitatively the performances of PM with respect to the straightforward alternative, a projective measurement exploiting, e.g., a polarising beam-splitter, we plot in Fig. <ref> the ratio R= u_PBS (P)/u(P)between the uncertainties on ⟨ P ⟩ (u(P) and u_PBS (P)respectively, considering u_PBS (P) the uncertainty associated with the measurement procedure saturating the Quantum Cramer-Rao bound, i.e. being equal to the inverse of the square root of the Quantum Fisher information <cit.>). We consider in both cases the same initial resources (photon number), taking into account the photons lost along the interaction-protection steps (see Methods). We consider two different scenarios:K=7 (yellow surface) and K=100 (blue surface) interaction-protection stages. One immediately notices that, in both cases, PM is almost always advantageous (R>1) with respect to the projective measurement, going below the R=1 plane (in magenta) only in presence of extremely weak interactions. In our experiment, with ξ∼0.4 and just K=7, a 10% advantage is already present for most of the possible states, even if the maximum for R corresponds to ξ∼1. For K=100, instead, the reasonably weak interaction ξ∼0.4 grants the maximum of the advantage (R>8.5 almost everywhere), while for stronger interaction the advantage is reduced to R<4. The advantage of the protective measurement technique stems from the very high survival probability of the protected photons. This comes from the fact that, in presence of a sequence of identical interaction-protection stages as in our scheme, the relative probability of losing a photon in a protection step decreases with the single photon advancing in the sequence, since it is more likely to find the photon close to the “right paths” created in our K interaction-protection steps (see Methods for theoretical details). It is also worth pointing out that our experiment is the first realisation of a “robust” WM<cit.> at single photon level.To conclude, our results demonstrate that a single-event detection can provide a reliable information regarding a certain property of a quantum system -the expectation value of the polarisation operator in our case- which was considered to be only statistical, belonging to an ensemble of identically prepared quantum system. This means that PM can be useful in practical situations where one wants to test an unknown state preparation procedure, taking advantage of the fact that both the state preparation and the state protection are performed exploiting the same projective measurement system (or equivalently a set of identical projective measurements as in our case). This is the first experimental realisation of protective measurements <cit.>. § ACKNOWLEDGEMENTSThis work has been supported by EMPIR-14IND05 “MIQC2”, and by the John Templeton Foundation (Grant No. 43467). E.C. was supported by ERC AdG NLST. L.V. acknowledges support of the Israel Science Foundation Grant No. 1311/14 and the German-Israeli Foundation for Scientific Research and Development Grant No. I-1275-303.14§ COMPETING FINANCIAL INTERESTS The authors declare no competing financial interests. § METHODS§.§ SetupOur experimental setup (Fig. <ref>) is composed of three parts. In the first one -the preparation stage- we produce single photons in well-defined polarisation states, by means of a heralded single-photon source (SPS) based on Type-I Parametric Down-Conversion (PDC) <cit.>, and a state filtering system. The spatial mode is close to a Gaussian of width 4.1 pixels.The second part contains a sequence of weak interactions and state protection mechanism, the latter based on polarisation filters.Finally, the photons are detected by a single-photon detector with a spatial resolution consisting of an array of single-photon detectors.Incidentally, the scheme for protective measurement has got some analogy with to the one realised in <cit.>, but of course these two papers, aimed at realising sequential weak measurements, do not consider any protection mechanism. Furthermore, in these cases two weak interactions were considered, while in the actual scheme we have realised a challenging sequence of 7 interactions in a row. The SPS is based on a 796 nm mode-locked Ti:Sapphire laser (repetition rate: 76 MHz), whose second harmonic emission pumps a 10 × 10 × 5 mm LiIO_3 non-linear crystal, in which correlated photons are produced by PDC. The idler photons (λ_i = 920 nm) are coupled to a single-mode fiber (SMF) and then addressed to a Silicon Single-Photon Avalanche Diode (SPAD), heralding the presence of the correlated signal photons (λ_s = 702 nm). These, after being SMF-coupled, are addressed to a launcher injecting them into the free-space optical path, where the protective measurement protocol is implemented.After the launcher, the heralded single photons are collimated by a telescopic system, and then prepared in the linear polarisation state |ψ_θ⟩ (by means of a calcite polariser followed by a half-wave plate and a polariser). We have estimated the quality of our single-photon emission with a Hanbury-Brown and Twiss interferometer, obtaining a value for the parameter α <cit.> (directly connected to the second-order Glauber autocorrelation function g^(2)) of 0.13 ± 0.01 without any background or dark count subtraction, that being largely below 1 testifies the quality of our single photon source.The Hamiltonian evolution of the quantum state is induced by exploitingbirefringence. In our optical path we can insert up to K=7 birefringent units, each of them composed of two different calcite crystals(the number K=7 of verification measurements was chosen from practical considerations approximating the ideal case of large K; because of losses originating from optical elements imperfections and because of the detector non-unit quantum efficiency and dark counts, further increasing K would result in a low photon survival probability, and consequently a low signal-to-noise ratio at the detector output). The first crystal of each element is a 2 mm long birefringent crystal whose extraordinary (e) optical axis lies in the Y-Z plane, with an angle of π/4 with respect to the z direction. Due to the spatial walk-off effect experienced by the horizontally-polarised photons (i.e. along the x direction), horizontal and vertical-polarisation paths get slightly separated along the x direction (coupling to the pointer variable).The second crystal of each unit is a 1.1 mm long birefringent crystal with the optical e-axis orthogonal to the beam (thus not contributing to the spatial walk-off) that nullifies, through phase compensation, the temporal walk-off introduced by the first one.The protection of the quantum state, implementing the quantum Zeno scheme, is realised by inserting a thin-film polariser after each birefringent unit, projecting the photons on the same polarisation of the initial state |ψ_θ⟩. Uhlmann's fidelities between reconstructed states and theoretical expected states always exceed 99%.At the end of the optical path, the photons are detected by a spatial-resolving single-photon detector prototype. This device is a two-dimensional array made of 32×32 “smart pixels” - each hosting a Silicon Single-Photon Avalanche Diode (SPAD) detector and its front-end electronics <cit.>, but we used only 32×20 pixels to avoid distortion due to dark counts in the regions with negligible photon detection probability). The SPAD array is gated with a 6 ns integration windows, triggered by the SPAD in the heralding arm in order to reduce the dark counts and improve the signal-to-noise ratio. Aside from the single-photon state |ψ_17π/60⟩=0.629|H⟩+0.777|V⟩, with which we obtained the data set presented in the paper, we tested our setup with other two states, namely |ψ_π/4⟩=1/√(2)(|H⟩+|V⟩) and | ψ_π/8⟩ = 0.924|H⟩+0.383|V⟩.The obtained results for these data sets, together with the one shown in the main paper, are summarized in Fig. <ref>. As immediate to see, in the unprotected case the photon counts accumulate around the positions x=± a, forming the distributions reported in panels (a), (b) and (c) for the states |ψ_π/4⟩, |ψ_17π/60⟩ and |ψ_π/8⟩, respectively. From these photon-counts distributions, we can evaluate the expectation value of each state polarisation, obtaining ⟨ P_π/4⟩ = -0.03(4), ⟨ P_17π/60⟩ = -0.21(2), ⟨ P_π/8⟩ = 0.72(2), all in excellent agreement with the theoretical expectations (⟨ P_π/4⟩=0, ⟨ P_17π/60⟩ = -0.208 and ⟨ P_π/8⟩ = 0.707, respectively).When we introduce the protection mechanism, instead, we notice (panels (d), (e) and (f)) that all the photons accumulate around a specific position, corresponding to x=a⟨ P ⟩: as stated in the main text, this means that each single photon carries information about the expectation value of its polarisation, granting the possibility of extracting such value even in a one-shot experiment with just a single photon. The expectation values obtained in the protected case are respectively ⟨ P_π/4⟩ = -0.012(14), ⟨ P_17π/60⟩ = -0.19(2) and ⟨ P_π/8⟩ = 0.72(2), in good agreement with the theory, as well as with the ones obtained without protecting the single photons.In order to obtain these values, the dark counts of the 32×32 SPAD array, responsible for the “floor” of counts in all the six plots, were properly evaluated and subtracted. §.§ Theoretical analysis In the following we will describe the theory behind the protective measurement technique implemented in our experiment to extract the expectation value of the photon polarisation P=|H⟩⟨ H | - |V⟩⟨ V | by means of a measurement performed on a single protected photon, where H and V are the horizontal and vertical polarisations. As explained in the paper, for doing this we take advantage of a spatial resolving detector, thus it is quite obvious to consider the space and polarisation degrees of freedom when we describe the wavefunction of our single photon, i.e. |Ψ_in⟩= |ψ⟩⊗ | f_x ⟩ with |ψ⟩ = cosθ | H ⟩ + sinθ | V ⟩ and |f_x ⟩ = ∫dx f(x) |x ⟩ where f(x) is a Gaussian curve whose square is normalised to one, namely f(x)=(2 πσ^2)^-1/4exp(-x^2/4 σ^2).The protective measurement consists of a sequence of identical interactions, exploiting the spatial walk-off in non-linear crystals (using a technique completely analogous to the one used in, e.g., <cit.>) corresponding to the unitary transformation U=exp(i g 𝐏⊗Π_H ) (being 𝐏 the momentum operator and g the von Neumann coupling constant between 𝐏 and the projector Π_H=|H⟩⟨ H|), and the protective measurement, performed by exploiting the polarisation projector Π_ψ= |ψ⟩⟨ψ |. Actually, this can be also described as a test on an unknown state preparation procedure exploiting a series of identical projective measurements, the first one really used to prepare (select) the single-photon state, while the other ones used as protective measurements. After K steps of this sequence consisting of a weak interaction followed by a protective measurement, the non-normalised output state of the single photons will be:|Ψ_out⟩= ( Π_θ U )^K |Ψ_in⟩= ( ⟨ H| Π_θ |H⟩ e^ i g 𝐏 + ⟨ V| Π_θ |V⟩1_x)^K|Ψ_in⟩. Thus, the probability that the protected single-photon survives after K interaction-verification steps is p_sur (K) = Tr[|Ψ_out⟩⟨Ψ_out | ], while the probability of finding the protected single-photon in a specific position x_0 of our spatial-resolving detector (given that it survived the verification process) isF_K (x_0) =p_sur (K)^-1 Tr[|x_0⟩⟨ x_0 |Ψ_out⟩⟨Ψ_out | ] =p_sur (K)^-1 (∑_n=0^KK!/n! (K-n)!⟨ H| Π_θ |H⟩^n⟨ V | Π_θ |V ⟩^K-n f(x_0 + n g))^2. As explained in the paper, the spatially-resolved detection of the protected single-photon provides an estimation of the value of P. A relevant question is related to the quality of this estimation, i.e. the uncertainty that can be associated with this estimation. This uncertainty is obviously associated to the uncertainty on the arrival position of one protected single photon, related to the probability distribution profile F_K (x). The uncertainty in the position is u(x)=√(ϵ(x^2) - ϵ(x)^2 ), with ϵ(x^n)=∫dx x^n F_K (x). Then, we note that there ia a correspondence between the value of ⟨ P ⟩ and the average position ϵ(x), where the protected single photon is detected, i.e., the H-polarisation corresponds to ⟨ P ⟩=1 and the position ϵ(x)=K g/2, while the V-polarisation corresponds to ⟨ P ⟩=-1 and the position ϵ(x)=-K g/2 (actually, the V-polarisation is not affected by the interaction according to the unitary interaction operator U, while the H-polarisation is shifted by K g, because of the fact that each interaction induces a shift g, according to Eq. (<ref>). Anyway, we decided to shift the whole X axis by -K g/2 for an easier comparison with the well-known Stern-Gerlach experiment). Thus, the uncertainty on ⟨ P ⟩ associated with the detection of a single protected photon can be obtained simply by re-scaling the spatial uncertainty u(x), i.e. u(P)=u(x) 2/K g.It is interesting to compare the performance of the protective measurement technique presented above in situations of strong and weak interaction, corresponding to g ≫σ and g ≪σ respectively, versus the usual technique involving a single strong measurement, for example exploiting a polarizing-beam-splitter (PBS). In the PBS measurement, starting from M initial photons in the polarisation state | ψ⟩ the probability distribution of observing m photons V-polarised (and, obviously, (M-m) H-polarised) is the binomial-one with probability parameter (cosθ)^2. Thus, the estimator of P is P= 2 m /M -1 and the uncertainty on this estimator can be evaluated as u_PBS (P)= √(⟨ P^2⟩ - ⟨ P ⟩^2)= |sin(2 θ)|/√(M). This uncertainty level is easily demonstrated to be the optimal one in terms of quantum Fisher information (indeed the quantum Fisher Information is ℋ= sin(2 θ)^-2), i.e. the one that saturates the quantum Cramer-Rao bound <cit.>.In order to provide a fair comparison between this PBS-based measurement and our protective-measurement-based one we should consider the two measurement approaches exploiting the same initial resources, i.e. the same number of initial prepared photons. In the protective measurement case we have considered the uncertainty associated with the detection of a protected single-photon, but the probability of survival of a K-step protective measurement process is p_sur (K), this means that to have one protected photon arriving at our ideal spatial resolving detector we need, on average, 1/p_sur (K) initial photons. To perform a fair comparison, we set M=1/p_sur (K) also in the PBS measurement case, and we define the ratio between the uncertainties R= u_PBS (P)/u(P). This is what we show in Fig. 4 of the main text, where R plotted versus the strength of the interaction (ξ in the main text called interaction strength corresponds to the ratio g/σ) and the (linear) polarisation state of the single photon | ψ⟩ (specifically, (cosθ)^2 is ⟨ψ | Π_H | ψ⟩), for K=100 and K=7 interactions. As already said, the protective measurement procedure is almost always advantageous (R>1) with respect to the PBS measurement, and it becomes disadvantageous only in the presence of extremely weak interactions (it is not shown inthe figure, but, e.g., also for K=100 and g/σ = 0.02 and |ψ⟩ = 2^-1/2(|H⟩+ |V⟩) we have R=0.996).In the case of K=7 (K=100) interactions, the maximum advantage is R ∼ 3 ( R ∼ 8.5) for ξ∼ 1 (ξ∼ 0.4), while a strong interaction reduces our factor to R ∼ 1.6 (R ∼ 4). We underline that, considering a weak interaction exploiting, as in our experimental configuration, birefringent crystals separating the H- and V-photons paths of 1.66 pixels, but with K=100 interaction-protection stages, one would observe a total separation of 166 pixels between H and V polarisations, an order of magnitude above the FWHM of the single-photons spatial mode (a scenario similar to the one of Fig. 2 in the main text).Here we want to stress that, even though projective measurement is an optimal measurement for the parameter P, i.e. a measurement procedure able to reach the quantum Cramer-Rao bound <cit.>, protective measurement allows obtaining an estimation of ⟨ P ⟩ far better than the one achievable with such method, surpassing this way the quantum Cramer-Rao bound itself (being the state preparation procedure embedded in the protection scheme). R can be understood also as a parameter identifying the resources needed to achieve the same level of uncertainties in the two measurement techniques: in fact, to achieve the same level of uncertainty, the initial number of photons needed in the PBS measurement is R^2, the one used in the protective measurement.The advantage of the protective measurement technique comes from the very high survival probability of the protected photons. As evident from Fig. SM3, indeed, with K=100 interactions andg/σ∼ 0.4 we have p_sur>0.57 even for the most lossy state. Also in the case of strong interaction g/σ = 6 the survival probability is surprisingly high (p_sur>0.05). To understand this, one should consider the situation in which the protective measurement is performed without weak interactions: one has just to replace the “weak” birefringent crystals by “strong” ones, or to use a beam of narrow width, such that the shift due to each crystal will be much larger than the width of the beam (the situation depicted in Fig. <ref>). Then, the readings of the detector provide the expectation value with a precision scaling as 1/√(K), where K is the number of interaction-protection stages. The 1/√(K) uncertainty scaling with the number of interactions K presents some analogy with the one of the projective measurement, where the uncertainty scales of a factor 1/√(N) with the number N of exploited photons.The advantage can be understood observing the fact that the average number of photons needed for observing a single protected photons, i.e. 1/p_sur (K), grows at a much slower pace with respect to K, as shown in Fig. <ref> (b). This comes from the fact that, in presence of a sequence of identical interaction-protection stages as in our scheme, the relative probability of losing a photon in a protection step because of unsuccessful protection measurement decreases with the single photon advancing in the sequence since photons are more likely on the “right path” (see Fig. <ref>).Obviously, the above considerations assume an ideal scenario where the the only source of losses in the protective measurement is represented by the projection measurement preforming the verification of the state, neglecting completely the optical losses induced by the real optical devices present in the experimental setup. In the specific case of our experiment, the optical losses greatly reduce the advantage discussed so far, but this is a technical limitation that can be, in principle, strongly reduced. 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"authors": [
"F. Piacentini",
"A. Avella",
"E. Rebufello",
"R. Lussana",
"F. Villa",
"A. Tosi",
"M. Gramegna",
"G. Brida",
"E. Cohen",
"L. Vaidman",
"I. P. Degiovanni",
"M. Genovese"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170627160046",
"title": "Determining the Quantum Expectation Value by Measuring a Single Photon"
} |
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fivecircleinversesansserif278E fivedeva096B fiveeighths215D fivegujarati0AEB fivegurmukhi0A6B fivehackarabic0665 fivehangzhou3025 fiveideographicparen3224 fiveinferior2085 fivemonospaceFF15 fiveoldstyle0035 fiveparen2478 fiveperiod248C fivepersian06F5 fiveroman2174 fivesuperior2075 fivethai0E55 fl0066 006C flat266D floorleft230A floorright230B florin0192 fmonospaceFF46 fmsquare3399 fofanthai0E1F fofathai0E1D follownotdbleqv2ABA follownotslnteql2AB6 followornoteqvlnt22E9 follows227B followsequal2AB0 followsorcurly227D followsorequal227F fongmanthai0E4F forall2200 forces22A9 forcesbar22AA fork22D4 four0034 fourarabic0664 fourbengali09EA fourcircle2463 fourcircleinversesansserif278D fourdeva096A fourgujarati0AEA fourgurmukhi0A6A fourhackarabic0664 fourhangzhou3024 fourideographicparen3223 fourinferior2084 fourmonospaceFF14 fournumeratorbengali09F7 fouroldstyle0034 fourparen2477 fourperiod248B fourpersian06F4 fourroman2173 foursuperior2074 fourteencircle246D fourteenparen2481 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hyphentwo2010 i0069 iacute00ED iacyrillic044F ibengali0987 ibopomofo3127 ibreve012D icaron01D0 icircle24D8 icircumflex00EE icyrillic0456 idblgrave0209 ideographearthcircle328F ideographfirecircle328B ideographicallianceparen323F ideographiccallparen323A ideographiccentrecircle32A5 ideographicclose3006 ideographiccomma3001 ideographiccommaleftFF64 ideographiccongratulationparen3237 ideographiccorrectcircle32A3 ideographicearthparen322F ideographicenterpriseparen323D ideographicexcellentcircle329D ideographicfestivalparen3240 ideographicfinancialcircle3296 ideographicfinancialparen3236 ideographicfireparen322B ideographichaveparen3232 ideographichighcircle32A4 ideographiciterationmark3005 ideographiclaborcircle3298 ideographiclaborparen3238 ideographicleftcircle32A7 ideographiclowcircle32A6 ideographicmedicinecircle32A9 ideographicmetalparen322E ideographicmoonparen322A ideographicnameparen3234 ideographicperiod3002 ideographicprintcircle329E ideographicreachparen3243 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phi103D5 phieuphacirclekorean327A phieuphaparenkorean321A phieuphcirclekorean326C phieuphkorean314D phieuphparenkorean320C philatin0278 phinthuthai0E3A phisymbolgreek03D5 phook01A5 phophanthai0E1E phophungthai0E1C phosamphaothai0E20 pi03C0 pi103D6 pieupacirclekorean3273 pieupaparenkorean3213 pieupcieuckorean3176 pieupcirclekorean3265 pieupkiyeokkorean3172 pieupkorean3142 pieupparenkorean3205 pieupsioskiyeokkorean3174 pieupsioskorean3144 pieupsiostikeutkorean3175 pieupthieuthkorean3177 pieuptikeutkorean3173 pihiragana3074 pikatakana30D4 pisymbolgreek03D6 piwrarmenian0583 planckover2pi210F planckover2pi1210F plus002B plusbelowcmb031F pluscircle2295 plusminus00B1 plusmod02D6 plusmonospaceFF0B plussmallFE62 plussuperior207A pmonospaceFF50 pmsquare33D8 pohiragana307D pointingindexdownwhite261F pointingindexleftwhite261C pointingindexrightwhite261E pointingindexupwhite261D pokatakana30DD poplathai0E1B postalmark3012 postalmarkface3020 pparen24AB precedenotdbleqv2AB9 precedenotslnteql2AB5 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qofhatafsegol05E7 05B1 qofhatafsegolhebrew05E7 05B1 qofhebrew05E7 qofhiriq05E7 05B4 qofhiriqhebrew05E7 05B4 qofholam05E7 05B9 qofholamhebrew05E7 05B9 qofpatah05E7 05B7 qofpatahhebrew05E7 05B7 qofqamats05E7 05B8 qofqamatshebrew05E7 05B8 qofqubuts05E7 05BB qofqubutshebrew05E7 05BB qofsegol05E7 05B6 qofsegolhebrew05E7 05B6 qofsheva05E7 05B0 qofshevahebrew05E7 05B0 qoftsere05E7 05B5 qoftserehebrew05E7 05B5 qparen24AC quarternote2669 qubuts05BB qubuts1805BB qubuts2505BB qubuts3105BB qubutshebrew05BB qubutsnarrowhebrew05BB qubutsquarterhebrew05BB qubutswidehebrew05BB question003F questionarabic061F questionarmenian055E questiondown00BF questiondownsmall00BF questiongreek037E questionmonospaceFF1F questionsmall003F quotedbl0022 quotedblbase201E quotedblleft201C quotedblmonospaceFF02 quotedblprime301E quotedblprimereversed301D quotedblright201D quoteleft2018 quoteleftreversed201B quotereversed201B quoteright2019 quoterightn0149 quotesinglbase201A quotesingle0027 quotesinglemonospaceFF07 r0072 raarmenian057C rabengali09B0 racute0155 radeva0930 radical221A radicalexF8E5 radoverssquare33AE radoverssquaredsquare33AF radsquare33AD rafe05BF rafehebrew05BF ragujarati0AB0 ragurmukhi0A30 rahiragana3089 rakatakana30E9 rakatakanahalfwidthFF97 ralowerdiagonalbengali09F1 ramiddlediagonalbengali09F0 ramshorn0264 rangedash2013 ratio2236 rbopomofo3116 rcaron0159 rcedilla0157 rcircle24E1 rcommaaccent0157 rdblgrave0211 rdotaccent1E59 rdotbelow1E5B rdotbelowmacron1E5D referencemark203B reflexsubset2286 reflexsuperset2287 registered00AE registersans00AE registerserif00AE reharabic0631 reharmenian0580 rehfinalarabicFEAE rehiragana308C rehyehaleflamarabic0631 FEF3 FE8E 0644 rekatakana30EC rekatakanahalfwidthFF9A resh05E8 reshdageshhebrewFB48 reshhatafpatah05E8 05B2 reshhatafpatahhebrew05E8 05B2 reshhatafsegol05E8 05B1 reshhatafsegolhebrew05E8 05B1 reshhebrew05E8 reshhiriq05E8 05B4 reshhiriqhebrew05E8 05B4 reshholam05E8 05B9 reshholamhebrew05E8 05B9 reshpatah05E8 05B7 reshpatahhebrew05E8 05B7 reshqamats05E8 05B8 reshqamatshebrew05E8 05B8 reshqubuts05E8 05BB reshqubutshebrew05E8 05BB reshsegol05E8 05B6 reshsegolhebrew05E8 05B6 reshsheva05E8 05B0 reshshevahebrew05E8 05B0 reshtsere05E8 05B5 reshtserehebrew05E8 05B5 revasymptequal22CD reversedtilde223D reviahebrew0597 reviamugrashhebrew0597 revlogicalnot2310 revsimilar223D rfishhook027E rfishhookreversed027F rhabengali09DD rhadeva095D rho03C1 rho103F1 rhook027D rhookturned027B rhookturnedsuperior02B5 rhosymbolgreek03F1 rhotichookmod02DE rieulacirclekorean3271 rieulaparenkorean3211 rieulcirclekorean3263 rieulhieuhkorean3140 rieulkiyeokkorean313A rieulkiyeoksioskorean3169 rieulkorean3139 rieulmieumkorean313B rieulpansioskorean316C rieulparenkorean3203 rieulphieuphkorean313F rieulpieupkorean313C rieulpieupsioskorean316B rieulsioskorean313D rieulthieuthkorean313E rieultikeutkorean316A rieulyeorinhieuhkorean316D rightangle221F rightanglene231D rightanglenw231C rightanglese231F rightanglesw231E righttackbelowcmb0319 righttriangle22BF rihiragana308A rikatakana30EA rikatakanahalfwidthFF98 ring02DA ringbelowcmb0325 ringcmb030A ringhalfleft02BF ringhalfleftarmenian0559 ringhalfleftbelowcmb031C ringhalfleftcentered02D3 ringhalfright02BE ringhalfrightbelowcmb0339 ringhalfrightcentered02D2 ringinequal2256 rinvertedbreve0213 rittorusquare3351 rlinebelow1E5F rlongleg027C rlonglegturned027A rmonospaceFF52 rohiragana308D rokatakana30ED rokatakanahalfwidthFF9B roruathai0E23 rparen24AD rrabengali09DC rradeva0931 rragurmukhi0A5C rreharabic0691 rrehfinalarabicFB8D rrvocalicbengali09E0 rrvocalicdeva0960 rrvocalicgujarati0AE0 rrvocalicvowelsignbengali09C4 rrvocalicvowelsigndeva0944 rrvocalicvowelsigngujarati0AC4 rsuperior0072 rtblock2590 rturned0279 rturnedsuperior02B4 ruhiragana308B rukatakana30EB rukatakanahalfwidthFF99 rupeemarkbengali09F2 rupeesignbengali09F3 rupiah20A8 ruthai0E24 rvocalicbengali098B rvocalicdeva090B rvocalicgujarati0A8B rvocalicvowelsignbengali09C3 rvocalicvowelsigndeva0943 rvocalicvowelsigngujarati0AC3 s0073 sabengali09B8 sacute015B sacutedotaccent1E65 sadarabic0635 sadeva0938 sadfinalarabicFEBA sadinitialarabicFEBB sadmedialarabicFEBC sagujarati0AB8 sagurmukhi0A38 sahiragana3055 sakatakana30B5 sakatakanahalfwidthFF7B sallallahoualayhewasallamarabicFDFA samekh05E1 samekhdageshFB41 samekhdageshhebrewFB41 samekhhebrew05E1 saraaathai0E32 saraaethai0E41 saraaimaimalaithai0E44 saraaimaimuanthai0E43 saraamthai0E33 saraathai0E30 saraethai0E40 saraiileftthaiF886 saraiithai0E35 saraileftthaiF885 saraithai0E34 saraothai0E42 saraueeleftthaiF888 saraueethai0E37 saraueleftthaiF887 sarauethai0E36 sarauthai0E38 sarauuthai0E39 satisfies22A8 sbopomofo3119 scaron0161 scarondotaccent1E67 scedilla015F schwa0259 schwacyrillic04D9 schwadieresiscyrillic04DB schwahook025A scircle24E2 scircumflex015D scommaaccent0219 sdotaccent1E61 sdotbelow1E63 sdotbelowdotaccent1E69 seagullbelowcmb033C second2033 secondtonechinese02CA section00A7 seenarabic0633 seenfinalarabicFEB2 seeninitialarabicFEB3 seenmedialarabicFEB4 segol05B6 segol1305B6 segol1f05B6 segol2c05B6 segolhebrew05B6 segolnarrowhebrew05B6 segolquarterhebrew05B6 segoltahebrew0592 segolwidehebrew05B6 seharmenian057D sehiragana305B sekatakana30BB sekatakanahalfwidthFF7E semicolon003B semicolonarabic061B semicolonmonospaceFF1B semicolonsmallFE54 semivoicedmarkkana309C semivoicedmarkkanahalfwidthFF9F sentisquare3322 sentosquare3323 seven0037 sevenarabic0667 sevenbengali09ED sevencircle2466 sevencircleinversesansserif2790 sevendeva096D seveneighths215E sevengujarati0AED sevengurmukhi0A6D sevenhackarabic0667 sevenhangzhou3027 sevenideographicparen3226 seveninferior2087 sevenmonospaceFF17 sevenoldstyle0037 sevenparen247A sevenperiod248E sevenpersian06F7 sevenroman2176 sevensuperior2077 seventeencircle2470 seventeenparen2484 seventeenperiod2498 seventhai0E57 sfthyphen00AD shaarmenian0577 shabengali09B6 shacyrillic0448 shaddaarabic0651 shaddadammaarabicFC61 shaddadammatanarabicFC5E shaddafathaarabicFC60 shaddafathatanarabic0651 064B shaddakasraarabicFC62 shaddakasratanarabicFC5F shade2592 shadedark2593 shadelight2591 shademedium2592 shadeva0936 shagujarati0AB6 shagurmukhi0A36 shalshelethebrew0593 sharp266F shbopomofo3115 shchacyrillic0449 sheenarabic0634 sheenfinalarabicFEB6 sheeninitialarabicFEB7 sheenmedialarabicFEB8 sheicoptic03E3 sheqel20AA sheqelhebrew20AA sheva05B0 sheva11505B0 sheva1505B0 sheva2205B0 sheva2e05B0 shevahebrew05B0 shevanarrowhebrew05B0 shevaquarterhebrew05B0 shevawidehebrew05B0 shhacyrillic04BB shiftleft21B0 shiftright21B1 shimacoptic03ED shin05E9 shindageshFB49 shindageshhebrewFB49 shindageshshindotFB2C shindageshshindothebrewFB2C shindageshsindotFB2D shindageshsindothebrewFB2D shindothebrew05C1 shinhebrew05E9 shinshindotFB2A shinshindothebrewFB2A shinsindotFB2B shinsindothebrewFB2B shook0282 sigma03C3 sigma103C2 sigmafinal03C2 sigmalunatesymbolgreek03F2 sihiragana3057 sikatakana30B7 sikatakanahalfwidthFF7C siluqhebrew05BD siluqlefthebrew05BD similar223C similarequal2243 sindothebrew05C2 siosacirclekorean3274 siosaparenkorean3214 sioscieuckorean317E sioscirclekorean3266 sioskiyeokkorean317A sioskorean3145 siosnieunkorean317B siosparenkorean3206 siospieupkorean317D siostikeutkorean317C six0036 sixarabic0666 sixbengali09EC sixcircle2465 sixcircleinversesansserif278F sixdeva096C sixgujarati0AEC sixgurmukhi0A6C sixhackarabic0666 sixhangzhou3026 sixideographicparen3225 sixinferior2086 sixmonospaceFF16 sixoldstyle0036 sixparen2479 sixperiod248D sixpersian06F6 sixroman2175 sixsuperior2076 sixteencircle246F sixteencurrencydenominatorbengali09F9 sixteenparen2483 sixteenperiod2497 sixthai0E56 slash002F slashmonospaceFF0F slong017F slongdotaccent1E9B slurabove2322 slurbelow2323 smile2323 smileface263A smonospaceFF53 sofpasuqhebrew05C3 softhyphen00AD softsigncyrillic044C sohiragana305D sokatakana30BD sokatakanahalfwidthFF7F soliduslongoverlaycmb0338 solidusshortoverlaycmb0337 sorusithai0E29 sosalathai0E28 sosothai0E0B sosuathai0E2A space0020 spacehackarabic0020 spade2660 spadesuitblack2660 spadesuitwhite2664 sparen24AE sphericalangle2222 square25A1 squarebelowcmb033B squarecc33C4 squarecm339D squarediagonalcrosshatchfill25A9 squaredot22A1 squarehorizontalfill25A4 squareimage228F squarekg338F squarekm339E squarekmcapital33CE squareln33D1 squarelog33D2 squaremg338E squaremil33D5 squareminus229F squaremm339C squaremsquared33A1 squaremultiply22A0 squareoriginal2290 squareorthogonalcrosshatchfill25A6 squareplus229E squaresolid25A0 squareupperlefttolowerrightfill25A7 squareupperrighttolowerleftfill25A8 squareverticalfill25A5 squarewhitewithsmallblack25A3 squiggleleftright21AD squiggleright21DD srsquare33DB ssabengali09B7 ssadeva0937 ssagujarati0AB7 ssangcieuckorean3149 ssanghieuhkorean3185 ssangieungkorean3180 ssangkiyeokkorean3132 ssangnieunkorean3165 ssangpieupkorean3143 ssangsioskorean3146 ssangtikeutkorean3138 ssuperior0073 st0073 0074 star22C6 sterling00A3 sterlingmonospaceFFE1 strokelongoverlaycmb0336 strokeshortoverlaycmb0335 subset2282 subsetdbl22D0 subsetdblequal2AC5 subsetnoteql228A subsetnotequal228A subsetorequal2286 subsetornotdbleql2ACB subsetsqequal2291 succeeds227B suchthat220B suhiragana3059 sukatakana30B9 sukatakanahalfwidthFF7D sukunarabic0652 summation2211 sun263C superset2283 supersetdbl22D1 supersetdblequal2AC6 supersetnoteql228B supersetnotequal228B supersetorequal2287 supersetornotdbleql2ACC supersetsqequal2292 svsquare33DC syouwaerasquare337C t0074 tabengali09A4 tackdown22A4 tackleft22A3 tadeva0924 tagujarati0AA4 tagurmukhi0A24 taharabic0637 tahfinalarabicFEC2 tahinitialarabicFEC3 tahiragana305F tahmedialarabicFEC4 taisyouerasquare337D takatakana30BF takatakanahalfwidthFF80 tatweelarabic0640 tau03C4 tav05EA tavdagesFB4A tavdageshFB4A tavdageshhebrewFB4A tavhebrew05EA tbar0167 tbopomofo310A tcaron0165 tccurl02A8 tcedilla0163 tcheharabic0686 tchehfinalarabicFB7B tchehinitialarabicFB7C tchehmedialarabicFB7D tchehmeeminitialarabicFB7C FEE4 tcircle24E3 tcircumflexbelow1E71 tcommaaccent0163 tdieresis1E97 tdotaccent1E6B tdotbelow1E6D tecyrillic0442 tedescendercyrillic04AD teharabic062A tehfinalarabicFE96 tehhahinitialarabicFCA2 tehhahisolatedarabicFC0C tehinitialarabicFE97 tehiragana3066 tehjeeminitialarabicFCA1 tehjeemisolatedarabicFC0B tehmarbutaarabic0629 tehmarbutafinalarabicFE94 tehmedialarabicFE98 tehmeeminitialarabicFCA4 tehmeemisolatedarabicFC0E tehnoonfinalarabicFC73 tekatakana30C6 tekatakanahalfwidthFF83 telephone2121 telephoneblack260E telishagedolahebrew05A0 telishaqetanahebrew05A9 tencircle2469 tenideographicparen3229 tenparen247D tenperiod2491 tenroman2179 tesh02A7 tet05D8 tetdageshFB38 tetdageshhebrewFB38 tethebrew05D8 tetsecyrillic04B5 tevirhebrew059B tevirlefthebrew059B tfm:cmbsy10/diamond2662 tfm:cmbsy10/heart2661 tfm:cmbsy5/diamond2662 tfm:cmbsy5/heart2661 tfm:cmbsy6/diamond2662 tfm:cmbsy6/heart2661 tfm:cmbsy7/diamond2662 tfm:cmbsy7/heart2661 tfm:cmbsy8/diamond2662 tfm:cmbsy8/heart2661 tfm:cmbsy9/diamond2662 tfm:cmbsy9/heart2661 tfm:cmmi10/phi03D5 tfm:cmmi10/phi103C6 tfm:cmmi12/phi03D5 tfm:cmmi12/phi103C6 tfm:cmmi5/phi03D5 tfm:cmmi5/phi103C6 tfm:cmmi6/phi03D5 tfm:cmmi6/phi103C6 tfm:cmmi7/phi03D5 tfm:cmmi7/phi103C6 tfm:cmmi8/phi03D5 tfm:cmmi8/phi103C6 tfm:cmmi9/phi03D5 tfm:cmmi9/phi103C6 tfm:cmmib10/phi03D5 tfm:cmmib10/phi103C6 tfm:cmmib5/phi03D5 tfm:cmmib5/phi103C6 tfm:cmmib6/phi03D5 tfm:cmmib6/phi103C6 tfm:cmmib7/phi03D5 tfm:cmmib7/phi103C6 tfm:cmmib8/phi03D5 tfm:cmmib8/phi103C6 tfm:cmmib9/phi03D5 tfm:cmmib9/phi103C6 tfm:cmsy10/diamond2662 tfm:cmsy10/heart2661 tfm:cmsy5/heart2661 tfm:cmsy6/diamond2662 tfm:cmsy6/heart2661 tfm:cmsy7/diamond2662 tfm:cmsy7/heart2661 tfm:cmsy8/diamond2662 tfm:cmsy8/heart2661 tfm:cmsy9/diamond2662 tfm:cmsy9/heart2661 tfm:eurb10/phi03D5 tfm:eurb10/phi103C6 tfm:eurb5/phi03D5 tfm:eurb5/phi103C6 tfm:eurb6/phi03D5 tfm:eurb6/phi103C6 tfm:eurb7/phi03D5 tfm:eurb7/phi103C6 tfm:eurb8/phi03D5 tfm:eurb8/phi103C6 tfm:eurb9/phi03D5 tfm:eurb9/phi103C6 tfm:eurm10/phi03D5 tfm:eurm10/phi103C6 tfm:eurm5/phi03D5 tfm:eurm5/phi103C6 tfm:eurm6/phi03D5 tfm:eurm6/phi103C6 tfm:eurm7/phi03D5 tfm:eurm7/phi103C6 tfm:eurm8/phi03D5 tfm:eurm8/phi103C6 tfm:eurm9/phi03D5 tfm:eurm9/phi103C6 tfm:fplmbi/phi03D5 tfm:fplmbi/phi103C6 tfm:fplmri/phi03D5 tfm:fplmri/phi103C6 tfm:lmbsy10/diamond2662 tfm:lmbsy10/heart2661 tfm:lmbsy5/diamond2662 tfm:lmbsy5/heart2661 tfm:lmbsy7/diamond2662 tfm:lmbsy7/heart2661 tfm:lmmi10/phi03D5 tfm:lmmi10/phi103C6 tfm:lmmi12/phi03D5 tfm:lmmi12/phi103C6 tfm:lmmi5/phi03D5 tfm:lmmi5/phi103C6 tfm:lmmi6/phi03D5 tfm:lmmi6/phi103C6 tfm:lmmi7/phi03D5 tfm:lmmi7/phi103C6 tfm:lmmi8/phi03D5 tfm:lmmi8/phi103C6 tfm:lmmi9/phi03D5 tfm:lmmi9/phi103C6 tfm:lmmib10/phi03D5 tfm:lmmib10/phi103C6 tfm:lmmib5/phi03D5 tfm:lmmib5/phi103C6 tfm:lmmib7/phi03D5 tfm:lmmib7/phi103C6 tfm:lmsy10/diamond2662 tfm:lmsy10/heart2661 tfm:lmsy5/diamond2662 tfm:lmsy5/heart2661 tfm:lmsy6/diamond2662 tfm:lmsy6/heart2661 tfm:lmsy7/diamond2662 tfm:lmsy7/heart2661 tfm:lmsy8/diamond2662 tfm:lmsy8/heart2661 tfm:lmsy9/diamond2662 tfm:lmsy9/heart2661 tfm:msam10/diamond2662 tfm:msam5/diamond2662 tfm:msam6/diamond2662 tfm:msam7/diamond2662 tfm:msam8/diamond2662 tfm:msam9/diamond2662 tfm:pxbmia/phi03D5 tfm:pxbmia/phi103C6 tfm:pxbsy/diamond2662 tfm:pxbsy/heart2661 tfm:pxbsya/diamond2662 tfm:pxmia/phi03D5 tfm:pxmia/phi103C6 tfm:pxsy/diamond2662 tfm:pxsy/heart2661 tfm:pxsya/diamond2662 tfm:pzdr/a12701 tfm:pzdr/a102721 tfm:pzdr/a100275E tfm:pzdr/a1012761 tfm:pzdr/a1022762 tfm:pzdr/a1032763 tfm:pzdr/a1042764 tfm:pzdr/a1052710 tfm:pzdr/a1062765 tfm:pzdr/a1072766 tfm:pzdr/a1082767 tfm:pzdr/a1092660 tfm:pzdr/a11261B tfm:pzdr/a1102665 tfm:pzdr/a1112666 tfm:pzdr/a1122663 tfm:pzdr/a1172709 tfm:pzdr/a1182708 tfm:pzdr/a1192707 tfm:pzdr/a12261E tfm:pzdr/a1202460 tfm:pzdr/a1212461 tfm:pzdr/a1222462 tfm:pzdr/a1232463 tfm:pzdr/a1242464 tfm:pzdr/a1252465 tfm:pzdr/a1262466 tfm:pzdr/a1272467 tfm:pzdr/a1282468 tfm:pzdr/a1292469 tfm:pzdr/a13270C tfm:pzdr/a1302776 tfm:pzdr/a1312777 tfm:pzdr/a1322778 tfm:pzdr/a1332779 tfm:pzdr/a134277A tfm:pzdr/a135277B tfm:pzdr/a136277C tfm:pzdr/a137277D tfm:pzdr/a138277E tfm:pzdr/a139277F tfm:pzdr/a14270D tfm:pzdr/a1402780 tfm:pzdr/a1412781 tfm:pzdr/a1422782 tfm:pzdr/a1432783 tfm:pzdr/a1442784 tfm:pzdr/a1452785 tfm:pzdr/a1462786 tfm:pzdr/a1472787 tfm:pzdr/a1482788 tfm:pzdr/a1492789 tfm:pzdr/a15270E tfm:pzdr/a150278A tfm:pzdr/a151278B tfm:pzdr/a152278C tfm:pzdr/a153278D tfm:pzdr/a154278E tfm:pzdr/a155278F tfm:pzdr/a1562790 tfm:pzdr/a1572791 tfm:pzdr/a1582792 tfm:pzdr/a1592793 tfm:pzdr/a16270F tfm:pzdr/a1602794 tfm:pzdr/a1612192 tfm:pzdr/a16227A3 tfm:pzdr/a1632194 tfm:pzdr/a1642195 tfm:pzdr/a1652799 tfm:pzdr/a166279B tfm:pzdr/a167279C tfm:pzdr/a168279D tfm:pzdr/a169279E tfm:pzdr/a172711 tfm:pzdr/a170279F tfm:pzdr/a17127A0 tfm:pzdr/a17227A1 tfm:pzdr/a17327A2 tfm:pzdr/a17427A4 tfm:pzdr/a17527A5 tfm:pzdr/a17627A6 tfm:pzdr/a17727A7 tfm:pzdr/a17827A8 tfm:pzdr/a17927A9 tfm:pzdr/a182712 tfm:pzdr/a18027AB tfm:pzdr/a18127AD tfm:pzdr/a18227AF tfm:pzdr/a18327B2 tfm:pzdr/a18427B3 tfm:pzdr/a18527B5 tfm:pzdr/a18627B8 tfm:pzdr/a18727BA tfm:pzdr/a18827BB tfm:pzdr/a18927BC tfm:pzdr/a192713 tfm:pzdr/a19027BD tfm:pzdr/a19127BE tfm:pzdr/a192279A tfm:pzdr/a19327AA tfm:pzdr/a19427B6 tfm:pzdr/a19527B9 tfm:pzdr/a1962798 tfm:pzdr/a19727B4 tfm:pzdr/a19827B7 tfm:pzdr/a19927AC tfm:pzdr/a22702 tfm:pzdr/a202714 tfm:pzdr/a20027AE tfm:pzdr/a20127B1 tfm:pzdr/a2022703 tfm:pzdr/a2032750 tfm:pzdr/a2042752 tfm:pzdr/a205276E tfm:pzdr/a2062770 tfm:pzdr/a212715 tfm:pzdr/a222716 tfm:pzdr/a232717 tfm:pzdr/a242718 tfm:pzdr/a252719 tfm:pzdr/a26271A tfm:pzdr/a27271B tfm:pzdr/a28271C tfm:pzdr/a292722 tfm:pzdr/a32704 tfm:pzdr/a302723 tfm:pzdr/a312724 tfm:pzdr/a322725 tfm:pzdr/a332726 tfm:pzdr/a342727 tfm:pzdr/a352605 tfm:pzdr/a362729 tfm:pzdr/a37272A tfm:pzdr/a38272B tfm:pzdr/a39272C tfm:pzdr/a4260E tfm:pzdr/a40272D tfm:pzdr/a41272E tfm:pzdr/a42272F tfm:pzdr/a432730 tfm:pzdr/a442731 tfm:pzdr/a452732 tfm:pzdr/a462733 tfm:pzdr/a472734 tfm:pzdr/a482735 tfm:pzdr/a492736 tfm:pzdr/a52706 tfm:pzdr/a502737 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tfm:zpzdr-reversed/a40272D tfm:zpzdr-reversed/a41272E tfm:zpzdr-reversed/a42272F tfm:zpzdr-reversed/a432730 tfm:zpzdr-reversed/a442731 tfm:zpzdr-reversed/a452732 tfm:zpzdr-reversed/a462733 tfm:zpzdr-reversed/a472734 tfm:zpzdr-reversed/a482735 tfm:zpzdr-reversed/a492736 tfm:zpzdr-reversed/a52706 tfm:zpzdr-reversed/a502737 tfm:zpzdr-reversed/a512738 tfm:zpzdr-reversed/a522739 tfm:zpzdr-reversed/a53273A tfm:zpzdr-reversed/a54273B tfm:zpzdr-reversed/a55273C tfm:zpzdr-reversed/a56273D tfm:zpzdr-reversed/a57273E tfm:zpzdr-reversed/a58273F tfm:zpzdr-reversed/a592740 tfm:zpzdr-reversed/a6271D tfm:zpzdr-reversed/a602741 tfm:zpzdr-reversed/a612742 tfm:zpzdr-reversed/a622743 tfm:zpzdr-reversed/a632744 tfm:zpzdr-reversed/a642745 tfm:zpzdr-reversed/a652746 tfm:zpzdr-reversed/a662747 tfm:zpzdr-reversed/a672748 tfm:zpzdr-reversed/a682749 tfm:zpzdr-reversed/a69274A tfm:zpzdr-reversed/a7271E tfm:zpzdr-reversed/a70274B tfm:zpzdr-reversed/a7125CF tfm:zpzdr-reversed/a72274D tfm:zpzdr-reversed/a7325A0 tfm:zpzdr-reversed/a74274F tfm:zpzdr-reversed/a752751 tfm:zpzdr-reversed/a7625B2 tfm:zpzdr-reversed/a7725BC tfm:zpzdr-reversed/a7825C6 tfm:zpzdr-reversed/a792756 tfm:zpzdr-reversed/a8271F tfm:zpzdr-reversed/a8125D7 tfm:zpzdr-reversed/a822758 tfm:zpzdr-reversed/a832759 tfm:zpzdr-reversed/a84275A tfm:zpzdr-reversed/a85276F tfm:zpzdr-reversed/a862771 tfm:zpzdr-reversed/a872772 tfm:zpzdr-reversed/a882773 tfm:zpzdr-reversed/a892768 tfm:zpzdr-reversed/a92720 tfm:zpzdr-reversed/a902769 tfm:zpzdr-reversed/a91276C tfm:zpzdr-reversed/a92276D tfm:zpzdr-reversed/a93276A tfm:zpzdr-reversed/a94276B tfm:zpzdr-reversed/a952774 tfm:zpzdr-reversed/a962775 tfm:zpzdr-reversed/a97275B tfm:zpzdr-reversed/a98275C tfm:zpzdr-reversed/a99275D thabengali09A5 thadeva0925 thagujarati0AA5 thagurmukhi0A25 thalarabic0630 thalfinalarabicFEAC thanthakhatlowleftthaiF898 thanthakhatlowrightthaiF897 thanthakhatthai0E4C thanthakhatupperleftthaiF896 theharabic062B thehfinalarabicFE9A thehinitialarabicFE9B thehmedialarabicFE9C thereexists2203 therefore2234 theta03B8 theta103D1 thetasymbolgreek03D1 thieuthacirclekorean3279 thieuthaparenkorean3219 thieuthcirclekorean326B thieuthkorean314C thieuthparenkorean320B thirteencircle246C thirteenparen2480 thirteenperiod2494 thonangmonthothai0E11 thook01AD thophuthaothai0E12 thorn00FE thothahanthai0E17 thothanthai0E10 thothongthai0E18 thothungthai0E16 thousandcyrillic0482 thousandsseparatorarabic066C thousandsseparatorpersian066C three0033 threearabic0663 threebengali09E9 threecircle2462 threecircleinversesansserif278C threedeva0969 threeeighths215C threegujarati0AE9 threegurmukhi0A69 threehackarabic0663 threehangzhou3023 threeideographicparen3222 threeinferior2083 threemonospaceFF13 threenumeratorbengali09F6 threeoldstyle0033 threeparen2476 threeperiod248A threepersian06F3 threequarters00BE threequartersemdashF6DE threeroman2172 threesuperior00B3 threethai0E53 thzsquare3394 tihiragana3061 tikatakana30C1 tikatakanahalfwidthFF81 tikeutacirclekorean3270 tikeutaparenkorean3210 tikeutcirclekorean3262 tikeutkorean3137 tikeutparenkorean3202 tilde02DC tildebelowcmb0330 tildecmb0303 tildecomb0303 tildedoublecmb0360 tildeoperator223C tildeoverlaycmb0334 tildeverticalcmb033E timescircle2297 tipehahebrew0596 tipehalefthebrew0596 tippigurmukhi0A70 titlocyrilliccmb0483 tiwnarmenian057F tlinebelow1E6F tmonospaceFF54 toarmenian0569 tohiragana3068 tokatakana30C8 tokatakanahalfwidthFF84 tonebarextrahighmod02E5 tonebarextralowmod02E9 tonebarhighmod02E6 tonebarlowmod02E8 tonebarmidmod02E7 tonefive01BD tonesix0185 tonetwo01A8 tonos0384 tonsquare3327 topatakthai0E0F tortoiseshellbracketleft3014 tortoiseshellbracketleftsmallFE5D tortoiseshellbracketleftverticalFE39 tortoiseshellbracketright3015 tortoiseshellbracketrightsmallFE5E tortoiseshellbracketrightverticalFE3A totaothai0E15 tpalatalhook01AB tparen24AF trademark2122 trademarksans2122 trademarkserif2122 tretroflexhook0288 triagdn25BC triaglf25C4 triagrt25BA triagup25B2 triangle25B3 triangledownsld25BC triangleinv25BD triangleleft25C1 triangleleftequal22B4 triangleleftsld25C0 triangleright25B7 trianglerightequal22B5 trianglerightsld25B6 trianglesolid25B2 ts02A6 tsadi05E6 tsadidageshFB46 tsadidageshhebrewFB46 tsadihebrew05E6 tsecyrillic0446 tsere05B5 tsere1205B5 tsere1e05B5 tsere2b05B5 tserehebrew05B5 tserenarrowhebrew05B5 tserequarterhebrew05B5 tserewidehebrew05B5 tshecyrillic045B tsuperior0074 ttabengali099F ttadeva091F ttagujarati0A9F ttagurmukhi0A1F tteharabic0679 ttehfinalarabicFB67 ttehinitialarabicFB68 ttehmedialarabicFB69 tthabengali09A0 tthadeva0920 tthagujarati0AA0 tthagurmukhi0A20 tturned0287 tuhiragana3064 tukatakana30C4 tukatakanahalfwidthFF82 turnstileleft22A2 turnstileright22A3 tusmallhiragana3063 tusmallkatakana30C3 tusmallkatakanahalfwidthFF6F twelvecircle246B twelveparen247F twelveperiod2493 twelveroman217B twentycircle2473 twentyhangzhou5344 twentyparen2487 twentyperiod249B two0032 twoarabic0662 twobengali09E8 twocircle2461 twocircleinversesansserif278B twodeva0968 twodotenleader2025 twodotleader2025 twodotleaderverticalFE30 twogujarati0AE8 twogurmukhi0A68 twohackarabic0662 twohangzhou3022 twoideographicparen3221 twoinferior2082 twomonospaceFF12 twonumeratorbengali09F5 twooldstyle0032 twoparen2475 twoperiod2489 twopersian06F2 tworoman2171 twostroke01BB twosuperior00B2 twothai0E52 twothirds2154 u0075 uacute00FA ubar0289 ubengali0989 ubopomofo3128 ubreve016D ucaron01D4 ucircle24E4 ucircumflex00FB ucircumflexbelow1E77 ucyrillic0443 udattadeva0951 udblacute0171 udblgrave0215 udeva0909 udieresis00FC udieresisacute01D8 udieresisbelow1E73 udieresiscaron01DA udieresiscyrillic04F1 udieresisgrave01DC udieresismacron01D6 udotbelow1EE5 ugrave00F9 ugujarati0A89 ugurmukhi0A09 uhiragana3046 uhookabove1EE7 uhorn01B0 uhornacute1EE9 uhorndotbelow1EF1 uhorngrave1EEB uhornhookabove1EED uhorntilde1EEF uhungarumlaut0171 uhungarumlautcyrillic04F3 uinvertedbreve0217 ukatakana30A6 ukatakanahalfwidthFF73 ukcyrillic0479 ukorean315C umacron016B umacroncyrillic04EF umacrondieresis1E7B umatragurmukhi0A41 umonospaceFF55 underscore005F underscoredbl2017 underscoremonospaceFF3F underscoreverticalFE33 underscorewavyFE4F union222A uniondbl22D3 unionmulti228E unionsq2294 universal2200 uogonek0173 uparen24B0 upblock2580 upperdothebrew05C4 uprise22CF upsilon03C5 upsilondieresis03CB upsilondieresistonos03B0 upsilonlatin028A upsilontonos03CD upslope29F8 uptackbelowcmb031D uptackmod02D4 uragurmukhi0A73 uring016F ushortcyrillic045E usmallhiragana3045 usmallkatakana30A5 usmallkatakanahalfwidthFF69 ustraightcyrillic04AF ustraightstrokecyrillic04B1 utilde0169 utildeacute1E79 utildebelow1E75 uubengali098A uudeva090A uugujarati0A8A uugurmukhi0A0A uumatragurmukhi0A42 uuvowelsignbengali09C2 uuvowelsigndeva0942 uuvowelsigngujarati0AC2 uvowelsignbengali09C1 uvowelsigndeva0941 uvowelsigngujarati0AC1 v0076 vadeva0935 vagujarati0AB5 vagurmukhi0A35 vakatakana30F7 vav05D5 vavdageshFB35 vavdagesh65FB35 vavdageshhebrewFB35 vavhebrew05D5 vavholamFB4B vavholamhebrewFB4B vavvavhebrew05F0 vavyodhebrew05F1 vcircle24E5 vdotbelow1E7F vector20D7 vecyrillic0432 veharabic06A4 vehfinalarabicFB6B vehinitialarabicFB6C vehmedialarabicFB6D vekatakana30F9 venus2640 verticalbar007C verticallineabovecmb030D verticallinebelowcmb0329 verticallinelowmod02CC verticallinemod02C8 vewarmenian057E vhook028B vikatakana30F8 viramabengali09CD viramadeva094D viramagujarati0ACD visargabengali0983 visargadeva0903 visargagujarati0A83 visiblespace2423 visualspace2423 vmonospaceFF56 voarmenian0578 voicediterationhiragana309E voicediterationkatakana30FE voicedmarkkana309B voicedmarkkanahalfwidthFF9E vokatakana30FA vparen24B1 vtilde1E7D vturned028C vuhiragana3094 vukatakana30F4 w0077 wacute1E83 waekorean3159 wahiragana308F wakatakana30EF wakatakanahalfwidthFF9C wakorean3158 wasmallhiragana308E wasmallkatakana30EE wattosquare3357 wavedash301C wavyunderscoreverticalFE34 wawarabic0648 wawfinalarabicFEEE wawhamzaabovearabic0624 wawhamzaabovefinalarabicFE86 wbsquare33DD wcircle24E6 wcircumflex0175 wdieresis1E85 wdotaccent1E87 wdotbelow1E89 wehiragana3091 weierstrass2118 wekatakana30F1 wekorean315E weokorean315D wgrave1E81 whitebullet25E6 whitecircle25CB whitecircleinverse25D9 whitecornerbracketleft300E whitecornerbracketleftverticalFE43 whitecornerbracketright300F whitecornerbracketrightverticalFE44 whitediamond25C7 whitediamondcontainingblacksmalldiamond25C8 whitedownpointingsmalltriangle25BF whitedownpointingtriangle25BD whiteleftpointingsmalltriangle25C3 whiteleftpointingtriangle25C1 whitelenticularbracketleft3016 whitelenticularbracketright3017 whiterightpointingsmalltriangle25B9 whiterightpointingtriangle25B7 whitesmallsquare25AB whitesmilingface263A whitesquare25A1 whitestar2606 whitetelephone260F whitetortoiseshellbracketleft3018 whitetortoiseshellbracketright3019 whiteuppointingsmalltriangle25B5 whiteuppointingtriangle25B3 wihiragana3090 wikatakana30F0 wikorean315F wmonospaceFF57 wohiragana3092 wokatakana30F2 wokatakanahalfwidthFF66 won20A9 wonmonospaceFFE6 wowaenthai0E27 wparen24B2 wreathproduct2240 wring1E98 wsuperior02B7 wturned028D wynn01BF x0078 xabovecmb033D xbopomofo3112 xcircle24E7 xdieresis1E8D xdotaccent1E8B xeharmenian056D xi03BE xmonospaceFF58 xparen24B3 xsuperior02E3 y0079 yaadosquare334E yabengali09AF yacute00FD yadeva092F yaekorean3152 yagujarati0AAF yagurmukhi0A2F yahiragana3084 yakatakana30E4 yakatakanahalfwidthFF94 yakorean3151 yamakkanthai0E4E yasmallhiragana3083 yasmallkatakana30E3 yasmallkatakanahalfwidthFF6C yatcyrillic0463 ycircle24E8 ycircumflex0177 ydieresis00FF ydotaccent1E8F ydotbelow1EF5 yeharabic064A yehbarreearabic06D2 yehbarreefinalarabicFBAF yehfinalarabicFEF2 yehhamzaabovearabic0626 yehhamzaabovefinalarabicFE8A yehhamzaaboveinitialarabicFE8B yehhamzaabovemedialarabicFE8C yehinitialarabicFEF3 yehmedialarabicFEF4 yehmeeminitialarabicFCDD yehmeemisolatedarabicFC58 yehnoonfinalarabicFC94 yehthreedotsbelowarabic06D1 yekorean3156 yen00A5 yenmonospaceFFE5 yeokorean3155 yeorinhieuhkorean3186 yerahbenyomohebrew05AA yerahbenyomolefthebrew05AA yericyrillic044B yerudieresiscyrillic04F9 yesieungkorean3181 yesieungpansioskorean3183 yesieungsioskorean3182 yetivhebrew059A ygrave1EF3 yhook01B4 yhookabove1EF7 yiarmenian0575 yicyrillic0457 yikorean3162 yinyang262F yiwnarmenian0582 ymonospaceFF59 yod05D9 yoddageshFB39 yoddageshhebrewFB39 yodhebrew05D9 yodyodhebrew05F2 yodyodpatahhebrewFB1F yohiragana3088 yoikorean3189 yokatakana30E8 yokatakanahalfwidthFF96 yokorean315B yosmallhiragana3087 yosmallkatakana30E7 yosmallkatakanahalfwidthFF6E yotgreek03F3 yoyaekorean3188 yoyakorean3187 yoyakthai0E22 yoyingthai0E0D yparen24B4 ypogegrammeni037A ypogegrammenigreekcmb0345 yr01A6 yring1E99 ysuperior02B8 ytilde1EF9 yturned028E yuhiragana3086 yuikorean318C yukatakana30E6 yukatakanahalfwidthFF95 yukorean3160 yusbigcyrillic046B yusbigiotifiedcyrillic046D yuslittlecyrillic0467 yuslittleiotifiedcyrillic0469 yusmallhiragana3085 yusmallkatakana30E5 yusmallkatakanahalfwidthFF6D yuyekorean318B yuyeokorean318A yyabengali09DF yyadeva095F z007A zaarmenian0566 zacute017A zadeva095B zagurmukhi0A5B zaharabic0638 zahfinalarabicFEC6 zahinitialarabicFEC7 zahiragana3056 zahmedialarabicFEC8 zainarabic0632 zainfinalarabicFEB0 zakatakana30B6 zaqefgadolhebrew0595 zaqefqatanhebrew0594 zarqahebrew0598 zayin05D6 zayindageshFB36 zayindageshhebrewFB36 zayinhebrew05D6 zbopomofo3117 zcaron017E zcircle24E9 zcircumflex1E91 zcurl0291 zdot017C zdotaccent017C zdotbelow1E93 zecyrillic0437 zedescendercyrillic0499 zedieresiscyrillic04DF zehiragana305C zekatakana30BC zero0030 zeroarabic0660 zerobengali09E6 zerodeva0966 zerogujarati0AE6 zerogurmukhi0A66 zerohackarabic0660 zeroinferior2080 zeromonospaceFF10 zerooldstyle0030 zeropersian06F0 zerosuperior2070 zerothai0E50 zerowidthjoinerFEFF zerowidthnonjoiner200C zerowidthspace200B zeta03B6 zhbopomofo3113 zhearmenian056A zhebrevecyrillic04C2 zhecyrillic0436 zhedescendercyrillic0497 zhedieresiscyrillic04DD zihiragana3058 zikatakana30B8 zinorhebrew05AE zlinebelow1E95 zmonospaceFF5A zohiragana305E zokatakana30BE zparen24B5 zretroflexhook0290 zstroke01B6 zuhiragana305A zukatakana30BA = 1 dummy `$=3format = hang, font = small, figurewithin = section pdftexgraphicx svgnamesxcolor = 9 = 1. positioning, patterns, arrows, calc[program = texindy] [program = texindy, name = symbols, title = Symbols]others.bib [label = my]my.bib *citetitle#1note-mark-format = #1.,text-format = , perpage = true, footnote = true @text#1#2 @marginpar @format @write@mark@note@mark@formatthe@snotez@mark @note@mark@sep#2 @format @write@mark@note@mark@formatthe@snotez@mark @note@mark@sep#2 |||||||||| ‖‖‖‖[2]⟨⟩#1,#2 {} {}{}⟨⟩⟨⟩⌈⌉⌊⌋ ` `,=,[1]` ="8000 #1 @rlay @rlay#1#2@skip -@th#1###2 wide 0pt--[1] @cev#1[ headfont = , headpunct = ., postheadspace = 0.5em, bodyfont = , spaceabove = , spacebelow = , qed = ]theorem[numberwithin=section, style=theorem, refname=main theorem, main theorems, Refname=Main Theorem, Main Theorems, name=Main Theorem]main-theorem [sibling=main-theorem, style=theorem, refname=theorem,theorems, Refname=Theorem,Theorems]theorem [sibling=main-theorem, style=theorem, refname=lemma,lemmata, Refname=Lemma,Lemmata]lemma [sibling=main-theorem, style=theorem, refname=corollary,corollaries, Refname=Corollary,Corollaries]corollary [ headfont = , headpunct = ., postheadspace = 0.5em, bodyfont = , spaceabove = , spacebelow = , qed = ]definition[sibling=main-theorem, style=definition, refname=definition,definitions, Refname=Definition,Definitions]definition [sibling=main-theorem, style=definition, refname=example,examples, Refname=Example,Examples]example [sibling=main-theorem, style=definition, refname=counterexample,counterexamples, Refname=Counterexample,Counterexamples]counterexample [sibling=main-theorem, style=definition, refname=remark,remarks, Refname=Remark,Remarks]remark [sibling=main-theorem, style=definition, refname=open problem,open problems, Refname=Open Problem,Open Problems, name=Open Problem]open-problem [ headfont = , headpunct = ., postheadspace = 0.5em, bodyfont = , spaceabove = , spacebelow = , qed =]proof[unnumbered, style=proof, refname=proof,proofs, Refname=Proof,Proofs]proof [unnumbered, style=proof, refname=proof sektch,proof sketches, Refname=Proof Sketch,Proof Sketches, name=Proof Sketch]proof-sketch [unnumbered, style=proof, refname=usage note,usage notes, Refname=Usage Note,Usage Notes, name=Usage Note]usage-note (#1,#2,#3,#4,#5,(#6,#7),#8)[shift=(#6,#7),rotate=#8] [fill=#4] (-3,-1) rectangle (-1,+1);[fill=#3] (-1,+1) rectangle (+1,+3);[fill=#1] (-1,-1) rectangle (+1,+1);[fill=#5] (-1,-3) rectangle (+1,-1);[fill=#2] (+1,-1) rectangle (+3,+1); (#1,#2,#3,#4,#5,(#6,#7)) [shift=(#6,#7)] /∠in 0/0,8/90(#1,#2,#3,#4,#5,(,0),∠) (#1,#2,#3,#4,#5,(#6,#7)) [shift=(#6,#7)] /∠in 0/0,8/90(#1,#2,#3,#4,#5,(0,-),∠) (#1,#2,#3,#4,#5,(#6,#7)) [shift=(#6,#7)] /∠in 0/0,8/90,16/180,24/270(#1,#2,#3,#4,#5,(,0),∠); (#1,#2,#3,#4,#5,(#6,#7)) [shift=(#6,#7)] /∠in 0/0,8/90,16/180,24/270(#1,#2,#3,#4,#5,(0,-),∠); (#1,#2,#3,#4,#5,(#6,#7)) [shift=(#6,#7)] /∠in 0/0,8/180(#1,#2,#3,#4,#5,(0,-),∠);/∠in 0/90,8/270(#1,#2,#3,#4,#5,(8,-),∠); (#1,(#2,#3)) [shift=(#2,#3)] (#1,white,white,white,white,(0,0),0) (#1,black,white,white,white,(8,0)) (#1,black,black,white,white,(40,0)) (#1,black,white,black,white,(72,0)) (#1,black,black,black,white,(88,0)) (#1,black,black,black,black,(120,0),0)(#1,(#2,#3)) [shift=(#2,#3)] (#1,white,white,white,white,(0,0),0) (#1,black,white,white,white,(8,0)) (#1,black,black,white,white,(16,0)) (#1,black,white,black,white,(24,0)) (#1,black,black,black,white,(32,0)) (#1,black,black,black,black,(40,0),0)(#1,(#2,#3)) [shift=(#2,#3)] (#1,white,white,white,white,(0,0),0) (#1,black,white,white,white,(8,0)) (#1,black,black,white,white,(24,0)) (#1,black,white,black,white,(32,0)) (#1,black,black,black,white,(48,0)) (#1,black,black,black,black,(64,0),0)(#1,(#2,#3)) [shift=(#2,#3)] (#1,white,white,white,white,(0,0),0) (#1,black,white,white,white,(8,0)) (#1,black,black,white,white,(16,0)) (#1,black,white,black,white,(32,0)) (#1,black,black,black,white,(40,0)) (#1,black,black,black,black,(48,0),0)(#1,(#2,#3)) [shift=(#2,#3)] (#1,white,white,white,white,(0,0),0) (#1,black,white,white,white,(8,0)) (#1,black,black,white,white,(24,0)) (#1,black,white,black,white,(40,0)) (#1,black,black,black,white,(48,0)) (#1,black,black,black,black,(64,0),0)(#1,(#2,#3),#4) [shift = (#2, #3), rotate = #4] (0, 0) – (#1, 0); (0, 0) – (0, #1); (0, 0) – (0, -#1);((#1,#2),#3) [shift = (#1, #2), scale = #3] [loosely dashed, tension = 0.5] plot[smooth cycle] coordinates (-0 + , -) (-0 + +̨ ł+ , -ȷ) (-0, -ȷ- -̨ ł- ) (-0 - ȷ- -̨ ł- , 0) (-0, ȷ+ +̨ ł+ ) (-0 + +̨ ł+ , ȷ) (-0 + , ) ; ((#1,#2),#3) [shift = (#1, #2), scale = #3] [densely dotted, tension = 0.5] plot[smooth cycle] coordinates (0 - 2*, -2*) (0 - -̨ ł- , -ȷ) (0, -ȷ- -̨ ł- ) (0 + +̨ ł+ , -ȷ) (0 + 2*, -2*) (0 + ȷ, --̨ ł- ) (0 + ȷ+ +̨ ł+ , 0) (0 + ȷ, +̨ ł+ ) (0 + 2*, 2*) (0 + +̨ ł+ , ȷ) (0, ȷ+ +̨ ł+ ) (0 - -̨ ł- , ȷ) (0 - 2*, 2*) ; ((#1,#2),#3,#4,#5,#6) [shift = (#1, #2), scale = #3, rotate = 90][densely dashed, tension = 0.5] plot[smooth cycle] coordinates (#4 - , 0) (#4, -#6) (0 - #5, -) (0 - -̨ ł- , -ȷ) (0, -ȷ- -̨ ł- ) (0 + +̨ ł+ , -ȷ) (0 + , -) (0 + ȷ, --̨ ł- ) (0 + ȷ+ +̨ ł+ , 0) (0 + ȷ, +̨ ł+ ) (0 + , ) (0 + +̨ ł+ , ȷ) (0, ȷ+ +̨ ł+ ) (0 - -̨ ł- , ȷ) (0 - #5, +) (#4, #6) ; ((#1,#2),#3) [shift = (#1, #2), scale = #3] i̊n 0, 180[rotate = ]̊ [loosely dashdotted, tension = 0.5] plot[smooth cycle] coordinates (-ȷ+ +̨ ł+ , 0) (-ȷ, +̨ ł+ ) (-ȷ- -̨ ł- , 0) (-ȷ, --̨ ł- ) ;; ((#1,#2),#3) ((#1,#2),#3); [shift = (#1, #2), scale = #3] i̊n 0, 180[shift = (0, -ȷ), rotate = ]̊ [dashdotted, tension = 0.5] plot[smooth cycle] coordinates (-+̨ ł+ /3, 0) (-,̨ ł+ /3) (--̨ ł- /3, 0) (-,̨ -ł- /3) ;[shift = (0, -ȷ- )̨, rotate = ]̊ [densely dashdotted, tension = 0.5] plot[smooth cycle] coordinates (-ł+ /3, 0) (-ł, /3) (-ł- /3, 0) (-ł, -/3) ;;((#1,#2),(#3,#4)) [step = 1cm, gray, very thin] (#1, #2) grid (#3, #4); [->] (#1, 0) – (#3, 0) node[right] x; [->] (0, #2) – (0, #4) node[above] y;((#1,#2),#3,#4) [shift = (#1, #2)] [#4] (0, #3) – (#3, 0) – (0, -#3) – (-#3, 0) – cycle; ((#1,#2),#3,#4) ((#1,#2),#3,solid); [shift = (#1, #2)] [loosely dashed] (0, #4) – (#4, 0) – (0, -#4); ((#1,#2),#3) [shift = (#1, #2)] in -#3, ..., #3abs() - #3 #3 - abs() in , ..., (, ) circle (0.1cm); ((#1,#2),#3) [shift = (#1, #2)] in -#3, ..., #3abs() - #3 #3 - abs() in , (, ) circle (0.1cm); ((#1,#2),#3) [shift = (#1, #2)] in -#3, ..., 0abs() - #3 #3 - abs() in , (, ) circle (0.1cm); ((#1,#2),#3) [shift = (#1, #2)] in 0, ..., #3abs() - #3 #3 - abs() in , (, ) circle (0.1cm);(#1,(#2,#3),#4) [shift = (#2, #3), rotate = #4] (0, 0) – (#1, 0); (0, 0) – (0, #1); (0, 0) – (0, -#1); 0.166 (0 + ı, 0) circle (); (0 + ı+ ȷ, 0) circle (); (0 + ȷ, 0 + ı) circle (); (0, 0) circle (); (0 + ȷ, 0 - ı) circle (); (0 + ı+ ȷ+ ,̨ 0) circle (); (0 + ı+ ,̨ 0 + ȷ) circle (); (0 + ı+ ,̨ 0 - ȷ) circle (); (0 + ȷ+ ,̨ 0 + ı) circle (); (0 + ,̨ 0 + ı+ ȷ) circle (); (0, 0 + ı) circle (); (0 - ı+ ,̨ 0 + ȷ) circle (); (0 - ı, 0) circle (); (0 - ı+ ,̨ 0 - ȷ) circle (); (0 + ȷ+ ,̨ 0 - ı) circle (); (0, 0 - ı) circle (); (0 + ,̨ 0 - ı- ȷ) circle ();0.166 (0 - ı- , 0 - ) rectangle (0 - ı+ , 0 + ); (0 - , 0 - ) rectangle (0 + , 0 + ); (0 - ȷ- , 0 + ı- ) rectangle (0 - ȷ+ , 0 + ı+ ); (0 - ı- ȷ- , 0 - ) rectangle (0 - ı- ȷ+ , 0 + ); (0 - ȷ- , 0 - ı- ) rectangle (0 - ȷ+ , 0 - ı+ ); (0 + ı- , 0 - ) rectangle (0 + ı+ , 0 + ); (0 + ı- -̨ , 0 + ȷ- ) rectangle (0 + ı- +̨ , 0 + ȷ+ ); (0 + ı- -̨ , 0 - ȷ- ) rectangle (0 + ı- +̨ , 0 - ȷ+ ); (0 - , 0 + ı- ) rectangle (0 + , 0 + ı+ ); (0 - -̨ , 0 + ı+ ȷ- ) rectangle (0 - +̨ , 0 + ı+ ȷ+ ); (0 - ȷ- -̨ , 0 + ı- ) rectangle (0 - ȷ- +̨ , 0 + ı+ ); (0 - ı- -̨ , 0 + ȷ- ) rectangle (0 - ı- +̨ , 0 + ȷ+ ); (0 - ı- ȷ- -̨ , 0 - ) rectangle (0 - ı- ȷ- +̨ , 0 + ); (0 - ı- -̨ , 0 - ȷ- ) rectangle (0 - ı- +̨ , 0 - ȷ+ ); (0 - , 0 - ı- ) rectangle (0 + , 0 - ı+ ); (0 - ȷ- -̨ , 0 - ı- ) rectangle (0 - ȷ- +̨ , 0 - ı+ ); (0 - -̨ , 0 - ı- ȷ- ) rectangle (0 - +̨ , 0 - ı- ȷ+ );0.166 (0, 0 - ı) circle (); (0 + ı, 0 - ȷ) circle (); (0, 0) circle (); (0 - ı, 0 - ȷ) circle (); (0, 0 - ı- ȷ) circle (); (0 + ı+ ȷ, 0 - )̨ circle (); (0 + ı, 0) circle (); (0 + ı, 0 - ȷ- )̨ circle (); (0 + ȷ, 0 + ı- )̨ circle (); (0, 0 + ı) circle (); (0 - ȷ, 0 + ı- )̨ circle (); (0 - ı, 0) circle (); (0 - ı- ȷ, 0 - )̨ circle (); (0 - ı, 0 - ȷ- )̨ circle (); (0 + ȷ, 0 - ı- )̨ circle (); (0 - ȷ, 0 - ı- )̨ circle (); (0, 0 - ı- ȷ- )̨ circle ();(#1) (0, 0) – node[above] a (ı, 0);(ȷ,(ı,0),0); [every path/.append style = #1] (,̨(ı+ȷ,0),0); (ł,(ı+ȷ+,̨0),0); (ł,(ı+ȷ,)̨,90); (ł,(ı+ȷ,-)̨,-90); (,̨(ı,ȷ),90); (ł,(ı,ȷ+)̨,90); (ł,(ı+,̨ȷ),0); (ł,(ı-,̨ȷ),180); (,̨(ı,-ȷ),-90); (ł,(ı,-ȷ-)̨,-90); (ł,(ı+,̨-ȷ),0); (ł,(ı-,̨-ȷ),180); (0, 0) – node[below] a^-1 (-ı, 0);(ȷ,(-ı,0),180); [every path/.append style = #1] (,̨(-ı-ȷ,0),180); (ł,(-ı-ȷ-,̨0),180); (ł,(-ı-ȷ,)̨,90); (ł,(-ı-ȷ,-)̨,-90); (,̨(-ı,ȷ),90); (ł,(-ı,ȷ+)̨,90); (ł,(-ı+,̨ȷ),0); (ł,(-ı-,̨ȷ),180); (,̨(-ı,-ȷ),-90); (ł,(-ı,-ȷ-)̨,-90); (ł,(-ı+,̨-ȷ),0); (ł,(-ı-,̨-ȷ),180); (0, 0) – node[left] b (0, ı);(ȷ,(0,ı),90); [every path/.append style = #1] (,̨(0,ı+ȷ),90); (ł,(0,ı+ȷ+)̨,90); (ł,(,̨ı+ȷ),0); (ł,(-,̨ı+ȷ),180); (,̨(ȷ,ı),0); (ł,(ȷ+,̨ı),0); (ł,(ȷ,ı+)̨,90); (ł,(ȷ,ı-)̨,-90); (,̨(-ȷ,ı),180); (ł,(-ȷ-,̨ı),180); (ł,(-ȷ,ı+)̨,90); (ł,(-ȷ,ı-)̨,-90); (0, 0) – node[right] b^-1 (0, -ı);(ȷ,(0,-ı),-90); [every path/.append style = #1] (,̨(0,-ı-ȷ),-90); (ł,(0,-ı-ȷ-)̨,-90); (ł,(,̨-ı-ȷ),0); (ł,(-,̨-ı-ȷ),180); (,̨(ȷ,-ı),0); (ł,(ȷ+,̨-ı),0); (ł,(ȷ,-ı+)̨,90); (ł,(ȷ,-ı-)̨,-90); (,̨(-ȷ,-ı),180); (ł,(-ȷ-,̨-ı),180); (ł,(-ȷ,-ı+)̨,90); (ł,(-ȷ,-ı-)̨,-90);[every path/.append style = dash pattern = on 1pt off 1pt](ȷ,(ı,0),0); (,̨(ı+ȷ,0),0); (ł,(ı+ȷ+,̨0),0); (ł,(ı+ȷ,)̨,90); (ł,(ı+ȷ,-)̨,-90); (,̨(ı,ȷ),90); (ł,(ı,ȷ+)̨,90); (ł,(ı+,̨ȷ),0); (ł,(ı-,̨ȷ),180); (,̨(ı,-ȷ),-90); (ł,(ı,-ȷ-)̨,-90); (ł,(ı+,̨-ȷ),0); (ł,(ı-,̨-ȷ),180); [every path/.append style = dash pattern = on 1pt off 1pt on off 1pt](ȷ,(-ı,0),180); (,̨(-ı-ȷ,0),180); (ł,(-ı-ȷ-,̨0),180); (ł,(-ı-ȷ,)̨,90); (ł,(-ı-ȷ,-)̨,-90); (,̨(-ı,ȷ),90); (ł,(-ı,ȷ+)̨,90); (ł,(-ı+,̨ȷ),0); (ł,(-ı-,̨ȷ),180); (,̨(-ı,-ȷ),-90); (ł,(-ı,-ȷ-)̨,-90); (ł,(-ı+,̨-ȷ),0); (ł,(-ı-,̨-ȷ),180); [every path/.append style = densely dotted](ȷ,(0,ı),90); (,̨(0,ı+ȷ),90); (ł,(0,ı+ȷ+)̨,90); (ł,(,̨ı+ȷ),0); (ł,(-,̨ı+ȷ),180); (,̨(ȷ,ı),0); (ł,(ȷ+,̨ı),0); (ł,(ȷ,ı+)̨,90); (ł,(ȷ,ı-)̨,-90); (,̨(-ȷ,ı),180); (ł,(-ȷ-,̨ı),180); (ł,(-ȷ,ı+)̨,90); (ł,(-ȷ,ı-)̨,-90);(ȷ,(0,-ı),-90); (,̨(0,-ı-ȷ),-90); (ł,(0,-ı-ȷ-)̨,-90); (ł,(,̨-ı-ȷ),0); (ł,(-,̨-ı-ȷ),180); (,̨(ȷ,-ı),0); (ł,(ȷ+,̨-ı),0); (ł,(ȷ,-ı+)̨,90); (ł,(ȷ,-ı-)̨,-90); (,̨(-ȷ,-ı),180); (ł,(-ȷ-,̨-ı),180); (ł,(-ȷ,-ı+)̨,90); (ł,(-ȷ,-ı-)̨,-90);(0, 0) – node[above] a (ı, 0); [every path/.append style = dotted] (ȷ,(ı,0),0);(,̨(ı+ȷ,0),0); (,̨(ı,ȷ),90); (,̨(ı,-ȷ),-90);(ł,(ı+ȷ+,̨0),0); (ł,(ı+ȷ,)̨,90); (ł,(ı+ȷ,-)̨,-90);(ł,(ı,ȷ+)̨,90); (ł,(ı+,̨ȷ),0); (ł,(ı-,̨ȷ),180);(ł,(ı,-ȷ-)̨,-90); (ł,(ı+,̨-ȷ),0); (ł,(ı-,̨-ȷ),180); (0, 0) – node[below] a^-1 (-ı, 0); [every path/.append style = dotted] (ȷ,(-ı,0),180); (,̨(-ı-ȷ,0),180); (,̨(-ı,ȷ),90); (,̨(-ı,-ȷ),-90);(ł,(-ı-ȷ-,̨0),180); (ł,(-ı-ȷ,)̨,90); (ł,(-ı-ȷ,-)̨,-90);(ł,(-ı,ȷ+)̨,90); (ł,(-ı+,̨ȷ),0); (ł,(-ı-,̨ȷ),180);(ł,(-ı,-ȷ-)̨,-90); (ł,(-ı+,̨-ȷ),0); (ł,(-ı-,̨-ȷ),180); (0, 0) – node[left] b (0, ı); [every path/.append style = dotted] (ȷ,(0,ı),90); (,̨(0,ı+ȷ),90); (,̨(ȷ,ı),0); (,̨(-ȷ,ı),180);(ł,(0,ı+ȷ+)̨,90); (ł,(,̨ı+ȷ),0); (ł,(-,̨ı+ȷ),180);(ł,(ȷ+,̨ı),0); (ł,(ȷ,ı+)̨,90); (ł,(ȷ,ı-)̨,-90);(ł,(-ȷ-,̨ı),180); (ł,(-ȷ,ı+)̨,90); (ł,(-ȷ,ı-)̨,-90); (0, 0) – node[right] b^-1 (0, -ı); [every path/.append style = dotted] (ȷ,(0,-ı),-90); (,̨(0,-ı-ȷ),-90); (,̨(ȷ,-ı),0); (,̨(-ȷ,-ı),180);(ł,(0,-ı-ȷ-)̨,-90); (ł,(,̨-ı-ȷ),0); (ł,(-,̨-ı-ȷ),180);(ł,(ȷ+,̨-ı),0); (ł,(ȷ,-ı+)̨,90); (ł,(ȷ,-ı-)̨,-90);(ł,(-ȷ-,̨-ı),180); (ł,(-ȷ,-ı+)̨,90); (ł,(-ȷ,-ı-)̨,-90); [shift = (0, 0), scale = 1] [dashed, tension = 0.5] plot[smooth cycle, tension = 0] coordinates (0 - , -) (0 + ı- , -) (0 + ı- , -ȷ- ) (0 + ı+ , -ȷ- ) (0 + ı+ , -) (0 + ı+ ȷ+ , -) (0 + ı+ ȷ+ , ) (0 + ı+ , ) (0 + ı+ , ȷ+ ) (0 + ı- , ȷ+ ) (0 + ı- , ) (0 - , ) ; (0, ı) circle (0.1cm); (0, -ı) circle (0.1cm); (-ı, 0) circle (0.1cm);(0, 0) – node[above] a (ı, 0);(ȷ,(ı,0),0);[every path/.append style = densely dotted] (,̨(ı+ȷ,0),0); (,̨(ı,ȷ),90); (,̨(ı,-ȷ),-90);(ł,(ı+ȷ+,̨0),0); (ł,(ı+ȷ,)̨,90); (ł,(ı+ȷ,-)̨,-90);(ł,(ı,ȷ+)̨,90); (ł,(ı+,̨ȷ),0); (ł,(ı-,̨ȷ),180);(ł,(ı,-ȷ-)̨,-90); (ł,(ı+,̨-ȷ),0); (ł,(ı-,̨-ȷ),180); (0, 0) – node[below] a^-1 (-ı, 0);(ȷ,(-ı,0),180);[every path/.append style = densely dotted] (,̨(-ı-ȷ,0),180); (,̨(-ı,ȷ),90); (,̨(-ı,-ȷ),-90);(ł,(-ı-ȷ-,̨0),180); (ł,(-ı-ȷ,)̨,90); (ł,(-ı-ȷ,-)̨,-90);(ł,(-ı,ȷ+)̨,90); (ł,(-ı+,̨ȷ),0); (ł,(-ı-,̨ȷ),180);(ł,(-ı,-ȷ-)̨,-90); (ł,(-ı+,̨-ȷ),0); (ł,(-ı-,̨-ȷ),180); (0, 0) – node[left] b (0, ı);(ȷ,(0,ı),90);[every path/.append style = densely dotted] (,̨(0,ı+ȷ),90); (,̨(ȷ,ı),0); (,̨(-ȷ,ı),180);(ł,(0,ı+ȷ+)̨,90); (ł,(,̨ı+ȷ),0); (ł,(-,̨ı+ȷ),180);(ł,(ȷ+,̨ı),0); (ł,(ȷ,ı+)̨,90); (ł,(ȷ,ı-)̨,-90);(ł,(-ȷ-,̨ı),180); (ł,(-ȷ,ı+)̨,90); (ł,(-ȷ,ı-)̨,-90); (0, 0) – node[right] b^-1 (0, -ı);(ȷ,(0,-ı),-90);[every path/.append style = densely dotted] (,̨(0,-ı-ȷ),-90); (,̨(ȷ,-ı),0); (,̨(-ȷ,-ı),180);(ł,(0,-ı-ȷ-)̨,-90); (ł,(,̨-ı-ȷ),0); (ł,(-,̨-ı-ȷ),180);(ł,(ȷ+,̨-ı),0); (ł,(ȷ,-ı+)̨,90); (ł,(ȷ,-ı-)̨,-90);(ł,(-ȷ-,̨-ı),180); (ł,(-ȷ,-ı+)̨,90); (ł,(-ȷ,-ı-)̨,-90); [shift = (0, 0), scale = 1] [dashed, tension = 0.5] plot[smooth cycle, tension = 0] coordinates (0 - ı- , -) (0 - , -) (0 - , -ı- ) (0 + , -ı- ) (0 + , -) (0 + ı- , -) (0 + ı- , -ȷ+ ) (0 + ı- -̨ , -ȷ+ ) (0 + ı- -̨ , -ȷ- ) (0 + ı- , -ȷ- ) (0 + ı- , -ȷ- -̨ ) (0 + ı+ , -ȷ- -̨ ) (0 + ı+ , -ȷ- ) (0 + ı+ +̨ , -ȷ- ) (0 + ı+ +̨ , -ȷ+ ) (0 + ı+ , -ȷ+ ) (0 + ı+ , -) (0 + ı+ ȷ- , -) (0 + ı+ ȷ- , --̨ ) (0 + ı+ ȷ+ , --̨ ) (0 + ı+ ȷ+ , -) (0 + ı+ ȷ+ +̨ , -) (0 + ı+ ȷ+ +̨ , ) (0 + ı+ ȷ+ , ) (0 + ı+ ȷ+ , +̨ ) (0 + ı+ ȷ- , +̨ ) (0 + ı+ ȷ- , ) (0 + ı+ , ) (0 + ı+ , ȷ- ) (0 + ı+ +̨ , ȷ- ) (0 + ı+ +̨ , ȷ+ ) (0 + ı+ , ȷ+ ) (0 + ı+ , ȷ+ +̨ ) (0 + ı- , ȷ+ +̨ ) (0 + ı- , ȷ+ ) (0 + ı- -̨ , ȷ+ ) (0 + ı- -̨ , ȷ- ) (0 + ı- , ȷ- ) (0 + ı- , ) (0 + , ) (0 + , ı+ ) (0 - , ı+ ) (0 - , ) (0 - ı- , ) ; (0, ı+ ȷ) circle (0.1cm); (ȷ, ı) circle (0.1cm); (-ȷ, ı) circle (0.1cm); (0, -ı- ȷ) circle (0.1cm); (ȷ, -ı) circle (0.1cm); (-ȷ, -ı) circle (0.1cm); (-ı- ȷ, 0) circle (0.1cm); (-ı, ȷ) circle (0.1cm); (-ı, -ȷ) circle (0.1cm);(0, 0) – node[above] a (ı, 0);(ȷ,(ı,0),0);(,̨(ı+ȷ,0),0); (,̨(ı,ȷ),90); (,̨(ı,-ȷ),-90);[every path/.append style = densely dotted] (ł,(ı+ȷ+,̨0),0); (ł,(ı+ȷ,)̨,90); (ł,(ı+ȷ,-)̨,-90);(ł,(ı,ȷ+)̨,90); (ł,(ı+,̨ȷ),0); (ł,(ı-,̨ȷ),180);(ł,(ı,-ȷ-)̨,-90); (ł,(ı+,̨-ȷ),0); (ł,(ı-,̨-ȷ),180); (0, 0) – node[below] a^-1 (-ı, 0);(ȷ,(-ı,0),180);(,̨(-ı-ȷ,0),180); (,̨(-ı,ȷ),90); (,̨(-ı,-ȷ),-90);[every path/.append style = densely dotted] (ł,(-ı-ȷ-,̨0),180); (ł,(-ı-ȷ,)̨,90); (ł,(-ı-ȷ,-)̨,-90);(ł,(-ı,ȷ+)̨,90); (ł,(-ı+,̨ȷ),0); (ł,(-ı-,̨ȷ),180);(ł,(-ı,-ȷ-)̨,-90); (ł,(-ı+,̨-ȷ),0); (ł,(-ı-,̨-ȷ),180); (0, 0) – node[left] b (0, ı);(ȷ,(0,ı),90);(,̨(0,ı+ȷ),90); (,̨(ȷ,ı),0); (,̨(-ȷ,ı),180); [every path/.append style = densely dotted] (ł,(0,ı+ȷ+)̨,90); (ł,(,̨ı+ȷ),0); (ł,(-,̨ı+ȷ),180);(ł,(ȷ+,̨ı),0); (ł,(ȷ,ı+)̨,90); (ł,(ȷ,ı-)̨,-90);(ł,(-ȷ-,̨ı),180); (ł,(-ȷ,ı+)̨,90); (ł,(-ȷ,ı-)̨,-90); (0, 0) – node[right] b^-1 (0, -ı);(ȷ,(0,-ı),-90);(,̨(0,-ı-ȷ),-90); (,̨(ȷ,-ı),0); (,̨(-ȷ,-ı),180);[every path/.append style = densely dotted] (ł,(0,-ı-ȷ-)̨,-90); (ł,(,̨-ı-ȷ),0); (ł,(-,̨-ı-ȷ),180);(ł,(ȷ+,̨-ı),0); (ł,(ȷ,-ı+)̨,90); (ł,(ȷ,-ı-)̨,-90);(ł,(-ȷ-,̨-ı),180); (ł,(-ȷ,-ı+)̨,90); (ł,(-ȷ,-ı-)̨,-90); i̊n 0, 90, 180[rotate = ]̊ (0, ı+ ȷ+ )̨ circle (0.1cm); (,̨ ı+ ȷ) circle (0.1cm); (-,̨ ı+ ȷ) circle (0.1cm);(ȷ, ı+ )̨ circle (0.1cm); (ȷ, ı- )̨ circle (0.1cm); (ȷ+ ,̨ ı) circle (0.1cm);(-ȷ, ı+ )̨ circle (0.1cm); (-ȷ, ı- )̨ circle (0.1cm); (-ȷ- ,̨ ı) circle (0.1cm); (#1,(#2,#3)) [shift = (#2, #3)] (0, 0) circle (1.333 * 0.1cm); (0, 0) edge node[auto] a (#1, 0); (0, 0) edge node[auto] a^-1 (-#1, 0); (0, 0) edge node[auto] b (0, #1); (0, 0) edge node[auto] b^-1 (0, -#1); (#1,#2,(#3,#4),#5) [shift = (#3, #4), rotate = #5] (0, 0) circle (1.333 * #1 * 0.1cm); (0, 0) – (#2, 0); (0, 0) – (-#1, 0); (0, 0) – (0, #2); (0, 0) – (0, -#2); (#1,#2,(#3,#4),#5) [shift = (#3, #4), rotate = #5, every path/.append style = densely dotted] (0, 0) – (#1, 0) (0, 0) – (-#1, 0); (#1, 0) – (#1, #2) (#1, 0) – (#1, -#2); (-#1, 0) – (-#1, #2) (-#1, 0) – (-#1, -#2); (#1,(#2,#3),#4) [shift = (#2, #3), rotate = #4, every path/.append style = densely dotted] (0, 0) – (#1, 0) (0, 0) – (0, #1) (0, 0) – (0, -#1); (#1,(#2,#3),#4) [shift = (#2, #3), rotate = #4] (#1, 0) circle (1.333 * #1 * 0.1cm); (0, 0) – (#1, 0); [densely dotted] (0, 0) – (0, #1) (0, 0) – (0, -#1); ((#1,#2),#3) [shift = (#1, #2), rotate = #3] (,̨ł,(0+ȷ+,̨0),0);(ł,,(0+ȷ,+̨ł),90); (ł,,(0+ȷ,--̨ł),-90);[shift = (0, 0)] [densely dotted] (0, 0) – (ȷ, 0);(,̨ł,(0+ȷ,0),90); (,(0+ȷ++̨ł,0),0); (,(0+ȷ+,̨ł),90); (,(0+ȷ+,̨-ł),-90); (,(0+ȷ+ł,)̨,0); (,(0+ȷ-ł,)̨,180); (,(0+ȷ+ł,-)̨,0); (,(0+ȷ-ł,-)̨,180);(#1,#2,(#3,#4),#5) [shift = (#3,#4), rotate = #5] (0, 0) circle (0.666 * #1 * 0.1cm); (0, 0) – (#1, 0); [densely dotted] (#1, 0) – (#1 + #2, 0) (#1, 0) – (#1, #2) (#1, 0) – (#1, -#2); (#1, #2) circle (0.666 * #2 * 0.1cm); (#1, -#2) circle (0.666 * #2 * 0.1cm); (#1,#2,#3,(#4,#5),#6) (#1,#2,(#4,#5),#6) [shift = (#4,#5), rotate = #6] (#1 + #2 + #3, 0) circle (0.666 * #3 * 0.1cm); (#1 + #2, 0) – (#1 + #2 + #3, 0); [densely dotted] (#1 + #2, 0) – (#1 + #2, #3) (#1 + #2, 0) – (#1 + #2, -#3); (#1 + #2, #3) circle (0.666 * #3 * 0.1cm); (#1 + #2, -#3) circle (0.666 * #3 * 0.1cm); [scale = 0.55, y = -1cm, arrow/.style = ->, shorten >= 0.2cm, shorten <= 0.2cm, composed/.style = densely dotted] (-3.5, 0) grid (4.5, 2); in -3, ..., 3at (+ 0.5, -0.5) ; at (+ 0.5, 1.5) ;in -4, ..., 3[arrow] (+ 1.5, 0.5) – (+ 0.5, 1.5); [arrow] (+ 0.5, 0.5) – (+ 1.5, 1.5);(-4, 0.5) edge[|->, bend right = 90] node[left] Δ (-4, 1.5); .5 [] [scale = 0.55, y = -1cm, arrow/.style = ->, shorten >= 0.2cm, shorten <= 0.2cm, composed/.style = densely dotted] (-3.5, 0) grid (4.5, 2); in -3, ..., 3at (+ 0.5, -0.5) ;in -3, ..., 4[arrow] (+ 0.5, 0.5) – (- 0.5, 1.5);(-4, 0.5) edge[|->, bend right = 90] node[left] Δ (-4, 1.5); .5 [] [scale = 0.55, y = -1cm, arrow/.style = ->, shorten >= 0.2cm, shorten <= 0.2cm, composed/.style = densely dotted] (-3.5, 0) grid (4.5, 2); in -3, ..., 3at (+ 0.5, -0.5) ;in -3, ..., 4[arrow] (- 0.5, 0.5) – (+ 0.5, 1.5);(-4, 0.5) edge[|->, bend right = 90] node[left] Δ' (-4, 1.5); .5 [] [scale = 0.55, y = -1cm, arrow/.style = ->, shorten >= 0.2cm, shorten <= 0.2cm, composed/.style = densely dotted] (-3.5, 0) grid (4.5, 2); in -3, ..., 3at (+ 0.5, -0.5) ;in -3, -1, ..., 3[arrow] (- 0.5, 0.5) – (+ 0.5, 1.5);in -3, -1, ..., 4[arrow] (+ 0.5, 0.5) – (- 0.5, 1.5);(-4, 0.5) edge[|->, bend right = 90] node[left] Δ” (-4, 1.5); .5 [] [scale = 0.55, y = -1cm, arrow/.style = ->, shorten >= 0.2cm, shorten <= 0.2cm, composed/.style = densely dotted] (-3.5, 0) grid (4.5, 2); in -3, ..., 3at (+ 0.5, -0.5) ;in -4, -3, -2, -1, 0, 2, 3[arrow] (+ 1.5, 0.5) – (+ 0.5, 1.5);[arrow] (1 - 0.5, 0.5) – (1 + 0.5, 1.5); (-4, 0.5) edge[|->, bend right = 90] node[left] Δ”' (-4, 1.5); [scale=0.06] (white,(0,0)) (black,(128,0))[scale=0.06] (white,(0,0)) (black,(72,0)) [scale=0.06] (white,(0,0)) (black,(56,0)) [scale=0.06] (white,(0,0)) (black,(48,0)) [scale=0.06] (white,(0,0)) (black,(72,0)) [scale = 0.55, y = -1cm, arrow/.style = ->, shorten >= 0.2cm, shorten <= 0.2cm, composed/.style = densely dotted] (-3.5, 0) grid (4.5, 3); in -3, ..., 3at (+ 0.5, -0.5) ;in -4, -3, -2, -1, 0, 2, 3in 0, 1[arrow] (+ 1.5, + 0.5) – (+ 0.5, + 1.5); in -4, -3, -2, -1, 2[arrow, composed] (+ 2.5, 0 + 0.5) – (+ 0.5, 1 + 1.5);in 0, 1[arrow] (1 - 0.5, + 0.5) – (1 + 0.5, + 1.5); [arrow, composed, shorten >= 0.175cm, shorten <= 0.175cm] (+ 0.5, 0.5) – (+ 0.5, 2.5);(-4, 0.5) edge[|->, bend right = 90] node[left] Δ”' (-4, 1.25); (-4, 1.75) edge[|->, bend right = 90] node[left] Δ”' (-4, 2.5); .5 [] [scale = 0.55, y = -1cm, arrow/.style = ->, shorten >= 0.2cm, shorten <= 0.2cm, composed/.style = densely dotted] (-3.5, 0) grid (4.5, 3); in -3, ..., 3at (+ 0.5, -0.5) ;in -4, ..., 3[arrow] (+ 1.5, 0.5) – (+ 0.5, 1.5);in -4, ..., 3[arrow] (+ 1.5, 1 + 0.5) – (+ 0.5, 1 + 1.5);in -4, ..., 2[arrow, composed] (+ 2.5, 0 + 0.5) – (+ 0.5, 1 + 1.5);(-4, 0.5) edge[|->, bend right = 90] node[left] Δ (-4, 1.25); (-4, 1.75) edge[|->, bend right = 90] node[left] Δ (-4, 2.5); .5 [] [scale = 0.55, y = -1cm, arrow/.style = ->, shorten >= 0.2cm, shorten <= 0.2cm, composed/.style = densely dotted] (-3.5, 0) grid (4.5, 3); in -3, ..., 3at (+ 0.5, -0.5) ; at (+ 0.5, 1.5) ; at (+ 0.5, 2.5) ;in 0, 1in -4, ..., 3[arrow] (+ 1.5, + 0.5) – (+ 0.5, + 1.5); [arrow] (+ 0.5, + 0.5) – (+ 1.5, + 1.5); in -4, ..., 2[arrow, composed] (+ 2.5, 0.5) – (+ 0.5, 2.5); [arrow, composed] (+ 0.5, 0.5) – (+ 2.5, 2.5); [arrow, composed, shorten >= 0.125cm, shorten <= 0.125cm] (+ 1.5, 0.5) – (+ 1.5, 2.5);(-4, 0.5) edge[|->, bend right = 90] node[left] Δ” (-4, 1.25); (-4, 1.75) edge[|->, bend right = 90] node[left] Δ” (-4, 2.5); .5 [] [scale = 0.55, y = -1cm, arrow/.style = ->, shorten >= 0.2cm, shorten <= 0.2cm, composed/.style = densely dotted] (-3.5, 0) grid (4.5, 3); in -3, ..., 3at (+ 0.5, -0.5) ; at (+ 0.5, 2.5) ;in -4, ..., 3[arrow] (+ 1.5, 0.5) – (+ 0.5, 1.5);in -4, ..., 3[arrow] (+ 1.5, 1.5) – (+ 0.5, 2.5); [arrow] (+ 0.5, 1.5) – (+ 1.5, 2.5);in -4, ..., 2[arrow, composed] (+ 2.5, 0 + 0.5) – (+ 0.5, 1 + 1.5);in -3, ..., 3[arrow, composed, shorten >= 0.125cm, shorten <= 0.125cm] (+ 0.5, 0 + 0.5) – (+ 0.5, 1 + 1.5);(-4, 0.5) edge[|->, bend right = 90] node[left] Δ (-4, 1.25); (-4, 1.75) edge[|->, bend right = 90] node[left] Δ” (-4, 2.5); .5 [] [scale = 0.55, y = -1cm, arrow/.style = ->, shorten >= 0.2cm, shorten <= 0.2cm, composed/.style = densely dotted] (-3.5, 0) grid (4.5, 3); in -3, ..., 3at (+ 0.5, -0.5) ; at (+ 0.5, 1.5) ; at (+ 0.5, 2.5) ;in -4, ..., 3[arrow] (+ 1.5, 1.5) – (+ 0.5, 2.5);in -4, ..., 3[arrow] (+ 1.5, 0.5) – (+ 0.5, 1.5); [arrow] (+ 0.5, 0.5) – (+ 1.5, 1.5);in -4, ..., 2[arrow, composed] (+ 2.5, 0 + 0.5) – (+ 0.5, 1 + 1.5);in -3, ..., 3[arrow, composed, shorten >= 0.125cm, shorten <= 0.125cm] (+ 0.5, 0 + 0.5) – (+ 0.5, 1 + 1.5);(-4, 0.5) edge[|->, bend right = 90] node[left] Δ” (-4, 1.25); (-4, 1.75) edge[|->, bend right = 90] node[left] Δ (-4, 2.5); .5 [] [scale = 0.55, y = -1cm, arrow/.style = ->, shorten >= 0.2cm, shorten <= 0.2cm, composed/.style = densely dotted] (-3.5, 0) grid (4.5, 3); in -3, ..., 3at (+ 0.5, -0.5) ;in -4, ..., 3[arrow] (+ 1.5, 0.5) – (+ 0.5, 1.5);in -4, -3, -2, -1, 0, 2, 3[arrow] (+ 1.5, 1 + 0.5) – (+ 0.5, 1 + 1.5);in -4, -3, -2, -1, 0, 2[arrow, composed] (+ 2.5, 0 + 0.5) – (+ 0.5, 1 + 1.5);[arrow] (1 - 0.5, 1.5) – (1 + 0.5, 2.5); [arrow, composed, shorten >= 0.125cm, shorten <= 0.125cm] (1.5, 0.5) – (1.5, 2.5); (-4, 0.5) edge[|->, bend right = 90] node[left] Δ (-4, 1.25); (-4, 1.75) edge[|->, bend right = 90] node[left] Δ' (-4, 2.5); .5 [] [scale = 0.55, y = -1cm, arrow/.style = ->, shorten >= 0.2cm, shorten <= 0.2cm, composed/.style = densely dotted] (-3.5, 0) grid (4.5, 3); in -3, ..., 3at (+ 0.5, -0.5) ;in -4, ..., 3[arrow] (+ 1.5, 1 + 0.5) – (+ 0.5, 1 + 1.5);in -4, -3, -2, -1, 0, 2, 3[arrow] (+ 1.5, 0.5) – (+ 0.5, 1.5);in -4, -3, -2, -1, 1, 2[arrow, composed] (+ 2.5, 0 + 0.5) – (+ 0.5, 1 + 1.5);[arrow] (1 - 0.5, 0.5) – (1 + 0.5, 1.5); [arrow, composed, shorten >= 0.125cm, shorten <= 0.125cm] (0.5, 0.5) – (0.5, 2.5); (-4, 0.5) edge[|->, bend right = 90] node[left] Δ' (-4, 1.25); (-4, 1.75) edge[|->, bend right = 90] node[left] Δ (-4, 2.5); .5 [] [scale = 0.55, y = -1cm, arrow/.style = ->, shorten >= 0.2cm, shorten <= 0.2cm, composed/.style = densely dotted] (-3.5, 0) grid (4.5, 3); in -3, ..., 3at (+ 0.5, -0.5) ; at (+ 0.5, 2.5) ;in -4, -3, -2, -1, 0, 2, 3[arrow] (+ 1.5, 0.5) – (+ 0.5, 1.5);in -4, ..., 3[arrow] (+ 1.5, 1.5) – (+ 0.5, 2.5); [arrow] (+ 0.5, 1.5) – (+ 1.5, 2.5);in -4, -3, -2, -1, 1, 2[arrow, composed] (+ 2.5, 0 + 0.5) – (+ 0.5, 1 + 1.5);in -3, -2, -1, 1, 3[arrow, composed, shorten >= 0.125cm, shorten <= 0.125cm] (+ 0.5, 0 + 0.5) – (+ 0.5, 1 + 1.5);[arrow, composed] (0.5, 0.5) – (2.5, 2.5); [arrow] (1 - 0.5, 0.5) – (1 + 0.5, 1.5); [arrow, composed, shorten >= 0.125cm, shorten <= 0.125cm] (0.4, 0.5) – (0.4, 2.5); [arrow, composed, shorten >= 0.125cm, shorten <= 0.125cm] (0.6, 0.5) – (0.6, 2.5); (-4, 0.5) edge[|->, bend right = 90] node[left] Δ' (-4, 1.25); (-4, 1.75) edge[|->, bend right = 90] node[left] Δ” (-4, 2.5); .5 [] [scale = 0.55, y = -1cm, arrow/.style = ->, shorten >= 0.2cm, shorten <= 0.2cm, composed/.style = densely dotted] (-3.5, 0) grid (4.5, 3); in -3, ..., 3at (+ 0.5, -0.5) ; at (+ 0.5, 1.5) ; at (+ 0.5, 2.5) ;in -4, -3, -2, -1, 0, 2, 3[arrow] (+ 1.5, 1.5) – (+ 0.5, 2.5);[arrow] (1 - 0.5, 1.5) – (1 + 0.5, 2.5); in -4, ..., 3[arrow] (+ 1.5, 0.5) – (+ 0.5, 1.5); [arrow] (+ 0.5, 0.5) – (+ 1.5, 1.5);in -4, -3, -2, -1, 0, 2[arrow, composed] (+ 2.5, 0 + 0.5) – (+ 0.5, 1 + 1.5);in -3, -2, -1, 0, 1, 2, 3[arrow, composed, shorten >= 0.125cm, shorten <= 0.125cm] (+ 0.5, 0 + 0.5) – (+ 0.5, 1 + 1.5);[arrow, composed] (-0.5, 0.5) – (1.5, 2.5); (-4, 0.5) edge[|->, bend right = 90] node[left] Δ” (-4, 1.25); (-4, 1.75) edge[|->, bend right = 90] node[left] Δ' (-4, 2.5);.5 [] [scale = 0.55] [shift = (0, 0)] (0, 0) – (4, 0); (0, 4) – (4, 4); [dashed] (0, 2) – (0, 4); [dashed] (4, 0) – (4, 4);[circle, fill, inner sep = 0pt, minimum size = 2pt] (m) at (0.5, 0) [label = -90:m] ; [circle, fill, inner sep = 0pt, minimum size = 2pt] (m') at (2.5, 0) [label = -90:m'] ;[very thick] (m) – (m'); [very thick, dashed] (0, 0) – (0, 2);[circle, fill, inner sep = 0pt, minimum size = 2pt] at (0.5, 2) ; [circle, fill, inner sep = 0pt, minimum size = 2pt] (a) at (0, 2) [label = 180:a] ;[shift = (5.5, 0)] (0, 0) – (4, 0); (0, 4) – (4, 4); [dashed] (0, 0) – (0, 3); [dashed] (4, 0) – (4, 4);4 / (2 * pi) [shift = (2, 4.5 + )̊] [dashdotted, domain = 90:270] plot (*̊ cos(), *̊ sin()); [dotted, domain = -90:90] plot (*̊ cos(), *̊ sin());[dashdotted] (-2, -)̊ – (0, -)̊; [dotted] (0, -)̊ – (2, -)̊; [circle, fill, inner sep = 0pt, minimum size = 2pt] (m) at (0.5, 0) [label = -90:m] ; [circle, fill, inner sep = 0pt, minimum size = 2pt] (m') at (2.5, 0) [label = -90:m'] ; [circle, fill, inner sep = 0pt, minimum size = 2pt] (mr) at (3.5, 0) [label = 90:ϱ_-1(m)] ;[very thick] (mr) – (m'); [very thick, dashed] (0, 4) – (0, 3);[circle, fill, inner sep = 0pt, minimum size = 2pt] at (0.5, 3) ; [circle, fill, inner sep = 0pt, minimum size = 2pt] (a') at (0, 3) [label = 180:a'] ;.5 [] [scale = 0.55] [shift = (0, 0)] (0, 0) – (4, 0); (0, 4) – (4, 4); [dashed] (0, 0) – (0, 4); [dashed] (4, 0) – (4, 4);[circle, fill, inner sep = 0pt, minimum size = 2pt] (m) at (0.5, 0) [label = -90:m] ; [yscale = 0.6] (Yl) at (2.5, 0) {; [yscale = 0.6] (Yr) at (3.75, 0) }; (Y) at (3.125, 0) [label = -90:Y] ;[rotate = 90, yscale = 0.6] (Bl) at (0, 2) {; [rotate = 90, yscale = 0.6] (Br) at (0, 3.25) }; (B) at (0, 2.625) [label = 180:B] ; (0.5, 2) – (0.5, 3.25);[shift = (5.5, 0)] (0, 0) – (4, 0); (0, 4) – (4, 4); [dashed] (0, 0) – (0, 4); [dashed] (4, 0) – (4, 4);4 / (2 * pi) [shift = (2, 4.5 + )̊] [dashdotted, domain = 90:270] plot (*̊ cos(), *̊ sin()); [dotted, domain = -90:90] plot (*̊ cos(), *̊ sin());[dashdotted] (-2, -)̊ – (0, -)̊; [dotted] (0, -)̊ – (2, -)̊; [circle, fill, inner sep = 0pt, minimum size = 2pt] (m) at (0.5, 0) [label = -90:m] ; [yscale = 0.6] (Yl) at (2.5, 0) {; [yscale = 0.6] (Yr) at (3.75, 0) }; (Y) at (3.125, 0) [label = -90:Y] ;[circle, fill, inner sep = 0pt, minimum size = 2pt] (mr) at (3.5, 0) [label = ϱ_-1(m)] ;[rotate = 90, yscale = 0.6] (Bl) at (0, 3) {; [rotate = 90, yscale = 0.6] (Br) at (0, 0.25) }; (B') at (0, 3.5) [label = 180:B'] ; (0.5, 3) – (0.5, 4); (0.5, 0) – (0.5, 0.25); .5 [] [scale = 0.55] [shift = (0, 0)] (0, 0) – (4, 0); (0, 4) – (4, 4); [dashed] (0, 0) – (0, 4); [dashed] (4, 0) – (4, 4);[yscale = 0.6] (Xl) at (0.5, 0) {; [yscale = 0.6] (Xr) at (1.75, 0) }; (X) at (1.125, 0) [label = -90:X] ; [circle, fill, inner sep = 0pt, minimum size = 2pt] (m') at (2.5, 0) [label = -90:m'] ; (0.5, 2) – (1.75, 0.75);[shift = (5, 0)] (0, 0) – (4, 0); (0, 4) – (4, 4); [dashed] (0, 0) – (0, 4); [dashed] (4, 0) – (4, 4);4 / (2 * pi) [shift = (2, 4.5 + )̊] [dashdotted, domain = 90:270] plot (*̊ cos(), *̊ sin()); [dotted, domain = -90:90] plot (*̊ cos(), *̊ sin());[dashdotted] (-2, -)̊ – (0, -)̊; [dotted] (0, -)̊ – (2, -)̊; [yscale = 0.6] (Xl) at (0.5, 0) {; [yscale = 0.6] (Xr) at (1.75, 0) }; (X) at (1.125, 0) [label = -90:X] ; [circle, fill, inner sep = 0pt, minimum size = 2pt] (m') at (2.5, 0) [label = -90:m'] ;[yscale = 0.6] (Xl') at (3.5, 0) }; [yscale = 0.6] (Xr') at (2.25, 0) {; (X') at (2.875, 0) [label = ϱ_-1(X)] ; (0.5, 3) – (1.5, 4); (1.5, 0) – (1.75, 0.25);.5 [] [scale = 0.55] [shift = (0, 0)] (0, 0) – (4, 0); (0, 4) – (4, 4); [dashed] (0, 0) – (0, 4); [dashed] (4, 0) – (4, 4);[yscale = 0.6] (Xl) at (0.5, 0) {; [yscale = 0.6] (Xr) at (1.75, 0) }; (X) at (1.125, 0) [label = -90:X] ;[yscale = 0.8] (Yl) at (3.25, 0) {; [yscale = 0.8] (Yr) at (3.75, 0) }; (Y) at (3.5, 0) [label = -90:Y] ;[fill] (0.5, 2.75) – (0.5, 3.25) – (1.75, 2) – (1.75, 1.5) – cycle;[shift = (5, 0)] (0, 0) – (4, 0); (0, 4) – (4, 4); [dashed] (0, 0) – (0, 4); [dashed] (4, 0) – (4, 4);4 / (2 * pi) [shift = (2, 4.5 + )̊] [dashdotted, domain = 90:270] plot (*̊ cos(), *̊ sin()); [dotted, domain = -90:90] plot (*̊ cos(), *̊ sin());[dashdotted] (-2, -)̊ – (0, -)̊; [dotted] (0, -)̊ – (2, -)̊; [yscale = 0.6] (Xl) at (0.5, 0) {; [yscale = 0.6] (Xr) at (1.75, 0) }; (X) at (1.125, 0) [label = -90:X] ;[yscale = 0.8] (Yl) at (3.25, 0) {; [yscale = 0.8] (Yr) at (3.75, 0) }; (Y) at (3.5, 0) [label = -90:Y] ;[yscale = 0.6] (Xl') at (3.5, 0) }; [yscale = 0.6] (Xr') at (2.25, 0) {; (X') at (2.875, 0) [label = ϱ_-1(X)] ;[fill] (0.5, 0.25) – (1.75, 1.5) – (1.75, 1) – (0.75, 0) – (0.5, 0) – cycle; [fill] (0.5, 4) – (0.5, 3.75) – (0.75, 4); [ρ = 1] [scale = 0.4] ((-5,-4),(5,4)); ((0,0),1,solid);((1,0),1,dashed); ((0,0),1); [ρ = 2] [scale = 0.4] ((-5,-4),(5,4)); ((0,0),2,solid);((1,0),2,dashed); ((0,0),2); [ρ = 3] [scale = 0.4] ((-5,-4),(5,4)); ((0,0),3,solid);((1,0),3,dashed); ((0,0),3); [scale = 0.4] ı3 ȷ1.5 0̨.̨7̨5̨ ł0.375(4, -3) edge[|->, bend right = 60] node[below] rotation about the midpoint of the edge labelled a (20, -3);[dash pattern = on 1pt off 1pt] (0, 0) – node[auto] a (ı, 0);[dash pattern = on 1pt off 1pt on off 1pt] (0, 0) – node[auto] a^-1 (-ı, 0);[densely dotted] (0, 0) – node[auto] b (0, ı);(0, 0) – node[auto] b^-1 (0, -ı);(4, 3) edge[|->, bend left] node[above] φ(180) (8, 3);[shift = (12, 0)][dash pattern = on 1pt off 1pt on off 1pt] (0, 0) – node[auto] a (ı, 0);[dash pattern = on 1pt off 1pt] (0, 0) – node[auto] a^-1 (-ı, 0);(0, 0) – node[auto] b (0, ı);[densely dotted] (0, 0) – node[auto] b^-1 (0, -ı);[rotate = 180] (16, 3) edge[|->, bend left] node[above] a · (20, 3); [shift = (24, 0)] [dash pattern = on 1pt off 1pt] (0, 0) – node[auto] a (ı, 0);[shift = (ı, 0)] [dash pattern = on 1pt off 1pt on off 1pt] (0, 0) – (ȷ, 0); (0, 0) – (0, ȷ); [densely dotted] (0, 0) – (0, -ȷ);[every path/.append style = dash pattern = on 1pt off 1pt on off 1pt] (,̨(ı+ȷ,0),0); (ł,(ı+ȷ+,̨0),0); (ł,(ı+ȷ,)̨,90); (ł,(ı+ȷ,-)̨,-90);(,̨(ı,ȷ),90); (ł,(ı,ȷ+)̨,90); (ł,(ı+,̨ȷ),0); (ł,(ı-,̨ȷ),180); [every path/.append style = densely dotted] (,̨(ı,-ȷ),-90); (ł,(ı,-ȷ-)̨,-90); (ł,(ı+,̨-ȷ),0); (ł,(ı-,̨-ȷ),180); [every path/.append style = dash pattern = on 1pt off 1pt] (0, 0) – node[auto] a^-1 (-ı, 0);(ȷ,(-ı,0),180); (,̨(-ı-ȷ,0),180); (ł,(-ı-ȷ-,̨0),180); (ł,(-ı-ȷ,)̨,90); (ł,(-ı-ȷ,-)̨,-90); (,̨(-ı,ȷ),90); (ł,(-ı,ȷ+)̨,90); (ł,(-ı+,̨ȷ),0); (ł,(-ı-,̨ȷ),180); (,̨(-ı,-ȷ),-90); (ł,(-ı,-ȷ-)̨,-90); (ł,(-ı+,̨-ȷ),0); (ł,(-ı-,̨-ȷ),180);(0, 0) – node[auto] b (0, ı);(ȷ,(0,ı),90); (,̨(0,ı+ȷ),90); (ł,(0,ı+ȷ+)̨,90); (ł,(,̨ı+ȷ),0); (ł,(-,̨ı+ȷ),180); (,̨(ȷ,ı),0); (ł,(ȷ+,̨ı),0); (ł,(ȷ,ı+)̨,90); (ł,(ȷ,ı-)̨,-90); (,̨(-ȷ,ı),180); (ł,(-ȷ-,̨ı),180); (ł,(-ȷ,ı+)̨,90); (ł,(-ȷ,ı-)̨,-90);(0, 0) – node[auto] b^-1 (0, -ı);(ȷ,(0,-ı),-90); (,̨(0,-ı-ȷ),-90); (ł,(0,-ı-ȷ-)̨,-90); (ł,(,̨-ı-ȷ),0); (ł,(-,̨-ı-ȷ),180); (,̨(ȷ,-ı),0); (ł,(ȷ+,̨-ı),0); (ł,(ȷ,-ı+)̨,90); (ł,(ȷ,-ı-)̨,-90); (,̨(-ȷ,-ı),180); (ł,(-ȷ-,̨-ı),180); (ł,(-ȷ,-ı+)̨,90); (ł,(-ȷ,-ı-)̨,-90);[scale = 0.4] ı3 ȷ1.5 0̨.̨7̨5̨ ł0.375(4, -3) edge[|->, bend right = 60] node[below] a b^-1·φ(180)() (20, -3);[dash pattern = on 1pt off 1pt] (0, 0) – node[auto] a (ı, 0);[dash pattern = on 1pt off 1pt on off 1pt] (0, 0) – node[auto] a^-1 (-ı, 0);[densely dotted] (0, 0) – node[auto] b (0, ı);(0, 0) – node[auto] b^-1 (0, -ı);(4, 3) edge[|->, bend left] node[above] φ(90) (8, 3);[shift = (12, 0)](0, 0) – node[auto] a (ı, 0);[densely dotted] (0, 0) – node[auto] a^-1 (-ı, 0);[dash pattern = on 1pt off 1pt] (0, 0) – node[auto] b (0, ı);[dash pattern = on 1pt off 1pt on off 1pt] (0, 0) – node[auto] b^-1 (0, -ı);[rotate = 90] (16, 3) edge[|->, bend left] node[above] a ·φ(90)(a^-1·) (20, 3); [shift = (24, 0)] (0, 0) – node[auto] a (ı, 0);(ȷ,(ı,0),0); (,̨(ı+ȷ,0),0); (ł,(ı+ȷ+,̨0),0); (ł,(ı+ȷ,)̨,90); (ł,(ı+ȷ,-)̨,-90); (,̨(ı,ȷ),90); (ł,(ı,ȷ+)̨,90); (ł,(ı+,̨ȷ),0); (ł,(ı-,̨ȷ),180); [shift = (ı, -ȷ)] [densely dotted] (0, 0) – (0, -)̨ (0, -)̨ – (0, --̨ ł) (0, -)̨ – (ł, -)̨ (0, -)̨ – (-ł, -)̨; [dash pattern=on 1pt off 1pt on off 1pt](0, 0) – (,̨ 0) (,̨ 0) – (+̨ ł, 0) (,̨ 0) – (,̨ ł) (,̨ 0) – (,̨ -ł); [dash pattern = on 1pt off 1pt] (0, 0) – (-,̨ 0) (-,̨ 0) – (--̨ ł, 0) (-,̨ 0) – (-,̨ ł) (-,̨ 0) – (-,̨ -ł); (0, 0) – node[auto] a^-1 (-ı, 0);(ȷ,(-ı,0),180); (,̨(-ı-ȷ,0),180); (ł,(-ı-ȷ-,̨0),180); (ł,(-ı-ȷ,)̨,90); (ł,(-ı-ȷ,-)̨,-90); (,̨(-ı,ȷ),90); (ł,(-ı,ȷ+)̨,90); (ł,(-ı+,̨ȷ),0); (ł,(-ı-,̨ȷ),180); (,̨(-ı,-ȷ),-90); (ł,(-ı,-ȷ-)̨,-90); (ł,(-ı+,̨-ȷ),0); (ł,(-ı-,̨-ȷ),180);(0, 0) – node[auto] b (0, ı);(ȷ,(0,ı),90); (,̨(0,ı+ȷ),90); (ł,(0,ı+ȷ+)̨,90); (ł,(,̨ı+ȷ),0); (ł,(-,̨ı+ȷ),180); (,̨(ȷ,ı),0); (ł,(ȷ+,̨ı),0); (ł,(ȷ,ı+)̨,90); (ł,(ȷ,ı-)̨,-90); (,̨(-ȷ,ı),180); (ł,(-ȷ-,̨ı),180); (ł,(-ȷ,ı+)̨,90); (ł,(-ȷ,ı-)̨,-90);(0, 0) – node[auto] b^-1 (0, -ı);(ȷ,(0,-ı),-90); (,̨(0,-ı-ȷ),-90); (ł,(0,-ı-ȷ-)̨,-90); (ł,(,̨-ı-ȷ),0); (ł,(-,̨-ı-ȷ),180); (,̨(ȷ,-ı),0); (ł,(ȷ+,̨-ı),0); (ł,(ȷ,-ı+)̨,90); (ł,(ȷ,-ı-)̨,-90); (,̨(-ȷ,-ı),180); (ł,(-ȷ-,̨-ı),180); (ł,(-ȷ,-ı+)̨,90); (ł,(-ȷ,-ı-)̨,-90);[ρ = 1] [scale = 0.4] ı3 ȷ1.5 0̨.̨7̨5̨ ł0.375 ł ; [ρ = 2] [scale = 0.4] ı3 ȷ1.5 0̨.̨7̨5̨ ł0.375 0.1875 ; [ρ = 3] [scale = 0.4] ı3 ȷ1.5 0̨.̨7̨5̨ ł0.375 0.09375 ; [] [scale = 0.65] ı3 ȷ1 0̨.̨3̨3̨3̨ ł0.166 ł (solid); ((-ı,0),1); [anchor = north east] at (-ı- ȷ, -ȷ) A_e_F_2;((0,0),3.25); [anchor = base] at (ı, -ı) A_a;[] [scale = 0.65] ı3 ȷ1 0̨.̨3̨3̨3̨ ł0.166 ł[every path/.style=draw=white] ((0,0),3.25); (solid); ((0,-ı),1); [anchor = north east] at (-ȷ, -ı- ȷ) B_e_F_2;((0,0),3,-1.4,,/6); [anchor = base] at (ı, ı) B_b;[] [scale = 0.65] ı3 ȷ1 0̨.̨3̨3̨3̨ ł0.166 ł (solid); ((-ı,0),1); [anchor = north east] at (-ı- ȷ, -ȷ) A_e_F_2· e_F_2;((ı,0),1); [anchor = north west] at (ı+ ȷ, -ȷ) A_a · a;((0,-ı),1); [anchor = north east] at (-ȷ, -ı- ȷ) B_e_F_2· e_F_2;((0,ı),1,-ı-ı-ȷ--̨ł,4*,/3); [anchor = south west] at (ȷ, ı+ ȷ) B_b · b; 58 (0, 0) rectangle (, 8); (, 0) rectangle (+ , 8);(M) at (/2, 8.5) M; (M) at (+ /2, 8.5) M;[fill, circle, inner sep = 1pt, label = left: m] (m) at (/2 + 0.625, 5.75) ;[draw, circle, inner sep = 15pt, label = above: F] (F) at (/2, 5.75) ; [draw, circle, inner sep = 15pt, label = above: F] (Fx) at (+ /2, 5.75) ;[draw, circle, inner sep = 27pt, label = above: m ∈ M(mE) ∩ F ≠∅] (Nl) at (/2, 2) ;[inner sep = 51pt, label = above: =] (Nl) at (/2, 2) ;[inner sep = 60pt, label = above: 𝒩_l(F')] (Nl) at (/2, 2) ; [fill, circle, inner sep = 0.25pt] (mxx) at (+ /2 + 0.625, 5.75) ; [draw, circle, inner sep = 12pt, dashed] (m lib N) at (+ /2 + 0.625, 5.75) ;[fill, circle, inner sep = 1pt] (m lib n) at (+ /2 + 1, 5.5) ;[draw, circle, inner sep = 27pt, label = above: 𝒩_r(F) = FE] (Nr) at (+ /2, 5.75) ; [fill, circle, inner sep = 1pt, label = right: m'] (m') at (+ /2 - 0.5, 2) ; [draw, circle, inner sep = 15pt, label = above: F'] (F') at (+ /2, 2) ; [draw, circle, inner sep = 15pt, label = above: F'] (F'x) at (/2, 2) ; [fill, circle, inner sep = 0.25pt] (m'xx) at (/2 - 0.5, 2) ; [draw, circle, inner sep = 12pt, densely dotted] (m lib N) at (/2 - 0.5, 2) ;[fill, circle, inner sep = 1pt] (lib n inv m') at (/2 - 0.875, 1.75) ; [draw, circle, inner sep = 12pt, dashed] (lib n inv m' lib N) at (/2 - 0.875, 1.75) ;[->] (m) edge node[above] e (m lib n) (lib n inv m') edge node[above] e' (m');47 in 0(, 0) rectangle (+ , 8); in 1, 2, ..., 7(, ) – (+ , ); in (, 0) rectangle (+ , 8); in 2, 4, 6(, ) – (+ , ); in + 0.125[gray, dashed] (, -0.125) rectangle (+ , 8 - 0.125); in 1.875, 3.875, 5.875[gray, dashed] (, ) – (+ , ); [very thick] (0, 4) – (, 4);(M) at (/ 2, 8.5) M; (M) at (+ / 2, 8.5) M; (psi M) at (-1, 6) ψ(M); (psi' M) at (-1, 2) ψ'(M);[fill, circle, inner sep = 1pt] (psi m) at (- 0.5, 6.5) ; (psi A n) at (1.45, 6.5) ψ(A_e) = A_ee; [fill, circle, inner sep = 1pt] (psi' m) at (- 0.5, 2.5) ; (psi' B n') at (1.65, 2.5) ψ'(B_e') = B_e' e'; [fill, circle, inner sep = 1pt, label = right: m] (m) at (+ 0.5, 5) ;(A n) at (+ / 2, 5) A_e; [gray] (B n') at (+ /2 + 1.25, 4.5) B_e';[->] (psi m) edge[out = -60, in = -180] node[above] ϕ (m) (m) edge[out = 130, in = 10] node[above] ψ (psi m) (m) edge[out = 70, in = 70] node[above] e (psi m) (psi' m) edge[out = 60, in = -180] node[below] ϕ (m) (m) edge[out = -130, in = -10] node[below] ψ' (psi' m) (m) edge[out = -70, in = -70] node[below] e' (psi' m); [scale = 0.7] ı3 ȷ1 0̨.̨3̨3̨3̨ ł0.166ł (solid); ((-ı,0),1); [anchor = north east] at (-ı- ȷ, -ȷ) A^-;((ı,0),1); [anchor = north west] at (ı+ ȷ, -ȷ) A^+;((0,-ı),1); [anchor = north east] at (-ȷ, -ı- ȷ) B^-;((0,ı),1,-ı-ı-ȷ--̨ł,4*,/3); [anchor = south west] at (ȷ, ı+ ȷ) B^+; (4, 3) edge[|->, bend left] node[above] ϕ (8, 3);[shift = (12, 0)] (solid);((-ı,0),1);((0,0),3.333); [anchor = north west] at (ı+ ȷ+ +̨ ł, -ȷ- -̨ ł) A^+· a^-1;((0,-ı),1);((0,0),3,-1.4,,/6); [anchor = south east] at (-ı- ȷ, ȷ+ +̨ ł) B^+· b^-1; [] [scale = 0.35] ı3 ȷ1 0̨.̨3̨3̨3̨ ł0.166ł (solid); ((-ı,0),1);((ı,0),1); (8, 3) edge[|->, bend right] node[above] ψ (4, 3); (4, -3) edge[|->, bend right] node[below] ϕ (8, -3);[shift = (12, 0)] (solid);((-ı,0),1); ((0,0),3.25); [] [scale = 0.35] ı3 ȷ1 0̨.̨3̨3̨3̨ ł0.166ł (solid);((0,-ı),1);((0,ı),1,-ı-ı-ȷ--̨ł,4*,/3);(8, 3) edge[|->, bend right] node[above] ψ' (4, 3); (4, -3) edge[|->, bend right] node[below] ϕ (8, -3);[shift = (12, 0)] (solid);((0,-ı),1); ((0,0),3,-1.4,,/6); [scale = 0.7] ı3 ȷ1 0̨.̨3̨3̨3̨ ł0.166ł (solid);((-ı,0),1);[anchor = north east] at (-ı- ȷ, -ȷ) A^-· e_F_2;((-ȷ,ı),0.333);[anchor = east] at (-ȷ- -̨ ł- ł, ı) A^-· b; ((ı+ȷ,0),0.333);[anchor = north west] at (ı+ ȷ+ ,̨ -)̨ A^+· a;((ȷ,ı),0.333);[anchor = west] at (ȷ+ +̨ ł+ ł, ı) A^+· b; ((0,-ı),1);[anchor = north east] at (-ȷ, -ı- ȷ) B^-· e_F_2;((ı,-ȷ),0.333);[anchor = north west] at (ı+ ,̨ -ȷ- )̨ B^-· a; ((ı,ȷ),0.333,-ı-ı-ȷ--̨ł,4*,/3);[anchor = south west] at (ı+ ,̨ ȷ+ )̨ B^+· a;((0,ı+ȷ),0.333,-ı-ȷ--̨ł-ı-ȷ--̨ł-ı-ȷ--̨ł-ı-ȷ--̨ł-ı-ȷ--̨ł-ı-ȷ,14*,/3);[anchor = south west] at (,̨ ı+ ȷ+ )̨ B^+· b; (8, 3) edge[|->, bend right] node[above] · e_F_2, · a, or · b (4, 3); (4, -3) edge[|->, bend right] node[below] ϕ (8, -3);[shift = (12, 0)] (solid); ((-ı,0),1); [anchor = north east] at (-ı- ȷ, -ȷ) A^-;((ı,0),1); [anchor = north west] at (ı+ ȷ, -ȷ) A^+;((0,-ı),1); [anchor = north east] at (-ȷ, -ı- ȷ) B^-;((0,ı),1,-ı-ı-ȷ--̨ł,4*,/3); [anchor = south west] at (ȷ, ı+ ȷ) B^+; [A^-(1,0)] [scale = 0.4] ((-4,-4),(4,4)); ((0,0),2,solid);((-1,0),2,dashed); ((-1,0),2);[A^-(0,0), (1,0)] [scale = 0.4] ((-4,-4),(4,4)); ((0,0),2,solid);((-1,0),2,dashed);((-1/2,0),1.5);[A^-(-1,0), (0,0), (1,0)] [scale = 0.4] ((-4,-4),(4,4)); ((0,0),2,solid);((-1,0),2,dashed); ((1,0),2,dashed); ((0,0),1);[A^-E] [scale = 0.4] ((-4,-4),(4,4)); ((0,0),2,solid);((-1,0),2,dashdotted); ((1,0),2,dashdotted); ((0,1),2,densely dotted); ((0,-1),2,densely dotted); ((0,0),1); [A^+(1,0)] [scale = 0.4] ((-4,-4),(4,4)); ((0,0),2,solid);((-1,0),2,dashed); ((-1,0),2);[A^+(0,0), (1,0)] [scale = 0.4] ((-4,-4),(4,4)); ((0,0),2,solid);((-1,0),2,dashed); ((0,0),2); ((-1,0),2);[A^+(-1,0), (0,0), (1,0)] [scale = 0.4] ((-4,-4),(4,4)); ((0,0),2,solid);((-1,0),2,dashed); ((1,0),2,dashed); ((0,0),2);((-1,0),2); ((1,0),2);[A^+E] [scale = 0.4] ((-4,-4),(4,4)); ((0,0),2,solid);((-1,0),2,dashdotted); ((1,0),2,dashdotted); ((0,1),2,densely dotted); ((0,-1),2,densely dotted); ((-1,0),2); ((1,0),2); ((0,1),2); ((0,-1),2); [_(1,0) A] [scale = 0.4] ((-4,-4),(4,4)); ((0,0),2,solid); [_(0,0), (1,0) A] [scale = 0.4] ((-4,-4),(4,4)); ((0,0),2,solid);((-1/2,0),2.5); [_(-1,0), (0,0), (1,0) A] [scale = 0.4] ((-4,-4),(4,4)); ((0,0),2,solid);((0,0),2);((-1,0),2); ((1,0),2);[_E A] [scale = 0.4] ((-4,-4),(4,4)); ((0,0),2,solid);((0,0),2); ((0,0),3);[A^-a] [scale = 0.4] ı3 ȷ1.5 0̨.̨7̨5̨ ł0.375(densely dotted);(0 - ı, 0) circle (0.1cm); (0, 0) circle (0.1cm); (0 - ȷ, 0 + ı) circle (0.1cm); (0 - ı- ȷ, 0) circle (0.1cm); (0 - ȷ, 0 - ı) circle (0.1cm); (0 + ı, 0) circle (0.1cm); (0 + ı- ,̨ 0 + ȷ) circle (0.1cm); (0 + ı- ,̨ 0 - ȷ) circle (0.1cm); (0, 0 + ı) circle (0.1cm); (0 - ,̨ 0 + ı+ ȷ) circle (0.1cm); (0 - ȷ- ,̨ 0 + ı) circle (0.1cm); (0 - ı- ,̨ 0 + ȷ) circle (0.1cm); (0 - ı- ȷ- ,̨ 0) circle (0.1cm); (0 - ı- ,̨ 0 - ȷ) circle (0.1cm); (0, 0 - ı) circle (0.1cm); (0 - ȷ- ,̨ 0 - ı) circle (0.1cm); (0 - ,̨ 0 - ı- ȷ) circle (0.1cm); [A^-a, b] [scale = 0.4] ı3 ȷ1.5 0̨.̨7̨5̨ ł0.375(densely dotted); (0 - ı, 0) circle (0.1cm); (0, 0) circle (0.1cm); (0 + ı, 0) circle (0.1cm); (0, 0 + ı) circle (0.1cm); (0, 0 - ı) circle (0.1cm); [A^-a, b, a^-1, b^-1] [scale = 0.4] ı3 ȷ1.5 0̨.̨7̨5̨ ł0.375(densely dotted);(0 - ı, 0) circle (0.1cm); (0, 0) circle (0.1cm); (0 + ı, 0) circle (0.1cm); (0, 0 + ı) circle (0.1cm); (0, 0 - ı) circle (0.1cm); [A^+a] [scale = 0.4] ı3 ȷ1.5 0̨.̨7̨5̨ ł0.375(densely dotted);(0 - ı, 0) circle (0.1cm); (0, 0) circle (0.1cm); (0 - ȷ, 0 + ı) circle (0.1cm); (0 - ı- ȷ, 0) circle (0.1cm); (0 - ȷ, 0 - ı) circle 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((ı,0),90); ((ı,0),-90);[label = 60:e_F_2 a b^2] at (ı, ȷ+ )̨ ;ı3 ȷ1.5 0̨.̨7̨5̨ ł0.375 0.1875 0.09375(0, 0) edge[draw = none] node[auto] a (ı, 0); (0, 0) edge[draw = none] node[auto] b (0, ı); (0, 0) edge[draw = none] node[auto] a^-1 (-ı, 0); (0, 0) edge[draw = none] node[auto] b^-1 (0, -ı);(ı,ȷ,,̨(0,0),0); (ł,,,(ı+ȷ+,̨0),0); (ı,ȷ,,̨(0,0),180); (ł,,,(-ı-ȷ-,̨0),180); [shift = (ı+ ȷ+ ,̨ 0)] [densely dotted] (0, 0) – (0, ł) (0, 0) – (0, -ł); (0, ł) circle (0.666 * ł* 0.1cm); (0, -ł) circle (0.666 * ł* 0.1cm);[shift = (-ı- ȷ- ,̨ 0)] [densely dotted] (0, 0) – (0, ł) (0, 0) – (0, -ł); (0, ł) circle (0.666 * ł* 0.1cm); (0, -ł) circle (0.666 * ł* 0.1cm); [densely dotted](0, 0) – (0, ı) (0, 0) – (0, -ı); (0, -ı) circle (0.666 * ȷ* 0.1cm);(,̨ł,,(ı,ȷ),0); (,̨ł,,(ı,ȷ),180); [densely dotted] (ı, ȷ) – (ı, ȷ+ )̨; (ı, ȷ+ )̨ circle (0.666 * *̨ 0.1cm); (,̨ł,,(-ı,ȷ),0); (,̨ł,,(-ı,ȷ),180); [densely dotted] (-ı, ȷ) – (-ı, ȷ+ )̨; (-ı, ȷ+ )̨ circle (0.666 * *̨ 0.1cm);(ȷ,,̨ł,(0,ı),0); (ȷ,,̨ł,(0,ı),180); [densely dotted] (0, ı) – (0, ı+ ȷ); (0, ı+ ȷ) circle (0.666 * *̨ 0.1cm); [The circles E_m, drawn solid, about m ∈m_0, S∪m_1, m_2, m_3, m_4∪c_1, c_2, …, c_8.]< g r a p h i c s >[The boundaries of the curved circular disks E'_m, drawn dotted, about m ∈m_0, S∪m_1, m_2, m_3, m_4.]< g r a p h i c s > [The boundaries of the curved circular disks E'_m, drawn dotted, about m ∈m_0, S∪m_1, m_2, m_3, m_4∪c_1, c_2, …, c_8.]< g r a p h i c s >[The circles E_m, drawn solid, and the boundaries of the curved circular disks E'_m, drawn dotted, about m ∈m_0, S∪m_1, m_2, m_3, m_4∪c_1, c_2, …, c_8.]< g r a p h i c s > [> = To] (m) at (0,0) m; [below right = of m] (x) ∙; [above right = of x] (t) t;[->] (m) edge[bend left] node[right] e' (x); [->] (x) edge[bend left] node[left] (g')^-1 G_0 (m);[->] (t) edge[bend left] node[right] e (x); [->] (t) edge[bend right] node[above] g (g')^-1 G_0 (m);[c]0.2[circle dotted/.style = dash pattern = on .05mm off 1.5pt, line cap = round] 0.40.651 - 0.51 - 0.52.73 (, ) rectangle (+ , + + )(, + ) rectangle (+ , + - - )(, + + ) rectangle (+ + , + - - )(+ , + + ) rectangle (+ - - , + - - );[line width = 0.5pt, dashed, pattern = north east lines, pattern color = gray] (, ) rectangle (+ , + ); in 1, ..., 2 in 1, ..., 3[fill = white] (, ) circle (1.25pt); in 0, 3, 4 in 0, ..., 4(, ) circle (1.25pt); in 1, 2 in 0, 4(, ) circle (1.25pt); in 0, ..., 4in 0, ..., 4(- , - ) rectangle (+ , + );[gray, dashdotted] (- , - ) rectangle (+ , + );[line width = 0.5pt, dashed] (, ) rectangle (+ , + );[line width = 0.75pt, circle dotted] (+ , + ) rectangle (+ - , + - ); [line width = 0.75pt, circle dotted] (+ + , + + ) rectangle (+ - - , + - - );[c]0.8 The whole space is M; the dots and circles are the elements of the tiling T; for each element t ∈ T, the region enclosed by the rectangle with solid border centred at t is the set tE and the region enclosed by the rectangle with dash-dotted border centred at t is the set tE'; the region enclosed by the rectangle with dashed border is F_i; the region enclosed by the largest rectangle with dotted border is F_i^-E; the region enclosed by the smallest rectangle with dotted border is (F_i^-E)^-E', which, in this depiction, is equal to F_i^-E”, where E” = E' · (G_0 · E); the circles are the elements of T_i = T ∩ F_i^-E, which, in this depiction, is equal to T_i' = T ∩ ((F_i^-E)^-E')^+E'; the hatched region is _E”^- F_i = F_i ∖ F_i^-E”. [scale = 0.5] ı3 ȷ1.5 0̨.̨7̨5̨ ł0.375(solid);(0 + ı+ ȷ, 0) circle (0.1666cm); (0 + ı, 0 + ȷ) circle (0.1666cm); (0 + ı, 0 - ȷ) circle (0.1666cm); (0 + ȷ, 0 + ı) circle (0.1666cm); (0, 0 + ı+ ȷ) circle (0.1666cm); (0 - ȷ, 0 + ı) circle (0.1666cm); (0 - ı, 0 + ȷ) circle (0.1666cm); (0 - ı- ȷ, 0) circle (0.1666cm); (0 - ı, 0 - ȷ) circle (0.1666cm); (0 + ȷ, 0 - ı) circle (0.1666cm); (0 - ȷ, 0 - ı) circle (0.1666cm); (0, 0 - ı- ȷ) circle (0.1666cm);(4, 3) edge[|->, bend left] node[above] Δ (8, 3);[shift = (12, 0)] (solid);(0 + ı, 0) circle (0.1666cm); (0, 0 + ı) circle (0.1666cm); (0 - ı, 0) circle (0.1666cm); (0, 0 - ı) circle (0.1666cm);[c]0.5[circle dotted/.style = dash pattern = on .05mm off 1.5pt, line cap = round] 0.40.71 - 0.51 - 0.52.73 in 0, ..., 4in 0, ..., 4[fill] (, ) circle (1pt);(- , - ) rectangle (+ , + );[dashed, pattern = north east lines] (, ) rectangle (+ , + );in 1, ..., 2in 1, ..., 3[fill = white] (- , - ) rectangle (+ , + );(, ) circle (1pt);[line width = 0.75pt, circle dotted] (+ , + ) rectangle (+ - , + - );[c]0.5 The whole space is M; the dots and circles are the elements of the tiling T; for each element t ∈ T, the region enclosed by the rectangle with solid border centred at t is the set tE; the region enclosed by the rectangle with dashed border is F_i; the region enclosed by the rectangle with dotted border is F_i^-E; the circles are the elements of T_i = T ∩ F_i^-E; the hatched region is F_i^* = F_i ∖ (⋃_t ∈ T_i tE).[scale = 0.5] ı3 ȷ1.5 0̨.̨7̨5̨ ł0.375 0.1875 0.09375 40i̊n 0, 120, 240[rotate = ]̊ (0, 0) – (ı, 0); i̊n , -[shift = (ı, 0), rotate = ]̊ (0, 0) – (ȷ, 0); i̊n , -[shift = (ȷ, 0), rotate = ]̊ (0, 0) – (,̨ 0); i̊n , -[shift = (,̨ 0), rotate = ]̊ (0, 0) – (ł, 0); i̊n , -[shift = (ł, 0), rotate = ]̊ (0, 0) – (, 0); i̊n , -[shift = (, 0), rotate = ]̊ (0, 0) – (, 0); (0, 0) circle (0.125); [below right = 0 and -0.1] at (0, 0) e_G; (ı, 0) circle (0.1); [above] at (ı, 0) g; [shift = (ı, 0), rotate = ] (ȷ, 0) circle (0.075); [below right = 0 and -0.2] at (ȷ, 0) g s;[shift = (ı, 0), rotate = -] (ȷ, 0) circle (0.075); [above right = 0 and -0.2] at (ȷ, 0) g s';[> = To] (m) at (0,0) m; [right = of m] (x) ∙; [below = of m] (y) ∙; [draw, shape = circle, minimum size = 2cm, above right = -0.7cm of x] (A) A;[->] (m) edge node[above] e' (x); [->] (m) edge node[left] e (y); [->] (y) edge[bend right] node[right] g^-1· e' (x);[> = To, x=1cm, y=1cm, circle dotted/.style = dash pattern = on .05mm off 1.5pt, line cap = round] 0.51̨5̧0̊.̊0̊6̊2̊5̊ [fill = black] (0,0) circle[radius = ]̊ node ;(0,0) circle[radius = 1 * ((2 * ) + +̨ 1)] node ;(0,0) circle[radius = 2 * ((2 * ) + +̨ 1)] node ; in 1,2,...,[fill = black] (* 360 / :̧ 1 * ((2 * ) + +̨ 1)) circle[radius = ]̊ node[below] m_i,; [circle dotted] (* 360 / :̧ 1 * ((2 * ) + +̨ 1)) circle[radius = ]; [dashdotted] (* 360 / :̧ 1 * ((2 * ) + +̨ 1)) circle[radius = + ]; [dashed] (* 360 / :̧ 1 * ((2 * ) + +̨ 1)) circle[radius = (2 * ) + ];[fill = black] (1.75 * 360 / :̧ 1.75 * ((2 * ) + +̨ 1)) circle[radius = ]̊ node[above] m –(1.75 * 360 / :̧ 1 * ((2 * ) + +̨ 1)) circle[radius = ]̊ node[below] m' –(2 * 360 / :̧ 1 * ((2 * ) + +̨ 1)) node[below] m_i, 2= m”;[c]0.5[circle dotted/.style = dash pattern = on .05mm off 1.5pt, line cap = round] 0.30̨.̨1̨1 - 0.51 - 0.52.73 in 0, ..., 4in 0, ..., 4[fill] (, ) circle (1pt);(- , - ) rectangle (+ , + );[dashdotted] (- - ,̨ - - )̨ rectangle (+ + ,̨ + + )̨;[dashed, pattern = north east lines] (, ) rectangle (+ , + );in 1, ..., 2in 1, ..., 3[fill = white] (- , - ) rectangle (+ , + );(, ) circle (1pt);[line width = 0.75pt, circle dotted] (+ + ,̨ + + )̨ rectangle (+ - - ,̨ + - - )̨;[c]0.5The whole space is M; the dots and circles are the elements of the set T; for each element t ∈ T, the region enclosed by the rectangle with solid border about t is the set (t, θ) and the region enclosed by the rectangle with dash-dotted border about t is the set (t, θ)^+κ; the region enclosed by the rectangle with dashed border is F; the region enclosed by the rectangle with dotted border is F^-(θ + κ); the circles are the elements of S = T ∩ F^-(θ + κ); the hatched region is the set F ∖ (⋃_s ∈ S(s, θ)).[c]/ 2[circle dotted/.style = dash pattern = on .05mm off 1.5pt, line cap = round] 0.40.71 - 0.551 - 0.552.83.1 in 1, ..., 2in 1, ..., 3(, ) circle (1pt); in 0, 3, 4in 0, ..., 4(, ) circle (1pt); in 1, 2in 0, 4(, ) circle (1pt); in 0, ..., 4in 0, ..., 4(- , - ) rectangle (+ , + );[gray, dashdotted] (- , - ) rectangle (+ , + ); [line width = 0.5pt, dashed] (, ) rectangle (+ , + );[line width = 0.75pt, circle dotted] (+ , + ) rectangle (+ - , + - );[line width = 0.75pt, circle dotted] (+ + , + + ) rectangle (+ - - , + - - );[c]/ 2 The whole space is M; the dots and circles are the elements of the set T; for each element t ∈ T, the region enclosed by the rectangle with solid border about t is the set (t, θ)^+κ and the region enclosed by the rectangle with dash-dotted border about t is the set (t, θ'); the region enclosed by the rectangle with dashed border is F; the region enclosed by the smallest rectangle with dotted border is F^-(θ + κ + θ') and the region enclosed by the largest rectangle with dotted border is F^-(θ + κ); the circles are the elements of S = T ∩ F^-(θ + κ). edge/.style = draw = black, vertex/.style = draw = black, thick, midpoint/.style = draw = black, thick, boundary/.style = draw = black, thick, find/.style = draw = black, dotted, very thick,reflected/.style = draw = black, densely dotted, thick, syncLeft/.style = pattern = horizontal lines, pattern color = black, draw = none,syncMiddle/.style = pattern = fivepointed stars, pattern color = black, draw = none, syncRight/.style = pattern = dots, pattern color = black, draw = none,slowed/.style = draw = black,freeze/.style = draw = black, dashed,traversingThaw/.style = draw = black, loosely dotted, thick, thawingThaw/.style = draw = black, dashed, divide0/.style = draw = black, solid, divideN/.style = draw = black, densely dotted, reflectedDivide/.style = draw = black, leftColour/.style = draw = mygreen, pattern color = mygreen, leftFill/.style = color = mygreen, rightColour/.style = draw = myblue, pattern color = myblue, rightFill/.style = color = myblue, overlayLeftColour/.style = draw = myred, pattern color = myred, overlayLeftFill/.style = color = myred, overlayRightColour/.style = draw = myviolet, pattern color = myviolet, overlayRightFill/.style = color = myviolet[A singularity of order 1.] (-1em)/2! [> = To, x = 1cm, y = -1cm] ł3ł+ ł (gg) at (0, 0);(dr) at (ł, ł);(rdl) at (0, 2 * ł); [divide0] (gg) – (dr); [reflectedDivide] (dr) – (rdl);in 1, 2, ..., 15((2/3)^) * ł; [divideN] (gg) – (,̌ 2 * ł- ); [vertex] (0, 0) – (0, 6) (3, 0) – (3, 6); [edge] (0, 0) – (3, 0) (0, 6) – (3, 6); [A singularity of order -1.] (-1em)/2! [> = To, x = 1cm, y = -1cm] ł3ł+ ł (gg) at (0, 0);(dr) at (ł, ł);(rdl) at (0, 2 * ł); [divide0] (0, 0) – (-1, 1); [reflectedDivide] (-1, 1) – (2, 4); (0, 0) – (0, 2) – (2, 4) – (2, 0) – cycle; in 1, 2, ..., 15((2/3)^) * ł; [divideN] (gg) – (,̌ 2 * ł- );[vertex] (-1, 0) – (-1, 4) (0, 0) – (0, 4) (2, 0) – (2, 4); [edge] (-1, 0) – (2, 0) (-1, 4) – (2, 4);[> = To, x = 1cm, y = -1cm] ł32 * ł [find] (0, 0) – (ł, ł); [reflected] (ł, ł) – (0, ); [slowed] (0, 0) – ((1/3) * , ); [midpoint] (0.5 * ł, 1.5 * ł) – (0.5 * ł, );[vertex] (0, 0) – (0, );[vertex] (ł, 0) – (ł, ); [edge] (0, 0) – (ł, 0);[edge] (0, ) – (ł, );[> = To, x = 1cm, y = -1cm]ł32ł+ 2 * ł+ [shift = (ł, -1.5), scale = 0.5] (0, 0) – (-2.1213, 2.1213); (0, 0) – (1.414, 1.414); (g) at (ł, 0);[find] (g) – (0, ł); [reflected] (0, ł) – (ł, 2 * ł); [slowed] (ł, 2 * ł) – (ł+ (1/3) * , ); [find] (g) – (, )̊; [reflected] (, )̊ – (ł, 2 * )̊; [slowed] (ł, 2 * )̊ – (0.5 * - (1/3) * 0.5 * , ); [midpoint] (0.5 * , - 0.5 * ) – (0.5 * , ); [vertex] (0, 0) – (0, ); [vertex] (g) – (ł, ); [vertex] (, 0) – (, ); [edge] (0, 0) – (g);[edge] (g) – (, 0); [edge] (0, ) – (ł, );[edge] (ł, ) – (, ); (#1,#2) ł#1ł+ ł (gg) at (0, 0);(dr) at (ł, ł);(rdl) at (0, 2 * ł); [divide0] (gg) – (dr); [reflectedDivide] (dr) – (rdl);in 1, 2, ..., 15((2/3)^) * ł; (b #2 ) at (,̌ 2 * ł- );(be #2 ) at (,̌ );[divideN] (gg) – (b #2 ); [boundary] (b #2 ) – (be #2 );(#1) ≤#1(≤,0);in 1, 2, ..., 15[shift = (b 0 )] - 1; ((1/3) * (2/3)^* ≤,);[> = To, x = 1cm, y = -1cm] (3); [vertex] (0, 0) – (0, 6) (3, 0) – (3, 6); [edge] (0, 0) – (3, 0) (0, 6) – (3, 6); [> = To, x = 1cm, y = -1cm] ł32ł+ ł+ ł+ [shift = (ł, -1.5), scale = 0.5] (0, 0) – (-2.1213, 2.1213); (0, 0) – (1.414, 1.414); (g) at (ł, 0);(lb) at (0, ł);(rb) at (, )̊;(lg) at (ł, 2 * ł);(rg) at (ł, 2 * )̊;(m) at (0.5 * , ł+ 0.5 * );(mlf) at (0.5 * ł, ł+ 0.5 * ł);(mlt) at (0.5 * ł, - 0.5 * ł);(mrf) at (ł+ 0.5 * ,̊ +̊ 0.5 * )̊;(mrt) at (ł+ 0.5 * ,̊ - 0.5 * )̊; (lfb) at ((mlf) + (- 0.5 * ł, 0.5 * ł));(rfb) at ((mrf) + (0.5 * ,̊ 0.5 * )̊);(lfg) at ((mlf) + (0.5 * ł, 0.5 * ł));(rfg) at ((mrf) + (- 0.5 * ,̊ 0.5 * )̊); (0, 0) – (0, 2 * ł) – (0.5 * ł, 1.5 * ł) – (ł, 2 * ł) – (ł, 0) – cycle; [xscale = -1, shift = (-ł, 0)] (ł); [freeze] (0.5 * ł, 1.5 * ł) – (0, 2 * ł) (0.5 * ł, 1.5 * ł) – (ł, 2 * ł);[shift = (0, - 2 * ł)] (0, 2 * ł) – (0.5 * ł, 1.5 * ł) – (ł, 2 * ł) – cycle; [xscale = -1, shift = (-ł, 0)] (ł); [thawingThaw] (0.5 * ł, 1.5 * ł) – (0, 2 * ł) (0.5 * ł, 1.5 * ł) – (ł, 2 * ł);[shift = (ł, 0)] (0, 0) – (0, 2 * )̊ – (0.5 * ,̊ 1.5 * )̊ – (,̊ 2 * )̊ – (,̊ 0) – cycle; ()̊;[freeze] (0.5 * ,̊ 1.5 * )̊ – (0, 2 * )̊ (0.5 * ,̊ 1.5 * )̊ – (,̊ 2 * )̊;[shift = (0, - 2 * )̊](0, 2 * )̊ – (0.5 * ,̊ 1.5 * )̊ – (,̊ 2 * )̊ – cycle; ()̊;[thawingThaw] (0.5 * ,̊ 1.5 * )̊ – (0, 2 * )̊ (0.5 * ,̊ 1.5 * )̊ – (,̊ 2 * )̊;[slowed] (g) – (mlf);[slowed] (g) – (mrf);[slowed] (ł, +̊ )̊ – (0.5 * , +̊ +̊ 1.5 * (ł- )̊); [white, semithick] (mlf) – (lfg) (mrf) – (rfg);[freeze] (mlf) – (lfg) (mlf) – (lfb) (mrf) – (rfb) (mrf) – (rfg); [traversingThaw] (m) – (mlt) (m) – (mrt); [white, semithick] (0, ) – (0.5 * ł, - 0.5 * ł) – (ł, ); [white, semithick] (, ) – (ł+ 0.5 * ,̊ - 0.5 * )̊ – (- ,̊ );[thawingThaw] (0.5 * ł, - 0.5 * ł) – (0, )(0.5 * ł, - 0.5 * ł) – (ł, ); [thawingThaw] (ł+ 0.5 * ,̊ - 0.5 * )̊ – (, )(ł+ 0.5 * ,̊ - 0.5 * )̊ – (- ,̊ ); [midpoint] (mlf) – (mlt) (mrf) – (mrt); [vertex] (0, 0) – (0, ) (g) – (ł, ) (, 0) – (, ); [edge] (0, 0) – (g) – (, 0) (0, ) – (ł, ) – (, ); [> = To, x = 1cm, y = -1cm]ł32ø1.333max(ł, )̊min(ł, )̊ł+ + + [shift = (ł, -1.5), scale = 0.5] (0, 0) – (-2.1213, 2.1213); (0, 0) – (1.414, 1.414); (g) at (ł, 0);(lb) at (0, ł);(rb) at (, )̊;(lg) at (ł, 2 * ł);(rg) at (ł, 2 * )̊;(m) at (0.5 * , + 0.5 * );(mlf) at (0.5 * ł, ł+ 0.5 * ł);(mlt) at (0.5 * ł, - 0.5 * ł);(mrf) at (ł+ 0.5 * ,̊ +̊ 0.5 * )̊;(mrt) at (ł+ 0.5 * ,̊ - 0.5 * )̊; (lfb) at ((mlf) + (- 0.5 * ł, 0.5 * ł));(rfb) at ((mrf) + (0.5 * ,̊ 0.5 * )̊);(lfg) at ((mlf) + (0.5 * ł, 0.5 * ł));(rfg) at ((mrf) + (- 0.5 * ,̊ 0.5 * )̊);[syncLeft] (g) – (lb) – (lfb) – (mlf) – (lg) – cycle; [syncRight] (g) – (rb) – (rfb) – (mrf) – (rg) – cycle; [syncLeft, thawingThaw] (0, ) – (0.5 * ł, - 0.5 * ł) – (ł, ); [syncRight, thawingThaw] (, ) – (ł+ 0.5 * ,̊ - 0.5 * )̊– (- ,̊ ); [find] (g) – (lb); [reflected] (lb) – (mlf); [find] (g) – (rb); [reflected] (rb) – (, + ); [slowed] (, + ) –(0.5 * , + + 1.5 * (- )); [slowed] (g) – (mlf);[slowed] (g) – (mrf);[midpoint] (mlf) – (mlt) (mrf) – (mrt); [traversingThaw] (m) – (mlt) (m) – (mrt); [freeze] (mlf) – (lfg) (mlf) – (lfb) (mrf) – (rfb) (mrf) – (rfg); [vertex] (0, 0) – (0, ) (g) – (ł, ) (, 0) – (, ); [edge] (0, 0) – (g) – (, 0) (0, ) – (ł, ) – (, ); [> = To, x = 1cm, y = -1cm]ł32ø1.333ł+ 2 *(g) at (0, 0);(v) at (ł, 0);(iv) at (ł, ł);(lb) at (ł, ł);(rb) at (, ł+ )̊;(lg) at (0, 2 * ł);(rv) at (ł, ł+ 2 * )̊;(m) at (0.5 * , + 0.5 * );(mlf) at (0.5 * ł, ł+ 0.5 * ł);(mlt) at (0.5 * ł, - 0.5 * ł);(mrf) at (ł+ 0.5 * ,̊ ł+ +̊ 0.5 * )̊;(mrt) at (ł+ 0.5 * ,̊ - 0.5 * )̊; (lfv) at ((mlf) + (0.5 * ł, 0.5 * ł));(rfb) at ((mrf) + (0.5 * ,̊ 0.5 * )̊);(lfg) at ((mlf) + (- 0.5 * ł, 0.5 * ł));(rfg) at ((mrf) + (- 0.5 * ,̊ 0.5 * )̊);[syncLeft] (g) – (lb) – (lfv) – (mlf) – (lg) – cycle; [syncRight] (iv) – (rb) – (rfb) – (mrf) – (rv) – cycle; [syncLeft, thawingThaw] (0, ) – (0.5 * ł, - 0.5 * ł) – (ł, ); [syncRight, thawingThaw] (, ) – (ł+ 0.5 * ,̊ - 0.5 * )̊– (- ,̊ ); [find] (g) – (lb); [reflected] (lb) – (mlf);[find] (iv) – (rb); [reflected] (rb) – (rv) – (m); [traversingThaw] (m) – (mlt); [traversingThaw] (m) – (mrt); [midpoint] (mlf) – (mlt); [midpoint] (mrf) – (mrt); [slowed] (g) – (mlf) – (/2, - /2);[slowed] (iv) – (mrf); [freeze] (mlf) – (lfg); [freeze] (mlf) – (lfv); [freeze] (mrf) – (rfb); [freeze] (mrf) – (rfg); [vertex] (v) – (ł, ); [vertex] (0, 0) – (0, ); [vertex] (, 0) – (, ); [edge] (0, 0) – (v);[edge] (v) – (, 0); [edge] (0, ) – (ł, );[edge] (ł, ) – (, ); (#1,#2,#3) [shift = #1](0, 3pt) rectangle (-3pt, -3pt); [#2, draw = black] (0, 0) circle (2pt); (0, -3pt) rectangle (3pt, 3pt); [#3, draw = black] (0, 0) circle (2pt); (#1,#2,#3,#4) [shift = #1] [#3, draw = none] (-1pt, -2pt) rectangle (1pt, 2pt); [black] (-1pt, -2pt) – (1pt, -2pt)(-1pt, 2pt) – (1pt, 2pt); [shift = (-1pt,0)] (0, 3pt) rectangle (-3pt, -3pt); [#2, draw = black] (0, 0) circle (2pt);[shift = (1pt,0)] (0, -3pt) rectangle (3pt, 3pt); [#4, draw = black] (0, 0) circle (2pt);[> = To, x = 1cm, y = -1cm]ł132ł+ +̱ + +̱(g) at (ł, 0);(m) at (/2, - /2); (mf2) at ((ł+ )̱/2, +̱ (ł+ )̱/2);(ml2) at (ł+ (+̱ )̊/2, 1.5 * (+̱ )̊);(ml) at (ł/2, 1.5 * ł);(mr) at (ł+ +̱ /̊2, +̱ 1.5 * )̊;(mb) at (ł+ /̱2, 1.5 * )̱; (mf2e) at ((ł+ )̱/2, - (ł+ )̱/2);(ml2e) at (ł+ (+̱ )̊/2, - (+̱ )̊/2);(mle) at (ł/2, - ł/2);(mre) at (ł+ +̱ /̊2, - /̊2);(mbe) at (ł+ /̱2, - /̱2); [syncLeft] (g) – ((ml) + (ł/2, ł/2)) – (ml) – ((ml) + (-ł/2, ł/2)) – (0, ł) – cycle; [syncMiddle] (g) – (ł+ ,̱ )̱ – ((mb) + (/̱2, /̱2)) – (mb) – ((mb) + (-/̱2, /̱2)) – cycle; [syncRight] (ł+ ,̱ )̱ – (, +̱ )̊ – ((mr) + (/̊2, /̊2)) – (mr) – ((mr) + (-/̊2, /̊2)) – cycle; [find] (g) – (0, ł); [reflected] (0, ł) – (ł, 2 * ł); [slowed] (ł, 2 * ł) – (m);[midpoint] (mf2) – (mf2e);[find] (g) – (ł+ ,̱ )̱; [reflected] (ł+ ,̱ )̱ – (mf2);[find] (ł+ ,̱ )̱ – (, +̱ )̊; [reflected] (, +̱ )̊ – (m);[slowed] (ł+ ,̱ )̱ – (mr); [midpoint] (mr) – (mre); [slowed] (g) – (ł/2, 1.5 * ł); [midpoint] (ml) – (mle); [slowed] (g) – (mb) – (ml2); [midpoint] (mb) – (mbe); [midpoint] (ml2) – (ml2e); [freeze] (ml) – ((ml) + (ł/2, ł/2)) (ml) – ((ml) + (-ł/2, ł/2)) (mb) – ((mb) + (/̱2, /̱2)) (mb) – ((mb) + (-/̱2, /̱2)) (mr) – ((mr) + (/̊2, /̊2)) (mr) – ((mr) + (-/̊2, /̊2)); [traversingThaw] (m) – (mf2e) (m) – (ml2e) (mf2e) – (mle) (mf2e) – (mbe) (ml2e) – (mbe) (ml2e) – (mre); [syncLeft] (mle) – ((mle) + (ł/2, ł/2)) – ((mle) + (-ł/2, ł/2)); [thawingThaw] (mle) – ((mle) + (ł/2, ł/2))(mle) – ((mle) + (-ł/2, ł/2)); [syncMiddle] (mbe) – ((mbe) + (/̱2, /̱2)) – ((mbe) + (-/̱2, /̱2)); [thawingThaw] (mbe) – ((mbe) + (/̱2, /̱2))(mbe) – ((mbe) + (-/̱2, /̱2)); [syncRight] (mre) – ((mre) + (/̊2, /̊2)) – ((mre) + (-/̊2, /̊2)); [thawingThaw] (mre) – ((mre) + (/̊2, /̊2))(mre) – ((mre) + (-/̊2, /̊2));((m),leftFill,overlayLeftFill,rightFill); ((mf2),leftFill,overlayLeftFill); ((mf2e),leftFill,overlayLeftFill); ((mr),rightFill,rightFill); ((mre),rightFill,rightFill); ((ml),leftFill,leftFill); ((mle),leftFill,leftFill); ((mb),overlayLeftFill,overlayLeftFill); ((mbe),overlayLeftFill,overlayLeftFill); ((ml2),overlayLeftFill,rightFill); ((ml2e),overlayLeftFill,rightFill);[vertex] (0, 0) – (0, ) (ł, 0) – (ł, ) (ł+ ,̱ 0) – (ł+ ,̱ ) (, 0) – (, ); [edge, leftColour] (0, 0) – (ł, 0) (0, ) – (ł, ); [edge, overlayLeftColour] (ł, 0) – (ł+ ,̱ 0) (ł, ) – (ł+ ,̱ ); [edge, rightColour] (ł+ ,̱ 0) – (, 0)(ł+ ,̱ ) – (, ); [> = To, x = 1cm, y = -1cm] [leftColour] (0, 0) – (-0.707, 0.707); [overlayLeftColour] (0, 0) – (2.1213, 2.1213); [rightColour] (2.1213, 2.1213) – (2.1213, 4.1213); ł32ø1.333ł+ ł+ ł+ł32ø1.333ł+ [vertex] (v) – (ł, ); [vertex, leftColour] (0, 0) – (0, ); [vertex, rightColour] (, 0) – (, ); [vertex, overlayLeftColour] (ł- ø, 0) – (ł- ø, ); [vertex, overlayRightColour] (ł+ ø, 0) – (ł+ ø, ); [edge, leftColour] (0, 0) – (v)(0, ) – (ł, );[edge, rightColour] (v) – (, 0) (ł, ) – (, );[edge, overlayLeftColour] ((v) - (0, 0.4pt)) – ((v) + (-ø, 0) - (0, 0.4pt))((v) + (0, ) - (0, 0.4pt)) – ((v) + (-ø, ) - (0, 0.4pt)); [edge, overlayRightColour] ((v) - (0, 0.4pt)) – ((v) + (ø, 0) - (0, 0.4pt)) ((v) + (0, ) - (0, 0.4pt)) – ((v) + (ø, ) - (0, 0.4pt));[> = To, x = 1cm, y = -1cm]; ł+ ł+ [shift = (0.8, -0.8), scale = 0.2] [leftColour] (0, 0) – (-2.1213, 2.1213); [overlayLeftColour] (0, 0) – (0, 1.333); [overlayRightColour] ((0, 0) + (0.4pt, 0)) – ((0, 1.333) + (0.4pt, 0)); [rightColour] (0, 0) – (1.414, 1.414); [shift = (4.3, -0.8), scale = 0.2] [leftColour] (0, 0) – (-2.1213, 2.1213); [overlayLeftColour] (0, 0) – (0, -1.333); [overlayRightColour] ((0, 0) + (0.4pt, 0)) – ((0, -1.333) + (0.4pt, 0)); [rightColour] (0, 0) – (1.414, 1.414); (v) at (ł, 0);(lb) at (0, ł);(rb) at (, )̊;(obl) at (ł- ø, ø);(obr) at (ł+ ø, ø);(lg) at (ł, 2 * ł);(rg) at (ł, 2 * )̊;(og) at (ł, 2 * ø);(m) at (0.5 * , ł+ 0.5 * );(m1) at (0.5 * (ł+ ø), ł+ 0.5 * (ł+ ø));(m2) at (ł- ø+ 0.5 * (+̊ ø), +̊ 0.5 * (+̊ ø));(m1t) at (0.5 * (ł+ ø), - 0.5 * (ł+ ø));(m2t) at (ł- ø+ 0.5 * (+̊ ø), - 0.5 * (+̊ ø));(mlf) at (0.5 * ł, ł+ 0.5 * ł);(mlt) at (0.5 * ł, - 0.5 * ł);(mrf) at (ł+ 0.5 * ,̊ +̊ 0.5 * )̊;(mrt) at (ł+ 0.5 * ,̊ - 0.5 * )̊;(mofl) at (ł- 0.5 * ø, ø+ 0.5 * ø);(motl) at (ł- 0.5 * ø, - 0.5 * ø);(mofr) at (ł+ 0.5 * ø, ø+ 0.5 * ø);(motr) at (ł+ 0.5 * ø, - 0.5 * ø); (lfb) at ((mlf) + (- 0.5 * ł, 0.5 * ł));(rfb) at ((mrf) + (0.5 * ,̊ 0.5 * )̊);(ofbl) at ((mofl) + (- 0.5 * ø, 0.5 * ø));(ofbr) at ((mofr) + (0.5 * ø, 0.5 * ø));(lfg) at ((mlf) + (0.5 * ł, 0.5 * ł));(rfg) at ((mrf) + (- 0.5 * ,̊ 0.5 * )̊);(ofg) at ((mofr) + (- 0.5 * ø, 0.5 * ø));[syncLeft, leftColour, draw = none] (v) – (lb) – (lfb) – (mlf) – (lg) – cycle; [syncRight, rightColour, draw = none] (v) – (rb) – (rfb) – (mrf) – (rg) – cycle;[syncMiddle, overlayLeftColour, draw = none] (v) – (obl) – (ofbl) – (mofl) – (og) – cycle; [syncMiddle, overlayRightColour, draw = none] (v) – (obr) – (ofbr) – (mofr) – (og) – cycle; [find, leftColour] (v) – (lb); [reflected, leftColour] (lb) – (m);[find, rightColour] (v) – (rb); [reflected, rightColour] (rb) – (rg); [find, overlayLeftColour] (v) – (obl); [reflected, overlayLeftColour] (obl) – (og); [slowed, rightColour] (og) – (m2); [find, overlayRightColour] (v) – (obr); [reflected, overlayRightColour] (obr) – (og); [slowed, leftColour] (og) – (m1); [thawingThaw, syncLeft, leftColour] (0, ) – (0.5 * ł, - 0.5 * ł) – (ł, ); [thawingThaw, syncRight, rightColour] (, ) – (ł+ 0.5 * ,̊ - 0.5 * )̊– (- ,̊ ); [thawingThaw, syncMiddle, overlayLeftColour] (ł, ) – (ł- 0.5 * ø, - 0.5 * ø)– (ł- ø, ); [thawingThaw, syncMiddle, overlayRightColour] (ł, ) – (ł+ 0.5 * ø, - 0.5 * ø)– (ł+ ø, ); [traversingThaw, leftColour] (m) – (m1t); [traversingThaw, leftColour] (m1t) – (mlt); [traversingThaw, leftColour] (m1t) – ((m1t) + (ł - 0.5 * (ł + ø), ł - 0.5 * (ł + ø)));[traversingThaw, overlayRightColour] ((m1t) + (ł - 0.5 * (ł + ø), ł - 0.5 * (ł + ø))) – (motr); [traversingThaw, leftColour] (m) – ((m) + (ł - 0.5 * (ł + )̊, ł - 0.5 * (ł + )̊));[traversingThaw, rightColour] ((m) + (ł - 0.5 * (ł + )̊, ł - 0.5 * (ł + )̊)) – (m2t);[traversingThaw, rightColour] (m2t) – (mrt);[traversingThaw, rightColour] (m2t) – ((m2t) + (-(ł - ø + 0.5 * (+̊ø) - ł), ł - ø + 0.5 * (+̊ø) - ł));[traversingThaw, overlayLeftColour] ((m2t) + (-(ł - ø + 0.5 * (+̊ø) - ł), ł - ø + 0.5 * (+̊ø) - ł)) – (motl);[slowed, leftColour] (v) – (mlf);[slowed, rightColour] (v) – (mrf);[slowed, overlayLeftColour] ((v) + (0.4pt, 0)) – ((mofl) + (0.4pt, 0));[slowed, overlayRightColour] ((v) - (0.4pt, 0)) – ((mofr) - (0.4pt, 0));[slowed, leftColour] (ł, +̊ )̊ – (0.5 * , +̊ +̊ 1.5 * (ł- )̊); [freeze, leftColour] (mlf) – (lfg) (mlf) – (lfb); [freeze, rightColour] (mrf) – (rfb)(mrf) – (rfg); [freeze, overlayLeftColour] (mofl) – (ofbl)(mofl) – (ofg); [freeze, overlayRightColour] (mofr) – (ofbr) (mofr) – (ofg); ((m),leftFill,rightFill); [midpoint, leftColour] (mlf) – (mlt);((mlf),leftFill,leftFill); ((mlt),leftFill,leftFill); [midpoint, rightColour] (mrf) – (mrt);((mrf),rightFill,rightFill); ((mrt),rightFill,rightFill); [midpoint, overlayLeftColour] (mofl) – (motl);((mofl),overlayLeftFill,overlayLeftFill); ((motl),overlayLeftFill,overlayLeftFill); [midpoint, overlayRightColour] (mofr) – (motr);((mofr),overlayRightFill,overlayRightFill); ((motr),overlayRightFill,overlayRightFill); [midpoint, leftColour] (m1) – (m1t);((m1),leftFill,overlayRightFill); ((m1t),leftFill,overlayRightFill); [midpoint, rightColour] (m2) – (m2t);((m2),overlayLeftFill,rightFill); ((m2t),overlayLeftFill,rightFill);; [> = To, x = 1cm, y = -1cm]ø+ ł+ ł+(v) at (ł, 0);(gl) at (ł- ø, 0);(gr) at (ł+ ø, 0);(lgv) at (ł, ø);(fov) at (ł, 2 * ø);(folg) at (ł- ø, 2 * ø);(forg) at (ł+ ø, 2 * ø);(mol) at (ł- ø/2, 1.5 * ø);(mor) at (ł+ ø/2, 1.5 * ø);let 1 = (mol) in coordinate (stol) at (1, - ø/2);let 1 = (mor) in coordinate (stor) at (1, - ø/2);(sv) at ((mol) + (ø/2, 3 * ø/2));(lvl) at ((lgv) + (-ł, ł));(rlv) at ((lvl) + (ł, ł));(ml) at ((lvl) + (ł/2, ł/2));(fll) at ((ml) + (-ł/2, ł/2));(flv) at ((ml) + (ł/2, ł/2));let 1 = (ml) in coordinate (stl) at (1, - ł/2);(lvr) at ((lgv) + (,̊)̊);(rfv) at ((lvr) + (-,̊)̊);(mr) at ((lvr) + (-/̊2, /̊2));(frr) at ((mr) + (/̊2, /̊2));(frv) at ((mr) + (-/̊2, /̊2));let 1 = (mr) in coordinate (str) at (1, - /̊2);(mlo) at ((ł+ ø)/2, 1.5 * (ł+ ø));(mro) at (ł- ø+ (ø+ )̊/2, 1.5 * (ø+ )̊);(m) at ((ł+ )̊/2, ø+ ł+ (ł+ )̊/2);(stlo) at ((ł+ ø)/2, - (ł+ ø)/2);(stro) at (ł- ø+ (ø+ )̊/2, - (ø+ )̊/2);(tv) at (ł, );(tr) at (, );(tl) at (0, );(tol) at (ł- ø, );(tor) at (ł+ ø, );let 1 = (m) in coordinate (tmhv) at (ł, 1 + 1 - łcm);let 1 = (stlo) in coordinate (tmlohv) at (ł, 1 + 1 - łcm);let 1 = (stro) in coordinate (tmrohv) at (ł, 1 - 1 + łcm);[syncLeft, leftColour, draw = none] (lgv) – (lvl) – (fll) – (ml) – (flv) – cycle; [syncRight, rightColour, draw = none] (lgv) – (lvr) – (frr) – (mr) – (frv) – cycle; [syncMiddle, overlayLeftColour, draw = none] (gl) – (lgv) – (fov) – (mol) – (folg) – cycle; [syncMiddle, overlayRightColour, draw = none] (gr) – (lgv) – (fov) – (mor) – (forg) – cycle; [syncLeft, leftColour, draw = none] (tl) – (stl) – (tv) – cycle; [syncRight, rightColour, draw = none] (tr) – (str) – (tv) – cycle; [syncMiddle, overlayLeftColour, draw = none] (tol) – (stol) – (tv) – cycle; [syncMiddle, overlayRightColour, draw = none] (tor) – (stor) – (tv) – cycle; [find, overlayLeftColour] (gl) – (lgv);[find, rightColour] (lgv) – (lvr);[reflected, overlayLeftColour] (lgv) – (mol);[slowed, rightColour] (lgv) – (mr);[reflected, rightColour] (lvr) – ((lvr) + (-,̊)̊);[slowed, leftColour] ((lvr) + (-,̊)̊) – (m); [find, overlayRightColour] (gr) – (lgv);[find, leftColour] (lgv) – (lvl);[reflected, overlayRightColour] (lgv) – (mor);[slowed, leftColour] (lgv) – (ml);[reflected, leftColour] (lvl) – (m);[slowed, overlayLeftColour] (gl) – (mol) – (sv); [slowed, rightColour] (sv) – (mro); [slowed, overlayRightColour] (gr) – (mor) – (sv); [slowed, leftColour] (sv) – (mlo); [midpoint, leftColour] (ml) – (stl) (mlo) – (stlo); [midpoint, rightColour] (mr) – (str)(mro) – (stro); [midpoint, overlayLeftColour] (mol) – (stol); [midpoint, overlayRightColour] (mor) – (stor); [freeze, leftColour] (ml) – (fll) (ml) – (flv); [freeze, rightColour] (mr) – (frr)(mr) – (frv); [freeze, overlayLeftColour] (mol) – (fov)(mol) – (folg); [freeze, overlayRightColour] (mor) – (fov) (mor) – (forg);[traversingThaw, leftColour] (m) – (stlo) – (stl); [traversingThaw, leftColour] (m) – (tmhv); [traversingThaw, rightColour] (tmhv) – (stro) – (str); [traversingThaw, leftColour] (stlo) – (tmlohv); [traversingThaw, overlayRightColour] (tmlohv) – (stor); [traversingThaw, rightColour] (stro) – (tmrohv); [traversingThaw, overlayLeftColour] (tmrohv) – (stol); [thawingThaw, leftColour] (stl) – (tl)(stl) – (tv); [thawingThaw, rightColour] (str) – (tr) (str) – (tv); [thawingThaw, overlayLeftColour] (stol) – (tol) (stol) – (tv); [thawingThaw, overlayRightColour] (stor) – (tor)(stor) – (tv); ((ml),leftFill,leftFill); ((stl),leftFill,leftFill); ((mr),rightFill,rightFill); ((str),rightFill,rightFill); ((mol),overlayLeftFill,overlayLeftFill); ((stol),overlayLeftFill,overlayLeftFill); ((mor),overlayRightFill,overlayRightFill); ((stor),overlayRightFill,overlayRightFill); ((mlo),leftFill,overlayRightFill); ((stlo),leftFill,overlayRightFill); ((mro),overlayLeftFill,rightFill); ((stro),overlayLeftFill,rightFill); ((m),leftFill,rightFill); (0, 1) circle (2pt) node[right] ; (0, -1) circle (2pt) node[below] v”; (-3, -1) circle (2pt) node[below] (p̂); (4, -1) circle (2pt) node[below] (p̂); (0, 1) – (0, -1) (-3, -1) – (0, -1) – (4, -1);(-2, 0.5) circle (2pt) node[left] v; (0, 0.5) circle (2pt) node[right] v'; [loosely dashed] (-2, 0.5) – (0, 0.5);(-2, -2) circle (2pt) node[below] v; (-1, -1) circle (2pt) node[above] v'; [thick, loosely dotted] (-2, -2) – (-1, -1);(3, -3) circle (2pt) node[below] v; (2, -1) circle (2pt) node[above] v'; [dashdotted] (3, -3) – (2, -1); [tree/.style = dashed, gray] (g) at (0, 0); (u1) at (1, 0);(u+) at (2, 0); (uk) at (3, 0); (hatv) at (4, 0); (w1) at (4.5, -0.25); (wl) at (5.25, -0.6125);(hatv1) at (4 + 5, 5/2);(hatv2) at (4 + 8, -8/2);(m) at (4 + 1.47987, -1.47987/2); [tree] (g) – ++(-0.8, -0.4) – ++(0, 0.8) – cycle; [tree] (g) – ++(-1, -0.7) – ++(0, 0.9) – cycle; [tree] (u1) – ++(-0.2, -1.5) – ++(0.4, 0) – cycle; [tree] (u1) – ++(-0.4, -1.7) – ++(0.4, 0) – cycle; [tree] (uk) – ++(-0.2, -2) – ++(0.4, 0) – cycle; [tree] (uk) – ++(-0.4, -2.2) – ++(0.4, 0) – cycle; [tree] (hatv) – ++(-0.2, -1.8) – ++(0.4, 0) – cycle; [tree] (hatv) – ++(-0.4, -2) – ++(0.4, 0) – cycle; [tree] (w1) – ++(-0.2, -1) – ++(0.4, 0) – cycle; [tree] (w1) – ++(-0.1, -1.1) – ++(0.4, 0) – cycle; [tree] (wl) – ++(-0.2, -2) – ++(0.4, 0) – cycle; [tree] (wl) – ++(-0.1, -2.1) – ++(0.4, 0) – cycle; [tree] (hatv) – ((hatv1) + (-1, 1)) – ((hatv1) - (-1, 1)) – cycle; [tree] (hatv) – ((hatv2) + (-1, -1)) – ((hatv2) - (-1, -1)) – cycle;[dotted, thick] (g) – (hatv); (hatv) – (hatv1) (hatv) – (hatv2);(g) circle (2pt) node[above] ; (u1) circle (2pt) node[above] u_1;(u+) circle (0pt) node[above] …; (uk) circle (2pt) node[above] u_k; (hatv) circle (2pt) node[above] v̂; (w1) circle (2pt) node[above right] w_1; (wl) circle (2pt) node[above right] w_ℓ; (hatv1) circle (2pt) node[above right] v̂_1; (hatv2) circle (2pt) node[below right] v̂_2; [draw = black, fill = white] (m) circle (2pt) node[below] ; [tree/.style = dashed, gray] (g) at (0, 0); [tree] (g) – ++(-0.8, -0.4) – ++(0, 0.8) – cycle; [tree] (g) – ++(-1, -0.7) – ++(0, 0.9) – cycle; (g) circle (2pt) node[above] ;(w) at (1.5, 0); [tree] (w) – ++(-0.2, -1.5) – ++(0.4, 0) – cycle; [tree] (w) – ++(-0.4, -1.7) – ++(0.4, 0) – cycle; (w) circle (2pt) node[above] w;(v) at (5, 0); (v) circle (2pt) node[below left] v; (v') at ((v) + (60: 0.5)); (v') circle (2pt) node[left] v'; (v”) at ((v) + (0: 1)); (v”) circle (2pt) node[below] v”; (v”') at ((v) + (-60: 1.5)); (v”') circle (2pt) node[left] v”';(g) – (w) (w) – (v) (v) – (v') (v) – (v”) (v) – (v”'); [tree/.style = dashed, gray] (g) at (0, 0); [tree] (g) – ++(-0.8, -0.4) – ++(0, 0.8) – cycle; [tree] (g) – ++(-1, -0.7) – ++(0, 0.9) – cycle; (g) circle (2pt) node[above] ;(w) at (1.5, 0); [tree] (w) – ++(-0.2, -1.5) – ++(0.4, 0) – cycle; [tree] (w) – ++(-0.4, -1.7) – ++(0.4, 0) – cycle; (w) circle (2pt) node[above] w;(v) at (5, 0); (v) circle (2pt) node[below left] v;(v') at ((v) + (60: 0.5)); [tree] (v') – ++(0.6, 1) – ++(0.4, 0) – cycle; [tree] (v') – ++(0.4, 1.1) – ++(0.5, 0) – cycle; (v') circle (2pt) node[left] v';(v”) at ((v) + (0: 1)); [tree] (v”) – ++(1, -0.3) – ++(0, 0.8) – cycle; [tree] (v”) – ++(1.2, -0.5) – ++(0, 0.9) – cycle; (v”) circle (2pt) node[below] v”;(v”') at ((v) + (-60: 1.5)); [tree] (v”') – ++(0.2, -1) – ++(0.4, 0) – cycle; [tree] (v”') – ++(0.4, -1.1) – ++(0.5, 0) – cycle; (v”') circle (2pt) node[left] v”';(g) – (w) (w) – (v) (v) – (v') (v) – (v”) (v) – (v”'); [tree/.style = dashed, gray] (g) at (0, 0); [tree] (g) – ++(-0.8, -0.4) – ++(0, 0.8) – cycle; [tree] (g) – ++(-1, -0.7) – ++(0, 0.9) – cycle; (g) circle (2pt) node[above] ;(v) at (3, 0); [tree] (v) – ++(7, -1) – ++(0, 1.7) – cycle; [tree] (v) – ++(-0.2, -1.5) – ++(0.4, 0) – cycle; [tree] (v) – ++(-0.4, -1.7) – ++(0.4, 0) – cycle; (v) circle (2pt) node[above] v;(w) at (2.5, 0); [tree] (w) – ++(-0.3, -0.7) – ++(0.5, 0) – cycle; (w) circle (2pt) node[above] w;(w') at (2, 0); [tree] (w') – ++(-0.6, -1) – ++(0.4, 0) – cycle; [tree] (w') – ++(-0.4, -1.1) – ++(0.5, 0) – cycle; (w') circle (2pt) node[above] w';(w”) at (0.5, 0); [tree] (w”) – ++(-1, -2.3) – ++(1.5, 0) – cycle; [tree] (w”) – ++(-0.8, -1.1) – ++(0.8, 0) – cycle; (w”) circle (2pt) node[above] w”;(g) – (v); [tree/.style = dashed, gray] (g) at (0, 0); [tree] (g) – ++(-0.8, -0.4) – ++(0, 0.8) – cycle; [tree] (g) – ++(-1, -0.7) – ++(0, 0.9) – cycle; (g) circle (2pt) node[above] ;(v) at (3, 0); [tree] (v) – ++(7, -1) – ++(0, 1.7) – cycle; [tree] (v) – ++(-0.2, -1.5) – ++(0.4, 0) – cycle; [tree] (v) – ++(-0.4, -1.7) – ++(0.4, 0) – cycle; (v) circle (2pt) node[above] v;(v') at (2, 0); 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[tree] (g) – ++(-0.8, -0.4) – ++(0, 0.8) – cycle; [tree] (g) – ++(-1, -0.7) – ++(0, 0.9) – cycle;[dotted, thick] (g) – (hatv) (hatv) – (v1) (hatv) – (hatv1) (v2) – (hatv2); (v1) – (m) (m) – (v2);(g) circle (2pt) node[above] ; (hatv) circle (2pt) node[above] v̂; (v1) circle (2pt) node[above] v_1; (v2) circle (2pt) node[above] v_2; (hatv1) circle (2pt) node[above right] v̂_1; (hatv2) circle (2pt) node[below right] v̂_2; [draw = black, fill = white] (m) circle (2pt) node[below] ;[british]di-rec-tion di-rec-tions where-up-on rel-e-vant au-tom-a-ton au-tom-a-ta tran-si-tion tran-si-tions pre-serv-ing a-me-na-ble a-me-na-bil-i-ty per-tur-ba-tion per-tur-ba-tions bound-a-ry bound-a-ries coun-ter-ex-am-ple e-qui-var-i-ant gen-er-at-ed gen-er-at-ing romanplain[-1cm]-3cm zur Erlangung des akademischen Grades einesDoktors der Naturwissenschaftender KIT-Fakultät für Informatikdes Karlsruher Instituts für Technologie (KIT)genehmigteDissertationvonaus Heilbronn Datum der mündlichen Prüfung: 8. Juni 2017Erster Gutachter: Prof. Dr. Jörn Müller-QuadeZweiter Gutachter: MCF Dr. Michel Coornaertempty[1]AbstractAbstract CHAPTER: ABSTRACTThe present dissertation studies cellular automata whose cell spaces are not as usual the integers or groups but merely sets that are acted transitively upon by groups. These cell spaces can be continuous like the real numbers acted upon by translations or the sphere acted upon by rotations, or discrete like vertex-transitive graphs acted upon by graph automorphisms or uniform tilings of the hyperbolic plane acted upon by tiling-respecting bijections. As usual all cells have a state which they change synchronously in discrete time steps depending on the states of neighbouring cells as described by a local transition function. This global behaviour is represented by the global transition function, which maps global states of the automaton, so-called global configurations, to such.Among others the following questions are investigated: Which properties of the local transition function are necessary and sufficient for the global transition function to be equivariant under translations (commute with the induced action on global configurations)? Is the composition of global transition functions itself a global transition function and if it is, of which cellular automaton (see <ref>)? Can global transition functions be pulled or pushed onto quotients, products, restrictions, and extensions of their cell spaces and if they can, how do the corresponding cellular automata change (see <ref>)? Are global transition functions for a well chosen topology (or uniformity) on the set of global configurations characterised by equivariance under translations and (uniform) continuity? Is the inverse of a bijective global transition function itself a global transition function (see <ref>)? On which cell spaces are global transition functions surjective if and only if they are pre-injective (see <ref>)? How can such cell spaces be characterised (see <ref>)? Can these questions be answered for restrictions of global transition functions to translation invariant and compact subsets of the set of global configurations (see <ref>)? Is there an optimal-time algorithm for the firing squad synchronisation problem on (continuous) graph-shaped cell spaces (see <ref>)?§ EQUIVARIANCE AND COMPOSITIONOn groups global transition functions are equivariant under translations and the composition of global transition functions is a global transition function. On general cell spaces this is only the case if the local transition function has certain symmetries. Broadly speaking, these symmetries swallow changes of orientation that can be caused by translations and they prevent that composition produces asymmetries that contradict the local and uniform inner workings of cellular automata.§ QUOTIENTS, PRODUCTS, RESTRICTIONS, AND EXTENSIONSIf the neighbourhood of a cellular automaton is included in some dimensions, then the automaton and its global transition function can be restricted to these dimensions and, analogously, both can be pulled onto the quotient of the cell space modulo the other dimensions. Conversely, a cellular automaton can be extended to a superset of its cell space and, if the cell space is a quotient, it can be pushed onto the set that underlies the quotient.§ CHARACTERISATION BY EQUIVARIANCE AND CONTINUITYBecause a global step of a cellular automaton is local and uniform in each cell, global transition functions of cellular automata on groups are characterised by equivariance under translations and (uniform) continuity with respect to a well chosen topology (or uniformity) on the set of global configurations. On general cell spaces such a characterisation is again only possible if the local transition function has certain symmetries. It follows from such a characterisation and topological theorems that the inverse of a bijective global transition function is itself a global transition function.§ EQUIVALENCE OF SURJECTIVITY AND PRE-INJECTIVITYMaps on finite sets are surjective if and only if they are injective. The set of global configurations is in general infinite but, as compensation, global transition functions are local and uniform. And indeed, global transition functions of cellular automata over the integers are surjective if and only if they are pre-injective, where pre-injectivity is essentially injectivity on global configurations with finite support. On groups this equivalence is only satisfied if the group is amenable, where amenability means in a sense that the group behaves like a finite group. On general cell spaces one can also find a notion of amenability such that the before-mentioned equivalence holds.§ RESTRICTIONS TO SHIFT SPACESSometimes it is reasonable to restrict the domain of a global transition function to certain global configurations. For example in the case that the input and the intermediate global configurations that occur in calculations performed by the automaton are of a certain form. Such sets of global configurations should for technical reasons be invariant under translations and compact and they are called shift spaces. Are such restrictions of global transition functions surjective if and only if they are pre-injective? Yes, at least if the cell space is amenable and the shift space is of finite type and strongly irreducible, which more or less means that it is generated by finitely many forbidden finite patterns and that two allowed finite patterns that are far enough apart can be embedded in a global configuration of the shift space.§ THE FIRING SQUAD SYNCHRONISATION PROBLEMA cellular automaton solves the firing squad synchronisation problem if, for each connected and finite region on which one cell is distinguished and whose other cells are in a quiescent state, after finitely many steps all cells of the region transit at the same time into a so-called fire state and this state does not occur before. In dimension one, on rectangles in dimension two, and for other special cases, there are optimal-time algorithms for this problem but not in general. In the present thesis, an optimal-time algorithm on (continuous) graph-shaped cell spaces is presented, which needs unbounded many states but uses these solely for simple geometrical constructions. ngerman [1]KurzfassungKurzfassung CHAPTER: KURZFASSUNGIn der vorliegenden Dissertation werden Zellularautomaten untersucht, deren Zellraum nicht wie üblich die ganzen Zahlen oder eine Gruppe ist sondern lediglich eine Menge auf der eine Gruppe transitiv operiert. Diese Zellräume können kontinuierlich sein, wie beispielsweise die reellen Zahlen mit der Operation durch Verschiebungen oder die Sphäre mit der Operation durch Drehungen, oder diskret, wie beispielsweise knoten-transitive Graphen mit der Operation durch Graphautomorphismen oder gleichförmige Parkettierungen der hyperbolischen Ebene mit der Operation durch parkettierungserhaltende Bijektionen. Wie üblich haben alle Zellen einen Zustand, den sie synchron, in diskreten Zeitschritten, in Abhängigkeit der Zustände benachbarter Zellen und wie von einer lokalen Überführungsfunktion vorgeschrieben, wechseln. Dieses globale Verhalten wird von der globalen Überführungsfunktion beschrieben, welche Gesamtzustände des Automaten, sogenannte globale Konfigurationen, auf ebensolche abbildet.Unter anderen werden die folgenden Fragestellungen behandelt: Unter welchen Bedingungen an die lokale Überführungsfunktion ist die globale Überführungsfunktion verschiebungsäquivariant (verträgt sich mit der induzierten Gruppenoperation auf globalen Konfigurationen)? Ist die Komposition zweier globaler Überführungsfunktionen wieder eine globale Überführungsfunktion und wenn ja, von welchem Zellularautomaten (siehe Kapitel <ref>)? Können globale Überführungsfunktionen auf Quotienten, Produkte, Einschränkungen und Erweiterungen ihrer Zellräume gezogen werden und wenn ja, wie wandeln sich dabei die zugehörigen Zellularautomaten (siehe Kapitel <ref>)? Sind globale Überführungsfunktionen nach geeigneter Wahl einer Topologie (oder Uniformität) auf der Menge der globalen Konfigurationen durch Verschiebungsäquivarianz und (gleichmäßige) Stetigkeit charakterisiert? Ist die Umkehrfunktion einer bijektiven globalen Überführungsfunktion wieder eine globale Überführungsfunktion (siehe Kapitel <ref>)? Auf welchen Zellräumen sind globale Überführungsfunktionen genau dann surjektiv, wenn sie prä-injektiv sind (siehe Kapitel <ref>)? Auf welche Weisen können derartige Zellräume charakterisiert werden (siehe Kapitel <ref>)? Können diese Fragen auch für Einschränkungen globaler Überführungsfunktionen auf verschiebungsinvariante und kompakte Teilmengen der Menge der globalen Konfigurationen beantwortet werden (siehe Kapitel <ref> und <ref>)? Gibt es einen Optimalzeitalgorithmus für das Firing Squad Synchronisation Problem auf (kontinuierlichen) graph-förmigen Zellräumen (siehe Kapitel <ref>)?§ VERSCHIEBUNGSÄQUIVARIANZ UND KOMPOSITIONAuf Gruppen sind globale Überführungsfunktionen verschiebungsäquivariant und die Komposition zweier globaler Überführungsfunktionen ist eine globale Überführungsfunktion. Auf allgemeineren Zellräumen ist dies nur dann der Fall, wenn die lokalen Überführungsfunktionen gewisse Symmetrien aufweisen. Salopp ausgedrückt schlucken diese Symmetrien Orientierungsänderungen, die bei Verschiebungen auftreten können, und sie verhindern, dass bei der Komposition Asymmetrien entstehen, die der lokalen und uniformen Arbeitsweise von Zellularautomaten widersprechen.§ QUOTIENTEN, PRODUKTE, EINSCHRÄNKUNGEN UND ERWEITERUNGENErstreckt sich die Nachbarschaft eines Zellularautomaten nur in einige Dimensionen, so ist dieser und seine globale Überführungsfunktion auf eben diese Dimensionen einschränkbar und analog können beide auf den Quotienten des Zellraums mit den anderen Dimensionen gezogen werden. Umgekehrt lässt sich ein Zellularautomat auf eine Obermenge seines Zellraums erweitern und, falls dieser ein Quotient ist, auf die dem Quotienten zugrundeliegende Menge.§ CHARAKTERISIERUNG DURCH VERSCHIEBUNGSÄQUIVARIANZ UND STETIGKEITDa ein globaler Schritt eines Zellularautomaten lokal und uniform in jeder Zelle ist, sind globale Überführungsfunktionen von Zellularautomaten auf Gruppen nach geeigneter Wahl einer Topologie (oder Uniformität) auf der Menge der globalen Konfigurationen durch Verschiebungsäquivarianz und (gleichmäßige) Stetigkeit charakterisiert. Auf allgemeineren Zellräumen ist eine solche Charakterisierung abermals nur unter zusätzlichen Bedingungen an die lokale Überführungsfunktion möglich. Insbesondere folgt aus einer solchen Charakterisierung und Sätzen der Topologie, dass die Umkehrfunktion einer bijektiven globalen Überführungsfunktion wieder eine globale Überführungsfunktion ist.§ ÄQUIVALENZ VON SURJEKTIVITÄT UND PRÄ-INJEKTIVITÄTAbbildungen auf endlichen Mengen sind genau dann surjektiv, wenn sie injektiv sind. Die Menge der globalen Konfigurationen ist im Allgemeinen unendlich, aber dafür sind globale Überführungsfunktionen lokal und uniform. Tatsächlich sind globale Überführungsfunktionen von Zellularautomaten auf den ganzen Zahlen genau dann surjektiv, wenn sie prä-injektiv sind, wobei Prä-Injektivität im Wesentlichen Injektivität auf globalen Konfigurationen mit endlichem Träger ist. Auf Gruppen gilt diese Äquivalenz nur dann, wenn die Gruppe mittelbar ist, wobei Mittelbarkeit in gewissem Sinne sicherstellt, dass sich die Gruppe wie eine endliche Gruppe verhält. Auf allgemeineren Zellräumen kann man ebenfalls einen derartigen Mittelbarkeitsbegriff so finden, dass die vorhin erwähnte Äquivalenz gilt.§ EINSCHRÄNKUNGEN AUF VERSCHIEBUNGSRÄUMEManchmal ist es sinnvoll den Definitionsbereich einer globalen Überführungsfunktion auf bestimmte globale Konfigurationen einzuschränken, beispielsweise dann, wenn die Eingaben und die bei der Berechnung auftretenden Zwischenergebnisse nur von einer gewissen Form sind. Solche Mengen globaler Konfigurationen sollten aus technischen Gründen verschiebungsinvariant und kompakt sein und werden Verschiebungsräume genannt. Auch für solche globale Überführungsfunktionen stellt sich die Frage ob sie genau dann surjektiv sind, wenn sie prä-injektiv sind. Dies ist zumindest dann der Fall, wenn der zugrundeliegende Zellraum mittelbar ist und der Verschiebungsraum streng irreduzibel ist, was in etwa bedeutet, dass man zwei erlaubte Muster, sofern man sie weit genug voneinander entfernt, in eine globale Konfiguration des Verschiebungsraums einbetten kann.§ THE FIRING SQUAD SYNCHRONISATION PROBLEMEin Zellularautomat löst das Firing Squad Synchronisation Problem, wenn für jedes zusammenhängende und endliche Gebiet auf dem eine Zelle ausgezeichnet ist und deren andere Zellen sich im Ruhezustand befinden, nach endlich vielen Schritten alle Zellen des Gebiets gleichzeitig in einen sogenannten Feuerzustand übergehen und dieser vorher nicht vorkommt. Im Eindimensionalen, auf Rechtecken im Zweidimensionalen, und für andere Spezialfälle gibt es Optimalzeitalgorithmen für dieses Problem, jedoch nicht im Allgemeinen. In dieser Arbeit wird erstmals ein Optimalzeitalgorithmus auf (kontinuierlichen) graph-förmigen Zellräumen vorgestellt, der zwar unbeschränkt viele Zustände benötigt, diese aber nur für einfache geometrische Konstruktionen verwendet.[1]Publicationspublications CHAPTER: PUBLICATIONSSome ideas and figures have appeared previously in the following publications:[my] *[heading=none] [1]AcknowledgementacknowledgementCHAPTER: ACKNOWLEDGEMENTSpecial thanks go to my unofficial supervisor Dr. Thomas Worsch, who always took the time to answer any — simple or hard, silly or smart, dull or exciting, pleasing or provocative — questions I had, for his excellent support and for keeping me motivated.scrheadings dummy [1]tableofcontents[section]chapter arabicCHAPTER: CELLULAR AUTOMATAAbstract. We introduce cellular automata whose cell spaces are left-homogeneous spaces; show that their global transition functions are equivariant, determined by their behaviour at one point, and closed under composition; and construct automata from others by taking quotients and products, restrictions and extensions. Examples of left-homogeneous spaces are spheres, Euclidean spaces, as well as hyperbolic spaces acted on by isometries; uniform tilings acted on by symmetries; vertex-transitive graphs, in particular, Cayley graphs, acted on by automorphisms; groups acting on themselves by multiplication; and integer lattices acted on by translations. Remark. Some parts of this chapter appeared in the paper *wacker:automata:2016<cit.> and they generalise parts of sections 1.1 to 1.5 of the monograph *ceccherini-silberstein:coornaert:2010<cit.> and parts of the paper *moriceau:2011<cit.>. Motivation. In the first chapter of the monograph Cellular Automata and Groups<cit.>, Tullio Ceccherini-Silberstein and Michel Coornaert develop the theory of cellular automata whose cell spaces are groups. Examples of groups are abound: The integer lattices and Euclidean spaces with addition (translation), the one-dimensional unit sphere embedded in the complex plane with complex multiplication (rotation), and the vertices of a Cayley graph with the group structure it encodes (graph automorphisms).Yet, there are many structured sets that do not admit a structure-preserving group structure. For example: Each Euclidean n-sphere, except for the zero-, one-, and three-dimensional, does not admit a topological group structure; and the Petersen graph does not admit an adjacency-preserving group structure on its vertices. However, these structured sets can be acted on by subgroups of their automorphism group by function application. For example Euclidean n-spheres can be acted on by rotations about their centres and graphs can be acted on by adjacency-preserving permutations of their vertices. Moreover, there are structured groups that have more symmetries than can be expressed by the group structure. The integer lattices and the Euclidean spaces under addition, for example, are groups, but addition expresses only their translational symmetries but not their rotational and reflectional ones. Though, they can be acted on by arbitrary subgroups of their symmetry groups, like the ones generated by translations and rotations.The general notion that encompasses these structure-preserving actions is that of a group set, that is, a set that is acted on by a group. A group set M acted on by G such that for each tuple (m, m') ∈ M × M there is a symmetry g ∈ G that transports m to m' is called left-homogeneous space and the action of G on M is said to be transitive. In particular, groups are left-homogeneous spaces — they act on themselves on the left by multiplication.In this chapter, we develop the theory of cellular automata whose cell spaces are left-homogeneous spaces. These cellular automata are defined so that their global transition functions are equivariant under the induced group action on global configurations. Depending on the choice of the cell space, these actions may be plain translations but also rotations and reflections. Exemplary for the first case are integer lattices that are acted on by translations; and for the second case Euclidean n-spheres that are acted on by rotations, but also the two-dimensional integer lattice that is acted on by the group generated by translations and the rotation by 90.Sébastien Moriceau defines and studies a more restricted notion of cellular automata over group sets in his paper *moriceau:2011<cit.>. He requires sets of states and neighbourhoods to be finite. His automata are the global transition functions of what we call semi-cellular automata with finite set of states and finite sufficient neighbourhood. For these he proves many results that are analogous to those in the present chapter, though he uses different techniques.His automata obtain the next state of a cell by translating the global configuration such that the cell is moved to the origin, restricting that configuration to the neighbourhood of the origin, and applying the local transition function to that local configuration. Our automata obtain the next state of a cell by determining the neighbours of the cell, observing the states of that neighbours, and applying the local transition function to that observed local configuration. The obtained states are the same, but the viewpoints are different, which manifests itself in proofs and constructions.To determine the neighbourhood of a cell we let the relative neighbourhood semi-act on the right on the cell. That right semi-action is to the left group action what right multiplication is to the corresponding left group multiplication. Many properties of cellular automata are a consequence of the interplay between properties of that semi-action, translations of global configurations, and rotations of local configurations. That semi-action allows us to define the notion of right amenability for left-homogeneous spaces (see <ref>) and to prove the Garden of Eden theorem for automata over such spaces (see <ref>), which states that each such automaton is surjective if and only if it is pre-injective. For example finitely right-generated left-homogeneous spaces of sub-exponential growth are right amenable, in particular, quotients of finitely generated groups of sub-exponential growth by finite subgroups acted upon by left multiplication. Contents. In <ref> we motivate the definition of semi-cellular and cellular automata on left-homogeneous spaces by a geometrical interpretation of traditional cellular automata on the two-dimensional integer lattice. In <ref> we introduce left group actions and our prime examples, which illustrate phenomena that cannot be encountered in groups acting on themselves on the left by multiplication. In <ref> we introduce coordinate systems, cell spaces, and right quotient set semi-actions that are induced by transitive left group actions and coordinate systems. In <ref> we introduce semi-cellular and cellular automata. In <ref> we show that a global transition function does not depend on the choice of coordinate system, is equivariant under the induced left group action on global configurations, is determined by its behaviour in the origin, and that the composition of two global transition functions is a global transition function. § INTRODUCTION Informally, a traditional two-dimensional cellular automaton is a regular grid of similar finite-state machines working in synchrony whose inputs are the states of neighbouring machines. Formally, it is a quadruple made up of the set M = ^2 of cells, a finite set Q of states, a finite subset N of ^2 — the (relative) neighbourhood (think of vectors) —, and a local transition function δ from Q^N to Q. The maps in Q^N are local configurations and the maps in Q^M are global configurations. The neighbourhood of a cell m is m + N and the local configuration that is observed by a cell m in a global configuration c is the mapc(m + )_NN→ Q,n↦ c(m + n).The global transition function is the mapΔ Q^M→ Q^M,c↦[m ↦δ(c(m + )_N)]. The state Δ(c)(m) is determined by applying the local transition function to the local configuration observed by m in c. Because the local transition function is the same for all cells, the map Δ is uniform. And, because the local configuration that is observed by a cell is determined by the states of its finitely many neighbours, the map Δ is local.The translations of global configurations are the maps t, for t ∈^2, where^2 × Q^M→ Q^M,(t, c)↦[m ↦ c((-t) + m)].The global transition function is equivariant under translations (because it is uniform), which means thatt ∈^2c ∈ Q^M Δ(tc) = t Δ(c),and it is continuous with respect to the prodiscrete topology on Q^M (because it is local), which means thatO ⊆ Q^MopenΔ^-1(O)is open.Conversely, each map Δ on Q^M with these two properties is the global transition function of a cellular automaton. This characterisation of global transition functions is known as Curtis-Hedlund-Lyndon theorem. It follows from this theorem and basic topology that the inverse of a bijective global transition function is itself a global transition function. Moreover, because global transition functions are uniform, they are determined by their behaviour in the origin; and, because the composition of two global transition functions is uniform and local, it is itself a global transition function.The cells of the cellular automaton can be restricted to the group that is generated by the neighbourhood. For example, if N is the set (-1, 0), (0, 0), (1, 0), then the quadruple made up of the set ×0 of cells, the set Q of states, the neighbourhood N, and the local transition function δ is a cellular automaton over ×0. Its global transition function is the map Δ'Q^×0→ Q^×0, c ↦ [(m_1, 0) ↦δ(c((m_1, 0) + )_N)] and the global transition function Δ of the original automaton is essentially the product ∏_z ∈Δ'. Similarly, because N has only the origin in common with the normal subgroup 0× of ^2, the original automaton induces one whose set of cells is the quotient group ^2(0×).Each subset X of the phase space Q^M has an entropy, which is a number that measures the complexity of X. More precisely, it is the asymptotic growth rate of the number of square-shaped patterns that occur in X for squares about the origin of increasing sizes. Because the boundaries of large enough squares in M are negligible with respect to the squares themselves, it follows from the locality of the global transition function Δ that the entropy of Δ(Q^M) is not greater than the one of Q^M. Using this one can show that Δ is surjective if and only if Δ(Q^M) has maximal entropy, and that Δ(Q^M) has maximal entropy if and only if Δ is pre-injective. This establishes the Garden of Eden theorem, which states that Δ is surjective if and only if it is pre-injective.Cellular automata are capable of synchronising all cells of a nice enough region, in the sense that a synchronisation process can be started in one cell and after some time all cells of the region go into the same predetermined state and this state did not occur before. This is achieved with a divide and conquer strategy: The region is divided in a regular fashion into smaller and smaller regions until no further division is possible, which happens for all regions at the same time and at which point all cells go into the predetermined state.The above definition of automata can be interpreted algebraically or geometrically. Algebraically, the cells M form a group under addition and the (relative) neighbourhood N is given such that the neighbourhood of the neutral element 0 is N itself. If above we replace ^2 by any group G, the element 0 by its neutral element, and the addition + by its operation, then we get the definition of cellular automata over G. Their global transition functions are still characterised by equivariance and continuity, invertible if and only if they are bijective, determined by their behaviour in the origin, closed under composition, and, if the group is amenable, then they are surjective if and only if they are pre-injective, where amenability broadly speaking means that there are subsets of G whose boundaries are negligible with respect to the subsets themselves.Geometrically, the cells M form a grid. This grid has the translational symmetries g +, for g ∈^2. They form a group under composition. This group is isomorphic to the group G = ^2 under addition. In other words, the translation vectors in G encode the translations of M. Hence, the group G acts on M on the left by translations, more formally, by (g, m) ↦ g + m. Dedicate the cell m_0 = 0 as the origin of M. The (relative) neighbourhood N is given such that the neighbourhood of the origin m_0 is N itself. For each cell m, there is a translation vector g_m_0, m such that g_m_0, m + m_0 = m, namely m; this property of the action of G on M is known as transitivity. The neighbourhood of m is the translation of N by g_m_0, m, namely g_m_0, m + N. A cell m uses these translations to observe the local configuration c(m + )_N in a global configuration c. Moreover, the action of G on M induces the actionof G on Q^M (translations of global configurations). That interpretation suggests the following generalisation of cellular automata. Let M be a set, let G be a group, and letbe a transitive left group action of G on M. The actioninduces a left group actionof G on Q^M. Moreover, let m_0 be an element of M, let g_m_0, m_0 be the neutral element of G, and, for each element m ∈ M ∖m_0, let g_m_0, m be an element of G such that g_m_0, m m_0 = m. If in the geometrical interpretation we use M as the grid, the group G as the translation vectors, the actionas the action of G on M by translations, namely +, the element m_0 as the origin, and, for each cell m, the element g_m_0, m as the dedicated translation vector of m, then we get a definition of semi-cellular automata over M. They have the prefix semi because their global transition functions are in general not equivariant under .For example, the right shift map on the one-dimensional grid, acted upon by the group of translations and reflections, is equivariant under translations but not under reflections. The reason is that translations leave the meanings of right and left, whereas reflections reverse them. That the right shift map is not equivariant under reflections can already be seen by the fact that its local transition function depends on the distinction between right and left, in other words, by the fact that it is not invariant under reflections. In general, to get equivariance of a global transition function under all symmetries, its local transition function must be invariant under certain symmetries. The stabiliser G_0 of m_0 is the set of all group elements that fix m_0 (think of rotations about m_0). Let us assume that N is invariant under G_0. Then, the restriction ofto G_0 induces a left group action ∙ of G_0 on Q^N (think of rotations of local configurations). A global transition function is equivariant underif and only if its local transition function is invariant under ∙. A semi-cellular automaton with the latter property is a cellular automaton. For these automata, the global transition function does not depend on the choice of g_m_0, m_m ∈ M, the Curtis-Hedlund-Lyndon theorem holds, and the other statements made above hold also. Actually, for many properties it is sufficient that the local transition function is invariant under the restriction of ∙ to G_0 ∩ H, where H is the subgroup of G generated by g_m_0, m m ∈ M.Note that we fix an origin m_0, because we need a reference cell for the (relative) neighbourhood N; we fix the group elements g_m_0, m_m ∈ M, because we need them to define the neighbourhood of each cell m as g_m_0, m N; we choose g_m_0, m_0 as the neutral element, because we want the neighbour of the origin that corresponds to the relative neighbour n to be n itself; we require the left group actionto be transitive, because otherwise there would be a cell m for which there would be no group element g such that gm_0 = m and hence we could not fix a group element g_m_0, m. Fixing m_0 and g_m_0, m_m ∈ M is necessary for the definition of global transition functions by local transition functions. However, it is not necessary for the characterisation of global transition functions by equivariance and continuity. We could regard N as the neighbourhood of the origin (a set of points) and g_m_0, m N as translating the neighbourhood of the origin to m. However, we regard N as a relative neighbourhood (a set of vectors) and mN = g_m_0, m N as adding the relative neighbourhood on the right to m (think of point-vector addition). Because we have chosen an origin m_0, there is a canonical bijection between M and the quotient set GG_0. Under the identification of M with GG_0, the relative neighbourhood N is a subset of GG_0. And, the map M × GG_0 → M, (m, 𝔤) ↦ g_m_0, m𝔤, is a right semi-action that semi-commutes with . We could have definedas a map with domain M × M, but to state its properties it is convenient to have the domain M × GG_0. In the definition ofwe identified M with GG_0 to regard 𝔤∈ GG_0 as an element of M. There is an equivalent definition that works without this identification. The transporter G_m, m' is the set of all group elements that transport m to m'. The quotient set GG_0 is the set of all transporters from m_0. Under the identification of M with GG_0, a (relative) neighbour n is the transporter G_m_0, n. And, under the identification of singleton sets with their only element, we have mn = g_m_0, m n = G_m, g_m_0, m n m = g_m_0, m n g_m_0, m^-1 m. In particular, because g_m_0, m_0 is the neutral element, we have m_0n = nm_0. So, we can think of n as a localised vector with initial cell m_0, of conjugating n with g_m_0, m as changing the initial cell to m, ofas a means to turn localised vectors into a cell by adding it to its initial cell, and ofas a means to add a localised vector with initial cell m_0 to any cell. And, we can defineby mn = g_m_0, m n g_m_0, m^-1 m.The right semi-actioncan be used to define the notion of right amenability for the left action . Right amenability is characterised by the existence of invariant finitely additive probability measures, the existence of invariant means, the existence of right Følner nets, and the non-existence of right paradoxical decompositions. This characterisation is known as Tarski-Følner theorem. In the case that the left actionis right amenable, the Garden of Eden theorem holds.Some properties of cellular automata still hold if the set of states and the neighbourhood are infinite. And, on continuous spaces, infinite and compact neighbourhoods are more natural than finite ones. Therefore, unless stated otherwise, we drop the usual finiteness requirements. However, in the uniform variant of the Curtis-Hedlund-Lyndon theorem, we require the neighbourhood to be compact (a generalisation of finiteness for continuous spaces); and in the topological variant, we require the set of states and the neighbourhood to be finite. § LEFT GROUP ACTIONSIntroduction. The symmetries of a geometric object are distance-preserving bijections on the object. The identity map is the trivial symmetry that maps each point to itself. The composition of two symmetries is again a symmetry. The symmetries under composition form a group. One says that this group (or some subgroup) acts on the geometric object by mapping points to points.On each point the identity map acts trivially and the composition of two symmetries acts in the same way as the right symmetry does followed by the left. The symmetries of a circle are the reflections about lines through its centre and the rotations about it. Note that the reflection about the centre of the circle is not missing, because it is identical to the rotation by 180. For each pair of points on the circle, there is a symmetry, even a rotation, that maps one point to the other. One says that the symmetries, even only the rotations, act transitively on the circle.Moreover, for each pair of points on the circle, there is at most, even exactly, one rotation that maps one point to the other. One says that the rotations act freely on the circle.However, the symmetries do not act freely on the circle, because for each pair of points, in addition to the rotation that maps one point to the other, the reflection about the line through the centre of the circle and the centres of the circular arcs connecting the two points does so too.For each point on the circle, the points it can be mapped to by a symmetry, called orbit of the point, is the circle; the symmetries that map the point to itself, called stabiliser of the point, are the identity map and the reflection about the line through the centre of the circle and the point; and the symmetries that map a point to another point, called transporter from the point to the other one, are the reflection about the line through the centre of the circle and the centres of the circular arcs connecting the points, and the rotation by the central angle between the points with the correct sign. The symmetries of a square are the four reflections (about the horizontal line through the centre of the square, the vertical line through the centre of the square, and the two diagonals) and the four rotations (by 0, 90, 180, and 270). For each pair of vertices of the square, there is a symmetry, even a rotation, that maps one vertex to the other. However, there is no symmetry that maps a vertex to a point on an edge and vice versa. Thus, the symmetries, even only the rotations, act transitively on the vertices; whereas the symmetries do not act transitively on the square itself.Therefore, it is often appropriate to regard pointy geometrical objects as graphs, that is, as their vertices equipped with structural information induced by their edges. Their symmetries are graph automorphisms, that is, bijections on the vertices that preserve adjacency.Contents. In <ref> we introduce left group actions and left group sets. In <ref> we introduce three examples of left group sets that we use to illustrate new notions throughout the present chapter. In <ref> we show in which sense left group actions act by symmetries. In <ref> we introduce restrictions of left group actions to subgroups. In <ref> we introduce faithfulness, freeness, transitivity, and regularity of left group actions. In <ref> we introduce left-homogeneous spaces and principal ones. In <ref> we introduce orbits, stabilisers, and transporters. In <ref> we show how stabilisers and transporters of two elements with the same orbit relate to each other. In <ref> we introduce quotient sets and note that the quotient set by the stabiliser of a point is the set of transporters from that point. In <ref> we introduce orbit spaces and note that they partition the set of points. In <ref> we introduce invariance of maps and sets under left group actions. In <ref> we introduce equivariance of maps from one group set to another; such maps are homomorphisms. In <ref> we show that the inverse of an equivariant and bijective map is again equivariant. In <ref> we show that a group acts transitively on each of its quotient sets on the left by multiplication. In <ref> we show that each transitive left group action is isomorphic to an action as in <ref>. In <ref> we show that each free and transitive left group action is isomorphic to a group multiplication, which we elaborate on in <ref>. In <ref> we note that each left action of an abelian group induces a faithful, free, and transitive left group action. And in <ref> we show that left group actions induce left group actions on maps over invariant subsets of points.Let M be a set, let G be a group, letbe a map from G × M to M, and let e_G be the neutral element of G. The mapis called left group action of G on Mleft group actionof G on Mgroup action of G on M!left[symbols]arrow right@, the group G is said to act on M on the left by G acts on M on the left by , and the triple M, G, is called left group setleft group set M, G,group set!left[symbols]Mcalligraphic@ℳ if and only ifm ∈ Me_Gm = m,andm ∈ Mg ∈ Gg' ∈ Gg g'm = g(g'm).[Group]Let G be a group. It acts on itself on the left by multiplication. [Plane]Let M be the Euclidean plane ^2 and let G be the special Euclidean group ^+(2), that is, the group generated by translations and rotations of M. The group G acts on M on the left by function application. This action is denoted by . [Sphere]Let M be the Euclidean unit 2-sphere, that is, the surface of the ball of radius 1 in three-dimensional Euclidean space, and let G be the rotation group. The group G acts on M on the left by function application, that is, by rotation about the centre of the sphere. This action is denoted by .Let M, M', and M” be three sets, let f be a map from M × M' to M”, and let m_0 and m_0' be two elements of M and M' respectively. The mapsM→ M”, partially applied map f(, m_0')[symbols]funderscorem0prime@f(, m_0')m↦ f(m, m_0'),andM'→ M”, partially applied map f(m_0, )[symbols]fm0underscore@f(m_0, )m'↦ f(m_0, m'),are denoted by f(, m_0') and f(m_0, ) respectively and called partially applied maps[symbols]underscore@. We will also use analogous notations for maps with more than two arguments and for maps that are written in infix notation. Let M be a set, let G be a group, and let (M) be the symmetric group of M. For each left group actionof G on M, the mapfG→(M),g↦ g ,is a group homomorphism. And, for each group homomorphism f from G to (M), the mapG × M→ M,(g, m)↦ f(g)(m),is a left group action.Let M and M' be two sets, let N and N' be two subsets of M and M' respectively, and let f be a map from M to M' such that f(N) ⊆ N'. * The mapf_N → N' N→ N', restriction f_N → N' of f to N and N'[symbols]harpoon up right N to N prime@_N → N'n↦ f(n),is called restriction of f to N and N'.* The mapf_NN→ M', domain restriction f_N of f to N[symbols]harpoon up right N@_Nn↦ f(n),is called domain restriction of f to N.Letbe a left group action of G on M and let H be a subgroup of G. The left group action _H × M of H on M is denoted by _H. Letbe a left group action of G on M. It is called * faithfulfaithful if and only ifg ∈ Gg' ∈ G ∖g m ∈ Mgm ≠ g'm; *freefree if and only ifg ∈ Gg' ∈ G( m ∈ Mgm = g'm)g = g'; *transitivetransitive if and only if the set M is non-empty andm ∈ Mm' ∈ Mg ∈ Ggm = m'; * regularregular if and only if it is both free and transitive.[Group]In the situation of <ref>, the left group action is faithful, transitive, and free. [Plane]In the situation of <ref>, the left group action is faithful and transitive but not free. [Sphere]In the situation of <ref>, the left group action is faithful and transitive but not free.Letbe a left group action of G on M and let P be an adjective. The group G is said to act Ply on M on the left by act Ply on M on the left byif and only if the actionis P.Let M, G, be a left group set. It is called left-homogeneous spaceleft-homogeneous spacehomogeneous space!left if and only if the actionis transitive. Let M, G, be a left-homogeneous space. It is called principalprincipalleft-homogeneous space!principalhomogeneous space!left!principal if and only if the actionis free. Letbe a left group action of G on M, and let m and m' be two elements of M. * The setGm = gmg ∈ Gorbit Gm of m under [symbols]Garrowrightm@Gmis called orbit of m under .* The setG_m = g ∈ Ggm = mstabiliser G_m of m under [symbols]Gm@G_mis called stabiliser of m under .* The setG_m, m' = g ∈ Ggm = m'transporter G_m, m' of m to m' under [symbols]Gmmprime@G_m, m'is called transporter of m to m' under .[Group]In the situation of <ref>, each orbit is G and each stabiliser is e_G.[Plane]In the situation of <ref>, for each point m ∈ M, its orbit is M and its stabiliser is the group of rotations about itself. [Sphere]In the situation of <ref>, for each point m ∈ M, its orbit is M and its stabiliser is the group of rotations about the line through the centre and itself.[Riemannian Symmetric Space]Let (M, ) be a Riemannian manifold, let (M) be the isometry group of M, and let the geodesic reflection at any point of M be an isometry. Then, M is geodesically complete and (M) acts transitively on M by function application (see theorems 1.4 and 1.3 of <cit.>). Examples of such Riemannian manifolds include: * The Euclidean space ^d of dimension d ∈_+ with the Euclidean metric. Its isometry group (M) is the Euclidean group (d) generated by translations and orthogonal linear maps. The stabiliser of the origin 0 is the orthogonal group (d).* The sphere ^d = v ∈^d+1v_2 = 1 of dimension d ∈_+ with the Riemannian metric induced by the dot product ·. Its isometry group (M) is the orthogonal group (d + 1). The stabiliser of the north pole 0, 0, …, 0, 1^ is (d) ⊆(d + 1).* The real hyperbolic space ^d = v ∈^d+1vv = -1, v_d+1 > 0 of dimension d ∈_+ with the Riemannian metric induced by the Lorentzian indefinite inner product^d+1×^d+1 →,(v,v')↦∑_i = 1^d v_i v'_i - v_i+1 v'_i+1.Its isometry group (M) is the group of future preserving Lorentz transformations (d,1)^+. The stabiliser of the north pole (0,0,…,0,1)^ is (d) ⊆(d, 1)^+.* The orthogonal group (d) = A ∈^d × d A^ A = I in dimension d ∈_+ with the Riemannian metric induced by the trace inner product^d × d×^d × d →,(A, A')↦ A^ A'.* Each compact Lie group M with bi-invariant Riemannian metric . Letbe a left group action of G on M, let m and m' be two elements of M that have the same orbit under , and let g be an element of G_m, m'. Then, G_m' = g G_m g^-1 and g G_m = G_m, m' = G_m' g.First, let g' ∈ G_m. Then,g g' g^-1 m' = g g'm = gm = m'.Hence, g g' g^-1∈ G_m'. In conclusion, g G_m g^-1⊆ G_m'. Secondly, let g' ∈ G_m'. Then, as above, g” = g^-1 g' g ∈ G_m. Hence, g' = g g” g^-1∈ g G_m g^-1. In conclusion, G_m'⊆ g G_m g^-1. To sum up, G_m' = g G_m g^-1. Moreover, g G_m = G_m' g.Thirdly, let g' ∈ G_m. Then, g g' ∈ G_m, m'. In conclusion, g G_m ⊆ G_m, m'. Lastly, let g' ∈ G_m, m'. Then, g” = g^-1 g' ∈ G_m. Hence, g' = g g”∈ g G_m. In conclusion, G_m, m'⊆ g G_m. To sum up, g G_m = G_m, m'. Let G be a group and let H be a subgroup of G. The setGH = g Hg ∈ Gquotient set GH of G by H[symbols]GmoduloH@GHis called quotient set of G by H. Letbe a left group action of G on M and let m be an element of M. The quotient set GG_m is equal to G_m, m' m' ∈ Gm.Letbe a left group action of G on M. The set[symbols]G modulo-reverse M@GM GM = Gmm ∈ Morbit space GM of [symbols]GmoduloMback@GMis called orbit space of .Let M be a set and let 𝔑 be a subset of the power set of M. The set 𝔑 is called * pairwise disjointpairwise disjoint!setpairwise disjoint if and only ifN ∈𝔑 N' ∈𝔑 (N ≠ N'N ∩ N' = ∅); * cover of Mcover of M!setcover of M if and only if ⋃_N ∈𝔑 N = M;* partition of Mpartition of M!setpartition of M if and only if it is pairwise disjoint and a cover of M.The orbit space ofis a partition of M.Let M be a set, let N be a subset of M, and let f be a map from M to M. The set N is called invariant under fset invariant under f if and only if f(N) ⊆ N. Let M and M' be two sets, let f be a map from M to M', and letbe a left group action of G on M. The map f is called -invariantinvariant map@-invariant map-invariant map if and only ifg ∈ Gm ∈ Mf(gm) = f(m).Letbe a left group action of G on M and let N be a subset of M. The set N is called -invariantinvariant set@-invariant set-invariant set if and only if GN ⊆ N. Let M and M' be two sets, let f be a map from M to M', let G and G' be two groups, let φ be a group homomorphism from G to G', and letand ' be two left group actions of G on M and of G' on M' respectively. The tuple (f, φ) is called* (, ')-equivariantequivariant tuple 10@(, ')-equivariant tuple(, ')-equivariant tuple if and only if g ∈ Gm ∈ Mf(gm) = φ(g) ' f(m); * -equivariantequivariant tuple 20@-equivariant tuple-equivariant tuple if and only if it is (, ')-equivariant, M = M', and = '.Let M and M' be two sets, let f be a map from M to M', let G be a group, and letand ' be two left group actions of G on M and M' respectively. The map f is called * (, ')-equivariantequivariant map 1@(, ')-equivariant map(, ')-equivariant map if and only if the tuple (f, _G) is (, ')-equivariant;* -equivariantequivariant map 2@-equivariant map-equivariant map if and only if it is (, ')-equivariant, M = M', and = '.In the situation of <ref>, let f and φ be bijective, and let (f, φ) be (, ')-equivariant. The tuple (f^-1, φ^-1) is (', )-equivariant.For each g' ∈ G' and each m' ∈ M',f^-1(g' ' m')= f^-1(φ(φ^-1(g')) ' f(f^-1(m')))= f^-1(f(φ^-1(g')f^-1(m')))= φ^-1(g')f^-1(m').In the situation of <ref>, let f be bijective and (, ')-equivariant. The inverse f^-1 is (', )-equivariant. This is a direct consequence of <ref>. Let G be a group and let H be a subgroup of G. The group G acts transitively on the quotient set GH on the left by· G × GH→ GH, left group action · of G on GH[symbols]dotcentre@·[symbols]centredot@·(g, g' H)↦ g g' H. The map · is well-defined, because, for each g ∈ G, each g_1' ∈ G, and each g_2' ∈ G,g_1' H = g_2' Hg g_1' H = g g_2' H.It is a left group action, because, for each g' H ∈ GH, e_G · g' H = g' H,and, for each g_1 ∈ G, each g_2 ∈ G, and each g' H ∈ GH,g_1 g_2 · g' H= g_1 g_2 g' H= g_1 · g_2 g' H= g_1 · (g_2 · g' H).It is transitive, because, for each g_1' H ∈ GH and each g_2' H ∈ GH,g_2' g_1'^-1· g_1' H = g_2' H.After choosing an element m_0, a left-homogeneous space is isomorphic to GG_m_0, G, ·, which is shown inLetbe a transitive left group action of G on M, let m_0 be an element of M, and let G_0 be the stabiliser of m_0 under . The mapι M→ GG_0, (, ·)-equivariant bijection ι from M to GG_0[symbols]iota@ιm↦ G_m_0, m,is (, ·)-equivariant and bijective. For each g ∈ G and each m ∈ M,ι(gm) = G_m_0, gm = g · G_m_0, m = g ·ι(m).Hence, ι is (, ·)-equivariant. Moreover, for each (m, m') ∈ M × M with m ≠ m', we have G_m_0, m≠ G_m_0, m'. Thus, ι is injective. Furthermore, for each g G_0 ∈ GG_0, we have G_m_0, gm_0 = g G_0. Therefore, ι is surjective.After choosing an element m_0, a principal left-homogeneous space is isomorphic to G, which is shown inLetbe a free and transitive left group action of G on M and let m_0 be an element of M. The mapι M→ G,m↦ g_m_0, m,where g_m_0, m∈ G_m_0, m,is (, ·)-equivariant and bijective.Because G_m_0, m = 1 and G_m_0 = e_G, the map ι is well-defined, the quotient set GG_m_0 can be identified with G, and the statement is a direct consequence of <ref>. [Principal]Let ℳ = M, G, be a principal left-homogeneous space. Then, for each element m ∈ M and each element m' ∈ M, there is one and only one element g_m, m'∈ G such that g_m, m' m = m', in particular, because e_Gm = m, we have g_m, m = e_G, and, because g_m, m'^-1 m' = m, we have g_m', m = g_m, m'^-1.Let m_0 be an element of M and let M be equipped with the group multiplicationM × M→ M,(m, m')↦ g_m_0, m g_m_0, m' g_m_0, m^-1 m(= g_m_0, m g_m_0, m' m_0).Then, the element m_0 is the neutral element and, for each element m ∈ M, the element g_m, m_0 m_0 is the inverse element of m. Moreover, the maps{ι M→ G,m↦ g_m_0, m, } and {ι^-1 G→ M,g↦ gm_0, }are group isomorphisms that are inverse to each other. Under the identification of M with G by either isomorphism, which depends on the arbitrary choice of m_0, the left group actionis the group multiplication of M and of G. In the words of John Baez: A torsor [principal left-homogeneous space] is like a group that has forgotten its identity<cit.>. [Normal]Let ℳ = M, G, be a left-homogeneous space whose stabilisers are normal subgroups of G, which is for example the case if the group G is abelian. Then, because the stabilisers are conjugate to each other, they are all equal and we denote them by G_0. The mapGG_0 × M→ M,(g G_0, m)↦ gm,is a faithful, free, and transitive left group action of GG_0 on M and the triple ℳ' = M, GG_0, is a principal left-homogeneous space. Moreover, the quotient group GG_0 is isomorphic to G if and only if the stabiliser G_0 is trivial, which is the case if and only if the actionis free. Letbe a left group action of G on M, let H be a subgroup of G, let N be a subset of M such that HN ⊆ N, and let Q be a set. The group H acts on Q^N on the left byH × Q^N→ Q^N,(h, f)↦ [n ↦ f(h^-1 n)]. The mapis well-defined, because HN ⊆ N. It is a left group action, because, for each f ∈ Q^N and each n ∈ N,(e_Hf)(n) = f(e_Hn) = f(n),and, for each h ∈ H, each h' ∈ H, each f ∈ Q^N, and each n ∈ N,(h h'f)(n)= f((h')^-1 h^-1 n)= f((h')^-1 (h^-1 n))= (h'f)(h^-1 n)= (h(h'f))(n). § RIGHT QUOTIENT SET SEMI-ACTIONSIntroduction. The rotations of a circle act on it on the left by function application, because a rotation by x degrees, ρ_x, followed by a rotation by y degrees, ρ_y, is the same as the rotation by y + x degrees, ρ_y ρ_x, symbolically, ρ_y(ρ_x ) = (ρ_y ρ_x). They also act on the circle on the right by function application, because a rotation by x degrees, ρ_x, followed by a rotation by y degrees, ρ_y, is the same as the rotation by x + y degrees, ρ_x ρ_y, symbolically, (ρ_x) ρ_y =(ρ_x ρ_y). These actions commute with each other, because a rotation by x degrees followed by a rotation by y degrees is the same as a rotation by y degrees followed by a rotation by x degrees, symbolically, (ρ_x ) ρ_y = ρ_x(ρ_y). More succinctly, rotations act on the right by function application, and the left and right actions commute, because rotations commute under composition. The symmetries of a circle act on it on the left by function application. However, they do not act on it on the right by function application, because, for example, the rotation by 90 followed by the reflection about the vertical line v through the centre of the circle is not the same as the reflection about v followed by the rotation by 90, symbolically, (ρ_90) ϱ_v ≠ (ρ_90ϱ_v). In a sense, the problem is that reflections treat different points differently: Some points stay put, others are reflected to points close by, and still others to points far away.To solve this, let us fix a point on the circle and call it origin (beware, do not mistake this point for the origin of the space the circle may be embedded in). We want to define the right group semi-action such that a symmetry acts on each point as it does on the origin. For example, under the right semi-action, if a symmetry stabilises the origin, then it shall stabilise each point; and, if a symmetry throws the origin to its opposite point, then so it shall do with each point. So, a symmetry semi-acts on the right on a point by first rotating the point to the origin, secondly acting with the symmetry on the left, and lastly undoing the first rotation, symbolically, m σ = (ρ_m σρ_m^-1)m, where ρ_m denotes the rotation that rotates the origin to m. Note that m_0 σ = σ m_0. And, that this semi-action agrees with the right group action of the rotations on the circle. In particular, it is transitive.The identity map semi-acts trivially on each point on the right, symbolically, m= m. However, in general, the composition of two symmetries semi-acts in a different way on the right than the first symmetry does followed by the second, symbolically, m(σς) ≠ (m σ) ς. Yet, it can be seen that the difference is little in the sense that there is a symmetry ς_0 that stabilises the origin and may depend on m and σ such that m(σς) = (m σ)(ς_0 ς). Because of this property, the mapis a semi-action. Note that only the identity map and the reflection about the line ℓ through the centre of the circle and the origin stabilise the origin; and that ς_0 is the identity map, if σ and ς are both rotations or both reflections, and the reflection about ℓ, otherwise.The right semi-action semi-commutes with the left action, which means that first acting on the left and then semi-acting on the right is almost the same as first semi-acting on the right and then acting on the left, where the defect is again a symmetry that stabilises the origin. Symbolically, (σ m) ς = σ (m(ς_0 ς)), where ς_0 stabilises the origin and may depend on m and σ.For a point m on the circle, there are two symmetries that map the origin to m, the rotation ρ_m and the (roto-)reflection ρ_m ϱ_ℓ, where ϱ_ℓ is the reflection that stabilises the origin. Because these two symmetries semi-act the same way on each point on the right, the semi-actionis not free. The elements of the quotient set of the symmetries of the circle by the stabiliser of the origin, namely , ϱ_ℓ, are the sets ρ_m, ρ_m ϱ_ℓ for points m. This quotient set semi-acts on the circle by m ρ_m, ρ_m ϱ_ℓ = m ρ_m. So, it acts in the same way asbut is free, which means that, if m Σ = mT, then Σ = T.Under the identification of the quotient set, which is even a quotient group, with the rotations, the right quotient set semi-actionis identical to the right group actionof the rotations on the circle we considered at the beginning of this introduction. However, while the symmetry group of the circle has this nice subgroup, namely the rotation group, that acts freely and transitively on it on the right, the symmetry groups of other geometrical objects do not have such subgroups. Nevertheless, we can construct a free and transitive right quotient set semi-action on these geometrical objects as we did for the circle.Contents. In <ref> we introduce coordinate systems for left-homogeneous spaces as tuples made up of an origin and, for each point, a group element (think of a coordinate) that transports the origin to that point. In <ref> we introduce cell spaces as left-homogeneous spaces equipped with coordinate systems. In <ref> we introduce bigness of subgroups with respect to a coordinate system as containing all coordinates. In <ref> we introduce right quotient set semi-actions induced by cell spaces of the quotient set of the group by the stabiliser of the origin on the points, which is to the left group action what right multiplication is to the corresponding left group multiplication. In <ref> we show that semi-actions are free and transitive. In <ref> we show that semi-actions semi-commute with their corresponding left group action and exhaust their defect with respect to this semi-commutativity in the origin. And in <ref> we show that under the identification of the quotient set with the points, left group actions on quotient sets by multiplication and right quotient set semi-actions can be expressed in terms of left group actions on points. Let M be a set, let I be a set, and let f be a map from I to M. The map f is called family m_i_i ∈ I of elements in M indexed by Ifamily of elements in M indexed by I and denoted by m_i_i ∈ I[symbols]miiinI@m_i_i ∈ I, where m_i = f(i), for i ∈ I.Let ℳ = M, G, be a left-homogeneous space, let m_0 be an element of M, let g_m_0, m_0 be the neutral element of G, and, for each element m ∈ M ∖m_0, let g_m_0, m be an element of G such that g_m_0, m m_0 = m. The tuple 𝒦 = m_0, g_m_0, m_m ∈ M is called coordinate system for ℳcoordinate system 𝒦 for ℳ[symbols]Kcalligraphic@𝒦; the element m_0 is called originorigin m_0[symbols]m0@m_0; for each element m ∈ M, the element g_m_0, m is called coordinate of mcoordinate g_m_0, m of m[symbols]gm0m@g_m_0, m; for each subgroup H of G, the stabiliser of the origin m_0 under _H, which is G_m_0∩ H, is denoted by H_0stabiliser H_0 of m_0 under _H[symbols]H0@H_0, in particular, the stabiliser G_m_0 is denoted by G_0stabiliser G_0 of m_0 under [symbols]G0@G_0. Let ℳ = M, G, be a left-homogeneous space and let 𝒦 = m_0, g_m_0, m_m ∈ M be a coordinate system for ℳ. The tuple ℛ = ℳ, 𝒦 is called cell spacecell space ℛ[symbols]Rcalligraphic@ℛ, each element m ∈ M is called cellcell m[symbols]m@m, and each element g ∈ G is called symmetrysymmetry g[symbols]g@g. [Group]In the situation of <ref>, let m_0 be the neutral element e_G of G and, for each element m ∈ G, let g_m_0, m be the only element in G such that g_m_0, m m_0 = m, namely m. The tuple 𝒦 = m_0, g_m_0, m_m ∈ G is a coordinate system for ℳ = G, G, · and the tuple ℛ = ℳ, 𝒦 is a cell space. [Plane]In the situation of <ref>, let m_0 be the origin (0,0)^ of M and, for each point m ∈ M, let g_m_0, m be the translation + m that translates m_0 to m. Note that g_m_0, m_0 is the identity map. The tuple 𝒦 = m_0, g_m_0, m_m ∈ M is a coordinate system for ℳ = M, G, and the tuple ℛ = ℳ, 𝒦 is a cell space. [Sphere]In the situation of <ref>, let m_0 be the north pole (0,0,1)^ of M and, for each point m ∈ M, let g_m_0, m be a rotation about an axis in the (x, y)-plane that rotates m_0 to m, which is unique unless m is the south pole. Note that g_m_0, m_0 is the identity map. The tuple 𝒦 = m_0, g_m_0, m_m ∈ M is a coordinate system for ℳ = M, G, and the tuple ℛ = ℳ, 𝒦 is a cell space. Let ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space and let H be a subgroup of G. The group H is called 𝒦-bigbig@𝒦-big𝒦-big if and only ifm ∈ Mg_m_0, m∈ H.[Group]In the situation of <ref>, because the set g_m_0, m m ∈ G is G, the only 𝒦-big subgroup of G is the group G. [Plane]In the situation of <ref>, because the set g_m_0, m m ∈ M is the set T of translations, the subgroup T of G is 𝒦-big; and, because G is the inner semi-direct product of the rotations R_0 about m_0 acting on T, each inner semi-direct product of a subgroup of R_0 acting on T is a 𝒦-big subgroup of G.[Sphere]In the situation of <ref>, because the set g_m_0, m m ∈ M generates G, the only 𝒦-big subgroup of G is the group G.The terms coordinate system and big are due to <cit.>. The subgroup of G that is generated by the set g_m_0, m m ∈ M of coordinates, is the smallest 𝒦-big subgroup of G, where smallest means that it is included in each 𝒦-big subgroup of G. In the remainder of this section, let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space.The mapM × GG_0→ M, right quotient set semi-actionof GG_0 on M with defect G_0[symbols]arrowleftunderscore@(m, g G_0)↦ g_m_0, m g g_m_0, m^-1 m(= g_m_0, m gm_0),is a right quotient set semi-action of GG_0 on M with defect G_0, which means that, for each 𝒦-big subgroup H of G,m ∈ MmG_0 = m,andm ∈ Mh ∈ Hh_0 ∈ H_0 𝔤∈ GG_0 mh ·𝔤 = (mh G_0)h_0 ·𝔤.For each m ∈ M,mG_0 = me_G G_0 = g_m_0, m e_Gm_0 = g_m_0, m m_0 = m.Let H be a 𝒦-big subgroup of G. Furthermore, let m ∈ M and let h ∈ H. Put h_0 = g_m_0, g_m_0, m hm_0^-1 g_m_0, m h. Because g_m_0, g_m_0, m hm_0^-1∈ H, g_m_0, m∈ H, andh_0m_0= g_m_0, g_m_0, m hm_0^-1 (g_m_0, m hm_0)= m_0,we have h_0 ∈ H_0. Moreover, for each g G_0 ∈ GG_0,mh · g G_0= mh g G_0= g_m_0, m h gm_0= g_m_0, g_m_0, m hm_0 h_0 gm_0= (g_m_0, m hm_0)h_0 g G_0= (mh G_0)h_0 · g G_0. The second property of the right quotient set semi-actionis equivalent to the following: For each 𝒦-big subgroup H of G,m ∈ Mh ∈ Hh_0 ∈ H_0 𝔤∈ GG_0(mh G_0) 𝔤 = mh h_0 ·𝔤.[Principal]In the situation of <ref>, the tuple 𝒦 = m_0, g_m_0, m_m ∈ M is a coordinate system for ℳ, the stabiliser of m_0 underand defect ofis the trivial subgroup of G, and, under the identification of GG_0 with G and of G with M as in <ref>, the induced semi-actionis the group multiplication on M from <ref> (note the similarity of their definitions).[Group]In the situation of <ref>, the stabiliser G_0 of the neutral element m_0 under · is the trivial subgroup e_G of G and, for each element m ∈ G and each element g ∈ G, we have mg G_0 = g_m_0, m g g_m_0, m^-1 m = m g m^-1 m = m g. Under the natural identification of GG_0 with G, the induced semi-actionis the right group action of G on itself by right multiplication. [Plane] In the situation of <ref>, the stabiliser G_0 of the origin m_0 underis the group of rotations about m_0. The special Euclidean group G is the inner semi-direct product of G_0 acting on the abelian group T of translations. Under the identification of GG_0 with T by t G_0 ↦ t, the induced semi-actionis the right group action of T on M by function application.[Sphere]In the situation of <ref>, the stabiliser G_0 of the north pole m_0 underis the group of rotations about the z-axis. An element g G_0 ∈ GG_0 semi-acts on a point m on the right by the induced semi-actionby first rotating m to m_0, g_m_0, m^-1 m = m_0, secondly rotating m_0 as prescribed by g, g g_m_0, m^-1 m = gm_0, and thirdly undoing the first rotation, g_m_0, m g g_m_0, m^-1 m = g_m_0, m (gm_0), in other words, by first changing the rotation axis of g such that the new axis stands to the line through the centre and m as the old one stood to the line through the centre and m_0, g_m_0, m g g_m_0, m^-1, and secondly rotating m as prescribed by this new rotation.Let N_0 be a subset of the sphere M, which we think of as a geometrical object on the sphere that has its centre at m_0, for example, a circle of latitude. The set N = g G_0 ∈ GG_0gm_0 ∈ N_0 = G_m_0, m m ∈ N_0 = g_m_0, m G_0m ∈ N_0 can be thought of as a realisation of N_0 in GG_0, because m_0N = g_m_0, m_0g_m_0, m m ∈ N_0 m_0 = N_0. Furthermore, for each point m ∈ M, the set mN = g_m_0, m N_0 has the same shape and size as N_0 but its centre at m. Note that N = ι(N_0), where ι is the bijection from <ref>. The maps{ι M→ GG_0,m↦ G_m_0, m, } and { m_0GG_0→ M, 𝔤 ↦ m_0 𝔤, }are inverse to each other and, under the identification of GG_0 with M by either of these maps,g ∈ G 𝔤∈ GG_0 ≃ Mg ·𝔤 = g 𝔤,andm ∈ M 𝔤∈ GG_0 ≃ Mm 𝔤 = g_m_0, m𝔤. According to <ref>, the map ι is bijective and, for each m ∈ M,m_0 ι(m)= m_0G_m_0, m= m_0g_m_0, m G_0= g_m_0, m_0 g_m_0, m m_0= e_Gm= m.Therefore, m_0= ι^-1. Moreover, according to <ref>, the map ι is (, ·)-equivariant. Hence, for each g ∈ G and each 𝔤∈ GG_0,g ·𝔤 = g ·ι(ι^-1(𝔤)) = ι(g ι^-1(𝔤)).And, for each m ∈ M and each g G_0 ∈ GG_0,mg G_0= g_m_0, m gm_0= g_m_0, m (gm_0)= g_m_0, mι^-1(G_m_0, gm_0)= g_m_0, mι^-1(g G_0).[Group]In the situation of <ref>, the maps m_0 and ι are the identity maps on G ≃ GG_0. [Plane]In the situation of <ref>, the maps m_0 and ι map translations to points, which encode translation vectors, and vice versa. [Sphere]In the situation of <ref>, under the identification of GG_0 with g_m_0, m m ∈ M by g_m_0, m G_0 ↦ g_m_0, m, the maps m_0 and ι map rotations to points, which encode rotation angles and axes in the (x, y)-plane, and vice versa. The semi-actionis similar toin m_0, which means thatg ∈ Gm_0g G_0 = gm_0.In particular,g ∈ G 𝔤∈ GG_0m_0g ·𝔤 = g(m_0 𝔤),andm ∈ M 𝔤∈ GG_0m 𝔤 = g_m_0, m (m_0 𝔤). The similarity follows from the fact that g_m_0, m_0 = e_G. And the other two properties follow from the fact thatis a left group action. The semi-actionis *freefree, which means that𝔤∈ GG_0 𝔤' ∈ GG_0 ( m ∈ Mm 𝔤 = m 𝔤') 𝔤 = 𝔤'; *transitivetransitive, which means that the set M is non-empty andm ∈ Mm' ∈ M 𝔤∈ GG_0m 𝔤 = m'.* Let m ∈ M and let m' ∈ M. Put m” = g_m_0, m^-1 m'. Becauseis transitive, there is a g ∈ G such that gm_0 = m”. Hence,mg G_0= g_m_0, m gm_0= g_m_0, m (gm_0)= g_m_0, m m”= g_m_0, m (g_m_0, m^-1 m')= e_Gm'= m'. * Let g G_0 and g' G_0 be two elements of GG_0, and let m be an element of M such that mg G_0 = mg' G_0. Then, g_m_0, m gm_0 = g_m_0, m g'm_0. Hence, gm_0 = g'm_0. Therefore, g^-1 g'm_0 = m_0. Thus, g^-1 g' ∈ G_0. In conclusion, g G_0 = g' G_0.[Group]In the situation of <ref>, the semi-action , being but right multiplication, is free and transitive. [Plane]In the situation of <ref>, the semi-action , being but translation, is free and transitive. [Sphere]In the situation of <ref>, under the identification of GG_0 with M by ι, according to <ref>, for each point m ∈ M, the map m is the rotation by g_m_0, m, which is injective and surjective, and therefore the semi-actionis free and transitive.The semi-action*semi-commutes with semi-commutes with commutes with@semi-commutes with , which means that, for each 𝒦-big subgroup H of G,m ∈ Mh ∈ Hh_0 ∈ H_0 𝔤∈ GG_0 (hm) 𝔤 = h(mh_0 ·𝔤); *exhausts its defect with respect to its semi-commutativity within m_0exhaust ones defect with respect to ones semi-commutativity within m_0, which means that g_0 ∈ G_0 𝔤∈ GG_0(g_0^-1 m_0) 𝔤 = g_0^-1 (m_0g_0 ·𝔤).Let H be a 𝒦-big subgroup of G. * Let h ∈ H and let m ∈ M. Put h_0 = g_m_0, m^-1 h^-1 g_m_0, hm. Because g_m_0, m^-1∈ H, g_m_0, hm^-1∈ H, andg_m_0, m^-1 h^-1 g_m_0, hm m_0= g_m_0, m^-1 h^-1 (hm)= g_m_0, m^-1 m= m_0,we have h_0 ∈ H_0. Moreover, for each g G_0 ∈ GG_0,(hm)g G_0= g_m_0, hm gm_0= h g_m_0, m h_0 gm_0= h(g_m_0, m h_0 gm_0)= h(mh_0 · g G_0). * For each g_0 ∈ G_0 and each g G_0 ∈ GG_0, because g_m_0, m_0 = e_G,(g_0^-1 m_0)g G_0= m_0g G_0= g_m_0, m_0 gm_0= g_0^-1 g_m_0, m_0 g_0 gm_0= g_0^-1 (g_m_0, m_0 g_0 gm_0)= g_0^-1 (m_0g_0 g G_0)= g_0^-1 (m_0g_0 · g G_0). [Group]In the situation of <ref>, because group multiplication is associative, the right multiplicationcommutes with the left multiplication . [Plane]In the situation of <ref>, because the group T of translations is abelian, the right semi-actioncommutes with the left action _T. Indeed,t ∈ T 𝔱∈ T(t _T ) 𝔱 = 𝔱(t()) = t(𝔱()) = t _T (𝔱).[Cayley Graph]Let M be a group, let S be a generating set of M, and let 𝒢 = M, E, λ be the coloured S-Cayley graph of M. The graph 𝒢 is edge-labelled and directed, its set of vertices M is the set underlying M, its set of edges E is the subset (m, m s)m ∈ M, s ∈ S of M × M, and its edge-labelling λ is the map E → S, (m, m s) ↦ s.The automorphism group of 𝒢, namelyG =gM → Mbijective m ∈ Mm' ∈ Ms ∈ S (m, m') ∈ E λ(m, m') = s (g(m), g(m')) ∈ E λ(g(m), g(m')) = s ,acts freely and transitively on M on the left by function application, which we denote by . For each element m ∈ M, left multiplication by m, which we denote by m ·, is the unique graph automorphism that maps the neutral element e_M to m (note that all graph automorphisms are of this form).The triple ℳ = M, G, is a principal left-homogeneous space and the tuple 𝒦 = e_M, m ·_m ∈ M is a coordinate system for ℳ. Under the identification of GG_0 with G, the induced semi-actionis the right group action of G on M given by (m, g) ↦ m · g(e_M). [Normal]In the situation of <ref>, let 𝒦 = m_0, g_m_0, m_m ∈ M be a coordinate system for ℳ. The tuple 𝒦' = m_0, g_m_0, m G_0_m ∈ M is the one and only coordinate system for ℳ' with origin m_0. Both coordinate systems induce the right quotient group actionof GG_0 on M given by (m, g G_0) ↦ g_m_0, m gm_0. In the case that the group G is abelian, this action is given by (m, g G_0) ↦ gm, which is, apart from the order of the arguments, identical to . In any case, under the identification of M with GG_0 by ι, the left and right quotient group actionsandare identical to the group multiplication of GG_0. And, the right quotient group actioncommutes with the left group action . § SEMI-CELLULAR, BIG-CELLULAR, AND CELLULAR AUTOMATAIntroduction. Let us consider the following discrete-time dynamical system on a circle whose points can be coloured black and white. In one time step a point becomes white if there is a white point nearby in the clockwise direction, where two points are said to be near each other if their arc distance is not greater than, say, one-hundredth of the circle's circumference; and otherwise retains its colour. The time evolution of that system is uniform, in the sense that each point determines its next colour by the same rule, and it is local, in the sense that each point determines its next colour by means of the colours of nearby points. Moreover, it is equivariant under rotations of the circle, but, despite its uniformity, it is not equivariant under reflections.For example, if the left side of the circle, without the top and bottom points, is white and the other points are black, then in one time step the top point stays black; but, if we first reflect the circle about the vertical line through the centre of the circle, secondly do one time step, and lastly reflect again, then the top point is white. The reason is that the local rule uses the direction of rotation, namely clockwise, which stays the same under rotations but changes under reflections. If the rule considered all nearby points regardless of whether they lie in the clockwise or anticlockwise direction, then time evolution would be equivariant under all symmetries of the circle.The time evolution of this dynamical system is the global transition function of a semi-cellular automaton whose cells are the points of the circle, whose states are the colours black and white, whose neighbourhood is the set of all nearby points (in either direction) of a designated point, and whose local transition function is the local rule for the designated point described above. Actually, the nearby points in the clockwise direction would be sufficient as neighbourhood, but those are not invariant under the reflection that stabilises the designated point and for technical reasons we want this invariance.Without designating a point, the neighbourhood can be described by all rotations that map a point to a point nearby and the local transition function by a map that maps a local configuration to the colour white if there is a white clockwise neighbour, where we call a neighbour clockwise if it maps a point to a point nearby in the clockwise direction; and otherwise to the colour of the identity map, which plays the role of the designated point. The neighbourhood of a point can be recovered by applying the (relative) neighbours to the point, in other words, by acting with the (relative) neighbours on the point on the right. As we have seen in the introduction of <ref>, this right action is equivalent to the right semi-action induced by the left action of the symmetries on the circle, where the rotations are identified with the quotient set of the symmetries of the circle by the stabiliser of a designated point. Contents. A semi-cellular automaton is made up of a cell space, a set of states, a neighbourhood (think of a disk about the origin as points or vectors), and a local transition function (see <ref>). The stabiliser of the origin acts on local configurations (think of rotations of disk-shaped patterns; see <ref>). The group acts on global configurations (think of rotations and translations of unbounded patterns; see <ref>). A cell observes a local configuration by first semi-acting on the right on itself with the (relative) neighbourhood to determine its neighbours (think of point-vector additions) and secondly reading the states of these neighbours (see <ref>); or, alternatively, by first translating the global configuration such that the cell is moved to the origin and secondly restricting this translated configuration to the neighbourhood (see <ref>). The global transition function applies the local transition function synchronously to the observed local configuration of each cell to determine its new state (see <ref> and <ref>). This function is equivariant under the action on global configurations (traditionally known as shift-invariance) if and only if the local transition function is invariant under the action on local configurations (see <ref>). If the latter is the case, then the automaton is a cellular automaton (see <ref>). And, if the local transition function is only invariant under a restriction of the action on local configurations to a big subgroup, then the automaton is a big-cellular automaton (see <ref>). Body. In this section, let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space.Let Q be a set, let N be a subset of GG_0 such that G_0 · N ⊆ N, and let δ be a map from Q^N to Q. The quadruple 𝒞 = ℛ, Q, N, δ is called semi-cellular automatonsemi-cellular automaton 𝒞automaton!semi-cellular[symbols]Ccalligraphic@𝒞, each element q ∈ Q is called statestate q[symbols]q@q, the set N is called neighbourhoodneighbourhood N[symbols]N@N, each element n ∈ N is called neighbourneighbour n[symbols]n@n, and the map δ is called local transition functionlocal transition function δtransition function!local[symbols]delta@δ.Under the identification of GG_0 with M by ι, the neighbourhood N is a subset of M such that G_0N ⊆ N [Group]In the situation of <ref>, the semi-cellular automata over ℛ are the usual cellular automata over the group G. [Plane]In the situation of <ref>, let Q be the set of real numbers, let M be identified with GG_0 by ι, let ε be a positive real number, let N be the open disk of radius ε about m_0, and letδ Q^N→ Q, ℓ ↦∂^2 ℓ/∂ x^2(m_0) + ∂^2 ℓ/∂ y^2(m_0),[t].41if ℓ is twice continuously differentiable on an open disk about m_0, 0,otherwise,where ∂^2 ℓ / ∂ x^2(m_0) and ∂^2 ℓ / ∂ y^2(m_0) are the second-order partial derivatives of ℓ by x and y at m_0. The quadruple 𝒞 = ℛ, Q, N, δ is a semi-cellular automaton. [Sphere]In the situation of <ref>, let Q be the set 0,1, let N_0 be the union of all circles of latitude between 45 and 90 north, which is a curved circular disk of radius π/4 with the north pole m_0 at its centre, let N be the set ι(N_0) = G_m_0, m m ∈ N_0, and let δ Q^N→ Q, ℓ ↦ 0,if ∀ n ∈ N ℓ(n) = 0,1,if ∃ n ∈ N ℓ(n) = 1.The quadruple 𝒞 = ℛ, Q, N, δ is a semi-cellular automaton. In the remainder of this section, let 𝒞 = ℛ, Q, N, δ be a semi-cellular automaton. Each map ℓ∈ Q^N is called local configurationlocal configuration ℓconfiguration!local[symbols]l@ℓ. The stabiliser G_0 acts on Q^N on the left by∙ G_0 × Q^N→ Q^N, left group action ∙ of G_0 on Q^N[symbols]bullet@∙(g_0, ℓ)↦ [n ↦ℓ(g_0^-1· n)]. Under the identification of GG_0 with M by ι, we have ∙ (g_0, ℓ) ↦ [n ↦ℓ(g_0^-1 n)]. The semi-cellular automaton 𝒞 is called big-cellular automaton 𝒞big-cellular automatonautomaton!big-cellularcellular automaton!big[symbols]Ccalligraphic@𝒞 if and only if there is a 𝒦-big subgroup H of G such that the local transition function δ is ∙_H_0-invariant. Let H be a 𝒦-big subgroup of G and let π be the canonical projection of Q^N onto H_0Q^N. The local transition function δ is ∙_H_0-invariant if and only if there is a map 𝔡 H_0Q^N → Q such that δ = 𝔡π, in other words, if and only if the map 𝔡 H_0Q^N → Q, H_0 ∙ℓ↦δ(ℓ), is well-defined. Because each 𝒦-big subgroup of G includes the 𝒦-big subgroup of G that is generated by the set g_m_0, m m ∈ M of coordinates, the semi-cellular automaton 𝒞 is a big-cellular automaton if and only if its local transition function δ is ∙_G_0 ∩g_m_0, m m ∈ M-invariant. The semi-cellular automaton 𝒞 is called cellular automaton 𝒞cellular automatonautomaton!cellular[symbols]Ccalligraphic@𝒞 if and only if its local transition function δ is ∙-invariant. [Group]In the situation of <ref>, the stabiliser G_0 of the neutral element m_0 is the trivial subgroup e_G of G. Therefore, each semi-cellular automaton over ℛ has a ∙-invariant local transition function and is hence a cellular automaton. [Plane]In the situation of <ref>, think of the neighbourhood N as being embedded in the (x,y)-plane of the three-dimensional Euclidean space and of local configurations as graphs extending into the z-direction. The rotations G_0 about the z-axis act on these graphs by ∙ by rotating them. For each rotation g_0 ∈ G_0 and each local configuration ℓ∈ Q^N, the map ℓ is twice continuously differentiable on an open disk about m_0 if and only if g_0 ∙ℓ is; if they both are, then δ(ℓ) = δ(g_0 ∙ℓ) by elementary calculus; otherwise, δ(ℓ) = 0 = δ(g_0 ∙ℓ). Hence, the local transition function δ is ∙-invariant. Therefore, the quadruple 𝒞 is a cellular automaton.[Sphere]In the situation of <ref>, think of 0 as black, 1 as white, and of local configurations as black-and-white patterns on N_0 = m_0N. The rotations G_0 about the z-axis act on these patterns by ∙ by rotating them. The local transition function δ maps the black pattern to 0 and all others to 1, which is invariant under rotations. Therefore, the quadruple 𝒞 is a cellular automaton. The set Q^M is called phase spacephase space Q^M[symbols]QM@Q^M and each map c ∈ Q^M is called global configurationglobal configuration cconfiguration!global[symbols]c@c. The group G acts on Q^M on the left byG × Q^M→ Q^M, left group actionof G on Q^M[symbols]arrowrightblack@(g, c)↦ [m ↦ c(g^-1 m)].For each cell m ∈ M, the set mN is called neighbourhood mN of mneighbourhood of mneighbourhood!of m[symbols]marrowleftunderscoreN@mN. And, for each global configuration c ∈ Q^M and each cell m ∈ M, the local configurationc(m )_NN→ Q, local configuration c(m )_N observed by m in cn↦ c(mn),is called observed by m in clocal configuration!observed by m in cconfiguration!local!observed by m in c. Under the identification of GG_0 with M by ι, according to the proof of the forthcoming <ref>, for each cell m ∈ M, we have c(m )_N = (g_m_0, m^-1 c)_N. Because the semi-actionis free (see <ref> of lemma <ref>), for each local configuration ℓ∈ Q^N and each cell m ∈ M, there is a global configuration c ∈ Q^M such that the local configuration observed by m in c is ℓ.The mapΔ Q^M→ Q^M, global transition function Δtransition function!global[symbols]Delta@Δc↦ [m ↦δ(n ↦ c(mn))],is called global transition function. Under the identification of GG_0 with M by ι, according to <ref>, we have Δ c ↦ [m ↦δ((g_m_0, m^-1 c)_N)].[Plane]In the situation of <ref>, the restriction of the global transition function Δ to the twice continuously differentiable maps is known as Laplace operatorLaplace operator. Recall that the neighbourhood N is an open disk of radius ε about m_0, where ε is a positive real number. The map Δ does not depend on the radius ε — it can be chosen arbitrarily small without affecting Δ. In other words, there is no smallest neighbourhood for cellular automata whose global transition functions are Δ.[Sphere]In the situation of <ref>, repeated applications of the global transition function of 𝒞 grows white regions on M. [Hyperbolic Game of Life] Sébastien Moriceau presents an adaptation of Conway's Game of Life cellular automaton on a tessellation of the hyperbolic plane in example 3.3 (a) in <cit.>. For each subset A of M and each global configuration c ∈ Q^M, the states of the cells A in Δ(c) depend at most ondepend at most on the states of the cells AN in c, which means thatA ⊆ Mc ∈ Q^Mc' ∈ Q^M c_AN = c'_ANΔ(c)_A = Δ(c')_A.The global transition function Δ is determined by its behaviour at the origin m_0determined by its behaviour at the origin m_0, which means thatc ∈ Q^Mm ∈ M Δ(c)(m) = Δ(g_m_0, m^-1 c)(m_0).Indeed, for each global configuration c ∈ Q^M and each cell m ∈ M, according to <ref>,Δ(c)(m)= δn ↦ c(mn)= δn ↦ c(g_m_0, m (m_0n))= δn ↦ (g_m_0, m^-1 c)(m_0n)= Δ(g_m_0, m^-1 c)(m_0). Let E be a subset of N. It is called sufficient neighbourhood Esufficient neighbourhoodneighbourhood!sufficient[symbols]E@E if and only ifℓ∈ Q^N ℓ' ∈ Q^N []ℓ_E = ℓ'_E δ(ℓ) = δ(ℓ'),in which case the mapη Q^E→ Q, ℓ_E↦δ(ℓ),where ℓ∈ Q^N such that ℓ_E = ℓ_E,is called sufficient local transition functionsufficient local transition function ηlocal transition function!sufficient[symbols]eta@η.The neighbourhood itself is a sufficient neighbourhood. And, if the local transition function depends on all neighbours, then the neighbourhood is the only one. In general, it is impossible to choose the neighbourhood such that the local transition function depends on all neighbours, because it may depend only on an arbitrarily small disk about the origin (as in <ref>) or it may depend on a neighbour n but not on g_0 · n for some stabiliser g_0 of the origin (as in the example in the introduction of <ref>). Note that, because G_0 · N ⊆ N, for each n ∈ N, we have g_0 · n ∈ N. So, the assumption G_0 · N ⊆ N may force the neighbourhood to be larger than is actually necessary to determine the next state of a cell.Recall that the right quotient set semi-actionsemi-commutes with the left group action , symbolically, (g^-1 m)n = g^-1 (mg_0 · n), where g_0 does not depend on n. Thus, the local configuration that is observed by a cell g^-1 m in a global configuration c is a rotation by g_0 of the local configuration that is observed by m in gc, symbolically, c((g^-1 m) ) = g_0^-1∙ ((gc)(m )) (see <ref>). Hence, because local transition functions of cellular automata are ∙-invariant, their global transition functions are -equivariant (see <ref>). Moreover, this property of observed local configurations is also essential in the proofs of other theorems of <ref>.Let m be an element of M, let g be an element of G, and let g_0 be an element of G_0 such thatn ∈ N(g^-1 m)n = g^-1 (mg_0 · n).For each global configuration c ∈ Q^M,[n ↦ c((g^-1 m)n)] = g_0^-1∙ [n ↦ (gc)(mn)]. For each global configuration c ∈ Q^M,[n ↦ c((g^-1 m)n)]= [n ↦ c(g^-1 (mg_0 · n))]= g_0^-1∙ [n ↦ c(g^-1 (mn))]= g_0^-1∙ [n ↦ (gc)(mn)].In the definition of a semi-cellular automaton, instead of a (relative) neighbourhood N and a local transition function δ, we could have used a neighbourhood N_0 of m_0 (see <ref>) and a local transition function δ_0 of m_0 (see <ref>). Then, in the definition of the global transition function, to determine the next state of a cell, we could have translated the global configuration such that the cell is translated to m_0, restricted this translation to the neighbourhood of m_0, and applied the local transition function of m_0 (see <ref>). Note that under the identification of GG_0 with M by ι, we have N = N_0 and δ = δ_0. The set N_0 = m_0N is called neighbourhood N_0 of m_0neighbourhood of m_0neighbourhood!of m_0[symbols]N0@N_0.The mapδ_0Q^N_0 → Q, local transition function δ_0 of m_0[symbols]delta0@δ_0 ℓ_0↦δ(n ↦ℓ_0(m_0n)),is called local transition function of m_0local transition function!of m_0. The global transition function Δ of 𝒞 is identical to the mapΔ_0Q^M→ Q^M,c↦ [m ↦δ_0((g_m_0, m^-1 c)_N_0)]. Let c ∈ Q^M and let m ∈ M. For each n = g G_0 ∈ N,mn= g_m_0, m gm_0= g_m_0, m g_m_0, m_0 gm_0= g_m_0, m (g_m_0, m_0 gm_0)= g_m_0, m (m_0n)and thusc(mn) = c(g_m_0, m (m_0n)) = (g_m_0, m^-1 c)(m_0n).Therefore,Δ(c)(m)= δ(n ↦ c(mn))= δ(n ↦ (g_m_0, m^-1 c)(m_0n))= δ_0(n_0 ↦ (g_m_0, m^-1 c)(n_0))= δ_0((g_m_0, m^-1 c)_N_0)= Δ_0(c)(m).In conclusion, Δ = Δ_0. [Left Shift Map]Let M be the one-dimensional integer lattice , let G be the group τ_tm ↦ t + mt ∈ of translations of M, and letbe the left group action of G on M by function application. Moreover, let m_0 be the origin 0 and, for each point m ∈ M, let g_m_0, m be the translation τ_m. Furthermore, let Q be the set 0, 1, let N be the set τ_1, let δ be the map Q^N → Q, ℓ↦ℓ(τ_1), and let G be identified with GG_0 by τ_t ↦τ_t G_0, where G_0 is the stabiliser τ_0 of the origin m_0 under . The triple ℳ = M, G, is a principal left-homogeneous space, the tuple 𝒦 = m_0, g_m_0, m_m ∈ M is a coordinate system for ℳ, the quadruple 𝒞 = ℳ, 𝒦, Q, N, δ is a cellular automaton whose global transition function Δ is the left shift mapleft shift mapshift map!left Q^M → Q^M, c ↦ c( + 1). Under the additional identification ofwith G by t ↦τ_t, the quadruple 𝒞 is a traditional cellular automaton, more precisely, the actionis the addition + on , the neighbourhood N is the set 1, and the local transition function δ is the map ℓ↦ℓ(1). § INVARIANCE, EQUIVARIANCE, DETERMINATION, AND COMPOSITION OF GLOBAL TRANSITION FUNCTIONSInvariance, Equivariance, Determination, and Composition Contents. In <ref> we show how to turn a big-cellular automaton over one coordinate system into an automaton over another system that has the same global transition function. Conversely, in <ref> we show that two big-cellular automata with the same global transition function are related as in <ref> except for superfluous neighbours. In <ref> we show that a global transition function does not depend on the choice of coordinates. In <ref> we show that a global transition function is _H-equivariant if and only if its local transition function is ∙_H_0-invariant. In <ref> we show that a global transition function is determined by its behaviour in the origin. And in <ref> we show that the composition of two global transition functions is a global transition function.§.§ Invariance Under Change of Coordinates of Global Transition Functions Summary. Conjugation by a group element g is a bijection from a quotient set GG_m to GG_gm (see <ref>). Under the identifications of such quotient sets with M, all these conjugations together are in a sense the left group action(see <ref>). They are used to relate the right quotient set semi-action induced by one coordinate system to the semi-action induced by another system (see <ref>) and to turn a semi-cellular automaton over one coordinate system into an automaton in another system that has the same global transition function (see <ref>). It follows from the specifics of the latter construction that a global transition function does not depend on the choice of coordinates (see <ref>). And it follows that the set of global transition functions of big-cellular automata over one coordinate system is the same as the one over another system (see <ref>).Conversely, if two big-cellular automata have identical global transition functions, then they are related as in <ref> except for superfluous neighbours (see <ref>). It follows that two such automata over coordinate systems with the same origin are the same except for superfluous neighbours. Letbe a left group action of G on M. The group G acts on ⋃_m ∈ M GG_m on the left byG ×⋃_m ∈ M GG_m→⋃_m ∈ M GG_m, left group actionof G on ⋃_m ∈ M GG_m[symbols]circle@(g, g' G_m)↦ g g' G_m g^-1 (= g g' g^-1 G_gm),such that, for each element g ∈ G and each element m ∈ M, the map(g )_GG_m → GG_gm GG_m→ GG_gm,g' G_m↦ gg' G_m,is bijective.First, let g ∈ G, let m ∈ M, let g' G_m ∈ GG_m, and let m' = gm. Then, according to <ref>, we have G_m' = g G_m g^-1. Hence,g g' G_m g^-1 = g g' (g^-1 g) G_m g^-1= (g g' g^-1) (g G_m g^-1)= g g' g^-1 G_m'∈ GG_m'.In conclusion, the mapsand (g )_GG_m → GG_gm are well-defined.Secondly, let g G_m ∈⋃_m ∈ M GG_m. Then, e_Gg G_m = g G_m. Moreover, for each g' ∈ G and each g”∈ G,g' g” g G_m= g' g” g G_m (g”)^-1 (g')^-1= g'g” g G_m (g”)^-1= g'(g” g G_m).In conclusion, the mapis a left group action.Thirdly, for each g ∈ G and each m ∈ M, the map (g )_GG_m → GG_gm is bijective, because its inverse is (g^-1)_GG_gm→ GG_m. For each element m_0 ∈ M, let ι_m_0 be the (, ·)-equivariant bijection from <ref>. For each element g ∈ G, each element m ∈ M, and each element m' ∈ M,g ι_m(m') = gG_m, m' = G_gm, gm' = ι_gm(gm').In this sense, the mapis the left group action . Let g be an element of G, let m be an element of M, and let N be a subset of GG_m such that G_m · N ⊆ N. Then, G_gm· (gN) ⊆ gN. Indeed,G_gm· (gN)= (g G_m g^-1) · g N g^-1= g (G_m · N) g^-1⊆ g N g^-1= gN.The group actionsand · commute in the sense given inLetbe a left group action of G on M, let m be an element of M, let g and g' be two elements of G, and let g” G_m be an element of GG_m. Then, g(g' · g” G_m) = g g' g^-1· (gg” G_m). Indeed,g(g' · g” G_m)= g g' g” g^-1 G_gm= g g' g^-1· (g g” g^-1 G_gm)= g g' g^-1· (gg” G_m).Let ℳ = M, G, be a left-homogeneous space, and let 𝒦 = m_0, g_m_0, m_m ∈ M and 𝒦' = m_0', g_m_0', m'_m ∈ M be two coordinate systems for ℳ. The right quotient set semi-actionsand ' of GG_0 and GG_0' on M are similarsimilar, which means that, for each 𝒦- and 𝒦'-big subgroup H of G and each element h ∈ H such that hm_0 = m_0',m ∈ Mh_0 ∈ H_0 𝔤' ∈ GG_0'm ' 𝔤' = mh_0 · (h^-1𝔤').Let H be a 𝒦- and 𝒦'-big subgroup of G, let h ∈ H such that hm_0 = m_0', and let m ∈ M. Put h_0 = g_m_0, m^-1 g_m_0', m' h. Then, h_0 ∈ H_0 and g_m_0', m' = g_m_0, m h_0 h^-1. Furthermore, let g G_0' ∈ GG_0'. Then,m ' g G_0'= g_m_0', m' g (g_m_0', m')^-1 m= g_m_0, m h_0 h^-1 g h h_0^-1 g_m_0, m^-1 m= mh_0 h^-1 g h h_0^-1 G_0.Thus, because h_0^-1 G_0 = G_0 and h G_0 h^-1 = hG_0 = G_0',m ' g G_0'= mh_0 · h^-1 g h G_0= mh_0 · h^-1 g h G_0 h^-1 h= mh_0 · h^-1 g G_0' h= mh_0 · (h^-1 g G_0').Letbe a left group action of G on M, let Q be a set, and letD = ⋃_m ∈ Mδ Q^N → QN ⊆ GG_mwithG_m · N ⊆ N.The group G acts on D on the left by⊗ G × D→ D, left group action ⊗ of G on D(g, δ Q^N → Q)↦[Q^gN → Q, ℓ'↦δ(n ↦ℓ'(gn)). ] According to <ref>, the map ⊗ is well-defined. Let (δ Q^N → Q) ∈ D. Then, e_G ⊗δ = δ. And, for each g ∈ G, each g' ∈ G, and each ℓ”∈ Q^g g'N,(g g' ⊗δ)(ℓ”)= δ(n ↦ℓ”(g g'n))= δ(n ↦ℓ”(g(g'n)))= δ(n ↦ [n' ↦ℓ”(gn')](g'n))= (g' ⊗δ)(n' ↦ℓ”(gn'))= (g ⊗ (g' ⊗δ))(ℓ”).In conclusion, ⊗ is a left group action. In the situation of <ref>, let H be a 𝒦- and 𝒦'-big subgroup of G, let 𝒞 = ℳ, 𝒦, Q, N, δ be a semi-cellular automaton such that δ is ∙_H_0-invariant, let N' be the set hN, and let δ' be the map h ⊗δ. The quadruple 𝒞' = ℳ, 𝒦', Q, N', δ' is a semi-cellular automaton whose local transition function is ∙_H_0'-invariant and whose global transition function is identical to the one of 𝒞.We have N' ⊆ GG_0' and, according to <ref>, we have G_0' · N' ⊆ N'. In conclusion, 𝒞' is a semi-cellular automaton.Moreover, for each h_0' ∈ H_0' and each ℓ' ∈ Q^N', according to <ref> and because δ is ∙_H_0-invariant,δ'(h_0' ∙ℓ')= δ(n ↦ (h_0' ∙ℓ')(hn))= δ(n ↦ℓ'((h_0')^-1· (hn)))= δ(n ↦ℓ'(h(h^-1 (h_0')^-1 h · n)))= δ((h^-1 h_0' h) ∙ [n ↦ℓ'(hn)])= δ(n ↦ℓ'(hn))= δ'(ℓ').In conclusion, δ' is ∙_H_0'-invariant.Furthermore, let c ∈ Q^M and let m ∈ M. According to <ref>, there is an h_0 ∈ H_0 such thatn' ∈ N'm ' n' = mh_0 · (h^-1 n').Therefore, because δ is ∙_H_0-invariant,Δ'(c)(m)= δ'(n' ↦ c(m ' n'))= δ'(n' ↦ c(mh_0 · (h^-1 n')))= δ(n ↦ c(mh_0 · (h^-1 (hn))))= δ(n ↦ c(mh_0 · n))= δ(h_0^-1∙ [n ↦ c(mn)])= δ(n ↦ c(mn))= Δ(c)(m).In conclusion, Δ' = Δ. In the situation of <ref>, let m_0 = m_0', let H be a 𝒦- and 𝒦'-big subgroup of G, and let 𝒞 = ℳ, 𝒦, Q, N, δ be a semi-cellular automaton such that δ is ∙_H_0-invariant. The global transition function of the semi-cellular automaton ℳ, 𝒦', Q, N, δ is identical to the one of 𝒞.This is a direct consequence of <ref> with h = e_G. Let ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space and let 𝒞 be a cellular automaton over ℳ, 𝒦. For each coordinate system 𝒦' = m_0', g_m_0', m'_m ∈ M for ℳ, there is a cellular automaton over ℳ, 𝒦' whose global transition function is identical to the one of 𝒞.This is a direct consequence of <ref> with H = G and h = g_m_0, m_0'. [Lattice]Let M be the one-dimensional integer lattice , let T be the group τ_tm ↦ t + mt ∈ of translations of M, let R be the set ϱ_tm ↦ t - mt ∈ of reflections of M, let G be the group T ∪ R of translations and reflections of M, and letbe the left group action of G on M by function application. The triple ℳ = M, G, is a left-homogeneous space and the stabiliser G_0 of the origin 0 underis the group τ_0, ϱ_0. [Logical Or]In the situation of <ref>, let 𝒦 be a coordinate system for ℳ, let Q be the set 0, 1, let N be the set -1, 1, let δ be the map Q^N → Q, ℓ↦ℓ(-1) ℓ(1), and let M be identified with GG_0 by ι m ↦ G_0, m. The quadruple 𝒞 = ℳ, 𝒦, Q, N, δ is a cellular automaton whose global transition function Δ is the map Q^M → Q^M, c ↦ c( - 1)c( + 1) (see <ref>). Note that, because the binary operatoris commutative, the local transition function δ is ∙-invariant and the global transition function Δ does not depend on the coordinate system.[Peculiar Shift Maps]In <ref>, if the assumption that the subgroup H of G is 𝒦-big or the one that the local transition function δ is ∙_H_0-invariant does not hold, then the global transition function of 𝒞' may be different from the one of 𝒞, which is illustrated by the following examples.In the situation of <ref>, let Q be the set 0, 1, let N be the set -1, 1, let δ be the map Q^N → Q, ℓ↦ℓ(1), and let M be identified with GG_0 by ι m ↦ G_0, m. Because δ distinguishes between left and right, more precisely, because δ(ϱ_0 ∙) ≠δ, the map δ is not ∙-invariant. And, the global transition functions of semi-cellular automata over ℳ with local transition function δ depend on the choice of coordinates, which is illustrated by the following examples. *The tuples 𝒦 = 0, τ_m_m ∈ M and 𝒦' = 0, ϱ_m_m ∈ M are two coordinate systems for ℳ. For each cell m ∈ M and each element 𝔤∈ GG_0 ≃ M, we have m 𝔤 = m + 𝔤 and m ' 𝔤 = m - 𝔤, in particular, the (actual) neighbour of m that corresponds to the (relative) right neighbour 1 in 𝒦 is the cell m + 1, which lies to the right of m, and in 𝒦' it is the cell m - 1, which lies to the left. The reason is that the coordinates τ_m_m ∈ M maintain the meanings of left and right, whereas the coordinates ϱ_m_m ∈ M reverse them. The quadruples 𝒞 = ℳ, 𝒦, Q, N, δ and 𝒞' = ℳ, 𝒦', Q, N, δ are two semi-cellular automata whose global transition functions Δ and Δ' are the left shift map c ↦ c( + 1) (see <ref>) and the right shift map c ↦ c( - 1) (see <ref>). The reason is that δ depends on the meanings of left and right, which are maintained bybut reversed by '.Note that 𝒦' is actually not a coordinate system, because, by definition, the coordinate of the origin must be the identity map τ_0. That requirement though could be discarded with minor changes to some statements. It was merely made for convenience.* The tuple 𝒦” = 0, τ_m_m ∈ 2 M×ϱ_m_m ∈ 2 M + 1 is a coordinate system for ℳ. The right quotient set semi-action of GG_0 ≃ M on M induced by ℳ, 𝒦” is the map” M × GG_0→ M,(m, 𝔤)↦ m + 𝔤,if m is even,m - 𝔤,if m is odd. In particular, the (actual) neighbour of a cell that corresponds to the (relative) neighbour 1 is the cell to its right, if it is even, and the one to its left, if it is odd. The quadruple 𝒞” = ℳ, 𝒦”, Q, N, δ is a semi-cellular automaton whose global transition function is the map Δ” Q^M→ Q^M,c↦ [m ↦{ c(m + 1), if m is even,c(m - 1), if m is odd,}].In one step, each even cell exchanges states with the odd cell to its right (see <ref>).*The tuple 𝒦”' = 0, τ_m_m ∈ M ∖1×ϱ_m_m ∈1 is a coordinate system for ℳ. The right quotient set semi-action of GG_0 ≃ M on M induced by ℳ, 𝒦”' is the map”'M × GG_0→ M,(m, 𝔤)↦ m + 𝔤,if m ≠ 1,m - 𝔤,if m = 1.The quadruple 𝒞”' = ℳ, 𝒦”', Q, N, δ is a semi-cellular automaton whose global transition function is the mapΔ”'Q^M→ Q^M,c↦ [m ↦{ c(m + 1), if m ≠ 1,c(m - 1), if m = 1,}].In one step, the cells 0 and 1 exchange states and each other cell takes the state from the cell to its right (see <ref>). The semi-cellular automata 𝒞, 𝒞', 𝒞”, and 𝒞”' have different global transition functions, although they are equal except for their coordinates, in particular, they are related as in the construction in <ref> with h = e_G. However, the group G is the only subgroup of G that is big with respect to each pair of the coordinate systems 𝒦, 𝒦', 𝒦”, and 𝒦”', but the local transition function δ is not ∙_G_0-invariant. And, the subgroups H of T are the only subgroups of G such that the local transition function δ is ∙_H_0-invariant, but those subgroups of G are only 𝒦-big and neither big with respect to 𝒦', 𝒦”, nor 𝒦”'.In the situation of <ref>, let H be a 𝒦- and 𝒦'-big subgroup of G, let 𝒞 = ℳ, 𝒦, Q, N, δ and 𝒞' = ℳ, 𝒦', Q, N', δ' be two semi-cellular automata such that δ and δ' are ∙_H_0-invariant and Δ and Δ' are identical, let N_* = N ∩ (h^-1 N'), let N'_* = (hN) ∩ N', letδ_*Q^N_* → Q, ℓ_*↦δ(ℓ),where ℓ∈ Q^N such that ℓ_N_* = ℓ_*,and letδ'_*Q^N'_* → Q, ℓ'_*↦δ'(ℓ'),where ℓ' ∈ Q^N' such that ℓ'_N'_* = ℓ'_*.The quadruples 𝒞_* = ℳ, 𝒦, Q, N_*, δ_* and 𝒞'_* = ℳ, 𝒦', Q, N'_*, δ'_* are two semi-cellular automata such that δ_* and δ'_* are ∙_H_0-invariant, and Δ_* = Δ = Δ' = Δ'_*. Moreover, N'_* = hN_*, δ'_* = h ⊗δ_*, and ℓ_* ∈ Q^N_*ℓ∈ Q^N ℓ' ∈ Q^N' (ℓ_N_* = ℓ_* ℓ'_N'_* = ℓ_*(h^-1)δ(ℓ) = δ_*(ℓ_*) = δ'_*(ℓ_*(h^-1)) = δ'(ℓ') ). First, we have h^-1 N' ⊆ GG_h^-1 m_0' = GG_0 and, according to <ref>, we have G_0 · (h^-1 N') ⊆ h^-1 N'. Therefore, G_0 · N_* = (G_0 · N) ∩ (G_0 · (h^-1 N')) ⊆ N ∩ (h^-1 N') = N_*. Analogously, G_0 · N'_* ⊆ N'_*. Moreover, hN_* = (hN) ∩ (h(h^-1 N')) = (hN) ∩ N' = N'_*.Secondly, let ℓ_* ∈ Q^N_*, let ℓ∈ Q^N, and let ℓ' ∈ Q^N' such that ℓ_N_* = ℓ_* and ℓ'_N'_* = ℓ_*(h^-1). Then, because m_0 and m_0 ' are injective, there are c, c' ∈ Q^M such that c(m_0 )_N = ℓ and c'(m_0 ' )_N' = ℓ' (see <ref>). And, according to <ref>, there is an h_0 ∈ H_0 such that𝔤' ∈ GG_0'm_0 ' 𝔤' = m_0h_0 · (h^-1𝔤').Thus, because h_0 · (h^-1 N') = h^-1 N',m_0 ' N' = m_0h_0 · (h^-1 N') = m_0h^-1 N'.And, because h_0 · N_* = N_*,m_0 ' N'_* = m_0h_0 · (h^-1 N'_*) = m_0h_0 · N_* = m_0N_*.Hence, because m_0 is injective,(m_0N) ∩ (m_0 ' N')= (m_0N) ∩ (m_0h^-1 N')= m_0(N ∩ (h^-1 N'))= m_0N_*= m_0 ' N'_*.Moreover, for each n_* ∈ N_*, because m_0h_0 · n_* = h_0(m_0n_*) (see <ref>),c(m_0n_*)= ℓ(n_*)= ℓ_*(n_*)= ℓ_*(h^-1 (hn_*))= ℓ'(hn_*)= c'(m_0 ' hn_*)= c'(m_0h_0 · n_*)= (h_0^-1 c')(m_0n_*).Hence, because m_0N_* = m_0 ' N'_*, we have c_m_0N_* = (h_0^-1 c')_m_0 ' N'_*. Therefore, because (m_0N) ∩ (m_0 ' N') = m_0 ' N'_*, there is a c”∈ Q^M such that c”_m_0N = c_m_0N and c”_m_0 ' N' = (h_0^-1 c')_m_0 ' N'. Thus, because Δ = Δ', and (h_0^-1 c')(m_0 )_N = h_0^-1∙ℓ', and δ' is ∙_H_0-invariant, δ(ℓ)= Δ(c)(m_0)= Δ(c”)(m_0)= Δ'(c”)(m_0)= Δ'(h_0^-1 c')(m_0)= δ'(h_0^-1∙ℓ')= δ'(ℓ').In conclusion, δ(ℓ) = δ(ℓ').Thirdly, it follows that, for each ℓ_* ∈ Q^N_*, each ℓ_1 ∈ Q^N, and each ℓ_2 ∈ Q^N such that ℓ_1_N_* = ℓ_* and ℓ_2_N_* = ℓ_*, we have δ(ℓ_1) = δ'(ℓ') = δ(ℓ_2), where ℓ' ∈ Q^N' such that ℓ'_N'_* = ℓ_*(h^-1). In conclusion, δ_* is well-defined and, analogously, δ'_* is well-defined. Fourthly, let ℓ_* ∈ Q^N_*. Then, there are ℓ∈ Q^N and ℓ' ∈ Q^N' such that ℓ_N_* = ℓ_* and ℓ'_N'_* = ℓ_*(h^-1). Hence, because δ'(ℓ') = δ(ℓ),(h^-1⊗δ'_*)(ℓ_*) = δ'_*(ℓ_*(h^-1)) = δ'(ℓ') = δ(ℓ) = δ_*(ℓ_*).In conclusion, h^-1⊗δ'_* = δ_* and hence δ'_* = h ⊗δ_*.Fifthly, let h_0 ∈ H_0 and let ℓ_* ∈ N_*. Then, there is an ℓ∈ Q^N such that ℓ_N_* = ℓ_*, in particular, (h_0 ∙ℓ)_N_* = h_0 ∙ℓ_*. Hence, because δ is ∙_H_0-invariant,δ_*(h_0 ∙ℓ_*) = δ(h_0 ∙ℓ) = δ(ℓ) = δ_*(ℓ_*).In conclusion, δ_* is ∙-invariant and, analogously, δ'_* is ∙-invariant.Lastly, for each c ∈ Q^M and each m ∈ M, by the definition of δ_*,Δ_*(c)(m)= δ_*(n_* ↦ c(mn_*))= δ(n ↦ c(mn))= Δ(c)(m).Therefore, Δ_* = Δ and, analogously, Δ'_* = Δ'. In conclusion, Δ_* = Δ = Δ' = Δ'_*. In the situation of <ref>, let m_0 = m_0', let H be a 𝒦- and 𝒦'-big subgroup of G, let 𝒞 = ℳ, 𝒦, Q, N, δ and 𝒞' = ℳ, 𝒦', Q, N', δ' be two semi-cellular automata such that δ and δ' are ∙_H_0-invariant and Δ and Δ' are identical, let N_* = N ∩ N', and letδ_*Q^N_* → Q, ℓ_*↦δ(ℓ),where ℓ∈ Q^N such that ℓ_N_* = ℓ_*,(ℓ_*↦δ'(ℓ'),where ℓ' ∈ Q^N' such that ℓ'_N_* = ℓ_*).The quadruples 𝒞_* = ℳ, 𝒦, Q, N_*, δ_* and 𝒞'_* = ℳ, 𝒦', Q, N_*, δ_* are two semi-cellular automata such that δ_* is ∙_H_0-invariant and Δ_* = Δ = Δ' = Δ'_*. Moreover,ℓ_* ∈ Q^N_*ℓ∈ Q^N ℓ' ∈ Q^N' (ℓ_N_* = ℓ_* = ℓ'_N_*δ(ℓ) = δ_*(ℓ_*) = δ'(ℓ')). This is a direct consequence of <ref> with h = e_G. [Peculiar Shift Maps] In <ref>, if the assumption that the subgroup H of G is 𝒦- and 𝒦'-big does not hold, then the construction may not work, which is illustrated by the following examples.In the situation of <ref> of <ref>, let δ' be the map Q^N → Q, ℓ↦ℓ(-1) and let 𝒞” be the semi-cellular automaton ℳ, 𝒦', Q, N, δ'. The local transition functions δ and δ' are ∙_T_0-invariant and the global transition functions Δ and Δ” are both the left shift map, but the subgroup T of G is only 𝒦-big but not 𝒦'-big. The construction in <ref> applied to 𝒞 and 𝒞” with h = e_G yields δ_* = δ and δ'_* = δ'. However, contrary to the statement in the theorem, we have δ_* ≠ e_G ⊗δ'_*. This could be the case because 𝒦' is actually not a coordinate system, which is why we also give the following counterexample.In the situation of <ref> of <ref>, let N' be the set 0, 2, let δ' be the map Q^N'→ Q, ℓ↦ℓ(0), and let M be identified with GG_1 by m ↦ G_1, m (we have identified M twice now, but it will be clear from the context which identification applies). The tuple 𝒦”' = 1, τ_m - 1_m ∈ 2 M + 1×ϱ_m + 1_m ∈ 2 M is a coordinate system for ℳ. The right quotient set semi-action of GG_1 ≃ M on M induced by ℳ, 𝒦”' is the map”'M × GG_1→ M,(m, 𝔤)↦ m - 1 + 𝔤,if m is odd,m + 1 - 𝔤,if m is even.The quadruple 𝒞”' = ℳ, 𝒦”', Q, N, δ' is a semi-cellular automaton whose global transition function is the mapΔ”'Q^M→ Q^M,c↦ [m ↦{ c(m - 1), if m is odd,c(m + 1), if m is even,}].The local transition functions δ and δ' are ∙_T_0-invariant and the global transition functions Δ” and Δ”' are identical, but the subgroup T of G is neither 𝒦”- nor 𝒦”'-big. The construction in <ref> applied to 𝒞” and 𝒞”' with h = τ_1 yields δ_* = δ and δ'_* = δ'. However, because (τ_1 ⊗δ)(ℓ') = δ(n ↦ℓ'(τ_1(n))) = ℓ'(τ_1(1)) = ℓ'(2), for ℓ' ∈ Q^N' (where we used <ref>), we have τ_1 ⊗δ_* ≠δ'_*, contrary to the statement in the theorem. §.§ Equivariance, Determination, and Composition of Global Transition Functions Summary. The local configuration that is observed by a translated cell is identical to a rotation of the one observed by the original cell in the reversely translated global configuration, symbolically, c((gm) ) = g_0 ∙ ((g^-1 c)(m )) (see <ref>). Hence, if the local transition function is invariant under rotations, then the global transition function is equivariant under translations (see <ref>). Local configurations can be embedded in global configurations and, through this embedding, the local transition function is identical to the global transition function evaluated at the origin, symbolically, δ(ℓ) = Δ(ℓ̅)(m_0). And, rotations of local configurations translate to rotations of global configurations about the origin, symbolically, δ(h_0 ∙ℓ) = Δ(h_0 ℓ̅)(m_0). Hence, if the global transition function is equivariant under rotations, then the local transition function is invariant under rotations (see <ref>). Let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space, let 𝒞 = ℛ, Q, N, δ be a semi-cellular automaton, let H be a subgroup of G, and let Δ_0 be an _H-equivariant map from Q^M to Q^M such thatc ∈ Q^M Δ_0(c)(m_0) = δ(n ↦ c(m_0n)).The local transition function δ is ∙_H_0-invariant.Let ℓ∈ Q^N and let h_0 ∈ H_0. Then, there is a c ∈ Q^M such that ℓ is observed by m_0 in c (see <ref>), that is to say, that n ∈ N ℓ(n) = c(m_0n).And, becauseexhausts its defect with respect to its semi-commutativity within m_0 (see <ref> of <ref>),n ∈ N(h_0^-1 m_0)n = h_0^-1 (m_0h_0 · n).Hence, according to <ref>,δ(h_0 ∙ℓ)= δ(h_0 ∙ [n ↦ c(m_0n)])= δ(h_0 ∙ [n ↦ c((h_0^-1 m_0)n)])= δ(n ↦ (h_0c)(m_0n))= Δ_0(h_0c)(m_0).And, because Δ_0 is _H-equivariant,Δ_0(h_0c)(m_0)= (h_0 Δ_0(c))(m_0)= Δ_0(c)(h_0^-1 m_0)= Δ_0(c)(m_0)= δ(ℓ).Put the last two chains of equalities together to see that δ(h_0 ∙ℓ) = δ(ℓ). In conclusion, δ is ∙_H_0-invariant. Let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space, let 𝒞 = ℛ, Q, N, δ be a semi-cellular automaton, and let H be a 𝒦-big subgroup of G. The local transition function δ is ∙_H_0-invariant if and only if the global transition function Δ is _H-equivariant.First, let δ be ∙_H_0-invariant. Furthermore, let h ∈ H, let c ∈ Q^M, and let m ∈ M. Then, becausesemi-commutes with(see <ref> of <ref>), there is an h_0 ∈ H_0 such thatn ∈ N(h^-1 m)n = h^-1 (mh_0 · n).Hence, according to <ref>,(h Δ(c))(m)= Δ(c)(h^-1 m)= δ(n ↦ c((h^-1 m)n))= δ(h_0^-1∙ [n ↦ (hc)(mn)])= δ(n ↦ (hc)(mn))= Δ(hc)(m).In conclusion, Δ is _H-equivariant.Secondly, let Δ be _H-equivariant. Then, according to <ref> and <ref>, the local transition function δ is ∙_H_0-invariant. Let 𝒞 be a semi-cellular automaton. It is a cellular automaton if and only if its global transition function is -equivariant.This is a direct consequence of <ref> with H = G. [Lattice]Let M be the integer lattice ^d of dimension d ∈_+, let G be the symmetry group of M, which is generated by translations, rotations, and reflections, letbe the transitive left group action of G on M by function application, let m_0 be the origin 0, and, for each point m ∈ M, let g_m_0, m be the translation that maps m_0 to m, namely m +.The tuple ℛ = M, G, , m_0, g_m_0, m_m ∈ M is a cell space. The stabiliser G_0 of m_0 underis generated by the rotations about and reflections through the origin. Under the identification of GG_0 with ^d by ι, the induced semi-actionis the map M ×^d → M, (m, t) ↦ m + t (the elements of ^d are translation vectors).In the remainder of this example, let d = 2. Then, for each cell m ∈ M, there are four and only four rotations about m, namely those by 0, 90, 180, and 270; and there are five and only five reflections through m, namely those about the vertical, horizontal, descending diagonal, and ascending diagonal line through m, and the point reflection.Let Q be the binary set 0, 1, let N be the subset v ∈^2 v≤ 1 of ^2 (note that G_0N ⊆ N), and let δ be a map from Q^N → Q. The quadruple 𝒞 = ℛ, Q, N, δ is a semi-cellular automaton. The neighbourhood N is called von Neumann neighbourhoodvon Neumann neighbourhoodneighbourhood!von NeumannNeumann neighbourhood von@von Neumann neighbourhood. The local configurations of Q^N are depicted in <ref>. * Let H be the 𝒦-big subgroup of G that is generated by the translations. Then, the stabiliser H_0 of m_0 under _H is the trivial subgroup of H, the local transition function δ is ∙_H_0-invariant, and the global transition function Δ of 𝒞 is _H-equivariant.Under the identification of (H, ) with (^2, +) by t + ↦ t, that equivariance follows directly from the associativity of +, more precisely, from -t + (m + n) = (-t + m) + n, for t ∈^2, m ∈ M, and n ∈ N. Indeed, for each translation vector t ∈^2, each global configuration c ∈ Q^M, and each cell m ∈ M,Δ(tc)(m)= δ(n ↦ c(-t + (m + n)))= δ(n ↦ c((-t + m) + n))= (t Δ(c))(m). * Let H be the 𝒦-big subgroup of G that is generated by the translations and reflections about the horizontal and vertical axis. Then, the stabiliser H_0 is generated by the latter, the local transition function δ is ∙_H_0-invariant if and only if it maps the local configurations that are in the same column in <ref> to the same state, which are exactly those that are in the same orbit under ∙_H_0. * Let H be the 𝒦-big subgroup of G that is generated by the translations and reflections about the ascending and descending diagonal line through the origin. Then, the stabiliser H_0 is generated by the latter, the local transition function δ is ∙_H_0-invariant if and only if it maps the local configurations that are in the same column in <ref> to the same state. * Let H be the 𝒦-big subgroup of G that is generated by the translations and rotations about the origin. Then, the stabiliser H_0 consists of the latter, the local transition function δ is ∙_H_0-invariant if and only if it maps the local configurations that are in the same column in <ref> to the same state. * Let H be the 𝒦-big subgroup of G that is generated by the translations and the point reflection through the origin. Then, the stabiliser H_0 consists of the latter, the local transition function δ is ∙_H_0-invariant if and only if it maps the local configurations that are in same column in <ref> to the same state. * Let H be the 𝒦-big subgroup of G that is generated by the translations, rotations and reflections. Then, H is the group G and the stabiliser H_0 is generated by the rotations about and reflections through the origin. However, because there are only two states, the orbits of ∙_H_0 are the same as if H_0 were generated only by the rotations about the origin. Hence, the local transition function δ is ∙_H_0-invariant if and only if it maps the local configurations that are in the same column in <ref> to the same state.In each case, the global transition function Δ is equivariant under the respective symmetries of H. Figuratively speaking, being equivariant under translations means being oblivious of global positions, being equivariant under rotations means being oblivious of orientation, and being equivariant under reflections means being oblivious of orientation-reversal. [Plane] In the situation of <ref>, let 𝒞 = ℛ, Q, N, δ be a semi-cellular automaton, let R be a subgroup of the group G_0 of rotations about m_0, and let T ⋊ R be the inner semi-direct product of R acting on T. The group T ⋊ R is 𝒦-big and the stabiliser (T ⋊ R)_0 of m_0 under _T ⋊ R is the group R. According to <ref>, the local transition function δ is ∙_R-invariant if and only if the global transition function Δ is _T ⋊ R-equivariant. In particular, because δ is ∙_e_G-invariant, the map Δ is _T-equivariant. Broadly speaking, the more invariant δ is, the more equivariant is Δ, and vice versa.[Peculiar Shift Maps] In <ref>, if the assumption that the subgroup H of G is 𝒦-big does not hold, then the stated equivalence may not hold, which is illustrated by the following example.In the situation of <ref> of <ref>, the subgroup T of G is not 𝒦”-big and, although the local transition function δ is ∙_T_0-invariant, the global transition function Δ” is not _T-equivariant. However, according to <ref>, even for a subgroup H of G that is not 𝒦-big, there is no semi-cellular automaton whose local transition function is not ∙_H_0-invariant, although its global transition function is _H-equivariant. If a map on global configurations is equivariant under translations and is determined by a local transition function at the origin, then, using translations from the origin to all other cells, we see that the map is determined by the local transition function at all cells, in other words, the map is the induced global transition function (see <ref>).Let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space, let 𝒞 = ℛ, Q, N, δ be a semi-cellular automaton, let Δ_0 be a map from Q^M to Q^M, and let H be a 𝒦-big subgroup of G. The following two statements are equivalent:*The local transition function δ is ∙_H_0-invariant and the global transition function of 𝒞 is Δ_0.*The map Δ_0 is _H-equivariant andc ∈ Q^M Δ_0(c)(m_0) = δ(n ↦ c(m_0n)).First, let δ be ∙_H_0-invariant and let Δ_0 be the global transition function of 𝒞. According to <ref>, the map Δ_0 is _H-equivariant and, according to <ref>, <ref> holds.Secondly, let Δ_0 be _H-equivariant and let <ref> hold. According to <ref>, the local transition function δ is ∙_H_0-invariant. Furthermore, let c ∈ Q^M and let m ∈ M. Put h = g_m_0, m^-1∈ H. Then,Δ_0(c)(m) = Δ_0(c)(h^-1 m_0) = (h Δ_0(c))(m_0).And, because Δ_0 is _H-equivariant,(h Δ_0(c))(m_0) = Δ_0(hc)(m_0) = δ(n ↦ (hc)(m_0n)).And, becausesemi-commutes with(see <ref> of <ref>), there is an h_0 ∈ H_0 such that, for each n ∈ N, we have (h^-1 m_0)n = h^-1 (m_0h_0 · n); and therefore, according to <ref>,δ(n ↦ (hc)(m_0n))= δ(h_0 ∙ [n ↦ c((h^-1 m_0)n)])= δ(h_0 ∙ [n ↦ c(mn)]).And, because δ is ∙_H_0-invariant,δ(h_0 ∙ [n ↦ c(mn)]) = δ(n ↦ c(mn)).Put the last four chains of equalities together to see that Δ_0(c)(m) = δ(n ↦ c(mn)). In conclusion, Δ_0 is the global transition function of 𝒞.Let 𝒞 be a cellular automaton with set of cells M and set of states Q and let Δ_0 be a map from Q^M to Q^M. The global transition function of 𝒞 is Δ_0 if and only if c ∈ Q^M Δ_0(c)(m_0) = δ(n ↦ c(m_0n)). This is a direct consequence of <ref> and theorem <ref> with H = G.[Peculiar Complement and Shift Maps] In <ref>, if the assumption that the subgroup H of G is 𝒦-big does not hold, then <ref> may not be equivalent to <ref>, which is illustrated by the following examples. First, let ℛ be the cell space , , +, 0, m_m ∈, let 𝒞 be the cellular automaton ℛ,2, 0, δℓ↦ℓ(0), let Δ_0 be the map ( 2)^→ ( 2)^, c ↦ c(m) + (m + 2), and let H be the subgroup 2 of , which is not 0, m_m ∈-big. The map Δ_0 is the identity map on 2 and the bitwise complement map on 2 + 1, where we call 0 + 2 and 1 + 2 complements of each other. The map Δ_0 is _H-equivariant and Δ_0()(0) = (0) = δ(n ↦(0n)). However, the global transition function of 𝒞 is not Δ_0 but the identity map on ( 2)^. Actually, the map Δ_0 is not the global transition function of any semi-cellular automaton. Secondly, in the situation of <ref> of <ref>, the subgroup T of G is not 𝒦”'-big and, although the local transition function δ is ∙_T_0-invariant and the global transition function of 𝒞”' is Δ”', the map Δ”' is not _T-equivariant.Multiplying the neighbourhoods of two cellular automata and chaining their local transition functions yields an automaton whose global transition function is the composition of the ones of the other two automata (see <ref>).Let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space, let 𝒞 = ℛ, Q, N, δ and 𝒞' = ℛ, Q, N', δ' be two semi-cellular automata, and let H be a 𝒦-big subgroup of G such that δ and δ' are ∙_H_0-invariant. Furthermore, let N” = g · n'n ∈ N, n' ∈ N', g ∈ nand letδ” Q^N” → Q, ℓ” ↦δ(n ↦δ'(n' ↦ℓ”(g_m_0, m_0n· n'))).The quadruple 𝒞” = ℛ, Q, N”, δ” is a semi-cellular automaton whose local transition function is ∙_H_0-invariant and whose global transition function is ΔΔ'.Because G_0 · N ⊆ N, we have G_0 · N”⊆ N”. And, because g_m_0, m_0n∈ n, the map δ” is well-defined. Therefore, the quadruple 𝒞” = ℛ, Q, N”, δ” is a semi-cellular automaton.Moreover, because δ and δ' are ∙_H_0-invariant, according to theorem <ref>, the maps Δ and Δ' are _H-equivariant and thus ΔΔ' also. And, because g_m_0, m_0 = e_G, n ∈ Nn' ∈ N'(m_0n)n' = m_0g_m_0, m_0n· n'.Hence, for each c ∈ Q^M,(ΔΔ')(c)(m_0)= δ(n ↦Δ'(c)(m_0n))= δ(n ↦δ'(n' ↦ c((m_0n)n')))= δ(n ↦δ'(n' ↦ c(m_0g_m_0, m_0n· n')))= δ”(n”↦ c(m_0n”)).Therefore, according to <ref>, the local transition function δ” is ∙_H_0-invariant and the global transition function of 𝒞” is ΔΔ'. Let 𝒞 and 𝒞' be two cellular automata over ℛ and ℛ' respectively. There is a cellular automaton whose global transition function is ΔΔ'.This is a direct consequence of <ref> and theorem <ref> with H = G.In <ref>, under the identification of GG_0 with M by ι, we have N” = gn'n ∈ N, n' ∈ N', g ∈ G_m_0, n and, for each neighbour n ∈ N and each neighbour n' ∈ N', we have g_m_0, m_0n· n' = g_m_0, n n'.That the local transition function δ” in <ref> is ∙_H_0-invariant can be shown directly as follows: Let h_0 ∈ H_0. Then, for each n ∈ N,h_0^-1 g_m_0, m_0n G_0 = h_0^-1· n = g_m_0, m_0h_0^-1· n G_0,and hence, because H is 𝒦-big, the element h_n,0 = g_m_0, m_0h_0^-1· n^-1 h_0^-1 g_m_0, m_0n∈ G_0 ∩ H = H_0 satisfies h_0^-1 g_m_0, m_0n = g_m_0, m_0h_0^-1· n h_n,0. Therefore, because δ and δ' are ∙_H_0-invariant, for each ℓ”∈ Q^N”,δ”(h_0 ∙ℓ”)= δ(n ↦δ'(n' ↦ (h_0 ∙ℓ”)(g_m_0, m_0n· n')))= δ(n ↦δ'(n' ↦ℓ”(h_0^-1 g_m_0, m_0n· n')))= δ(n ↦δ'(n' ↦ℓ”(g_m_0, m_0h_0^-1· n· (h_n,0· n'))))= δ(n ↦δ'(h_n,0^-1∙ [n' ↦ℓ”(g_m_0, m_0h_0^-1· n· n')]))= δ(n ↦δ'(n' ↦ℓ”(g_m_0, m_0h_0^-1· n· n')))= δ(h_0 ∙ [n ↦δ'(n' ↦ℓ”(g_m_0, m_0n· n'))])= δ(n ↦δ'(n' ↦ℓ”(g_m_0, m_0n· n')))= δ”(ℓ”).In conclusion, δ” is ∙_H_0-invariant. [Peculiar Shift Maps]In <ref>, if the assumption that the subgroup H of G is 𝒦-big or the one that the local transition functions δ and δ' are ∙_H_0-invariant does not hold, then the global transition function of 𝒞” may not be ΔΔ', which is illustrated by the following example.Let 𝒞”' = ℳ, 𝒦”', Q, N, δ be the semi-cellular automata from <ref> of <ref>. The subgroup T of G is not 𝒦”'-big (and the local transition function δ is not ∙_G_0-invariant) and although the local transition function δ is ∙_T_0-invariant (and although the subgroup G of G is 𝒦”'-big), the map (Δ”')^2, which is depicted in <ref>, is not a global transition function of a semi-cellular automaton, in particular, not of the one the construction in <ref> applied to 𝒞”' yields.Broadly speaking, for each global configuration c ∈ Q^M and each cell m ∈ M, the state(Δ”')^2(c)(m) =c(m + 2),if m ∉0, 1,c(m),if m ∈0, 1,depends in an asymmetric way on the cell that cannot be induced uniformly by a local transition function and a coordinate system. And, the construction in <ref> applied to 𝒞”' yields the semi-cellular automaton 𝒞” = ℳ, 𝒦”', Q, -2, -1, 0, 1, 2, δ”ℓ”↦ℓ”(0) whose global transition function is the identity map on Q^M. Clearly, the square (Δ”')^2 is not the identity map on Q^M. First, suppose that there is a coordinate system 𝒦 = m_0, g_m_0, m_m ∈ M for ℳ and there is a semi-cellular automaton 𝒞” = ℳ, 𝒦, Q, N”, δ” whose global transition function is (Δ”')^2. Let m ∈ M ∖0, 1 and let c ∈ Q^M such that c(m + 2) ≠ c(m), for example, m = 4 and c ∈ Q^M with c ≡ 0 on M ∖6 and c ≡ 1 on 6. Then, letting c' = g_m_0, 0 g_m_0, m^-1 c, according to <ref>,c(m + 2)= (Δ”')^2(c)(m)= δ”((g_m_0, m^-1 c)_N”)= δ”((g_m_0, 0^-1 g_m_0, 0 g_m_0, m^-1 c)_N”)= δ”((g_m_0, 0^-1 c')_N”)= (Δ”')^2(c')(0)= c'(0)= c(g_m_0, m g_m_0, 0^-1 0)= c(m),which contradicts that c(m + 2) ≠ c(m). In conclusion, there is no such coordinate system and semi-cellular automaton.Secondly, the construction in <ref> applied to 𝒞”' yields the semi-cellular automaton 𝒞” with the neighbourhood N” = N + N = -2, -1, 0, 1, 2 and the local transition function δ” such that, for each local configuration ℓ”∈ Q^N”,δ”(ℓ”)= δ(n ↦δ(n' ↦ℓ”(g_0, n”'n'))) = ℓ”(g_0, 1”'1)= ℓ”(ϱ_1(1))= ℓ”(0), where 0, g_0, m”'_m ∈ M is the coordinate system 𝒦”'. In conclusion, the global transition function of 𝒞” is the identity map on Q^M.[Peculiar Shift Maps and Logical Or] In the situation of <ref>, let 𝒦 and 𝒦' be the two coordinate systems 0, τ_m_m ∈ M and 0, τ_m_m ∈ M ∖1×ϱ_m_m ∈1 for ℳ, let 𝒦” be a coordinate system for ℳ, let Q be the set 0, 1, let N be the set -1, 1, let δ be the map Q^N → Q, ℓ↦ℓ(1), let δ” be the map Q^N → Q, ℓ↦ℓ(-1) ℓ(1), and let M be identified with GG_0 by ι m ↦ G_0, m. The semi-cellular automaton 𝒞 = ℳ, 𝒦, Q, N, δ is the one from <ref> of <ref> whose global transition function Δ is the left shift map, the semi-cellular automaton 𝒞' = ℳ, 𝒦', Q, N, δ is the one from <ref> of <ref> whose global transition function Δ' is the left shift map with one defect in cell 1, and the cellular automaton 𝒞” = ℳ, 𝒦”, Q, N, δ” is the one from <ref> whose global transition function Δ” is a pairwise logical or map (it does not depend on the choice of 𝒦”).The group T of translations is 𝒦-big and, if we choose 𝒦 for 𝒦”, it is also 𝒦”-big; and the local transition functions δ and δ” are ∙_T_0-invariant. The construction in <ref> yields the following: The square Δ^2 is the global transition function of the semi-cellular automaton ℳ, 𝒦, Q, -2, -1, 0, 1, 2, ℓ↦ℓ(2) (see <ref>), the square (Δ”)^2 is the one of the cellular automaton ℳ, , Q, -2, -1, 0, 1, 2, ℓ↦ℓ(-2) ℓ(0) ℓ(2) (see <ref>), and the compositions Δ”Δ and ΔΔ” are the one of the semi-cellular automaton ℳ, 𝒦, Q, -2, -1, 0, 1, 2, ℓ↦ℓ(0) ℓ(2) (see <ref>). Explicitly, for each global configuration c ∈ Q^M and each cell m ∈ M, we have Δ^2(c)(m) = c(m + 2), and (Δ”)^2(c)(m) = c(m - 2)c(m)c(m + 2), and (Δ”Δ)(c)(m) = (ΔΔ”)(c)(m) = c(m)c(m + 2).There is no 𝒦'-big subgroup H of G such that the local transition function δ' is ∙_H_0-invariant. And indeed, the square (Δ')^2 (see <ref>), the compositions Δ' Δ and ΔΔ' (see <ref>), and the composition Δ”Δ' (see <ref>) are not global transition functions of semi-cellular automata, which is shown for (Δ')^2 in <ref>. Nevertheless, the composition Δ' Δ” (see <ref>) is the global transition function of the semi-cellular automaton ℳ, 𝒦', Q, -2, -1, 0, 1, 2, ℓ↦ℓ(0) ℓ(2); the reason is that, broadly speaking, the global transition function Δ”, which is applied first to global configurations, does not have a defect that could be propagated by Δ'. Explicitly, for each global configuration c ∈ Q^M and each cell m ∈ M,(Δ')^2(c)(m) =c(m + 2),if m ∉0, 1,c(m),if m ∈0, 1, (Δ' Δ)(c)(m) =c(m + 2),if m ≠ 1,c(m),if m = 1, (ΔΔ')(c)(m) =c(m + 2),if m ≠ 0,c(m),if m = 0, (Δ”Δ')(c)(m) =c(m)c(m + 2),if m ∉0, 2,c(m),if m = 0,c(m - 2)c(m + 2),if m = 2,and(Δ' Δ”)(c)(m) =c(m)c(m + 2),if m ≠ 1,c(m - 2)c(m),if m = 1.Broadly speaking, the states of the first four compositions depend in an asymmetric way on the cell that cannot be induced uniformly by a local transition function and a coordinate system; whereas the state of the last composition depends on the state of the cell itself and on the state of its second to the right or to the left neighbour and this reversal of orientation can be achieved by using translations in the one case and reflections in the other.It is notable that Δ and Δ' are induced by the same local transition function but by different coordinates (the ones for Δ are even the translations); and, if we choose 𝒦 for 𝒦”, that Δ' and Δ” are induced by the same coordinates but different local transition functions (the one for Δ” is even ∙-invariant). CHAPTER: QUOTIENTS, PRODUCTS, RESTRICTIONS, EXTENSIONS, AND DECOMPOSITIONS OF SPACES, AUTOMATA AND GLOBAL TRANSITION FUNCTIONSQuotients, Products, Restrictions, Extensions, ... Abstract. We introduce and study periodicity of global configurations, and quotients, products, restrictions, extensions, and decompositions of left-homogeneous spaces, coordinate systems, semi-cellular and cellular automata, and global transition functions of cellular automata. Remark. This chapter generalises parts of sections 1.6 and 1.7 of the monograph *ceccherini-silberstein:coornaert:2010<cit.>. Introduction. The orientation preserving symmetries of an infinite circular cylinder are the rotations about its axis and the translations along its axis. These symmetries and its subgroups act on the cylinder itself and also on patterns over the cylinder.The patterns over the cylinder that are invariant under rotations are precisely those that are unicoloured on intersections of planes that are perpendicular to the axis, which are circles. Such a pattern can squeezed to a pattern over the axis, which is essentially the cylinder modulo rotations. Conversely, such a squeezed pattern can be stretched to the original pattern over the cylinder. So, there is a one-to-one correspondence between patterns that are invariant under rotations and patterns over the axis. When we squeeze the cylinder to its axis, we squeeze its symmetries to the translations, which are essentially the symmetries modulo rotations.Analogously, the patterns that are invariant under translations are precisely those that are unicoloured along the axis. Such a pattern can be squashed to a pattern over the circle that has the same radius as the cylinder, which is essentially the cylinder modulo translations. Conversely, such a squashed pattern can can be stretched to the original pattern over the cylinder. So, there is a one-to-one correspondence between patterns that are invariant under translations and patterns over the circle. When we squash the cylinder to the circle, we squash its symmetries to the rotations, which are essentially the symmetries modulo translations.A map on patterns over the cylinder that is equivariant under rotations (or translations), maps patterns that are invariant under rotations (or translations) to such patterns. It can be squeezed (or squashed) to a map on patterns over the axis (or the circle). Conversely, such a squeezed (or squashed) map can be stretched to a map on patterns over the cylinder, which however may not be the original map. These constructions can for example be performed with global transition functions of cellular automata over the cylinder.If the neighbourhood of a semi-cellular automaton over the cylinder is such that the neighbours of a cell lie on a circle (or on the line through the cell that is parallel to the axis), then its global transition function can be restricted to the circle (or to the line) and this restriction can in turn be extended to the cylinder by repeating it along the axis (or in a circle around the axis). The restriction itself is a global transition function and its extension is the original function. As the original function consists of multiple copies of the restriction, it is injective, surjective, or bijective if and only if the restriction has the respective property.Contents. In <ref> we introduce and study the notion of periodicity for global configurations: Those of the same period are in a bijective relationship with the global configurations over the orbit under the period and periodicity is preserved by global transition functions of cellular automata. In <ref> we introduce quotients of left group sets, coordinate systems, cell spaces, semi-cellular and cellular automata, and global transition functions. In <ref> we introduce naïve products of semi-cellular automata; they are called naïve because naïve products of cellular automata are in general not cellular automata and because their global transition function depends on many arbitrary choices that have to be made. In <ref> we introduce products of cellular automata and products of global transition functions by subgroups that relate in a certain way to the stabiliser. In <ref> we introduce restrictions of left group sets, coordinate systems, cell spaces, semi-cellular and cellular automata, and global transition functions. In <ref> we introduce extensions of semi-cellular automata and global transition functions. In <ref> we show that global transition functions of some cellular automata are products of restrictions of themselves to subgroups that relate in a certain way to the stabiliser. Preliminary Notions. A left group set is a triple M, G,, where M is a set, G is a group, andis a map from G × M to M, called left group action of G on M, such that G →(M), g ↦ [g ], is a group homomorphism. The actionis transitive if M is non-empty and for each m ∈ M the map m is surjective; and free if for each m ∈ M the map m is injective. For each m ∈ M, the set Gm is the orbit of m, the set G_m = ( m)^-1(m) is the stabiliser of m, and, for each m' ∈ M, the set G_m, m' = ( m)^-1(m') is the transporter of m to m'.A left-homogeneous space is a left group set ℳ = M, G, such thatis transitive. A coordinate system for ℳ is a tuple 𝒦 = m_0, g_m_0, m_m ∈ M, where m_0 ∈ M and, for each m ∈ M, we have g_m_0, m m_0 = m. The stabiliser G_m_0 is denoted by G_0. The tuple ℛ = ℳ, 𝒦 is a cell space. The set g G_0g ∈ G of left cosets of G_0 in G is denoted by GG_0. The map M × GG_0 → M, (m, g G_0) ↦ g_m_0, m gm_0 is a right semi-action of GG_0 on M with defect G_0, which means thatm ∈ MmG_0 = m,andm ∈ Mg ∈ Gg_0 ∈ G_0 𝔤' ∈ GG_0 mg ·𝔤' = (mg G_0)g_0 ·𝔤'.It is transitive, which means that the set M is non-empty and for each m ∈ M the map m is surjective; and free, which means that for each m ∈ M the map m is injective; and semi-commutes with , which means thatm ∈ Mg ∈ Gg_0 ∈ G_0 𝔤' ∈ GG_0 (gm) 𝔤' = g(mg_0 ·𝔤'). A semi-cellular automaton is a quadruple 𝒞 = ℛ, Q, N, δ, where ℛ is a cell space; Q, called set of states, is a set; N, called neighbourhood, is a subset of GG_0 such that G_0 · N ⊆ N; and δ, called local transition function, is a map from Q^N to Q. A local configuration is a map ℓ∈ Q^N and a global configuration is a map c ∈ Q^M. The stabiliser G_0 acts on Q^N on the left by ∙ G_0 × Q^N → Q^N, (g_0, ℓ) ↦ [n ↦ℓ(g_0^-1· n)], and the group G acts on Q^M on the left by G × Q^M → Q^M, (g, c) ↦ [m ↦ c(g^-1 m)]. The global transition function of 𝒞 is the map Δ Q^M → Q^M, c ↦ [m ↦δ(n ↦ c(mn))]. A sufficient neighbourhood of 𝒞 is a subset E of N such that, for each ℓ∈ Q^N and each ℓ' ∈ Q^N with ℓ_E = ℓ'_E, we have δ(ℓ) = δ(ℓ').A cellular automaton is a semi-cellular automaton 𝒞 = ℛ, Q, N, δ such that the local transition function δ is ∙-invariant, which means that, for each g_0 ∈ G_0, we have δ(g_0 ∙) = δ(). Its global transition function is -equivariant, which means that, for each g ∈ G, we have Δ(g ) = g Δ(). (See <ref>.) Remark. Because the notation in the present chapter is rather heavy, I only present the statements and proofs for semi-cellular and cellular but not for big-cellular automata, which would require the introduction of a further subgroup. Moreover, for lack of a better option, I overloaded some notation, but it should be clear from the context what is meant.§ PERIODICITYLet M, G, be a left group set, let H be a subgroup of G, let Q be a set, and let c be a map from M to Q. The map c is called H-periodicperiodic@H-periodicH-periodic if and only ifh ∈ Hhc = c. Let M, G, be a left group set, let H be a subgroup of G, and let Q be a set. The set _H(Q^M) of H-periodic maps in Q^Mset of all H-periodic maps in Q^M is denoted by _H(Q^M)[symbols]PerHQM@_H(Q^M).The set _H(Q^M) is the set of fixed points of the left group action _H × Q^M.[Extreme Cases] The set _e_G(Q^M) is equal to Q^M and the set _G(Q^M) is equal to the set of constant maps from M to Q.[Cylinder]Let _42 be the additive cyclic group 42 of order 42, let M be the set _42×, let G be the additive group ^2, letbe the left group action of G on M by ((r_1, t_2), (m_1 + 42, m_2)) ↦ ((r_1 + m_1) + 42, t_2 + m_2), let H be the normal subgroup 0× of G, and let Q be the binary set 0, 1. The triple ℳ = M, G, is a left-homogeneous space.Geometrically, the set M is a discrete cylinder with axis a. Its rotations about a can be encoded by _42 and its translations along a can be encoded by . The group G is a cover of these encodings, where its first componentcovers _42. The rotational and translational symmetries themselves are (r_1, t_2)(r_1, t_2) ∈ G. Pictorially, maps from M to Q are black-and-white patterns over the cylinder M.For each black-and-white pattern cM → Q, it is H-periodic if and only if t_2 ∈ (m_1 + 42, m_2) ∈ M c(m_1 + 42, m_2 + t_2) = c(m_1 + 42, m_2),in other words, if and only if the pattern c is unicoloured along the axis a.Therefore, the set _H(Q^M) is equal toc ∈ Q^Mm_1 + 42∈_42 c(m_1 + 42, )is constant.The limit map of a convergent sequence of H-periodic maps with respect to the prodiscrete topology is H-periodic, which is shown in Let M, G, be a left group set, let H be a subgroup of G, and let Q be a set. Equip the set Q^M with the prodiscrete topology. The set _H(Q^M) is closed in Q^M.According to <ref>, the set Q^M is Hausdorff, and, according to <ref>, the group actionis continuous. Let g ∈ G. Then, the map g is continuous. Thus, the mapϕ Q^M→ Q^M × Q^M,c↦ (g , c),is continuous. Hence, because the diagonal D = (c,c)c ∈ Q^M is closed in Q^M × Q^M, its preimage under ϕ, namelyϕ^-1(D) = c ∈ Q^Mgc = c,is closed in Q^M. In conclusion, because the intersection of closed sets is closed, the set_H(Q^M) = ⋂_h ∈ Hc ∈ Q^Mhc = cis closed in Q^M. [Cylinder]In the situation of <ref>, the limit patterns of convergent sequences of black-and-white patterns over the cylinder M that are unicoloured along its axis a are unicoloured along a, in other words, the set _H(Q^M) is closed. If we first project points to their orbits under H and secondly map those orbits to states, the resulting map is H-periodic, because all points in the same orbit under H are mapped to the same state, which is shown inLet M, G, be a left group set, let H be a subgroup of G, let HM be the orbit space of _H × M, let Q be a set, let π be the canonical projection from M onto HM, and let c_ H be a map from HM to Q. The map c_ Hπ is H-periodic.For each h ∈ H and each m ∈ M,[]h(c_ Hπ)(m)= (c_ Hπ)(h^-1 m)= c_ H[]H(h^-1 m)= c_ H[](H h^-1)m= c_ H(Hm)= (c_ Hπ)(m).In conclusion, the map c_ Hπ is H-periodic. All H-periodic maps can be constructed as in <ref>, which is shown inLet M, G, be a left group set, let H be a subgroup of G, let HM be the orbit space of _H × M, let Q be a set, and let π be the canonical projection from M onto HM. The mapπ_*Q^HM →_H(Q^M), bijection π_* from Q^HM to _H(Q^M)[symbols]pistar@π_*c_ H ↦ c_ Hπ,is bijective.Note that, according to <ref>, the map π_* is well-defined.First, let c_ H, c_ H' ∈ Q^HM such that c_ H≠ c'_ H. Then, there is an Hm ∈ HM such that c_ H(Hm) ≠ c'_ H(Hm). Hence,π_*(c_ H)(m) = c_ H(π(m)) ≠ c'_ H(π(m)) = π_*(c'_ H)(m).Therefore, π_*(c_ H) ≠π_*(c'_ H). In conclusion, the map π_* is injective.Secondly, let c ∈_H(Q^M). Then, for each h ∈ H and each m ∈ M,c(hm) = (h^-1 c)(m) = c(m).Hence, the mapc_ H HM→ Q,Hm↦ c(m),is well-defined. Moreover, π_*(c_ H) = c. In conclusion, the map π_* is surjective. It follows that the number of H-periodic maps is equal to the number of maps from the set of orbits under H, as stated in Let M, G, be a left group set, let H be a subgroup of G such that the orbit space HM of _H × M is finite, and let Q be a finite set. The set _H(Q^M) is finite and _H(Q^M) = Q^HM.This is a direct consequence of <ref>. [Cylinder]In the situation of <ref>, identify the orbit space HM with the discrete circle _42 by H + m_1 + 42, m_2↦ m_1 + 42 (note that H + (m_1 + 42, m_2) = m_1 + 42×). Then, the canonical projection π is the map M →_42, (m_1 + 42, m_2) ↦ m_1 + 42, which is the projection onto the first component. Hence, for each black-and-white pattern c_ H over _42, the pattern c_ Hπ over M is unicoloured along the axis a. And, the map π_* extends black-and-white patterns over _42 unicolouredly along a to patterns over M. And, there are precisely 2^42 black-and-white patterns that are unicoloured along a. Equivariant maps preserve periodicity, as stated in Let M, G, and M', G', ' be two left group sets, let H be a subgroup of G, let Q be a set, let Δ be a map from Q^M to Q^M', and let φ be a group homomorphism from G to G' such that the tuple (Δ, φ) is (, ')-equivariant. Then, Δ(_H(Q^M)) ⊆_φ(H)(Q^M').Let c ∈_H(Q^M). Then, for each h ∈ H,φ(h) ' Δ(c) = Δ(hc) = Δ(c).Thus, Δ(c) is φ(H)-periodic. Global transition functions of cellular automata preserve periodicity, as stated in Let Δ be the global transition function of a cellular automaton over M, G, and let H be a subgroup of G. The set _H(Q^M) is invariant under Δ.According to <ref>, the map Δ is -equivariant, in other words, (Δ, _G) is (, )-equivariant. Hence, according to lemma <ref>, we have Δ(_H(Q^M)) ⊆_H(Q^M).[Cylinder]In the situation of <ref>, let r + 42 be an element of _42 and let Δ be the -equivariant map Q^M → Q^M, c ↦ c( + (r + 42, 0)), which rotates black-and-white patterns over the cylinder M about its axis a. The map Δ maps patterns that are unicoloured along a to such patterns. Shifts preserve periodicity under normal groups, as stated in Let M, G, be a left group set, let H be a normal subgroup of G, and let Q be a set. The set _H(Q^M) is -invariant.Let g ∈ G and let c ∈_H(Q^M). Moreover, let h ∈ H. Then, because g H = H g, there is an h' ∈ H such that h g = g h'. Hence,h(gc) = h gc = g h'c = g(h'c) = gc.Therefore, gc ∈_H(Q^M). In conclusion, _H(Q^M) is -invariant. Quotient groups act on periodic maps on the left, as stated in Let M, G, be a left group set, let H be a normal subgroup of G, and let Q be a set. The map__H GH ×_H(Q^M)→_H(Q^M), left group action __H of GH on _H(Q^M)[symbols]arrow right black periodic H@__H(g H, c)↦ gc,is a left group action of GH on _H(Q^M).The map __H is well-defined, because, according to lemma <ref>, for each g ∈ G, each h ∈ H, and each c ∈_H(Q^M), g hc = g(hc) = gc ∈_H(Q^M).And it is a left group action, because, for each g H ∈ GH and each g' H ∈ GH, we have g H g' H = g g' H, andis a left group action. [Cylinder]In the situation of <ref>, the subgroup H of G is normal and we identify the quotient group GH withby (r_1, t_2) + H ↦ r_1 (note that (r_1, t_2) + H = r_1×). Then, the left group actionis the map G × Q^M → Q^M, ((r_1, t_2), c) ↦ c( - (r_1 + 42),- t_2). It rotates and translates black-and-white patterns over the cylinder M about and along its axis a. In particular, it leaves the set of patterns that are unicoloured along a invariant. Moreover, the left quotient group action __H is the map ×_H(Q^M) →_H(Q^M), (r_1, c) ↦ c( - (r_1 + 42), ). It rotates black-and-white patterns over the cylinder M that are unicoloured along its axis about its axis. Restrictions of equivariant maps to periodic patterns are equivariant under induced quotient group actions, as stated in Let M, G, and M', G', ' be two left group sets, let H be a normal subgroup of G, let Q be a set, let Δ be a map from Q^M to Q^M', and let φ be a surjective group homomorphism from G to G' such that the tuple (Δ, φ) is (, ')-equivariant. The restriction Δ__H(Q^M) →_H(Q^M') is (__H, '__φ(H))-equivariant.According to <ref>, the restriction Δ__H = Δ__H(Q^M) →_H(Q^M') is well-defined. And, because H is normal in G and φ is a surjective group homomorphism, φ(H) is a normal subgroup of G' and thus '__φ(H) is well-defined. Moreover, for each g H ∈ GH and each c ∈_H(Q^M),Δ__H(g H __H c)= Δ(gc)= φ(g) ' Δ(c)= φ(g) φ(H) '__φ(H)Δ__H(c).In conclusion, Δ__H is (__H, '__φ(H))-equivariant.This holds in particular for global transition functions of cellular automata, as stated inLet Δ be the global transition function of a cellular automaton over M, G,, let H be a normal subgroup of G, and let Q be a set. The restriction Δ__H(Q^M) →_H(Q^M) is __H-equivariant.According to <ref>, the map Δ is -equivariant, in other words, (Δ, _G) is (, )-equivariant. Hence, according to lemma <ref>, the tuple (Δ__H(Q^M) →_H(Q^M), _G) is (__H, __H)-equivariant, in other words, Δ__H(Q^M) →_H(Q^M) is __H-equivariant. [Cylinder] In the situation of <ref>, the map Δ is -equivariant, which means, according to <ref>, that it is equivariant under rotations about the axis a of the cylinder M and under translations along a. So, its restriction to black-and-white patterns that are unicoloured along a is still equivariant under rotations about a, which is but __H-equivariance. § QUOTIENTS Quotient groups act on orbit spaces on the left, as stated in Let ℳ = M, G, be a left group set, let H be a normal subgroup of G, let HM be the orbit space of _H × M, and let_ H GH × HM→ HM, left group action _ H of GH on HM[symbols]arrow right modulo H@_ H(g H, Hm)↦ H(gm).The triple quotient ℳ H of ℳ by Hℳ H = HM, GH, _ H is a left group set and is called quotient of ℳ by H[symbols]MmoduloH@ℳ H.First, let g H, g' H ∈ GH such that g H = g' H and let Hm, Hm' ∈ HM such that Hm = Hm'. Then, there is an h ∈ H such that m = hm'. And, because g H = g' H = H g', there is an h' ∈ H such that g h = h' g'. Hence,H(gm)= H []g(hm')= H(g hm')= H(h' g'm')= H []h'(g'm')= H h'(g'm')= H(g'm').In conclusion, the map _ H is well-defined.Secondly, for each Hm ∈ HM,e_GH_ H (Hm)= e_G H _ H (Hm)= H(e_Gm)= Hm.And, for each g H ∈ GH, each g' H ∈ GH, and each Hm ∈ HM,g H · g' H _ H (Hm)= g g' H _ H (Hm)= H(g g'm)= H []g(g'm)= g H _ H[]H(g'm)= g H _ H[]g' H(Hm).In conclusion, the map _ H is a left group action. [Cylinder] In the situation of <ref>, the subgroup H is normal in G, we identify the orbit space HM with the discrete circle _42 by H + (m_1 + 42, m_2) ↦ m_1 + 42 (note that H + (m_1 + 42, m_2) = m_1 + 42×), and we identify the quotient group GH withby (r_1, t_2) + H ↦ r_1 (note that (r_1, t_2) + H = r_1×). The left quotient group action _ H is the map ×_42→_42, (r_1, m_1 + 42) ↦ (r_1 + m_1) + 42 and the quotient of ℳ by H is the triple _42, , _ H.Quotients of left-homogeneous spaces are left-homogeneous spaces, which is shown in Let ℳ = M, G, be a left-homogeneous space and let H be a normal subgroup of G. The quotient of ℳ by H is a left-homogeneous space.For each Hm ∈ HM and each Hm' ∈ HM, there is a g ∈ G such that gm = m', and hence g H _ H (Hm) = Hm'. In conclusion, the group action _ H is transitive. [Cylinder]In the situation of <ref>, the group set ℳ is a left-homogeneous space and so is the quotient ℳ H. Let G be a group and let G_0 and H be two subgroups of G. The set g_0 Hg_0 ∈ G_0⊆ GH is denoted by G_0HG_0H[symbols]G0moduloH@G_0H.If H is normal in G, then G_0H is a subgroup of GH.Stabilisers in quotients are quotients of stabilisers, as stated inLet HM, GH, _ H be a quotient of M, G, by H and let m be an element of M. The stabiliser (GH)_Hm of Hm under _ H is G_mH.First, let g_m H ∈ G_mH. Then,g_m H _ H (Hm) = H(g_mm) = Hm.Hence, g_m H ∈ (GH)_Hm. In conclusion, G_mH ⊆ (GH)_Hm.Secondly, let g H ∈ (GH)_Hm. Then,Hm = g H _ H (Hm) = H(gm).Hence, there is an h ∈ H such that hm = gm. Thus, h^-1 gm = m. Therefore, g_m = h^-1 g ∈ G_m and g = h g_m. Because H is normal in G,g H = h g_m H = H h g_m = H g_m = g_m H ∈ G_mH.In conclusion, (GH)_Hm⊆ G_mH. Altogether, (GH)_Hm = G_mH. [Cylinder]In the situation of <ref>, for each element (m_1 + 42, m_2) ∈ M, the stabiliser of (m_1, m_2) underis the subgroup 42×0 of G and the stabiliser of m_1 ≃ H(m_1, m_2) under _ H is the subgroup 42 of ≃ GH. Quotient actions on quotient patterns are essentially the original actions on periodic patterns, as stated in Let HM, GH, _ H be a quotient of M, G, by H, let Q be a set, and let π be the canonical projection from M onto HM. For each symmetry g ∈ G, each map c_ H∈ Q^HM, and each point m ∈ M,(g H _ H c_ H)(Hm) = []g π_*(c_ H)(m). For each g ∈ G, each c_ H∈ Q^HM, and each m ∈ M,(g H _ H c_ H)(Hm)= c_ H[]g^-1 H _ H (Hm)= c_ H[]H(g^-1 m)= (c_ Hπ)(g^-1 m)= π_*(c_ H)(g^-1 m)= []g π_*(c_ H)(m). The projection π_* is equivariant, as stated inLet HM, GH, _ H be a quotient of M, G, by H, let Q be a set, and let π be the canonical projection from M onto HM. The map π_* is (_ H, __H)-equivariant.For each g H ∈ GH, each c_ H∈ Q^HM, and each m ∈ M,π_*(g H _ H c_ H)(m)= (g H _ H c_ H)[]π(m)= (g H _ H c_ H)(Hm)= c_ H[]g^-1 H _ H (Hm)= c_ H[]H(g^-1 m)= c_ H[]π(g^-1 m)= π_*(c_ H)(g^-1 m)= []g π_*(c_ H)(m)= []g H __Hπ_*(c_ H)(m).Given a right inverse of the canonical projection onto the orbit space, a coordinate system induces one on the quotient space, as stated in Let ℳ = M, G, be a left-homogeneous space, let 𝒦 = m_0, g_m_0, m_m ∈ M be a coordinate system for ℳ, let ℛ be the cell space ℳ, 𝒦, let H be a normal subgroup of G, and let ρ be a right inverse of the canonical projection π M → HM such that ρ(Hm_0) = m_0. The tuple quotient 𝒦 (H, ρ) of 𝒦 by H and ρ𝒦 (H, ρ) = Hm_0, g_m_0, ρ(Hm)_Hm ∈ HM is a coordinate system for ℳ H and is called quotient of 𝒦 by H and ρ[symbols]KmoduloHrcalligraphic@𝒦 (H, ρ). And, the tuple quotient ℛ (H, ρ) of ℛ by H and ρℛ (H, ρ) = ℳ H, 𝒦 (H, ρ) is a cell space and is called quotient of ℛ by H and ρ[symbols]RmoduloHrcalligraphic@ℛ (H, ρ).Because ρ(Hm_0) = m_0 and g_m_0, m_0 = e_G, we have g_m_0, ρ(Hm_0) H = e_G H = e_GH. And, for each Hm ∈ HM,g_m_0, ρ(Hm) H _ H (Hm_0)= H(g_m_0, ρ(Hm) m_0)= H ρ(Hm)= Hm.In conclusion, 𝒦 (H, ρ) is a coordinate system for ℳ H and ℛ (H, ρ) is a cell space. [Cylinder]In the situation of <ref>, the tuple 𝒦 = (0 + 42, 0), (m_142, m_2)_(m_1 + 42, m_2) ∈ M is a coordinate system for M, G,, where 42 denotes the remainder of the Euclidean division by 42; the tuple ℛ = ℳ, 𝒦 is a cell space; the canonical projection π from M onto HM ≃_42 is given by (m_1 + 42, m_2) ↦ m_1 + 42; a right inverse ρ of π is given by m_1 + 42↦ (m_1 + 42, 0); the quotient of 𝒦 by H and ρ is the tuple 0 + 42, m_142_m_1 ∈, which is a coordinate system for ℳ H. The right semi-actions induced by quotients of cell spaces can be expressed in terms of the right semi-actions induced by the cell spaces themselves, as stated inLet ℛ (H, ρ) be a quotient of M, G, , m_0, g_m_0, m_m ∈ M by H and ρ. The right quotient set semi-action of (GH)(G_0H) on HM with defect G_0H induced by ℛ (H, ρ) is_ (H, ρ) HM × (GH)(G_0H)→ HM, right quotient set semi-action _ (H, ρ) induced by ℛ (H, ρ)[symbols]arrow left scored modulo H rho@_ (H, ρ) []Hm, g H (G_0H) ↦ H []ρ(Hm)g G_0. According to <ref>, we have (GH)_Hm_0 = G_0H. Moreover, for each Hm ∈ HM and each g H (G_0H) ∈ (GH)(G_0H),(Hm) _ (H, ρ) g H (G_0H)= H []ρ(Hm)g G_0= H []g_m_0, ρ(Hm) g g_m_0, ρ(Hm)^-1ρ(Hm)= g_m_0, ρ(Hm) g g_m_0, ρ(Hm)^-1 H _ H[]H []ρ(Hm)= g_m_0, ρ(Hm) H · g H · g_m_0, ρ(Hm)^-1 H _ H (Hm).Hence, _ (H, ρ) is well-defined and the right quotient set semi-action induced by ℛ (H, ρ). [Cylinder]In the situation of <ref>, identify the quotient set GG_0 with _42× by (r_1, t_2) + G_0 ↦ (r_1 + 42, t_2). Then, the right quotient set semi-actioninduced by ℛ is the map M × (_42×) → M, ((m_1 + 42, m_2), (r_1 + 42, t_2)) ↦ ((m_1 + r_1) + 42, m_2 + t_2), which is addition in the direct product _42×.Recall that the quotient group GH is identified with . Hence, its subgroup G_0H is identified with 42 and the quotient set (GH)(G_0H) is identified with _42. Also recall that the orbit space HM is identified with _42. Therefore, the right quotient set semi-action _ (H, ρ) induced by ℛ (H, ρ) is the map _42×_42→_42, (m_1 + 42, r_1 + 42) ↦ (m_1 + r_1) + 42, which is addition in _42. The quotient of a cell space by a group that includes the stabiliser of the origin does not depend on the chosen right inverse of the projection, as stated in Let ℛ (H, ρ) be a quotient of ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M by H and ρ such that the stabiliser G_0 is included in H. The quotient ℛ (H, ρ) and the right quotient set semi-action _ (H, ρ) do not depend on ρ.Let m, m' ∈ M such that Hm = Hm'. Then, there is an h ∈ H such that hm = m'. Thus, h g_m_0, m∈ G_m_0, m'. Hence, because G_m_0, m' = G_0 g_m_0, m', there is a g_0 ∈ G_0 such that h g_m_0, m = g_0 g_m_0, m'. Therefore, because H is normal, h H = H, and H g_0 = H,g_m_0, m H = H g_m_0, m = H h g_m_0, m= H g_0 g_m_0, m' = H g_m_0, m' = g_m_0, m' H.Hence, for each m ∈ M, we have g_m_0, m H = g_ρ(Hm),m_0 H. Therefore, the coordinates g_m_0, ρ(Hm) H, for Hm ∈ HM, do not depend on ρ. In conclusion, ℛ (H, ρ) and _ (H, ρ) do not depend on ρ.Because G_0 is included in H, the stabiliser G_0H of Hm_0 under _ H is trivial, hence the left group action _ H is free, and therefore ℳ H is a principal left-homogeneous space and 𝒦 (H, ρ) is its unique coordinate system for the origin m_0. The neighbourhood of a semi-cellular automaton over M is a subset of GG_0 and the neighbourhood of one over HM is a subset of (GH)(G_0H). There is a canonical projection from GG_0 onto (GH)(G_0H), which under suitable identifications of GG_0 with M and of (GH)(G_0H) with HM is the canonical projection from M onto HM, as stated inLet G be a group, let G_0 be a subgroup of G, and let H be a normal subgroup of G. The mapϖ GG_0→ (GH)(G_0H), canonical projection ϖ of GG_0 onto (GH)(G_0H)[symbols]pivar@ϖ[symbols]pivar@ϖg G_0↦ g H (G_0H),is well-defined, is surjective, and is called canonical projection from GG_0 onto (GH)(G_0H).Let g G_0, g' G_0 ∈ GG_0 such that g G_0 = g' G_0. Then, g^-1 g' ∈ G_0. Hence, g^-1 H · g' H = g^-1 g' H ∈ G_0H. Therefore, g H (G_0H) = g' H (G_0H). In conclusion, ϖ is well-defined. Moreover, because, for each g H (G_0H) ∈ (GH)(G_0H), we have ϖ(g G_0) = g H (G_0H), the map ϖ is surjective. [Cylinder]In the situation of <ref>, the canonical projection ϖ is the map _42×→_42, (r_1 + 42, t_2) ↦ r_1 + 42, which is the projection to the first component. Semi-cellular automata can be projected onto ones over orbit spaces, as shown in Let ℛ (H, ρ) be a quotient of M, G, , m_0, g_m_0, m_m ∈ M by H and ρ, let 𝒞 = ℛ, Q, N, δ be a semi-cellular or cellular automaton, let ϖ be the canonical projection from GG_0 onto (GH)(G_0H), letN_ H = ϖ(N) = g H (G_0H)g G_0 ∈ N,and letδ_ H Q^N_ H → Q, ℓ_ H ↦δ[]n ↦ℓ_ H[]ϖ(n)= δ[]g G_0 ↦ℓ_ H[]g H (G_0H).The quadruple quotient 𝒞 (H, ρ) of 𝒞 by H and ρ𝒞 (H, ρ) = ℛ (H, ρ), Q, N_ H, δ_ H is a semi-cellular or cellular automaton respectively and is called quotient of 𝒞 by H and ρ[symbols]CmoduloHrcalligraphic@𝒞 (H, ρ).For each g_0 ∈ G_0 and each g G_0 ∈ N, we have g_0 g G_0 ∈ N. Hence,(G_0H) · N_ H = g_0 Hg_0 ∈ G_0·g H (G_0H)g G_0 ∈ N= g_0 H · g H (G_0H)g_0 ∈ G_0, g G_0 ∈ N= g_0 g H (G_0H)g_0 ∈ G_0, g G_0 ∈ N⊆g H (G_0H)g G_0 ∈ N= N_ H.Therefore, the quadruple 𝒞 (H, ρ) is a semi-cellular automaton. From now on, let 𝒞 be a cellular automaton. Furthermore, let g_0 H ∈ G_0H and let ℓ_ H∈ Q^N_ H. Then,δ_ H(g_0 H ∙_ Hℓ_ H)= δ[]g G_0 ↦ (g_0 H ∙_ Hℓ_ H)(ϖ(g G_0))= δ[]g G_0 ↦ (g_0 H ∙_ Hℓ_ H)(g H (G_0H))= δ[]g G_0 ↦ℓ_ H(g_0^-1 H · g H (G_0H))= δ[]g G_0 ↦ℓ_ H(g_0^-1 g H (G_0H))= δ[]g G_0 ↦ℓ_ H(ϖ(g_0^-1 g G_0))= δ[]g_0 ∙[]g G_0 ↦ℓ_ H(ϖ(g G_0))= δ[]g G_0 ↦ℓ_ H(ϖ(g G_0))= δ_ H(ℓ_ H).Hence, δ_ H is ∙_ H-invariant. In conclusion, 𝒞 (H, ρ) is a cellular automaton. [Cylinder]In the situation of <ref>, let Q be the binary set 0, 1, let z be an integer, let N be the singleton set (-1 + 42, z), let δ be the ∙-invariant map Q^N → Q, ℓ↦ℓ(-1 + 42, z), and let 𝒞 be the cellular automaton ℛ, Q, N, δ. The global transition function Δ of 𝒞 is a shift over M. For example, if z = -1, then it is the diagonal shift from the bottom-left to the top-right.The quotient 𝒞 (H, ρ) is the quadruple ℛ (H, ρ), Q, N_ H, δ_ H, where the neighbourhood N_ H is the singleton set -1 + 42 and the local transition function δ_ H is the map Q^N_ H→ Q, ℓ_ H↦ℓ_ H(-1 + 42). The global transition function Δ_ H of 𝒞 (H, ρ) is the shift from left to right over _42. The global transition function of the quotient of a semi-cellular automaton can be expressed in terms of the global transition function of the automaton, as shown inLet ℛ (H, ρ) be a quotient of M, G, , m_0, g_m_0, m_m ∈ M by H and ρ, let 𝒞 = ℛ, Q, N, δ be a semi-cellular automaton, and let π be the canonical projection from M onto HM. The global transition function Δ_ H of 𝒞 (H, ρ) is identical to π_*^-1Δ__H(Q^M) →_H(Q^M)π_*, in particular, it does not depend on ρ and it is uniquely determined by M, G,, H, and Δ.Let Δ__H = Δ__H(Q^M) →_H(Q^M). Then, for each c_ H∈ Q^HM and each Hm ∈ HM,Δ_ H(c_ H)(Hm)= δ_ H(n_ H↦ c_ H((Hm) _ (H, ρ) n_ H))= δ_ H(g H (G_0H) ↦ c_ H((Hm) _ (H, ρ) g H (G_0H)))= δ(g G_0 ↦ c_ H((Hm) _ (H, ρ) g H (G_0H)))= δ(g G_0 ↦ c_ H(H(ρ(Hm)g G_0)))= δ(n ↦ c_ H(H(ρ(Hm)n)))= δ(n ↦ (c_ Hπ)((ρ(Hm)n)))= Δ(c_ Hπ)(ρ(Hm))= Δ(π_*(c_ H))(ρ(Hm))= Δ__H(π_*(c_ H))(ρ(Hm))= π_*^-1(Δ__H(π_*(c_ H)))(H ρ(Hm))= π_*^-1(Δ__H(π_*(c_ H)))(Hm)= (π_*^-1Δ__Hπ_*)(c_ H)(Hm).Therefore, Δ_ H = π_*^-1Δ__Hπ_*, which does not depend on ρ. Let 𝒞 be a cellular automaton. Then, according to <ref>, the restriction Δ__H(Q^M) →_H(Q^M) is __H-equivariant, and, according to <ref>, the map π_* is (_ H, __H)-equivariant. Hence, because Δ_ H is identical to π_*^-1Δ__H(Q^M) →_H(Q^M)π_*, the global transition function Δ_ H is _ H-equivariant. Let Δ be the global transition function of a semi-cellular or cellular automaton over M, G, and let H be a normal subgroup of G. The map quotient Δ_ H of Δ by HΔ_ H = π_*^-1Δ__H(Q^M) →_H(Q^M)π_* is the global transition function of a semi-cellular or cellular automaton over HM, GH, _ H and is called quotient of Δ by H[symbols]DeltamoduloH@Δ_ H. [Cylinder]In the situation of <ref>, according to <ref>, the canonical projection π is the map M →_42, (m_1 + 42, m_2) ↦ m_1 + 42 and the map π_* is the bijection Q^_42→_H(Q^M), c_ H↦ c_ Hπ. Hence, the inverse π_*^-1 is the bijection _H(Q^M) → Q^_42, c ↦ c(, 0). Therefore, global transition function Δ_ H is the map Q^_42→ Q^_42, c_ H↦Δ(c_ Hπ)(, 0).§ NAÏVE PRODUCTS Given a right inverse of the canonical projection from GG_0 onto (GH)(G_0H), a semi-cellular automaton over a quotient space can be extended to the original space, as shown inLet ℛ (H, ρ) be a quotient of M, G, , m_0, g_m_0, m_m ∈ M by H and ρ, let 𝒞_ H = ℛ (H, ρ), Q, N_ H, δ_ H be a semi-cellular automaton, let κ be a right inverse of the canonical projection ϖ GG_0 → (GH)(G_0H), let N = G_0 ·κ(N_ H), let ϰ = κ_N_ H→ N, and letδ Q^N→ Q, ℓ ↦δ_ H[]n_ H↦ℓ[]κ(n_ H)[]= δ_ H(ℓϰ).The quadruple naïve product 𝒞 of 𝒞_ H by H and κ𝒞 = ℛ, Q, N, δ is a semi-cellular automaton and is called naïve product of 𝒞_ H by H and κproduct of 𝒞_ H by H and κ!naïve. The subscript H of 𝒞_ H, N_ H, and δ_ H shall suggest that 𝒞_ H is a semi-cellular automaton over the quotient ℛ (H, ρ). It shall not mean that 𝒞_ H is the quotient of a semi-cellular automaton by H. [Cylinder]In the situation of <ref>, let κ be a right inverse of the canonical projection ϖ_42×→_42, let 𝒞_κ be the naïve product of 𝒞_ H by H and κ, and let Δ_κ be the global transition function of 𝒞. * If κ is given by r_1 + 42↦ (r_1 + 42, z), then 𝒞_κ = 𝒞 and Δ_κ = Δ.* If κ is given by r_1 + 42↦ (r_1 + 42, 0), then Δ_κ is the shift from left to right over M.* If κ is given by r_1 + 42↦ (r_1 + 42, r_142), then Δ_κ is the diagonal shift from the bottom-left to the top-right over M.* If κ is given by r_1 + 42↦ (r_1 + 42, -(r_142)), then Δ_κ is the diagonal shift from the top-left to the bottom-right over M.So, it depends on the choice of κ, whether the naïve product of the quotient of 𝒞 is again 𝒞 or not. If the right inverse κ is in a certain sense equivariant, then the product of a cellular automaton is a cellular automaton, as stated inIn the situation of <ref>, let 𝒞_ H be a cellular automaton and let κ be such thatg_0 ∈ G_0 𝔤_ H∈ (GH)(G_0H)g_0 ·κ(𝔤_ H) = κ(g_0 H ·_ H𝔤_ H).The naïve product 𝒞 of 𝒞_ H by H and κ is a cellular automaton.Let g_0 ∈ G_0 and let ℓ∈ Q^N. Then, for each n_ H∈ N_ H, because <ref> holds,((g_0 ∙ℓ) ϰ)(n_ H)= (g_0 ∙ℓ)(κ(n_ H))= ℓ(g_0^-1·κ(n_ H))= ℓ(κ(g_0^-1 H ·_ H n_ H))= (ℓϰ)(g_0^-1 H ·_ H n_ H)= (g_0 H ∙_ H (ℓϰ))(n_ H).Thus, (g_0 ∙ℓ) ϰ = g_0 H ∙_ H (ℓϰ). Hence, because δ_ H is ∙_ H-invariant,δ(g_0 ∙ℓ)= δ_ H((g_0 ∙ℓ) ϰ)= δ_ H(g_0 H ∙_ H (ℓϰ))= δ_ H(ℓϰ)= δ(ℓ).Therefore, δ is ∙-invariant. In conclusion, 𝒞 is a cellular automaton. In the situation of <ref>, if, for each g_0 ∈ G_0, we have g_0 ·κ() = κ(), then 𝒞 is a cellular automaton, regardless of whether 𝒞_ H is one or not. [Cylinder]In the situation of <ref>, because the group G is abelian, for each element (t_0 + 42, 0) ∈ G_0, we have (t_0 + 42, 0) ·κ() = κ(), and hence, according to <ref>, the naïve product 𝒞_κ is a cellular automaton. However, <ref> of <ref>, which in this example is equivalent to(t_0 + 42, 0) ∈ G_0r_1 + 42∈_42≃ (GH)(G_0H) (t_0 + 42, 0) ·κ(r_1 + 42) = κ((t_0 + r_1) + 42),holds if and only if the projection of κ to the second component is constant, which need not be the case. Therefore, although 𝒞_κ is a cellular automaton, the map κ need not satisfy <ref>. If <ref> of <ref> does not hold, then the naïve product may not be a cellular automaton. For example, if we factor by a normal subgroup H of G that includes the stabiliser G_0, then all local transition functions δ_ H are ∙_ H-invariant but their products are in general not ∙-invariant, which is illustrated in[Tree] Let G be the free group over a, b, where a ≠ b, let G_0 be the subgroup of G that is generated by a, let M be the quotient set GG_0, letbe the transitive left group action of G on M by left multiplication, let 𝒦 = m_0, g_m_0, m_m ∈ M be a coordinate system for ℳ = M, G,, and let ℛ be the cell space ℳ, 𝒦.Moreover, let H be the kernel of the group homomorphism φ G → 2 given by a ↦ 2, b ↦ 1 + 2, let ρ be a right inverse of the canonical projection π M → HM such that ρ(Hm_0) = m_0, and let κ be a right inverse of the canonical projection ϖ GG_0 → (GH)(G_0H).Furthermore, let Q be the binary set 0, 1, let N_ H be the singleton set b H (G_0H), let δ_ H be the map Q^N_ H→ Q, ℓ_ H↦ℓ_ H(b H (G_0H)), let 𝒞_ H be the semi-cellular automaton ℛ (H, ρ), Q, N_ H, δ_ H, and let 𝒞 = ℛ, Q, N, δ be the naïve product of 𝒞_ H by H and κ.The group H is a normal subgroup of G of index 2; the orbit space HM is the double quotient set H(GG_0), which we identify with GH by H (g G_0) ↦ g H; the quotient group GH is equal to H, b H; the left group action _ H and the right quotient set semi-action _ (H, ρ) it induces are identical to the group multiplication of GH; and the global transition function of 𝒞_ H is the shift operator on Q^GH. The stabiliser G_0 of m_0 underis a subgroup of H, the stabiliser G_0H of m_0 H under _ H is trivial, we identify the quotient set (GH)(G_0H) with GH by g H (G_0H) ↦ g H, the local transition function δ_ H is trivially ∙_ H-invariant, and the automaton 𝒞_ H is a cellular automaton.Because ϖ(κ(b H)) = b H, there is an element g ∈ G that is represented by a reduced word in which the symbol b occurs such that κ(b H) = g G_0. Thus, for each element g_0 ∈ G_0 ∖e_G, we have g_0 ·κ(b H) ≠κ(b H), in particular, <ref> of <ref> does not hold. Hence, for each element g_0 ∈ G_0 ∖e_G, there is a local configuration ℓ∈ Q^N such that ℓ(g_0^-1·κ(b H)) ≠ℓ(κ(b H)) and thereforeδ(g_0 ∙ℓ) =δ_ H(n_ H↦ℓ(g_0^-1·κ(n_ H))) =ℓ(g_0^-1·κ(b H)) ≠ℓ(κ(b H)) =δ_ H(n_ H↦ℓ(κ(n_ H))) =δ(ℓ).More specifically, if κ is the map GH → GG_0, H ↦ G_0, b H ↦ b G_0, and ℓ is the local configuration N → Q, b G_0 ↦ 0, a^k b G_0 ↦ 1, for k ∈∖0, then δ(a ∙ℓ) = 1 ≠ 0 = δ(ℓ). In conclusion, the local transition function δ is not ∙-invariant and hence the naïve product 𝒞 is not a cellular automaton.The image of a global configuration and a cell under the global transition function of the naïve product of a semi-cellular automaton can be expressed in terms of the global transition function of the original automaton, as stated in Let ℛ (H, ρ) be a quotient of M, G, , m_0, g_m_0, m_m ∈ M by H and ρ, let 𝒞 = ℛ, Q, N, δ be a naïve product of 𝒞_ H by H and κ, let c be a global configuration of Q^M, let m be an element of M, let f_m be the map ((Hm) _ (H, ρ))^-1 HM → (GH)(G_0H), and letc_ H, m HM→ Q,Hm'↦ c(m κ(f_m(Hm'))).Then, Δ(c)(m) = Δ_ H(c_ H, m)(Hm).Because _ (H, ρ) is free and transitive, (Hm) _ (H, ρ) is injective and surjective. Hence, f_m is well-defined and therefore c_ H, m is also well-defined. Moreover,Δ(c)(m)= δ(n ↦ c(mn))= δ_ H(n_ H↦ c(m κ(n_ H)))= δ_ H(n_ H↦ c_ H, m((Hm) _ (H, ρ) n_ H))= Δ_ H(c_ H, m)(Hm). [Cylinder]In the situation of <ref>, let κ be given by r_1 + 42↦ (r_1 + 42, 0), let c be a global configuration of Q^M, let m = (m_1 + 42, m_2) be an element of M, and let f_m and c_ H, m be as in <ref>. Recall that, according to <ref>, the semi-action _ (H, ρ) is addition in _42 and the semi-actionis addition in _42×. Hence, the map f_m is the map _42→_42, m_1' + 42↦ (m_1' - m_1) + 42, and the global configuration c_ H, m is the map _42→ Q, m_1' + 42↦ c(m_1' + 42, m_2), which is essentially the restriction of c to _42×m_2. The quotient of the naïve product of a semi-cellular automaton is the original automaton, as shown inLet ℛ (H, ρ) be a quotient of M, G, , m_0, g_m_0, m_m ∈ M by H and ρ, let 𝒞_ H = ℛ (H, ρ), Q, N_ H, δ_ H be a semi-cellular automaton, let 𝒞 be a naïve product of 𝒞_ H by H and κ, and let 𝒞 (H, ρ) = ℛ (H, ρ), Q, N_ H', δ_ H' be a quotient of 𝒞 = ℛ, Q, N, δ by H and ρ. Then, 𝒞 (H, ρ) = 𝒞_ H.For each g_0 ∈ G_0 and each g G_0 ∈ GG_0, ϖ(g_0 · g G_0) = g_0 g H (G_0H) = g_0 H ·_ H g H (G_0H) = g_0 H ·_ Hϖ(g G_0). Hence, N_ H'= ϖ(N)= ϖ(G_0 ·κ(N_ H))= g_0 Hg_0 ∈ G_0·_ Hϖ(κ(N_ H))= (G_0H) ·_ H N_ H= N_ H.Therefore, for each ℓ_ H∈ Q^N_ H,δ_ H'(ℓ_ H)= δ(n ↦ℓ_ H(ϖ(n)))= δ_ H(n_ H↦ℓ_ H(ϖ(κ(n_ H))))= δ_ H(n_ H↦ℓ_ H(n_ H))= δ_ H(ℓ_ H).In conclusion, 𝒞 (H, ρ) = 𝒞_ H.According to <ref>, the naïve product of the quotient of a semi-cellular automaton 𝒞 may not be 𝒞.§ PRODUCTS The naïve product by H and κ of a cellular automaton is in general not a cellular automaton (see <ref>). However, as we have seen in <ref>, if the map κ is in a certain sense equivariant (see <ref>), then the naïve product by H and κ of a cellular automaton is again a cellular automaton and is simply called product by H and κ (see <ref>). If the group H is G_0-producible (see <ref>), then such maps κ can be constructed as in <ref>. The global transition function of the product of a cellular automaton does not depend on the coordinate system, in particular, the right inverse κ can in a sense be chosen independently of the origin (see <ref>). Let G be a group, and let G_0 and H be two subgroups of G. The group H is called G_0-producibleproducible@G_0-producibleG_0-producible if and only ifg ∈ G(g G_0 g^-1) ∩ (G_0 H) ⊆ G_0. If the group G is abelian or the group H is included in G_0, then the group H is G_0-producible.Letbe a transitive left group action of G on M, let m_0 be an element of M, and let H be a subgroup of G. The group H is G_m_0-producible if and only ifm ∈ MG_m ∩ (G_m_0 H) ⊆ G_m_0. This is a direct consequence of <ref>.Let G be a group, let G_0 be a subgroup of G, let H be a normal subgroup of G, and let g be an element of G. The group H is G_0-producible if and only if it is (g G_0 g^-1)-producible.For reasons of symmetry, it suffices to prove one implication. To this end, let H be G_0-producible. Furthermore, let g' ∈ G. Then, because H is normal in G, we have g G_0 g^-1 H = g G_0 H g^-1. And, because H is G_0-producible, we have (g^-1 g' g G_0 g^-1 (g')^-1 g) ∩ (G_0 H) ⊆ G_0. Hence,(g' g G_0 g^-1 (g')^-1) ∩ (g G_0 g^-1 H)= g [](g^-1 g' g G_0 g^-1 (g')^-1 g) ∩ (G_0 H) g^-1⊆ g G_0 g^-1.In conclusion, H is g G_0 g^-1-producible. Let G be a group and let H be a subgroup of G. The subgroup H is called malnormalmalnormal if and only ifg ∈ G ∖ H(g H g^-1) ∩ H = e_G. The subgroups e_G and G of G are the only ones that are both normal and malnormal. We can use semi-direct products to construct groups with producible subgroups, as shown in Let F and H be two groups, let G_0 be a malnormal subgroup of F, and let φ be a group homomorphism from F to (H) such that G_0 ⊆(φ), and let G = H ⋊_φ F be the outer semi-direct product of F acting on H by φ. The normal subgroup H ×e_F of G is (e_H× G_0)-producible and, if there is a tuple (g_0, f) ∈ G_0 × (F ∖ G_0) such that f g_0 f^-1∉ G_0, then e_H× G_0 is not normal in G.Let (h, f) ∈ G. Then,(h, f) · (e_H× G_0) · (h, f)^-1= (h, f) · (e_H× G_0) · (φ(f^-1)(h^-1), f^-1)= (h φ(f g_0 f^-1)(h^-1), f g_0 f^-1)g_0 ∈ G_0.And, because the codomain of φ is (H),(e_H× G_0) · (H ×e_F)= (φ(g_0)(h), g_0)h ∈ H, g_0 ∈ G_0= H × G_0. First, let it be the case that f ∈ G_0. Then, f g_0 f^-1∈ G_0. Hence, because G_0 ⊆(φ), we have φ(f g_0 f^-1) =. Thus, h φ(f g_0 f^-1)(h^-1) = e_H. Therefore, (h, f) · (e_H× G_0) · (h, f)^-1⊆e_H× G_0. Secondly, let it be the case that f ∉ G_0. Then, because G_0 is malnormal in F, we have f g_0 f^-1 = e_F or f g_0 f^-1∉ G_0. Hence, we have (h φ(f g_0 f^-1)(h^-1), f g_0 f^-1) = (e_H, e_F) or f g_0 f^-1∉ G_0. Therefore, ((h, f) · (e_H× G_0) · (h, f)^-1) ∩ (e_H× G_0) · (H ×e_F) ⊆(e_H, e_F). In either case, ((h, f) · (e_H× G_0) · (h, f)^-1) ∩ (e_H× G_0) · (H ×e_F) ⊆e_H× G_0. In conclusion, the group H ×e_F is (e_H× G_0)-producible.Moreover, if there is a tuple (g_0, f) ∈ G_0 × (F ∖ G_0) such that f g_0 f^-1∉ G_0, then, according to the second case above, (e_H, f g_0 f^-1) ∈ (e_H, f) · (e_H× G_0) · (e_H, f)^-1, and therefore (e_H, f) · (e_H× G_0) · (e_H, f)^-1⊈e_H× G_0. In conclusion, e_H× G_0 is not normal in G. We can use direct products to construct groups with producible subgroups, as shown inLet F and H be two groups, let G_0 be a subgroup of F, and let G be the direct product H × F. The normal subgroup H ×e_F of G is (e_H× G_0)-producible and, if G_0 is not normal in F, then e_H× G_0 is not normal in G.The proof is omitted here. It is similar to the one of <ref>.[Thick Tree] Let F be the free group over a, b, where a ≠ b, let H be the additive cyclic group 7 of order 7, let G_0 be the non-normal subgroup of F that is generated by a, let G be the direct product H × F, letbe the transitive left group action of G on G(e_H× G_0) by left multiplication, and identify G(e_H× G_0) with H × (FG_0), H ×e_F with H, and (e_H× G_0) with G_0. The normal subgroup H of G is G_0-producible and the subgroup G_0 of G is not normal. As we have seen in <ref>, if the product of a cellular automaton is made by a right inverse κ that is in a certain sense equivariant, which we give a name below, then that product is itself a cellular automaton. Let G be a group, let G_0 be a subgroup of G, let H be a normal subgroup of G, let ϖ be the canonical projection from GG_0 onto (GH)(G_0H), and let κ be a right inverse of ϖ. The map κ is called G_0-equivariantequivariantG0@G_0-equivariantG_0-equivariant if and only if g_0 ∈ G_0 𝔤_ H∈ (GH)(G_0H) g_0 ·κ(𝔤_ H) = κ(g_0 H ·_ H𝔤_ H).[Thick Tree]In the situation of <ref>, recall that GG_0 is identified with H × (FG_0), and identify GH with F and G_0H with G_0. Then, the canonical projection ϖ from GG_0 onto (GH)(G_0H) is the projection to the second component, namely (h, f G_0) ↦ f G_0. Hence, for each right inverse κ of ϖ, it is G_0-equivariant if and only ifz ∈ f G_0 ∈ FG_0a^z ·κ(f G_0) = κ(a^z · f G_0),which is the case if and only if, for each element f G_0 ∈ FG_0, the projection to the first component of the restriction κ_G_0 · f G_0 is constant. In particular, for each element h ∈ H, the map κ FG_0 → H × (FG_0), f G_0 ↦ (h, f G_0), is a G_0-equivariant right inverse of ϖ. If H is a G_0-producible and normal subgroup of G, then we can construct a G_0-equivariant right inverse of the canonical projection from GG_0 onto (GH)(G_0H), as shown in Let G be a group, let G_0 be a subgroup of G, let H be a G_0-producible and normal subgroup of G, and let ϖ be the canonical projection from GG_0 onto (GH)(G_0H). Furthermore, let X be the quotient set (GH)(G_0H), let ∼ be the equivalence relation on X given byx ∈ Xx' ∈ X *x ∼ x'g_0 ∈ G_0g_0 H ·_ H x = x',let x_ii ∈ I be a transversal of X ∼, and let ξ be a right inverse of ϖ. The mapκ X→ GG_0,g_0 H ·_ H x_i↦ g_0 ·ξ(x_i),is a G_0-equivariant right inverse of ϖ.Subproof of well-definedness The equivalence classes [x_i]_∼, for i ∈ I, partition X and, for each i ∈ I and each x ∈ [x_i]_∼, there is a g_0 ∈ G_0 such that x = g_0 H ·_ H x_i. Hence, for each x ∈ X, there is a unique i ∈ I and a g_0 ∈ G_0 such that x = g_0 H ·_ H x_i. Therefore, if, for each i ∈ I, each g_0 ∈ G_0, and each g_0' ∈ G_0 such that g_0 H ·_ H x_i = g_0' H ·_ H x_i, we have g_0 ·ξ(x_i) = g_0' ·ξ(x_i), then κ is well-defined.Let i ∈ I, let g_0 ∈ G_0, and let g_0' ∈ G_0 such that g_0 H ·_ H x_i = g_0' H ·_ H x_i. Then, there is a g ∈ G such that ξ(x_i) = g G_0. Thus, x_i = ϖ(ξ(x_i)) = g H (G_0H). Hence, with g_0” = g_0^-1 g_0',g_0 H ·_ H x_i = g_0' H ·_ H x_i g_0” H ·_ H x_i = x_i g_0” g H (G_0H) = g H (G_0H) g^-1 g_0” g H (G_0H) = G_0H g_0 ∈ G_0g^-1 g_0” g H = g_0 H g_0 ∈ G_0g^-1 g_0” g ∈ g_0 H g^-1 g_0” g ∈ G_0 H.And, because g^-1 g_0” g ∈ G_g^-1 m and H is G_0-producible,g^-1 g_0” g ∈ G_0 Hg^-1 g_0” g ∈ G_0.And,g^-1 g_0” g ∈ G_0 g_0” g G_0 = g G_0 g_0”·ξ(x_i) = ξ(x_i) g_0 ·ξ(x_i) = g_0' ·ξ(x_i).In conclusion, because g_0 H ·_ H x_i = g_0' H ·_ H x_i, we have g_0 ·ξ(x_i) = g_0' ·ξ(x_i).Subproof of right inverseness Let x ∈ X. Then, there is a g_0 ∈ G_0 and there is an i ∈ I such that x = g_0 H ·_ H x_i. And, there is a g ∈ G such that ξ(x_i) = g G_0. Hence,(ϖκ)(x)= ϖ(κ(x))= ϖ(g_0 ·ξ(x_i))= ϖ(g_0 g G_0)= g_0 g H (G_0H)= g_0 H ·_ H g H (G_0H)= g_0 H ·_ Hϖ(g G_0)= g_0 H ·_ Hϖ(ξ(x_i))= g_0 H ·_ H x_i= x.In conclusion, κ is a right inverse of ϖ.Subproof of G_0-equivariance Let x ∈ X and let g_0 ∈ G_0. Then, there is a g_0' ∈ G_0 and there is an i ∈ I such that x = g_0' H ·_ H x_i. Hence,g_0 ·κ(x)= g_0 · (g_0' ·ξ(x_i))= g_0 g_0' ·ξ(x_i)= κ(g_0 g_0' H ·_ H x_i)= κ(g_0 H ·_ H (g_0' H ·_ H x_i))= κ(g_0 H ·_ H x).In conclusion, κ is G_0-equivariant.Let G be a group, let G_0 be a subgroup of G, let H be a G_0-producible and normal subgroup of G, and let ϖ be the canonical projection from GG_0 onto (GH)(G_0H). There is a G_0-equivariant right inverse of ϖ.This is a direct consequence of <ref>, because there is a right inverse ξ of ϖ.Each G_0-equivariant right inverse κ of ϖ is of the form as in <ref>. Indeed, let κ be such a right inverse, let x_ii ∈ I be a transversal of X ∼, and let ξ be identical to κ. Then, for each g_0 ∈ G_0 and each i ∈ I, we have κ(g_0 H ·_ H x_i) = g_0 ·κ(x_i) = g_0 ·ξ(x_i). [Thick Tree]In the situation of <ref>, broadly speaking, given a right inverse ξ of ϖ and representatives f_i G_0, for i ∈ I, of the orbits G_0 · f G_0, for f G_0 ∈ FG_0, forcing the images of the elements of G_0 · f_i G_0 under ξ to have the same first component as the image of f_i G_0 under ξ, yields a G_0-equivariant right inverse κ of ϖ. The naïve product by a G_0-equivariant right inverse is a cellular automaton, which we give a name in Let ℛ (H, ρ) be a quotient of M, G, , m_0, g_m_0, m_m ∈ M by H and ρ, let κ be a G_0-equivariant right inverse of the canonical projection ϖ GG_0 → (GH)(G_0H), and let 𝒞_ H = ℛ (H, ρ), Q, N_ H, δ_ H be a cellular automaton. The naïve product product 𝒞 of 𝒞_ H by H and κ𝒞 of 𝒞_ H by H and κ is a cellular automaton and is called product of 𝒞_ H by H and κ.According to <ref>, the semi-cellular automaton 𝒞 is a cellular automaton. [Thick Tree]In the situation of <ref>, let 𝒦 = e_G G_0, g_e_G G_0, g G_0_g G_0 ∈ GG_0 be a coordinate system for ℳ = GG_0, G,, let ρ be the right inverse of the canonical projection π H × (FG_0) ≃ GG_0 → H(GG_0) ≃ FG_0 given by ρ(f G_0) = (0 + 7, f G_0), and let κ be the right inverse of the canonical projection ϖ H × (FG_0) ≃ GG_0 → (GH)(G_0H) ≃ FG_0 given by f G_0 ↦ (0 + 7, f G_0) (note that the right inverses ρ and κ are identical).Furthermore, let Q be the binary set 0, 1; let N_ H be the (G_0 ≃ G_0H)-invariant subset a^z · b G_0z ∈ of FG_0 ≃ (GH)(G_0H); let δ_ H be the ∙_ H-invariant mapQ^N_ H → Q, ℓ_ H ↦ 0,if ∑_n_ H∈ N_ Hℓ_ H(n_ H) < ∞,1,otherwise;let 𝒞_ H be the cellular automaton ℳ, 𝒦 (H, ρ), Q, N_ H, δ_ H; and let Δ_ H be the global transition function of 𝒞_ H.The product of 𝒞_ H by H and κ has the neighbourhood N = 0 + 7× N_ H; the local transition function δ Q^N → Q, ℓ↦ 0, if ∑_n ∈ Nℓ(n) < ∞, and ℓ↦ 1, otherwise; and the product ∏_h ∈ HΔ_ H for global transition function Δ. Let M, G, be a left group set, let H be a normal subgroup of G, and, for each point m ∈ M, let ϖ_m be the canonical projection from GG_m onto (GH)(G_mH). * For each point m ∈ M, the set of all G_m-equivariant right inverses of ϖ_m is denoted by 𝔨_m[symbols]Km fraktur@𝔨_m𝔨_m.* The union of the sets 𝔨_m, for m ∈ M, is denoted by 𝔨[symbols]K fraktur@𝔨𝔨.The group acts on the set of all equivariant right inverses, as shown inLet M, G, be a left group set and let H be a normal subgroup of G. The group G acts on 𝔨 on the left by⊡ G ×𝔨 →𝔨,(g, κ_m)↦[ (GH)(G_gm H)→ GG_gm,x↦ g κ_m(g^-1 H _ H x), ]such that, for each symmetry g ∈ G and each point m ∈ M, the map(g ⊡)_𝔨_m →𝔨_gm𝔨_m→𝔨_gm, κ_m↦ g ⊡κ_m,is bijective. For each m ∈ M, let X_m = (GH)(G_mH) = (GH)(GH)_Hm. Then, for each m ∈ M, we have X_gm = (GH)(GH)_g H ·_H (Hm). Moreover, for each m ∈ M, let ϖ_m be the canonical projection from GG_m onto (GH)(G_mH).First, let g ∈ G, let m ∈ M, and let κ_m ∈𝔨_m. Then, for each x ∈ X_gm, according to <ref>, we have g^-1 H _ H x ∈ X_m and hence g κ_m(g^-1 H _ H x) ∈ GG_gm. Therefore, the map g ⊡κ_m is well-defined.Moreover, for each x ∈ X_gm, there is a g' ∈ G such that κ_m(g^-1 H _ H x) = g' G_m and henceϖ_gm[](g ⊡κ_m)(x) = ϖ_gm[]g κ_m(g^-1 H _ H x)= ϖ_gm(gg' G_m)= ϖ_gm(g g' g^-1 G_gm)= g g' g^-1 H (GH)_H(gm)= g H _ H g' H (GH)_Hm= g H _ Hϖ_m(g' G_m)= g H _ Hϖ_m[]κ_m(g^-1 H _ H x)= g H _ H (g^-1 H _ H x)= x.Therefore, the map g ⊡κ_m is a right inverse of ϖ_gm.Furthermore, for each g_gm∈ G_gm and each x ∈ X_gm, according to <ref> and because g^-1 g_gm g ∈ G_m and κ_m is G_m-equivariant,(g ⊡κ_m)(g_gm H ·_ H x)= g κ_m[]g^-1 H _ H (g_gm H ·_ H x)= g κ_m[]g^-1 g_gm g H ·_ H (g^-1 H _ H x)= g []g^-1 g_gm g ·κ_m(g^-1 H _ H x)= g_gm·[]g κ_m(g^-1 H _ H x)= g_gm· (g ⊡κ_m)(x).Therefore, the right inverse g ⊡κ_m is G_gm-equivariant. Altogether, the right inverse g ⊡κ_m is an element of 𝔨_gm. In conclusion, the maps ⊡ and (g ⊡)_𝔨_m →𝔨_gm are well-defined.Secondly, for each κ∈𝔨, we have e_G ⊡κ = κ. And, for each g ∈ G, each g' ∈ G, each κ∈𝔨, and each x ∈ X_m,(g g' ⊡κ)(x)= g g' κ[](g')^-1 g^-1 H _ H x= g []g' κ((g')^-1 H _ H (g^-1 H _ H x))= g(g' ⊡κ)(g^-1 H _ H x)= (g ⊡ (g' ⊡κ))(x),and hence g g' ⊡κ = g ⊡ (g' ⊡κ). In conclusion, the map ⊡ is a left group action. [Thick Tree]In the situation of <ref>, let (h, f) be an element of G. Then, the subgroup G_0 ≃e_H× G_0 of G is the stabiliser of e_G G_0 ≃e_H× G_0 underand its conjugate f G_0 f^-1≃e_H× (f G_0 f^-1) is the stabiliser of (h, f) G_0 ≃h× (f G_0) under . Hence, the right inverse (h, f) ⊡κ of the canonical projection ϖ_(h, f) G_0 H × (F(f G_0 f^-1)) ≃ G(f G_0 f^-1) → (GH)(f G_0 f^-1 H) ≃ F(f G_0 f^-1) is given by f' f G_0 f^-1↦ (0 + 7, f' f G_0 f^-1). The orbit of a right inverse contains one for each stabiliser, which is shown in Let M, G, be a left-homogeneous space and let H be a normal subgroup of G. Then,κ∈𝔨 m ∈ Mg ∈ Gg ⊡κ∈𝔨_m.Let κ∈𝔨 and let m ∈ M. Then, there is an m' ∈ M such that κ∈𝔨_m'. And, becauseis transitive, there is a g ∈ G such that gm' = m. Hence, according to <ref>, we have g ⊡κ∈𝔨_gm' = 𝔨_m. The global transition function of the product of a cellular automaton does not depend on the coordinate system, which is shown in Let ℳ = M, G, be a left-homogeneous space, let 𝒦 = m_0, g_m_0, m_m ∈ M and 𝒦' = m_0', g_m_0', m'_m ∈ M be two coordinate systems for ℳ, let H be a normal subgroup of G, and let ρ and ρ' be two right inverses of the canonical projection π M → HM such that ρ(Hm_0) = m_0 and ρ'(Hm_0') = m_0'Furthermore, let 𝒞_ H = ℳ, 𝒦 (H, ρ), Q, N_ H, δ_ H and 𝒞_ H' = ℳ, 𝒦' (H, ρ'), Q, N_ H', δ_ H' be two cellular automata such that the global transition function Δ_ H of 𝒞_ H is identical to the one, namely Δ_ H', of 𝒞_ H'.Moreover, let κ be an element of 𝔨_m_0 and let κ' be an element of 𝔨_m_0' such that there is an element g ∈ G_m_0, m_0' that satisfies g ⊡κ = κ', let 𝒞 be the product of 𝒞_ H by H and κ, and let 𝒞' be the product of 𝒞_ H' by H and κ'.The global transition function Δ of 𝒞 is identical to the one, namely Δ', of 𝒞'.Becauseand ' are similar (see <ref>), for each m ∈ M, there is a g_m,0' ∈ G_0' such that𝔤∈ GG_0m 𝔤 = m ' g_m,0' · (g 𝔤).And, because Δ_ H = Δ_ H', according to <ref>, there are cellular automata 𝒞_ H, * = ℳ, 𝒦 (H, ρ), Q, N_ H, *, δ_ H, * and 𝒞_ H, *' = ℳ, 𝒦' (H, ρ'), Q, N_ H, *', δ_ H, *' such that δ_ H, * = g^-1 H ⊗_ Hδ_ H, *' andℓ_ H∈ Q^N_ Hδ_ H(ℓ_ H) = δ_ H, *(ℓ_ H_N_ H, *), ℓ_ H' ∈ Q^N_ H'δ_ H'(ℓ_ H') = δ_ H, *'(ℓ_ H'_N_ H, *'). Let c ∈ Q^M and let m ∈ M. Then, according to <ref>,Δ(c)(m)= δ(n ↦ c(mn))= δ_ H(n_ H↦ c(m κ(n_ H)))= δ_ H(n_ H↦ c(m ' g_m,0' · (g κ(n_ H)))).Thus, according to <ref>,Δ(c)(m) = δ_ H(n_ H↦ c(m ' g(g^-1 g_m,0' g ·κ(n_ H)))).Thus, because κ is G_0-equivariant,Δ(c)(m)= δ_ H(n_ H↦ c(m ' g κ(g^-1 g_m,0' g · n_ H)))= δ_ H((g^-1 g_m,0' g)^-1∙_ H [n_ H↦ c(m ' g κ(n_ H))]).Thus, because δ_ H is ∙_ H-invariant,Δ(c)(m) = δ_ H(n_ H↦ c(m ' g κ(n_ H))).Thus, according to <ref>,Δ(c)(m) = δ_ H, *(n_ H, *↦ c(m ' g κ(n_ H, *))).Thus, because δ_ H, * = g^-1 H ⊗_ Hδ_ H, *',Δ(c)(m)= (g^-1 H ⊗_ Hδ_ H, *')(n_ H, *↦ c(m ' g κ(n_ H, *)))= δ_ H, *'(n_ H, *' ↦ c(m ' g κ(g^-1 H _ H n_ H, *')))= δ_ H, *'(n_ H, *' ↦ c(m ' (g ⊡κ)(n_ H, *'))).Thus, because g ⊡κ = κ',Δ(c)(m) = δ_ H, *'(n_ H, *' ↦ c(m ' κ'(n_ H, *'))).Thus, according to <ref>,Δ(c)(m)= δ_ H(n_ H' ↦ c(m ' κ'(n_ H')))= δ'(n ↦ c(m ' n))= Δ'(c)(m).In conclusion, Δ = Δ'. The notion of product of a global transition function is well-defined and we give it a name in Let ℳ = M, G, be a left-homogeneous space, let H be a normal subgroup of G, let Δ_ H be the global transition function of a cellular automaton over ℳ H, and let K be an element of the orbit space G 𝔨 under ⊡. The global transition function Δ of a product of 𝒞_ H by H and κ – where 𝒦 = m_0, g_m_0, m_m ∈ M is a coordinate system for ℳ, ρ is a right inverse of the canonical projection π M → HM such that ρ(Hm_0) = m_0, 𝒞_ H is a cellular automaton over ℳ, 𝒦 (H, ρ) whose global transition function is Δ_ H, and κ is an element of K ∩𝔨_m_0 – is the global transition function of a cellular automaton over ℳ and is called product of Δ_ H by H and Kproduct Δ of Δ_ H by H and K.According to <ref>, the global transition function Δ does not depend on the choice of 𝒦, ρ, 𝒞_ H, and κ; and it is hence well-defined. [Thick Tree]In the situation of <ref>, the product Δ of Δ_ H by H and G ⊡κ is the product ∏_h ∈ HΔ_ H, which does neither depend on 𝒦, nor on ρ, nor on κ. The quotient of the product of a global transition function is identical to the original global transition function, as shown inLet ℳ = M, G, be a left-homogeneous space, let H be a normal subgroup of G, let Δ_ H be the global transition function of a cellular automaton over ℳ H, let Δ be a product of Δ_ H by H and K, and let Δ_ H' be the quotient of Δ by H. Then, Δ_ H' = Δ_ H.This is a direct consequence of <ref>. The product of the quotient of a global transition function may not be the original global transition function, as illustrated in[Thick Tree]In the situation of <ref>, let 𝒦 = e_G G_0, g_e_G G_0, g G_0_g G_0 ∈ GG_0 be a coordinate system for ℳ = GG_0, G,; let Q be the binary set 0, 1; let N be the subset -1 + 7, 0 + 7, 1 + 7×e_F G_0 of GG_0 (note that G_0 · N ⊆ N); let δ be the ∙-invariant map Q^N → Q, ℓ↦ 0, if ∑_n ∈ Nℓ(n) < 3/2, and ℓ↦ 1, otherwise, which is known as majority rulemajority rulerule!majority; let 𝒞 be the cellular automaton ℳ, 𝒦, Q, N, δ; and let Δ be the global transition function of 𝒞, which realises the majority rule on each of the discrete circles H ×f G_0, for f G_0 ∈ FG_0. Each quotient of 𝒞 by H has the neighbourhood N_ H = e_F G_0, the local transition function δ_ H Q^e_F G_0→ Q, ℓ_ H↦ℓ_ H(e_F G_0), and the identity map on Q^FG_0 for global transition function Δ_ H. And, each product of Δ_ H by H is the identity map on Q^GG_0, which is not Δ.§ RESTRICTIONS There is a canonical bijection from HG_0 to HH_0, which is given inLet G be a group, let G_0 and H be two subgroups of G, and let H_0 be the subgroup G_0 ∩ H of H. The mapζ HG_0→ HH_0, canonical bijection ζ from HG_0 to HH_0[symbols]zeta@ζh G_0↦ h H_0,is well-defined, is bijective, is ·-equivariant, and is called canonical bijection from HG_0 to HH_0.For each h ∈ H and each h' ∈ H,h G_0 = h' G_0 h^-1 h' ∈ G_0 h^-1 h' ∈ H_0 h H_0 = h' H_0.Hence, ζ is well-defined and bijective. Moreover, for each h ∈ H and each h' G_0 ∈ HG_0, we have h ·ζ(h' G_0) = h h' H_0 = ζ(h · h' G_0). Therefore, ζ is ·-equivariant. Let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space and let H be a normal subgroup of G. Then,m ∈ MHm = m(HG_0). For each m ∈ M, because g_m_0, m^-1 H = H g_m_0, m^-1,Hm= g_m_0, m g_m_0, m^-1 Hm= g_m_0, m H g_m_0, m^-1 m= m(HG_0).Restrictions of left-homogeneous spaces, coordinate systems, and cell spaces are introduced in the two forthcoming definitions. Let ℳ = M, G, be a left group set, let m_0 be an element of M, and let H be a subgroup of G. The triple ℳ_m_0, H = Hm_0, H, _H × (Hm_0) → Hm_0restriction ℳ_m_0, H of ℳ at m_0 to H is a left-homogeneous space and is called restriction of ℳ at m_0 to H[symbols]Mm0Hcalligraphicharpoon@ℳ_m_0, H.Let ℳ = M, G, be a left-homogeneous space, let 𝒦 = m_0, g_m_0, m_m ∈ M be a coordinate system for ℳ, let ℛ be the cell space ℳ, 𝒦, and let H be a subgroup of G such that g_m_0, hm_0 h ∈ H⊆ H. The tuple 𝒦_H = m_0, g_m_0, hm_0_hm_0 ∈ Hm_0restriction 𝒦_H of 𝒦 to H is a coordinate system for ℳ_m_0, H and is called restriction of 𝒦 to H[symbols]KHcalligraphicharpoon@𝒦_H. And, the tuple ℛ_H = ℳ_m_0, H, 𝒦_Hrestriction ℛ_H of ℛ to H is a cell space and is called restriction of ℛ to H[symbols]RHcalligraphicharpoon@ℛ_H. [Sphere]Let M be the Euclidean unit 2-sphere, let G be the rotation group, letbe the left group action of G on M by function application, let m_0 be the north pole (0,0,1)^ of M and, for each point m ∈ M, let g_m_0, m be a rotation about an axis in the (x, y)-plane that rotates m_0 to m, which is unique unless m is the south pole, ℛ = ℳ, 𝒦 be the cell space M, G, , m_0, g_m_0, m_m ∈ M (this is the situation of <ref>).Furthermore, let a be the axis of the rotation g_m_0, S, where S is the south pole (0, 0, -1)^, and let H be the subgroup of G that consists of the rotations about a. The set Hm_0 is a great circle through the north and the south pole, the abelian group H is the group of orientation-preserving symmetries of this circle, the restriction _H × (Hm_0) → Hm_0 is the free left group action of H on Hm_0 by function application, and we have g_m_0, hm_0 hm_0 ∈ Hm_0⊆ H. Hence, the restriction of ℳ at m_0 to H is isomorphic to the abelian groupand the restriction of 𝒦 to H is the unique coordinate system for ℳ_m_0, H whose origin is m_0. In the remainder of this section, let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space, let H be a subgroup of G, and let ℛ_H = M__H, H, __H, m_0, g_m_0, m__H_m__H∈ M__H be the restriction of ℛ to H. The right quotient set semi-actions of a cell space and its restriction relate as shown inThe stabiliser of m_0 under __H is H_0 = G_0 ∩ H and the right quotient set semi-action induced by ℛ_H is__H M__H× HH_0→ M__H,(m__H, h H_0)↦ m__H h G_0. The subgroup H_0 of H is the stabiliser of m_0 under __H. Moreover, according to <ref>, for each h ∈ H, the coset h G_0 is uniquely determined by h H_0. And, for each m__H∈ M__H and each h H_0 ∈ HH_0,m__H__H h H_0= m__H h G_0= g_m_0, m__H h g_m_0, m__H^-1 m__H= g_m_0, m__H h g_m_0, m__H^-1__H m__H∈ M__H.In conclusion, the map __H is well-defined and it is the right quotient set semi-action induced by ℛ_H. Restrictions of (semi-)cellular automata are introduced in Let 𝒞 = ℛ, Q, N, δ be a semi-cellular or cellular automaton with a sufficient neighbourhood that is included in HG_0, let E be the biggest sufficient neighbourhood of 𝒞 such that H_0 · E ⊆ E and E ⊆ HG_0, let η be the sufficient local transition function from Q^E to Q, let ζ be the canonical bijection from HG_0 to HH_0, letN__H = ζ(E)(= h H_0h G_0 ∈ E),and letδ__H Q^N__H → Q, ℓ__H ↦ηe ↦ℓ__H(ζ(e)) (= ηh G_0 ↦ℓ__H(h H_0)).The quadruple 𝒞_H = ℛ_H, Q, N__H, δ__Hrestriction 𝒞_H of 𝒞 to H is a semi-cellular or cellular automaton respectively, and is called restriction of 𝒞 to H[symbols]CHcalligraphicharpoon@𝒞_H. First, because H_0 ·ζ(E) = ζ(H_0 · E) and H_0 · E ⊆ E, we have H_0 · N__H⊆ N__H. In conclusion, the quadruple 𝒞_H is a semi-cellular automaton.Secondly, let 𝒞 be a cellular automaton. Furthermore, let h_0 ∈ H_0. Then, because ζ is ·-equivariant and η is ∙_H_0-invariant, for each ℓ__H∈ Q^N__H,δ__H(h_0 ∙__Hℓ__H)= η[]e ↦ (h_0 ∙__Hℓ__H)(ζ(e)= η[]e ↦ℓ__H(h_0^-1·ζ(e)= η[]e ↦ℓ__H(ζ(h_0^-1· e)= η[]h_0 ∙[]e ↦ℓ__H(ζ(e))= η[]e ↦ℓ__H(ζ(e))= δ__H(ℓ__H).In conclusion, δ__H is ∙__H-invariant and hence 𝒞_H is a cellular automaton. [Sphere]In the situation of <ref>, let h be one of the two rotations about a by 1, let Q be the set 0, 1, let E be the singleton set h G_0, let N be the set G_0E, let η be the map Q^E → Q, ℓ_E ↦ℓ_E(h G_0), and let δ be the map Q^N → Q, ℓ↦η(ℓ_E). The restriction of the semi-cellular automaton 𝒞 = ℛ, Q, N, δ to H is the cellular automaton 𝒞_H = ℛ_H, Q, E, η whose global transition function is the rotation map h^-1__H. The global transition functions of a semi-cellular automaton and its restriction relate as shown inLet 𝒞_H be a restriction of 𝒞 = ℛ, Q, N, δ to H. Then,c ∈ Q^M Δ__H(c_M__H) = Δ(c)_M__H. Let η Q^E → Q be a sufficient local transition function of 𝒞 such that H_0 · E ⊆ E and E ⊆ HG_0, and let ζ be the canonical bijection from HG_0 to HH_0. Furthermore, let c ∈ Q^M. Then, for each m__H∈ M__H, according to <ref>,Δ__H(c_M__H)(m__H)= δ__H[]n__H↦ c_M__H(m__H__H n__H)= η[]e ↦ c_M__H(m__H__Hζ(e))= η[]e ↦ c_M__H(m__H e)= η[]e ↦ c(m__H e)= δ[]n ↦ c(m__H n)= Δ(c)(m__H).In conclusion, Δ__H(c_M__H) = Δ(c)_M__H. Restrictions of global transition functions of cellular automata are introduced inLet ℳ = M, G, be a left-homogeneous space, let H be a subgroup of G, let m_0 be an element of M, and let Δ Q^M → Q^M be the global transition function of a cellular automaton over ℳ with sufficient neighbourhood E such that E ⊆ HG_0. Furthermore, let π be the canonical projection from Q^M onto Q^Hm_0 and let ρ be a right inverse of π. The map Δ__m_0,H = πΔρrestriction Δ__m_0, H of Δ at m_0 to H is the global transition function of a cellular automaton over Hm_0, H, _H × (Hm_0) → Hm_0, does not depend on ρ, and is called restriction of Δ at m_0 to H[symbols]Deltam0Hharpoonm@Δ__m_0,H.There is a coordinate system 𝒦 with origin m_0 and a cellular automaton 𝒞 = ℳ, 𝒦, Q, N, δ with sufficient neighbourhood E whose global transition function is Δ. Moreover, there is a coordinate system 𝒦' = m_0, g_m_0, m'_m ∈ M for ℳ such that, for each m ∈ Hm_0, we have g_m_0, m' ∈ H. And, according to <ref>, the global transition function of the cellular automaton 𝒞' = ℳ, 𝒦', Q, N, δ is Δ. And, the set E' = H_0 · E is a sufficient neighbourhood of 𝒞' such that H_0 · E' ⊆ E' and E' ⊆ HG_0. Hence, according to <ref>, the map Δ__m_0,H is the global transition function of the restriction of 𝒞' to H, which does not depend on ρ.§ EXTENSIONS Extensions of semi-cellular automata are introduced in Let 𝒞__H = ℛ_H, Q, N__H, δ__H be a semi-cellular automaton, let ζ be the canonical bijection from HG_0 to HH_0, letN = G_0 ·ζ^-1(N__H) = g_0 h G_0g_0 ∈ G_0, h H_0 ∈ N__H,and letδ Q^N→ Q, ℓ ↦δ__H[]n__H↦ℓ(ζ^-1(n__H)) (= δ__H[]h H_0 ↦ℓ(h G_0)).The quadruple 𝒞 = ℛ, Q, N, δextension 𝒞 of 𝒞__H to G is a semi-cellular automaton and is called extension of 𝒞__H to G[symbols]CHcalligraphicharpoon@𝒞__H. [Sphere]In the situation of <ref>, the extension of the cellular automaton 𝒞_H to G is the semi-cellular automaton 𝒞. The global transition functions of a semi-cellular automaton and its extension relate as shown inLet 𝒞 be an extension of 𝒞__H = ℛ_H, Q, N__H, δ__H to G. Then,c ∈ Q^M Δ(c)_M__H = Δ__H(c_M__H),andc ∈ Q^Mm ∈ M Δ(c)(m) = Δ__H((g_m_0, m^-1 c)_M__H)(m_0). Let ζ be the canonical bijection from HG_0 to HH_0.First, let c ∈ Q^M. Then, for each m__H∈ M__H, according to <ref>,Δ(c)(m__H)= δ[]n ↦ c(m__H n)= δ__H[]n__H↦ c(m__Hζ^-1(n__H))= δ__H[]n__H↦ c(m__H__H n__H)= δ__H[]n__H↦ c_M__H(m__H__H n__H)= Δ__H(c_M__H)(m__H).In conclusion, Δ(c)_M__H = Δ__H(c_M__H).Secondly, for each c ∈ Q^M and each m ∈ M, according to <ref>,Δ(c)(m) = Δ(g_m_0, m^-1 c)(m_0) = Δ__H((g_m_0, m^-1 c)_M__H)(m_0).The global transition functions of a semi-cellular automaton and the restriction of its extension are identical as shown in Let 𝒞 be an extension of 𝒞__H = ℛ_H, Q, N__H, δ__H to G and let 𝒞_H = ℛ_H, Q, N__H', δ__H' be the restriction of 𝒞 to H. The global transition functions of 𝒞__H and 𝒞_H are identical.This is a direct consequence of <ref>. The global transition functions of a semi-cellular automaton and the extension of its restriction are identical as shown in Let 𝒞_H be a restriction of 𝒞 = ℛ, Q, N, δ to H and let 𝒞' = ℛ, Q, N', δ' be the extension of 𝒞_H to G. The global transition functions of 𝒞 and 𝒞' are identical.For each c ∈ Q^M and each m ∈ M, according to <ref>, <ref>, and <ref>,Δ'(c)(m)= Δ__H((g_m_0, m^-1 c)_M__H)(m_0)= Δ(g_m_0, m^-1 c)(m_0)= Δ(c)(m).In conclusion, Δ' = Δ.The extension of the restriction of a semi-cellular automaton and the restriction of the extension of a semi-cellular automaton is in general not the automaton we started out with, because in the first case superfluous neighbours may be removed by the restriction that are not re-added by the extension and in the second case superfluous neighbours may be added by the extension that are not removed by the restriction. The extension of a cellular automaton is in general not a cellular automaton. However, if the local transition function is in a certain sense invariant under ∙, then the extension is a cellular automaton, which is shown in Let 𝒞__H = ℛ_H, Q, N__H, δ__H be a semi-cellular automaton, let ζ be the canonical bijection from HG_0 to HH_0, let N be the set G_0 ·ζ^-1(N__H), let ξ be the restriction of ζ^-1 to N__H→ N, such thatg_0 ∈ G_0 ℓ∈ Q^N δ__H[](g_0 ∙ℓ) ξ = δ__H(ℓξ). The semi-cellular automaton 𝒞__H and its extension 𝒞 to G are cellular automata.Let h_0 ∈ H_0 and let ℓ__H∈ Q^N__H. Then, because ξ is ·-equivariant, for each n__H∈ N__H,(h_0 ∙__Hℓ__H)(n__H)= ℓ__H(h_0^-1· n__H)= (ℓ__Hξ^-1)[]ξ(h_0^-1· n__H)= (ℓ__Hξ^-1)[]h_0^-1·ξ(n__H)= []h_0 ∙ (ℓ__Hξ^-1)[]ξ(n__H)= [][]h_0 ∙ (ℓ__Hξ^-1)ξ(n__H).Thus, h_0 ∙__Hℓ__H = (h_0 ∙ (ℓ__Hξ^-1)) ξ. Hence, according to <ref>,δ__H(h_0 ∙__Hℓ__H)= δ__H[](h_0 ∙ (ℓ__Hξ^-1)) ξ= δ__H[](ℓ__Hξ^-1) ξ= δ__H(ℓ__H).Therefore, δ__H is ∙__H-invariant. In conclusion, 𝒞__H is a cellular automaton.Moreover, according to <ref>, for each g_0 ∈ G_0 and each ℓ∈ Q^N,δ(g_0 ∙ℓ)= δ__H[]n__H↦ (g_0 ∙ℓ)(ζ^-1(n__H))= δ__H[](g_0 ∙ℓ) ξ= δ__H(ℓξ)= δ(ℓ).Hence, δ is ∙-invariant. In conclusion, 𝒞 is a cellular automaton. If G_0 is included in H, then G_0 = H_0, N = N__H, δ = δ__H, ξ = _N, and <ref> just states that δ is ∙-invariant.Extensions of global transition functions of cellular automata are introduced in Let ℳ = M, G, be a left-homogeneous space, let m_0 be an element of M, let H be a subgroup of G that includes G_0, and let Δ__H Q^Hm_0→ Q^Hm_0 be the global transition function of a cellular automaton over ℳ_m_0, H. Furthermore, let 𝒦 = m_0, g_m_0, m_m ∈ M be a coordinate system for ℳ such that g_m_0, hm_0 hm_0 ∈ Hm_0⊆ H. The mapΔ Q^M→ Q^M, extension Δ of Δ__H at m_0 to Gc↦ [m ↦Δ__H((g_m_0, m^-1 c)_Hm_0)(m_0)],is the global transition function of a cellular automaton over ℳ, does not depend on the coordinates g_m_0, m_m ∈ M, and is called extension of Δ__H at m_0 to G.According to <ref>, there is a cellular automaton 𝒞__H = ℳ_m_0, H, 𝒦_H, Q, N, δ whose global transition function is Δ__H. According to <ref>, its extension to G is the cellular automaton 𝒞 = ℳ, 𝒦, Q, N, δ. According to <ref>, the map Δ is the global transition function of 𝒞. According to <ref>, this global transition function does not depend on the coordinates g_m_0, m_m ∈ M and hence neither does Δ.§ DECOMPOSITIONS Given a subgroup H of G, the orbit space of _M × HG_0 does in general not partition M (see <ref>). However, if H satisfies the property G_0 · (HG_0) ⊆ HG_0, then the aforementioned orbit space partitions M (see <ref>). For such a group H and a semi-cellular automaton whose neighbourhood is included in HG_0, the phase space is the product of configurations on orbits and the global transition function of the automaton is the product of its restrictions, in a certain sense, to orbits (see <ref>). The latter restrictions are conjugations of the restriction of the global transition function to H as introduced above (compare <ref>) and hence the global transition function is injective, surjective, or bijective if and only if its restriction to H has the respective property (see <ref>).Let G be a group, and let G_0 and H be two subgroups of G. The group H is called (G_0, G_0)-invariantinvariantG0G0@(G_0, G_0)-invariant(G_0, G_0)-invariantG0G0invariant@(G_0, G_0)-invariant if and only ifG_0 · (HG_0) ⊆ HG_0. Let G be a group, let G_0 be a subgroup of G, and let H be a normal subgroup of G. The group H is (G_0, G_0)-invariant.Let g_0 ∈ G_0 and let h ∈ H. Then, because H is normal in G, we have g_0 H = H g_0. Hence, there is an h' ∈ H such that g_0 h = h' g_0. Therefore, g_0 h G_0 = h' g_0 G_0 = h' G_0. In conclusion, G_0 · (HG_0) ⊆ HG_0.Let G be a group, and let G_0 and H be two subgroups of G such that G_0 ⊆ H. The group H is (G_0, G_0)-invariant.Let g_0 ∈ G_0 and let h ∈ H. Then, because G_0 ⊆ H, we have g_0 h ∈ H. Hence, g_0 h G_0 ∈ HG_0. In conclusion, G_0 · (HG_0) ⊆ HG_0.For a (G_0, G_0)-invariant subgroup H of G, the orbit space of _M × HG_0 partitions M, which is shown inLet ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space, let H be a (G_0, G_0)-invariant subgroup of G, and let ℌ be the quotient set HG_0. The orbit space m ℌ m ∈ M partitions M.Let ∼ be the binary relation on M given bym ∈ Mm' ∈ Mm ∼ m'm ∈ m' ℌ. First, let m ∈ M. Then, m = mG_0. Thus, m ∈ m ℌ. Hence, m ∼ m. In conclusion, ∼ is reflexive.Secondly, let m and m' ∈ M such that m ∼ m'. Then, there is an h' ∈ H such that m = m'h' G_0. And, there is a g_0' ∈ G_0 such that𝔤∈ GG_0m'h' ·𝔤 = (m'h' G_0)g_0' ·𝔤.Put h = (h')^-1. Then,m' = m'G_0= m'h' h G_0= (m'h' G_0)g_0' · h G_0= mg_0' · h G_0.And, because H is (G_0, G_0)-invariant, there is an h” G_0 ∈ℌ such that g_0' · h G_0 = h” G_0. Hence, m' = mh” G_0 ∈ m ℌ. Therefore, m' ∼ m. In conclusion, ∼ is symmetric.Thirdly, let m, m', m”∈ M such that m ∼ m' and m' ∼ m”. Then, there are h', h”∈ H such that m = m'h' G_0 and m' = m” h” G_0. Hence, m = (m” h” G_0)h' G_0. And, there is a g_0”∈ G_0 such that𝔤∈ GG_0(m” h” G_0)g_0”·𝔤 = m” h”·𝔤.Put g = (g_0”)^-1 h'. Then,m= (m” h” G_0)h' G_0= (m” h” G_0)g_0”· g G_0= m” h”· g G_0.And, because H is (G_0, G_0)-invariant, there is an h G_0 ∈ℌ such that g G_0 = (g_0”)^-1 h' G_0 = h G_0. Therefore, m = m” h”· h G_0 ∈ m”ℌ. Hence, m ∼ m”. In conclusion, ∼ is transitive.Altogether, ∼ is an equivalence relation. In conclusion, M ∼ = m ℌ m ∈ M partitions M.[Sphere]In the situation of <ref>, under the identification of M with GG_0 by ι, the set HG_0 is the great circle Hm_0 and the set G_0 · (HG_0) is the union ⋃_g_0 ∈ G_0 g_0(Hm_0) of the rotations of the great circle Hm_0 about the vertical axis, which is the whole sphere M. It follows that the subgroup H of G is not (G_0, G_0)-invariant.Moreover, for each point m ∈ M, the set m(HG_0) is the great circle g_m_0, m (Hm_0) through m, which is equal to Hm_0 if and only if the rotation axis of g_m_0, m is a, which in turn holds if and only if m lies on Hm_0. As each two different great circles intersect in precisely two points, the orbit space m(HG_0)m ∈ M does not partition the sphere M although it covers it. Let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space, let H be a (G_0, G_0)-invariant subgroup of G, and let ℌ be the quotient set HG_0. Then,m ∈ M(m ℌ) ℌ' ⊆ m ℌ. Let m ∈ M, let h G_0 ∈ℌ, and let h' G_0 ∈ℌ. Then, becausehas defect G_0, there is a g_0 ∈ G_0 such that (mh G_0)h' G_0 = mh g_0 h' G_0. And, because H is (G_0, G_0)-invariant, there is an h”∈ H such that g_0 h' G_0 = h” G_0. Therefore, (mh G_0)h' G_0 = mh h” G_0 ∈ m ℌ. In conclusion, (m ℌ) ℌ' ⊆ m ℌ.For a (G_0, G_0)-invariant subgroup H of G and a semi-cellular automaton 𝒞 whose neighbourhood is included in HG_0, the partition of M by the orbit space of _M × HG_0 induces a partition of the phase space Q^M and a partition of the global transition function of 𝒞 as shown in Let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space, let 𝒞 = ℛ, Q, N, δ be a semi-cellular automaton, and let Δ be the global transition function of 𝒞. Furthermore, let H be a (G_0, G_0)-invariant subgroup of G such that N ⊆ HG_0, let ℌ be the quotient set HG_0, and let m_ii ∈ I be a transversal of m ℌ m ∈ M. Moreover, for each index i ∈ I, letΔ_m_i ℌ Q^m_i ℌ → Q^m_i ℌ,c_m_i ℌ ↦[]m_i 𝔥↦δ[]n ↦ c_m_i ℌ((m_i 𝔥)n).Then, M = _i ∈ I m_i ℌ, Q^M = ∏_i ∈ I Q^m_i ℌ, and Δ = ∏_i ∈ IΔ_m_i ℌ.Becauseis a right group semi-action of GG_0 on M with defect G_0 and H is a (G_0, G_0)-invariant subgroup of G, according to <ref>, the family m ℌ_m ∈ M partitions M. Therefore, the transversal m_ii ∈ I is well-defined, M = _i ∈ I m_i ℌ, and Q^M = ∏_i ∈ I Q^m_i ℌ. Let i ∈ I. Then, for each m_i 𝔥∈ m_i ℌ and each n ∈ N, because H is (G_0, G_0)-invariant and N ⊆ℌ, according to <ref>, we have (m_i 𝔥)n ∈ m_i ℌ. In conclusion, Δ_m_i ℌ is well-defined.Moreover, for each c_m_i ℌ∈ Q^m_i ℌ and each extension c ∈ Q^M of c_m_i ℌ, we have Δ_m_i ℌ(c_m_i ℌ)(m) = Δ(c)(m). In conclusion, Δ = ∏_i ∈ IΔ_m_i ℌ. The maps Δ_m_i ℌ, for i ∈ I, are conjugate to each other as shown inIn the situation of <ref>, let 𝒞 be a cellular automaton, let m and m' be two elements of M, letϕ_m,m' m ℌ → m' ℌ,m 𝔥 ↦ m' 𝔥,and letϕ_m,m'^*Q^m' ℌ → Q^m ℌ,c_m' ℌ ↦ c_m' ℌϕ_m,m'.Then, Δ_m' ℌ = (ϕ_m,m'^*)^-1Δ_m ℌϕ_m,m'^*.Becauseis free, the map ϕ_m,m' is well-defined and bijective. Hence, the map ϕ_m,m'^* is also well-defined and bijective.Let c ∈ Q^M and let m 𝔥∈ m ℌ. Then(ϕ_m,m'^* Δ_m' ℌ)(c_m' ℌ)(m 𝔥)= ϕ_m,m'^*[]Δ_m' ℌ(c_m' ℌ)(m 𝔥)= []Δ_m' ℌ(c_m' ℌ) ϕ_m,m'(m 𝔥)= Δ_m' ℌ(c_m' ℌ)(m' 𝔥)= Δ(c)(m' 𝔥).And, because m' 𝔥 = g_m_0, m' g_m_0, m^-1 (m 𝔥) and Δ is -equivariant, Δ(c)(m' 𝔥)= Δ(c)[](g_m_0, m g_m_0, m'^-1)^-1 (m 𝔥)= []g_m_0, m g_m_0, m'^-1Δ(c)(m 𝔥)= Δ(g_m_0, m g_m_0, m'^-1 c)(m 𝔥).And, because, for each m 𝔥' ∈ m ℌ,(g_m_0, m g_m_0, m'^-1 c)(m 𝔥')= c[]g_m_0, m' g_m_0, m^-1 (m 𝔥')= c(m' 𝔥')= c_m' ℌ(m' 𝔥')= c_m' ℌ[]ϕ_m,m'(m 𝔥')= (c_m' ℌϕ_m,m')(m 𝔥'),we have (g_m_0, m g_m_0, m'^-1 c)_m ℌ = (c ϕ_m,m')_m ℌ, and henceΔ(g_m_0, m g_m_0, m'^-1 c)(m 𝔥)= Δ_m ℌ[](g_m_0, m g_m_0, m'^-1 c)_m ℌ(m 𝔥)= Δ_m ℌ(c_m ℌϕ_m,m')(m 𝔥)= Δ_m ℌ[]ϕ_m,m'^*(c_m ℌ)(m 𝔥)= (Δ_m ℌϕ_m,m'^*)(c_m ℌ)(m 𝔥). Altogether,(ϕ_m,m'^* Δ_m' ℌ)(c_m ℌ)(m 𝔥) = (Δ_m ℌϕ_m,m'^*)(c_m ℌ)(m 𝔥).Therefore, ϕ_m,m'^* Δ_m' ℌ = Δ_m ℌϕ_m,m'^*. In conclusion, Δ_m' ℌ = (ϕ_m,m'^*)^-1Δ_m ℌϕ_m,m'^*. Because of the above decomposition of Δ, the conjugacy of the maps Δ_m_i ℌ, for i ∈ I, and the equality of Δ_m_0 ℌ to the restriction of Δ to H, properties of the latter restriction translate to properties of Δ as stated in In the situation of <ref>, let 𝒞 be a cellular automaton, let 𝒞_H be the restriction of 𝒞 to H, and let Δ__H be the global transition function of 𝒞_H. The global transition function Δ is injective, surjective, or bijective if and only if the global transition function Δ__H is injective, surjective, or bijective respectively.According to <ref>, we have m_0 ℌ = Hm_0.Thus, Q^m_0 ℌ = Q^Hm_0. Hence, for each c_H ∈ Q^m_0 ℌ = Q^Hm_0 and each extension c ∈ Q^M of c_H, according to <ref> and <ref>,Δ_m_0 ℌ(c_H) = Δ(c)_m_0 ℌ = Δ(c)_Hm_0 = Δ__H(c_H).Therefore, Δ_m_0 ℌ = Δ__H.Moreover, according to <ref>, the map Δ is the product of Δ_m_i ℌ, for i ∈ I, and thus injective, surjective, or bijective if and only if all Δ_m_i ℌ, for i ∈ I, have the respective property. And, for each i ∈ I, according to <ref>, the map Δ_m_i ℌ is conjugate to Δ_m_0 ℌ and hence injective, surjective, or bijective if and only if Δ_m_0 ℌ has the respective property. Therefore, Δ is injective, surjective, or bijective if and only if Δ_m_0 ℌ (= Δ__H) has the respective property. Let ℳ = M, G, be a left-homogeneous space, let m_0 be an element of M, let H be a (G_0, G_0)-invariant subgroup of G, let Δ be the global transition function of a cellular automaton over ℳ with a sufficient neighbourhood E such that E ⊆ HG_0, and let Δ__H be the restriction of Δ at m_0 to H. The global transition function Δ is injective, surjective, or bijective if and only if the global transition function Δ__H is injective, surjective, or bijective respectively.This is a direct consequence of <ref> with the proof of <ref>. [Cylinder]Let M be the infinite circular cylinder () ×, let G be the additive group ^2, letbe the left group action of G on M by ((t_1, t_2), (m_1 + , m_2)) ↦ ((t_1 + m_1) + , t_2 + m_2), let ℳ be the left-homogeneous space M, G,, let 𝒦 be the coordinate system (0 + , 0), ((m_1), m_2)_(m_1 + , m_2) ∈ M for ℳ, where (m_1) denotes the fractional part of m_1, let Q be the binary set 0, 1, let N be the singleton set (0 + , -1), let δ be the ∙-invariant map Q^N → Q, ℓ↦ℓ(0 + , -1), let M be identified with GG_0 by ι, let 𝒞 be the cellular automaton ℳ, 𝒦, Q, N, δ, and let H be the normal and hence (G_0, G_0)-invariant subgroup 0× of G.The global transition function Δ of 𝒞 is the shift map (0 + , 1) along the axis of M, that is, the map Q^M → Q^M, c ↦ [(m_1 + , m_2) ↦ c((m_1 + , m_2 - 1))]. Under the canonical identification of 0 + × with , its restriction Δ__H at (0 + , 0) to H is the shift map Q^→ Q^, c__H↦ [m_2 ↦ c__H(m_2 - 1)]. Under the canonical identifications ofwith m_1 + ×, for m_1 + ∈, the map Δ is the product ∏_m_1 + ∈Δ__H. As the map Δ__H is bijective, so is Δ.Everything that has been done with and said about cellular automata in this chapter could have been done with and said about big-cellular automata under suitable assumptions. We chose not to do so, because the presentation would have been even more cumbersome. CHAPTER: CURTIS-HEDLUND-LYNDON THEOREMSAbstract. We prove a topological as well as a uniform variant of the Curtis-Hedlund-Lyndon theorem for big-cellular automata with compact sufficient neighbourhoods over tame left-homogeneous spaces. The latter states that a map on the phase space is the global transition function of such an automaton if and only if it is equivariant and uniformly continuous. It follows from topological theorems that such an automaton is invertible if and only if its global transition function is a uniform isomorphism, which, in a more special setting, is equivalent to being bijective. Remark. Some parts of this chapter appeared in the paper *wacker:automata:2016<cit.> and they generalise parts of sections 1.2, 1.8, 1.9, and 1.10 of the monograph *ceccherini-silberstein:coornaert:2010<cit.>. Summary. The prodiscrete topology on Q^M has for a base the cylinders c ∈ Q^Mc_F = b, for b ∈ Q^F and F ⊆ M finite; and the prodiscrete uniformity on Q^M has for a base the cylinders (c, c') ∈ Q^M × Q^Mc_F = c'_F, for F ⊆ M finite. In the case that Q is finite, a topological variant of the Curtis-Hedlund-Lyndon theorem states that a map from Q^M to Q^M is the global transition function of a cellular automaton with finite sufficient neighbourhood if and only if it is equivariant and continuous; and in any case, a uniform variant of the Curtis-Hedlund-Lyndon theorem states that a map from Q^M to Q^M is the global transition function of a cellular automaton with compact sufficient neighbourhood if and only if it is equivariant and uniformly continuous. The finiteness of the sufficient neighbourhood stems from the finiteness of F in the cylinders and the qualifier sufficient is needed because the neighbourhood itself may have to be infinite due to the requirement to be invariant under left multiplication by G_0.In more detail, let M be equipped with a topology and equip GG_0 with the topology induced by m_0. The uniformity of discrete convergence on compacta on Q^M has for a base the cylinders (c, c') ∈ Q^M × Q^Mc_K = c'_K, for K ⊆ M compact. If the right semi-actionmaps compacta to sets included in compacta, which is called semi-tameness, then the uniform variant of the Curtis-Hedlund-Lyndon theorem holds. It follows that cellular automata with compact sufficient neighbourhoods are invertible if and only if their global transition functions are equivariant uniform isomorphisms, which is, in the case that Q is finite and M carries the discrete topology, equivalent to being equivariant, continuous, and bijective. All these statements also hold for big-cellular automata if we choose corresponding notions of equivariance.The Curtis-Hedlund-Lyndon theorem is a famous theorem by Morton Landers Curtis, Gustav Arnold Hedlund, and Roger Conant Lyndon from 1969 and was published in the paper *hedlund:1969<cit.>. Contents. In <ref> we prove a topological variant of the Curtis-Hedlund-Lyndon theorem, which characterises global transition functions of big-cellular automata by equivariance and continuity. In <ref> we introduce tameness and semi-tameness of left-homogeneous spaces, which are essential in the proof of the uniform variant of the Curtis-Hedlund-Lyndon theorem. In <ref> we introduce properness and semi-properness of left-homogeneous spaces, which are sufficient conditions for semi-tameness. In <ref> we prove a uniform variant of the Curtis-Hedlund-Lyndon theorem, which characterises global transition functions of big-cellular automata by equivariance and uniform continuity. And in <ref> we characterise invertibility of big-cellular automata. Preliminary Notions. A left group set is a triple M, G,, where M is a set, G is a group, andis a map from G × M to M, called left group action of G on M, such that G →(M), g ↦ [g ], is a group homomorphism. The actionis transitive if M is non-empty and for each m ∈ M the map m is surjective; and free if for each m ∈ M the map m is injective. For each m ∈ M, the set Gm is the orbit of m, the set G_m = ( m)^-1(m) is the stabiliser of m, and, for each m' ∈ M, the set G_m, m' = ( m)^-1(m') is the transporter of m to m'.A left-homogeneous space is a left group set ℳ = M, G, such thatis transitive. A coordinate system for ℳ is a tuple 𝒦 = m_0, g_m_0, m_m ∈ M, where m_0 ∈ M and, for each m ∈ M, we have g_m_0, m m_0 = m. The stabiliser G_m_0 is denoted by G_0. The tuple ℛ = ℳ, 𝒦 is a cell space. The set g G_0g ∈ G of left cosets of G_0 in G is denoted by GG_0. The map M × GG_0 → M, (m, g G_0) ↦ g_m_0, m gm_0 is a right semi-action of GG_0 on M with defect G_0, which means thatm ∈ MmG_0 = m,andm ∈ Mg ∈ Gg_0 ∈ G_0 𝔤' ∈ GG_0 mg ·𝔤' = (mg G_0)g_0 ·𝔤'.It is transitive, which means that the set M is non-empty and for each m ∈ M the map m is surjective; and free, which means that for each m ∈ M the map m is injective; and semi-commutes with , which means thatm ∈ Mg ∈ Gg_0 ∈ G_0 𝔤' ∈ GG_0 (gm) 𝔤' = g(mg_0 ·𝔤'). A semi-cellular automaton is a quadruple 𝒞 = ℛ, Q, N, δ, where ℛ is a cell space; Q, called set of states, is a set; N, called neighbourhood, is a subset of GG_0 such that G_0 · N ⊆ N; and δ, called local transition function, is a map from Q^N to Q. A local configuration is a map ℓ∈ Q^N and a global configuration is a map c ∈ Q^M. The stabiliser G_0 acts on Q^N on the left by ∙ G_0 × Q^N → Q^N, (g_0, ℓ) ↦ [n ↦ℓ(g_0^-1· n)], and the group G acts on Q^M on the left by G × Q^M → Q^M, (g, c) ↦ [m ↦ c(g^-1 m)]. The global transition function of 𝒞 is the map Δ Q^M → Q^M, c ↦ [m ↦δ(n ↦ c(mn))]. A sufficient neighbourhood of 𝒞 is a subset E of N such that, for each ℓ∈ Q^N and each ℓ' ∈ Q^N with ℓ_E = ℓ'_E, we have δ(ℓ) = δ(ℓ').A subgroup H of G is 𝒦-big if the set g_m_0, m m ∈ M is included in H. A big-cellular automaton is a semi-cellular automaton 𝒞 = ℛ, Q, N, δ such that, for some 𝒦-big subgroup H of G, the local transition function δ is ∙_G_0 ∩ H-invariant, which means that, for each h_0 ∈ G_0 ∩ H, we have δ(h_0 ∙) = δ(). Its global transition function is _H-equivariant, which means that, for each h ∈ H, we have Δ(h ) = h Δ(). A cellular automaton is a big-cellular automaton whose local transition function is ∙-invariant. (See <ref>.)In the present chapter, we assume that the reader is familiar with the basics of the theories of topological and uniform spaces. A recapitulation of the required basics is given in <ref>. § TOPOLOGICAL CURTIS-HEDLUND-LYNDON THEOREMContents. In <ref> we introduce the prodiscrete topology and a generalisation. In <ref> we show that the phase space is Hausdorff and compact, and that the left group action on it is continuous. In <ref> we show that the global transition function of each big-cellular automaton with a finite sufficient neighbourhood is continuous and hence has a closed image. And in <ref> we prove a generalised topological variant of the Curtis-Hedlund-Lyndon theorem. Body. The prodiscrete topology is introduced inLet M be a set and let Q be a set. Equip Q with the discrete topology and Q^M = ∏_m ∈ M Q with the product topology. This topology on Q^M is called prodiscreteprodiscrete!topologyprodiscrete topologytopology!prodiscrete. The prodiscrete topology on Q^M is the coarsest topology such that, for each element m ∈ M, the projectionπ_mQ^M→ Q, π_m, for m ∈ M[symbols]pim@π_mc↦ c(m),is continuous. Thus, it has for a subbase the setsπ_m^-1(q) = c ∈ Q^Mc(m) = q,for q ∈ Q and m ∈ M.Hence, it has for a base the sets⋂_m ∈ Fπ_m^-1(b(m)) = c ∈ Q^Mc_F = b,for b ∈ Q^F and F ⊆ M finite.Therefore, for each map c ∈ Q^M, the sets, called cylinders,(c, F) = c' ∈ Q^Mc'_F = c_F,for F ⊆ M finite, cylinders (c, F), for c ∈ Q^M and F ⊆ M finite[symbols]CylcF@(c, F)constitute a neighbourhood base of c. A generalisation of the prodiscrete topology is introduced in Letbe a left group action of G on M, let L be a subgroup of G, and let Q be a set. The group L acts on M on the left by _L × M. Let m_ii ∈ Im_i, for i ∈ I[symbols]mi@m_i be a transversal of the orbit space of _L × M. Then, M = _i ∈ I Lm_i and Q^M = ∏_i ∈ I Q^Lm_i. For each index i ∈ I, equip Q^Lm_i with the discrete topology, and equip Q^M with the product topology. This topology on Q^M is called (, L)-prodiscreteprodiscrete topology@(, L)-prodiscrete topology(, L)-prodiscrete topologytopology!(, L)-prodiscrete.The (, e_G)-prodiscrete topology is but the prodiscrete topology.The (, L)-prodiscrete topology on Q^M is the coarsest topology such that, for each element m ∈ M, the projectionπ_Lm Q^M→ Q^Lm, π_Lm, for m ∈ M[symbols]piLarrowrightm@π_Lmc↦ c_Lm,is continuous. Thus, it has for a subbase the setsπ_Lm^-1(b) = c ∈ Q^Mc_Lm = b,for b ∈ Q^Lm and m ∈ M.Hence, it has for a base the sets⋂_m ∈ Fπ_Lm^-1(b_Lm) = c ∈ Q^Mc_LF = b,for b ∈ Q^LF and F ⊂ M finite.Therefore, for each map c ∈ Q^M, the sets, called cylinders,(c, LF) = c' ∈ Q^Mc'_LF = c_LF,for F ⊆ M finite, cylinders (c, LF), for c ∈ Q^M and F ⊆ M finite[symbols]CylcLarrowrightM0@(c, LF)constitute a neighbourhood base of c. The phase space is Hausdorff and totally disconnected, which is shown in Letbe a left group action of G on M, let L be a subgroup of G, and let Q be a set. The set Q^M, equipped with the (, L)-prodiscrete topology, is Hausdorff and totally disconnected.Let m_ii ∈ I be a transversal of the orbit space of _L × M. Then, for each index i ∈ I, the set Q^Lm_i, equipped with the discrete topology, is Hausdorff and totally disconnected. Therefore, according to <ref>, the set Q^M = ∏_i ∈ I Q^Lm_i, equipped with the product topology, is Hausdorff and totally disconnected. If the set of states is finite, then the phase space is compact, which is shown in Letbe a left group action of G on M, let L be a finite subgroup of G, and let Q be a finite set. The set Q^M, equipped with the (, L)-prodiscrete topology, is compact.Let m_ii ∈ I be a transversal of the orbit space of _L × M. Then, for each index i ∈ I, because Q^Lm_i≤Q^L < ∞, the set Q^Lm_i is finite and hence, equipped with the discrete topology, it is compact. Therefore, according to Tychonoff's <ref>, the set Q^M = ∏_i ∈ I Q^Lm_i, equipped with the product topology, is compact. The left group action on the phase space is continuous, which is shown in Letbe a left group action of G on M, let Q be a set, and let Q^M be equipped with the prodiscrete topology. The left group actionis continuouscontinuous left group action, which means that, for each element g ∈ G, the map g is continuous.Let g ∈ G and letϕ_gQ^M→ Q^M,c↦ gc.Furthermore, let m ∈ M. Then, for each c ∈ Q^M,(π_m ϕ_g)(c)= (gc)(m)= c(g^-1 m)= π_g^-1 m(c).Thus, the map π_m ϕ_g is identical to π_g^-1 m and is hence continuous. Therefore, according to <ref>, the map ϕ_g = g is continuous. In conclusion, the actionis continuous. The global transition function of a big-cellular automaton is continuous, which is shown inLet ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space, let H be a 𝒦-big subgroup of G, let 𝒞 = ℛ, Q, N, δ be a semi-cellular automaton with ∙_H_0-invariant local transition function δ and finite sufficient neighbourhood E, let L be a subgroup of H, and let Q^M be equipped with the (, L)-prodiscrete topology. The global transition function Δ of 𝒞 is continuous.Let c ∈ Q^M. Furthermore, let O be an open neighbourhood of Δ(c). Then, there is a finite subset F of M such that (Δ(c), LF) ⊆ O. And, because E is finite, the set FE is finite.Let c' ∈(c, L(FE)), let f ∈ F, and let l ∈ L. Then, because L ⊆ H and Δ is _H-equivariant by <ref>,Δ(c')(lf)= (l^-1Δ(c'))(f)= Δ(l^-1 c')(f)= δ(n ↦ (l^-1 c')(fn))= δ(n ↦ c'(l(fn)))and analogouslyΔ(c)(lf) = δ(n ↦ c(l(fn))).And, because c' is in (c, L(FE)),e ∈ Ec'(l(fe)) = c(l(fe)).Hence, because E is a sufficient neighbourhood, we have Δ(c')(lf) = Δ(c)(lf). Therefore, Δ(c') ∈(Δ(c), LF). Thus,Δ((c, L(FN))) ⊆(Δ(c), LF) ⊆ O.In conclusion, the global transition function Δ is continuous.Let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space, let 𝒞 = ℛ, Q, N, δ be a semi-cellular automaton with finite sufficient neighbourhood, and let Q^M be equipped with the prodiscrete topology. As in the proof of <ref>, one can show that the global transition function Δ of 𝒞 is continuous. If the set of states is finite, then the image of the global transition function of a big-cellular automaton is closed, which is shown inIn the situation of <ref>, let Q and L be finite. The set Δ(Q^M) is closed in Q^M.According to <ref>, the global transition function Δ is continuous. And, according to <ref>, the phase space Q^M is compact. Hence, the set Δ(Q^M) is a compact subset of Q^M. Moreover, according to <ref>, the phase space Q^M is Hausdorff. Therefore, the compact set Δ(Q^M) is in particular closed.In the situation of <ref>, as in the proof of <ref>, one can show that Δ(Q^M) is closed in Q^M. If the set of states is finite, then a map on the phase space is the global transition function of a big-cellular automaton with finite neighbourhood if and only if the map is equivariant and continuous, which is shown inLet ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space, let Q be a finite set, let Δ be a map from Q^M to Q^M, let H be a 𝒦-big subgroup of G, let L be a finite subgroup of H, and let Q^M be equipped with the (, L)-prodiscrete topology. The following two statements are equivalent: *The map Δ is the global transition function of a semi-cellular automaton over ℛ with ∙_H_0-invariant local transition function and finite sufficient neighbourhood.*The map Δ is _H-equivariant and continuous. First, let Δ be the global transition function of a semi-cellular automaton 𝒞 = ℛ, Q, N, δ with ∙_H_0-invariant local transition function δ and finite sufficient neighbourhood E. Then, according to <ref>, the map Δ is _H-equivariant, and, according to <ref>, it is continuous.Secondly, let Δ be _H-equivariant and continuous. Then, the mapΛ = π_Lm_0Δ Q^M→ Q^Lm_0,c↦Δ(c)_Lm_0,is continuous.For a moment, let c ∈ Q^M. Then, because Q^Lm_0 is equipped with the discrete topology, the preimage Λ^-1(Λ(c)) is open. Hence, there is a finite subset F_c of M such that (c, LF_c) ⊆Λ^-1(Λ(c)). In other words,c' ∈(c, LF_c) Λ(c') = Λ(c). Moreover, because c ∈(c, LF_c), for c ∈ Q^M, the sets (c, LF_c), for c ∈ Q^M, constitute an open cover of Q^M. Hence, because the space Q^M is compact by <ref>, there is a finite subset C of Q^M such thatQ^M = ⋃_c ∈ C(c, LF_c). Furthermore, because C and L are finite, the set E_0 = ⋃_c ∈ C LF_c is a finite subset of M and the set E = (m_0 )^-1(E_0) is a finite subset of GG_0. Let N = G_0 · E. Then, G_0 · N ⊆ N.For a while, let c and c' be two global configurations of Q^M such that c_m_0E = c'_m_0E. Then, according to <ref>, there is an element c_0 ∈ C such that c ∈(c_0, LF_c_0). Hence, because LF_c_0⊆ E_0 = m_0E,c'_LF_c_0 = c_LF_c_0 = c_0_LF_c_0.Thus, c' ∈(c_0, LF_c_0). Therefore, according to <ref>, we have Λ(c) = Λ(c_0) = Λ(c'). Hence, because m_0 ∈ Lm_0,Δ(c)(m_0) = Λ(c)(m_0) = Λ(c')(m_0) = Δ(c')(m_0).Therefore, because E ⊆ N, there is a map δ Q^N → Q such thatc ∈ Q^M Δ(c)(m_0) = δ(n ↦ c(m_0n)).The quadruple 𝒞 = ℳ, Q, N, δ is a semi-cellular automaton with finite sufficient neighbourhood E. Conclude with <ref> that δ is ∙_H_0-invariant and that Δ is the global transition function of 𝒞. The Curtis-Hedlund-Lyndon theorem presented above is a generalisation of a known theorem for cellular automata over groups.In the case that L = e_G, <ref> is essentially theorem 6.7 in <cit.> and, if additionally M = G andis the group multiplication of G, then it is theorem 1.8.1 in <cit.>. [Group <cit.>] In theorem <ref>, if the assumption that the set Q of states is finite does not hold, then <ref> does not follow from <ref>, which is illustrated by the following example. Let G be an infinite group, let Q be the set G, and let Δ be the map Q^G → Q^G, c ↦ [g ↦ c(g · c(g))]. According to example 1.8.2 in <cit.>, the map Δ is -equivariant and continuous but not the global transition function of a cellular automaton over G with finite neighbourhood.Note that, even if the set Q is infinite, <ref> follows from <ref>.§ TAMENESS AND SEMI-TAMENESS OF SPACESIntroduction. The symmetries of a circle, namely rotations and (roto-)reflections, act on it on the left by function application. A rotation is uniquely determined by its angle and a (roto-)reflection can be uniquely identified by an angle with respect to a designated line. Hence, the symmetries of the circle can be identified with the disjoint union of the angles, say, A ×1, -1 = 0, 360×1, -1. We do the identification by (a, r) ↦ρ_a ϱ_r, where ρ_a is the rotation by a degrees, ϱ_1 is the identity map, and ϱ_-1 is the reflection about the vertical line through the centre of the circle. This identification induces the group structure (a, r) + (a', r') ↦ (a + r · a'360, r · r') on A ×1, -1 and the left group action (a, r)m = ρ_a(ϱ_r(m)) of A ×1, -1 on the circle. Geometrically, the group A ×1, -1 is made up of two circles of circumference 360. The geometry on each circle induces a topology on it and both topologies together induce the product topology on A ×1, -1. It can be shown that addition and inversion in A ×1, -1 are continuous, and that the left group actionis continuous also. After all, the action just rotates and reflects points, so points that are close stay close. As we have seen in the introduction of <ref>, the right quotient set semi-action induced bycan be identified with the right group action of the rotations on the circle. Under the identification of rotations with angles, this action is ma = ρ_a(m), which is continuous. Hence, if we act with a compact subset of angles on a compact subset of points by that action, then we get a compact subset of points. This is not the case for all right semi-actions, but it is for those induced by so called tame left group actions. This property is of interest for cellular automata, because it implies that, if an automaton has a compact (relative) neighbourhood, then the actual neighbourhood of each cell is compact, even the union of all actual neighbourhoods of a compact subset of points is compact. For our purposes, being included in a compact set is sufficient, which is already implied by semi-tameness. Contents. In <ref> we equip GG_0 with the topology of M. In <ref> we introduce tameness and semi-tameness. In <ref> we note that the action of a group on a discrete space is tame and semi-tame. In <ref> we illustrate that (semi-)tameness of cell spaces depends on the coordinate system. In <ref> we introduce topological groups and group sets. In <ref> we show that a cell space with a continuous coordinate map is tame. And in <ref> we illustrate that continuity of coordinate maps is not necessary for semi-tameness. Body. We equip GG_0 with the topology of M inLet ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space and let M be equipped with a topology. Equip GG_0 with the topology induced by m_0topology on GG_0.A cell space is (semi-)tame if its right semi-action maps compacta to (sets included in) compacta. This property is essential in the proof of <ref>; there it is used to show that global transition functions of big-cellular automata are uniformly continuous.Let M be a topological space and let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space. The cell space ℛ is called * tametame cell spacecell space!tametame!cell space if and only if, for each compact subset K of M and each compact subset E of GG_0, the set KE is a compact subset of M;* semi-tamesemi-tame cell spacetame!semi-cell space!semi-tamesemi-tame!cell space if and only if, for each compact subset K of M and each compact subset E of GG_0, the set KE is included in a compact subset of M.A left-homogeneous space is (semi-)tame if all its right semi-actions map compacta to (sets included in) compacta.Let ℳ be a left-homogeneous space. It is called tametame/semi-tame left-homogeneous spaceleft-homogeneous space!tamehomogeneous space!tametame!left-homogeneous space or semi-tametame!semi-left-homogeneous space!semi-tamehomogeneous space!semi-tamesemi-tame!left-homogeneous space if and only if, for each coordinate system 𝒦 for ℳ, the cell space ℳ, 𝒦 is tame or semi-tame respectively.Each tame cell space and each tame left-homogeneous space is semi-tame. If M is equipped with the discrete topology, then the left-homogeneous space ℳ is tame. Tameness and semi-tameness of cell spaces depend on the coordinate system, which is illustrated by[Affine Space]Let M be the one-dimensional Euclidean space , let G be the group α_t, d→, r ↦ t + d · rt ∈ andd ∈∖0 of invertible affine transformations of M, which is called affine group of Maffine group of M, and letbe the left group action of G on M by function application. The triple ℳ = M, G, is a left-homogeneous space and the stabiliser G_0 of the origin 0 underis the group α_0, d d ∈∖0 of invertible linear transformations of M. Identify GG_0 with M by ι m ↦ G_0, m.* The tuple 𝒦 = 0, α_m, 1_m ∈ M is a coordinate system for ℳ such that = +. Hence, because the map + is continuous, for each compact subset K of M and each compact subset E of GG_0 ≃ M, the set KE = K + E is compact. Therefore, the cell space ℳ, 𝒦 is tame. * The tuple 𝒦' = 0, α_m, 1_m ∈ M ∖-1, 1×α_m, 2_m ∈-1, 1 is a coordinate system for ℳ such that(m, 𝔤) ↦ m + 𝔤,if m ∉-1, 1,m + 2 𝔤,if m ∈-1, 1.Hence, for each compact subset K of M and each compact subset E of GG_0 ≃ M, the set KE is included in the compact set K + 2 E but it need not be compact itself. For example, although the bounded and closed interval 0, 1 and the singleton set 1 are compact, the set 0, 11 = 1, 2∪3 is bounded but not compact. Therefore, the cell space ℳ, 𝒦' is semi-tame but not tame. *The tuple 𝒦” = 0, α_m, 1/m_m ∈ M ∖0×α_m, 1_m ∈0 is a coordinate system for ℳ such that(m, 𝔤) ↦ m + 1/m·𝔤,if m ≠ 0,m + 𝔤,if m = 0.The map fM ∖0→ M, m ↦ m + 1/m, is continuous and thus it maps intervals to intervals, in particular, because f(1) = 2 and lim_m ↓ 0 = ∞, we have f(0, 1) = 2, ∞. Hence, although the sets 0, 1 and 1 are compact, the set 0, 11 = 1∪ f(0, 1) = 1∪2, ∞ is unbounded, in particular, it is not included in a compact set. Therefore, the cell space ℳ, 𝒦” is not semi-tame.There are similar examples for the d-dimensional Euclidean space ^d acted on by the affine group ^d ⋊(d, ) and for the d-dimensional integer lattice ^d acted on by the symmetric group (^d) of bijective maps on ^d, for d ∈_+. A topological group is a group equipped with a topology such that multiplication and inversion are continuous.Let G be a group equipped with a topology. It is called topologicaltopological grouptopological!groupgroup topological@topological group if and only if the maps{ G × G→ G,(g, g')↦ g g', } and { G→ G,g↦ g^-1, }are continuous, where G × G is equipped with the product topology. A group set is topological if its group action is continuous.Let M be a topological space, let G be a topological group, let ℳ = M, G, be a left group set, and let G × M be equipped with the product topology. The group set ℳ is called topologicaltopological left group settopological!left group setleft group set!topologicalgroup set!topological if and only if the mapis continuous. Equipping a bare left group set with the discrete topology yields a topological group set.Let ℳ = M, G, be a left group set. Equip M and G with their respective discrete topology. The group set ℳ is topological.The product topology on G × M and the one on M × M are discrete. Hence, each subset of G × M and each of M × M is open. Thus, the mapis continuous. In conclusion, the group set ℳ is topological. A topological cell space is tame if its coordinate map is continuous. Let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a topological cell space such that the coordinate mapcoordinate map χ[symbols]chi@χ χ M → G, m ↦ g_m_0, m, is continuous. The mapis continuous and the space ℛ is tame.According to the definition of the product topology on M × GG_0, the projections π_1M × GG_0 → M, (m, 𝔤) ↦ m and π_2M × GG_0 → M, (m, 𝔤) ↦𝔤, are continuous. And, according to the universal property of the product topology on M × M, because the maps χπ_1 and π_2 are continuous, the map ψ M × GG_0 → M × M, (m, 𝔤) ↦ (g_m_0, m, 𝔤) = ((χπ_1)(m, 𝔤), π_2(m, 𝔤)) is continuous. Hence, under the identification of GG_0 with M by ι, becauseis continuous, the map = ψ is continuous. Therefore, the mapmaps compacta to compacta, in particular, for each compact subset K of M and each compact subset E of GG_0, the set KE is compact. In conclusion, the space ℛ is tame.Continuity of coordinate maps is not necessary for semi-tameness, which is illustrated by[Affine Space]In the situation of <ref>, the set ∖0 is a group under multiplication and the setis a group under addition. Let φ be the group homomorphism ∖0→(), d ↦ [t ↦ d · t], let ⋊_φ (∖0) be the outer semi-direct product of ∖0 acting onby φ, and let ⋊_φ (∖0) be equipped with the subspace metric and subspace topology induced by ×.The group multiplication of ⋊_φ (∖0) is the continuous map ((t, d), (t', d')) ↦ (t + d · t', d · d') and the group inversion of ⋊_φ (∖0) is the continuous map (t, d) ↦ (-t/d, 1/d). Hence, the group ⋊_φ (∖0) is topological. Moreover, the map ⋊_φ (∖0) → G, (t, d) ↦α_t, d, is a group isomorphism and, under the identification of ⋊_φ (∖0) with the group G by that isomorphism, which we shall henceforth do, the left group actionis the continuous map ((t, d), r) ↦ t + d · r. Therefore, the left group set ℳ is topological. * The coordinate map χ m ↦ (m, 1) induced by 𝒦 is continuous. * The coordinate mapχ'm ↦ (m, 1),if m ∉-1, 1,(m, 2),if m ∈-1, 1,induced by 𝒦' is not continuous. However, as we have already seen, the cell space ℳ, 𝒦' is semi-tame. Hence, continuity of the coordinate map is sufficient but not necessary for semi-tameness. * The coordinate mapχ” m ↦ (m, 1/m),if m ≠ 0,(m, 1),if m = 0,induced by 𝒦” is not continuous. Is there a tame cell space whose coordinate map is not continuous?Is there a semi-tame topological left-homogeneous space for which there is no coordinate system such that the coordinate map is continuous? I conjecture that the sphere acted upon by rotations is such a space. § PROPERNESS AND SEMI-PROPERNESS OF LEFT HOMOGENEOUS SPACESIntroduction. In the situation of the introduction of <ref>, the action map ofis α (A ×1, -1) × M → M × M, ((a, r), m) ↦ ((a, r)m, m), where M denotes the circle. Its domain is, topologically, the disjoint union of two tori equipped with the product topology; and its codomain is, topologically, a torus. The preimage of (m', m) under α consists of two tuples, one in each torus, namely ((a, 1), m) and ((a', -1), m), where a is the angle such that ρ_a(m) = m' and a' is the angle such that ρ_a'(ϱ_-1(m)) = m' (see <ref>); the preimage of Y ×m, where Y is an arc of the circle, consists of two similar paths in the poloidal or toroidal direction, one in each torus, of the form (B ×1) ×m and (B' ×-1) ×m, where B and B' are arcs of A of the same length (see <ref>); the preimage of m'× X, where X is an arc of the circle, consists of two similar paths in the diagonal direction, one in each torus, of the form ((a_m, 1), m)m ∈ X and ((a_m', -1), m)m ∈ X, where a_m is the angle such that ρ_a_m(m) = m' and a_m' is the angle such that ρ_a_m'(ϱ_-1(m)) = m' (see <ref>); and, the preimage of Y × X, where X and Y are arcs of the circle, consist of two similar ribbons in the diagonal direction, one in each torus (see <ref>). It can be shown that the preimages of compact subsets of the torus under the action map are compact. A left group action with such an action map is called proper. And, if each transversal of the preimages of the elements of compact subsets are only included in compact subsets of M, it is called semi-proper. For each coordinate system, properness implies almost tameness and semi-properness implies semi-tameness.Contents. In <ref> we introduce transversals of sets of sets. In <ref> we introduce properness and semi-properness of maps and actions. In <ref> we present examples of proper and semi-proper actions. In <ref> we show that for semi-proper actions each transversal of the transporters from a compact set to a compact set is included in a compact set. In <ref> we show that the action of a discrete group on a discrete space is semi-proper. In <ref> we show that each transitive semi-proper action is semi-tame for each coordinate system. In <ref> we present an example that demonstrates that for the previous property semi-properness is not necessary. And in <ref> we present an action that is not semi-proper but semi-tame for some but not all coordinate systems. Body. A transversal of a set of sets is a set that contains one element from each set. Let M be a set, let 𝔏 be a subset of the power set of M, and let T be a subset of M. The set T is called transversal of 𝔏transversal T of 𝔏 if and only if there is a surjective map f 𝔏→ T such that for each set A ∈𝔏 we have f(A) ∈ A. The image of a compact set under a continuous map is compact, but the preimage is in general not. A proper map is a continuous map whose preimages of compact sets are compact. And, a semi-proper map is one whose transversals of its fibres are included in compact sets.Let M and M' be two topological spaces and let f be a continuous map from M to M'. The map f is called * properproper mapproper!mapmap!proper if and only if, for each compact subset K of M', its preimage f^-1(K) is a compact subset of M; * semi-propersemi-proper mapproper!semi-semi-proper!mapmap!semi-proper if and only if, for each compact subset K of M', each transversal of f^-1(k)k ∈ K is included in a compact subset of M.A topological group set is proper if its action map is proper, and semi-proper if its action map is semi-proper. If a group action is proper as a map, then its action map is proper; the converse though is false (see problem 21-1 in <cit.>).Let ℳ = M, G, be a topological left group set, and let G × M and M × M be equipped with their respective product topology. The group set ℳ is called * properproper left group setproper!left group setleft group set!propergroup set!proper if and only if its so-called action mapα G × M→ M × M, action map αmap!action[symbols]alpha@α(g, m)↦ (gm, m),is proper;* semi-propersemi-proper left group setproper!semi-semi-proper!left group setleft group set!semi-propergroup set!semi-proper if and only if its action map is semi-proper.Each proper map is semi-proper and each proper left group set is semi-proper. There are numerous examples of proper and of semi-proper group sets. The Curtis-Hedlund-Lyndon theorem though is mostly interesting for non-compact ones. Therefore, we are especially interested in non-compact and proper or semi-proper homogeneous spaces.[Proper] * Each left group set M, G, with finite stabilisers, where M and G are equipped with their respective discrete topology, is proper.* Let M be a Hausdorff topological space, let G be a Lie group, and letbe a transitive and continuous left group action of G on M. According to example 2 in paragraph 1.1 in chapter 3 in <cit.>, the left-homogeneous space M, G, is proper if and only if its stabilisers are compact.* Let M be a Riemannian manifold, let G be the isometry group of M, and letbe the left group action of G on M by function application. According to example 3 in paragraph 1.1 in chapter 3 in <cit.>, the left group set M, G, is proper. * Let M be a manifold, let G be a compact Lie group, and letbe a continuous left group action of G on M. According to corollary 21.6 in <cit.>, the left group set M, G, is proper.* Let G be a Lie group, let H be a compact subgroup of G, and letbe the transitive left group action of G on GH by left multiplication. According to theorem 21.17 in <cit.>, the quotient space GH is a smooth manifold and the left group actionis smooth. Because H is compact, all stabilisers ofare compact and hence, according to <ref>, the left-homogeneous space GH, G, is proper. For example:* Let n be a positive integer. The general linear group (n, ) is a non-compact Lie group, the orthogonal group (n, ) is a maximal compact subgroup of (n, ), and the special orthogonal group (n, ) is a compact subgroup of (n, ). Hence, the left-homogeneous spaces (n, ) (n, ), (n, ), · and (n, ) (n, ), (n, ), · are proper. * The special linear group (2, ) is a non-compact Lie group and the special orthogonal group (2, ) is a maximal compact subgroup of (2, ). Hence, the left-homogeneous space (2, ) (2, ), (2, ), · is proper.The upper half-plane = x +y ∈ x ∈, y ∈_> 0 equipped with the Poincaré metric is the Poincaré half-plane, a model of the hyperbolic plane. The isometries of the Poincaré half-plane are the Möbius transformationsμ_a, b, c, d →,z↦a z + b/c z + d, where a, b, c, and d are four real numbers satisfying a d - b c ∈-1, 1. Note that because μ_a, b, c, d = μ_-a, -b, -c, -d, this naming of Möbius transformations is not unique.The group (2, ) acts transitively and smoothly, but not faithfully, on the Poincaré half-plane by(2, ) × →,(( a bc d ), z)↦μ_a, b, c, d(z).Note that because the entries a, b, c, and d of a matrix in (2, ) satisfy a d - b c = 1, this is an action by orientation-preserving isometries. The stabiliser ofunderis (2, ). According to theorem 21.18 in <cit.>, there is an (·, )-equivariant diffeomorphism from (2, ) (2, ) to . In particular, the left-homogeneous space , (2, ), is proper. * Let n be a positive integer. The Euclidean group (n) is a non-compact Lie group and the orthogonal group (n, ) is a compact subgroup of (n). Hence, the left-homogeneous space (n) (n, ), (n), · is proper.The group (n) acts transitively and smoothly on the Euclidean space ^n by function application denoted by . The stabiliser of 0 underis (n, ). According to theorem 21.18 in <cit.>, there is an (·, )-equivariant diffeomorphism from (n) (n, ) to ^n. In particular, the left-homogeneous space ^n, (n), is proper.* According to the first paragraph of section 6 in <cit.>, the indefinite unitary group (n, 1) acts properly and transitively on the (2n + 1)-dimensional anti-de Sitter space ^2n + 1. * Let M be the vertices of a uniform tiling of the Euclidean or hyperbolic plane, let G be the symmetry group of the tiling, letbe the transitive left group action of G on M by function application, and let M and G be equipped with their respective discrete topology. The left-homogeneous space M, G, is proper. [Semi-Proper] * According to <ref>, each left group set from <ref> is semi-proper.* According to the forthcoming <ref>, each left group set M, G,, where M and G are equipped with their respective discrete topology, is semi-proper.*Let ℳ = M, G, be a left group set, let M and G be equipped with their respective discrete topology, let m be an element of M such that the stabiliser of m underis infinite. According to the forthcoming <ref>, the group set ℳ is semi-proper. However, because the singleton set (m, m) is compact (that is, finite) and its preimage under the action map of ℳ is G_m ×m is not compact (that is, not finite), the group set ℳ is not proper. For example: * The group ^2 acts transitively onon the left by⊕^2 × →,((x, y), z) ↦ z + x.Each element ofhas the infinite stabiliser 0×. Hence, the left-homogeneous space ^2, , ⊕ is semi-proper but not proper.* Let F_2 be the free group over a, b, where a ≠ b, let φ be the homomorphism from F_2 tothat is uniquely determined by φ(a) = 1 and φ(b) = 0, and letbe the transitive left group action of F_2 ongiven by (g, z) ↦φ(g) + z. For each integer z ∈, the stabiliser of z is φ^-1(0), which is the infinite set of all elements of F_2 that can be written as products of a, b, a^-1, and b^-1 with the same number of occurrences of a and a^-1. Hence, the left-homogeneous space F_2, , is semi-proper but not proper. * Let S be an infinite set, let F be the free group over S, let M be the uncoloured S-Cayley graph of F, let G be the graph automorphisms of M, and letbe the transitive left group action of G on M by function application. For each element m ∈ M, because there are S-many outgoing edges to isomorphic subgraphs, the stabiliser of m is infinite. Hence, the left-homogeneous space G, M, is semi-proper but not proper. The union of the transporters from a compact set to a compact set under a proper group action is compact, and each transversal of the transporters from a compact set to a compact set under a semi-proper group action is included in a compact set.Let ℳ = M, G, be a left group set, and let K and K' be two compact subsets of M. *If ℳ is proper, then the set⋃_(k, k') ∈ K × K' G_k', kis a compact subset of G.*If ℳ is semi-proper, then each transversal of the setG_k', k (k, k') ∈ K × K'is included in a compact subset of G.According to Tychonoff's <ref>, and because product and subspace topologies behave well with each other, the set K × K' is a compact subset of M × M. * Let ℳ be proper. Then, because the action map α of ℳ is proper, the preimageα^-1(K × K') = (g, k') ∈ G × K'gk' ∈ Kis compact. Hence, because the canonical projection π of G × M onto G is continuous, the imageπ(α^-1(K × K')) = g ∈ Gk' ∈ K'gk' ∈ Kis compact. This set is just⋃_k ∈ K⋃_k' ∈ K'g ∈ Ggk' = k = ⋃_(k, k') ∈ K × K' G_k', k. * Let ℳ be semi-proper. Then, because the action map α of ℳ is semi-proper and the preimage of each tuple (k, k') ∈ K × K' under α is G_k', k×k', each transversal ofG_k', k×k' (k, k') ∈ K × K'is included in a compact subset of G × M. Hence, because the canonical projection π of G × M onto G is continuous, each transversal ofG_k', k (k, k') ∈ K × K'is included in a compact subset of G.Let M, G, be a proper left group set and let m be an element of M. The stabiliser G_m of m underis a compact subset of G.This is a direct consequence of <ref> of <ref> with K = K' = m.The stabilisers of semi-proper group sets need not be compact, as <ref> of <ref> illustrates. Equipping a bare left group set with the discrete topology yields a semi-proper group set. If its stabilisers are finite, then it is even proper; but if they are infinite, then it is not proper (see <ref>).Let ℳ = M, G, be a left group set. Equip M and G with their respective discrete topology. The group set ℳ is semi-proper. And, if its stabilisers are finite, then it is even proper.According to <ref>, the group set ℳ is topological. And, because the product topology on G × M and the one on M × M are discrete, each subset of G × M and each of M × M is compact if and only if it is finite.First, let K be a finite subset of M × M. Then, because the set α^-1(k)k ∈ K, where α is the action map of ℳ, is finite, so is each of its transversals. Thus, the map α is semi-proper. In conclusion, the group set ℳ is semi-proper.Secondly, let the stabilisers of ℳ be finite. Furthermore, let K be a finite subset of M × M. Then, for each tuple (m', m) ∈ K, either the transporter G_m, m' is empty, and hence finite, or there is an element g ∈ G_m, m' and thus G_m, m' = g G_m, and therefore G_m, m' is finite; thus, in either case, the preimage α^-1((m', m)) = G_m, m'×m is finite. Hence, the preimage α^-1(K) = ⋃_(m', m) ∈ Kα^-1((m', m)) is finite. Therefore, the action map α is proper. In conclusion, the group set ℳ is proper. Each proper left-homogeneous space is almost tame and each semi-proper left-homogeneous space is semi-tame (see <ref>). However, (semi-)properness is not necessary for (semi-)tameness (see <ref>).Let ℳ = M, G, be a left-homogeneous space. If it is semi-proper, then it is semi-tame. And, if it is proper, then it is almost tame, in the sense that, for each compact subset K of M and each compact subset E of GG_0 such that G_0 · E ⊆ E, the set KE is a compact subset of M.First, let ℳ be semi-proper. Furthermore, let 𝒦 = m_0, g_m_0, m_m ∈ M be a coordinate system for ℳ, let K be a compact subset of M, and let E be a compact subset of GG_0. Then, because ke = g_m_0, k (m_0e), for k ∈ K and e ∈ E,KE = ⋃_k ∈ K g_m_0, k (m_0E) = g_m_0, k k ∈ K (m_0E).Both the singleton set m_0 and the set K are compact. Thus, according to <ref> of <ref>, the transversal g_m_0, k k ∈ K of G_m_0, k (m_0, k) ∈m_0× K is included in a compact subset of G. Hence, the product g_m_0, k k ∈ K× (m_0E) is included in a compact subset of G × M. Therefore, becauseis continuous, the image g_m_0, k k ∈ K (m_0E) is included in a compact subset of M. Thus, the set KE is included in a compact subset of M. Hence, the cell space ℳ, 𝒦 is semi-tame. In conclusion, the homogeneous space ℳ is semi-tame.Secondly, let ℳ be proper. Furthermore, let 𝒦 = m_0, g_m_0, m_m ∈ M be a coordinate system for ℳ, let K be a compact subset of M, and let E be a compact subset of GG_0 such that G_0 · E ⊆ E. Then, because kg_0 · e = g_m_0, k g_0(m_0e), for k ∈ K, g_0 ∈ G_0, and e ∈ E,KE= ⋃_k ∈ K kG_0 · E= ⋃_k ∈ K g_m_0, k G_0(m_0E)= (⋃_k ∈ K G_m_0, k)(m_0E).The set K is compact and so is the singleton set m_0. Thus, according to <ref> of <ref>, the set ⋃_k ∈ K G_m_0,k is compact. And, because E is compact, the set m_0E is compact. Hence, the product (⋃_k ∈ K G_m_0,k) × (Hm_0) is compact. Therefore, becauseis continuous, the image (⋃_k ∈ K G_m_0,k)(Hm_0) is compact. In conclusion, the set KE is compact. We take a closer look at almost tameness in Let ℳ = M, G, be a left-homogeneous space and let it be called almost tamealmost tametame!almost if and only if it is topological and, for each coordinate system 𝒦 = m_0, g_m_0, m_m ∈ M for ℳ, the stabiliser G_0 is compact, and, for each compact subset K of M and each compact subset E of GG_0 such that G_0 · E ⊆ E, the set KE is a compact subset of M.If ℳ is proper, then it is almost tame (this follows from <ref> and <ref>). And, if it is almost tame, then it is semi-tame (this follows from the fact that, if E is a compact subset of GG_0, then G_0 · E is a compact subset of GG_0, where we use that ℳ is topological and that G_0 is compact). However, for tameness and semi-tameness we do not need ℳ to be topological, and there is no simple relationship between tameness and almost tameness. There are left-homogeneous spaces that are not semi-proper but nevertheless tame, which is illustrated by[Euclidean Space]For each d ∈_+, equip ^d with the Euclidean topology. Note that, for each d ∈_+ and each d' ∈_+, the product topology on ^d ×^d' is the Euclidean topology on ^d + d'.The groups ^2 andunder addition are topological. And, the group ^2 acts continuously and transitively onon the left by ((x, y), z) ↦ x + z. Each stabiliser underis 0×. And, each right quotient set semi-action of ^2(0×) oninduced byis (z, (x, y) + 0×) ↦ z + x.Under the identification ofwith ^2(0×) by the homeomorphic map ι x ↦x×, the semi-actionis the continuous map (z, x) ↦ z + x from ^2 to . Recall that images of compact subsets under continuous maps are compact. Hence, for each compact subset K ofand each compact subset E of , because K × E is a compact subset of ^2, the set KE is a compact subset of . Therefore, the left-homogeneous space , ^2, is tame.The action map ofis α ((x, y), z) ↦ (x + z, z). The diagonal K of the unit square in ^2, namely (z, z)z ∈0, 1, is compact. Moreover, for each element (z, z) ∈ K, its preimage under α is (0×) ×z. Hence, the set T = ((0, 0), 0)∪((0, 1/z), z)z ∈0, 1 is a transversal of α^-1((z, z))(z, z) ∈ K. The transversal T is not included in a compact subset of ^2 ×. Therefore, the left-homogeneous space , ^2, is not semi-proper.Analogously, the group ^2 acts on the compact topological circleon the left by ((x, y), z + ) ↦ (x + z) + and the left-homogeneous space , ^2, is tame but not semi-proper.There are left-homogeneous spaces that are not semi-proper but, if equipped with certain coordinate systems, nevertheless tame or semi-tame, which is illustrated by[Affine Space]In the situation of <ref>, the left-homogeneous space ℳ is not semi-proper, but, for some coordinate systems 𝒦, the cell space ℳ, 𝒦 is tame or semi-tame. Indeed, the action map α is ((t, d), m) ↦ (t + d · m, m), the preimage α^-1(0, 1×1) is ((m' - d, d), 1)d ∈∖0 m' ∈0, 1, and an unbounded transversal of that preimage is ((-1, 1), 1)∪((m' - 1/m', 1/m'), 1)m' ∈0, 1, which is not included in a compact set. § UNIFORM CURTIS-HEDLUND-LYNDON THEOREM Introduction. A metric space is a uniform space and a uniform space is a topological space. The uniformity induced by a metric is generated by the set of entourages that contains for each positive size the entourage of open balls of that size about each point (think of a family of neighbourhoods, one for each point, that are comparable in size). The topology induced by a uniformity is the set that contains the neighbourhoods of all points that can be extracted from the entourages (by throwing all neighbourhoods in one pot we lose the ability to compare the sizes of neighbourhoods of different points).A map on a metric space is continuous if, by bringing two points sufficiently close together, their images can be brought arbitrarily close together; and it is uniformly continuous if the sufficient closeness of two points that is needed to get a certain closeness of their images is the same for all pairs of points. In topological terms, continuity means that preimages of open sets are open; and in uniform terms, uniform continuity means that preimages of entourages are entourages. The concept of uniform continuity cannot be expressed in topological terms, because the sizes of neighbourhoods of different points are not comparable.The global transition function of a traditional cellular automaton is uniform and local (see <ref>). It is uniform because all cells use the same local transition function, and local because the neighbourhood is finite. Uniformity is equivalent to equivariance under translations of global configurations and locality to continuity with respect to the prodiscrete topology on global configurations.If the set of cells is discrete, then locality is naturally expressed by requiring neighbourhoods to be finite and characterised by uniform continuity, if the set of states is infinite, and continuity, otherwise, where the set of global configurations is equipped with the prodiscrete uniformity or topology, which is generated by cylinders with finite bases. And, if the set of cells is continuous, then locality is naturally expressed by requiring neighbourhoods to be compact and characterised by uniform continuity, where the set of global configurations is equipped with the uniformity that is generated by cylinders with compact bases. In the first case, if the set of states is finite, then continuity suffices in the characterisation; in the second case though, if the neighbourhood is infinite, then even for finite sets of states, it seems that continuity is not sufficient in the characterisation. Moreover, it may seem more natural to choose a bounded and closed neighbourhood. However, boundedness is only defined on metric spaces. And, according to the Heine-Borel theorem, on Euclidean spaces, compactness is characterised by boundedness and closedness. So, compactness seems and turns out to be a good alternative. For the characterisation, if the set of states is infinite, then we require the left group action to be semi-tame. The reason is that for a global transition function to be continuous, for each cylinder ℭ with a compact base K, there must be a cylinder ℭ' with a compact base whose image is included in ℭ. Because the new states of the cells in K are uniquely determined by the current states of cells in KN, the base of the cylinder ℭ' must include KN. Hence, for there to be such a cylinder ℭ', the set KN must be included in a compact set, which is the case if the left group action is semi-tame.Contents. Given a semi-tame topological structure on the set of cells, we equip the set of global configurations with a uniform and a topological structure (see <ref>) that generalise the prodiscrete topology and uniformity (see <ref>), and prove a uniform and a topological variant of the Curtis-Hedlund-Lyndon theorem. In <ref>, the uniform variant, we show that global transition functions are characterised by -equivariance and uniform continuity. And in its <ref>, the topological variant, that under the assumption that the set of states is finite, they are characterised by -equivariance and continuity. Note that, if the set of cells is compact or finite respectively, then uniform continuity or continuity are insubstantial (see <ref>). Body. We equip the set of global configurations with the topology and the uniformity generated by cylinders with compact bases in Let M be a topological space and let Q be a set. * The topology on Q^M that has for a subbase the sets𝔈(K, b), for b ∈ Q^K and K ⊆ M compact 𝔈(K, b) = c ∈ Q^Mc_K = b, [symbols]EKbfraktur@𝔈(K, b)forb ∈ Q^KandK ⊆ Mcompact,is called topology of discrete convergence on compactatopology of discrete convergence on compactaof discrete convergence on compacta!topologydiscrete convergence on compacta@of discrete convergence on compactadiscrete convergence on compacta!topology@topology.* The uniformity on Q^M that has for a subbase the sets𝔈(K), for K ⊆ M compact 𝔈(K) = (c, c') ∈ Q^M × Q^Mc_K = c'_K, [symbols]EKfraktur@𝔈(K)forK ⊆ Mcompact,is called uniformity of discrete convergence on compactauniformity of discrete convergence on compactaof discrete convergence on compacta!uniformitydiscrete convergence on compacta!uniformity@uniformity. Because finite intersections of compact sets are compact, the topology and uniformity of discrete convergence on compacta have for a base the sets 𝔈(K, b), for b ∈ Q^K and K ⊆ M compact, and 𝔈(K), for K ⊆ M compact, respectively.The topology induced by the uniformity of discrete convergence on compacta on Q^M is the topology of discrete convergence on compacta on Q^M. If the topological space M is compact, then the topology and uniformity of discrete convergence on compacta on Q^M are the discrete topology and uniformity on Q^M.Because each finite subset of M is compact, the topology and uniformity of discrete convergence on compacta on Q^M are finer than the prodiscrete topology and uniformity on Q^M. The prodiscrete uniformity is introduced in <ref> of <ref> and a base for it is given in <ref>.Let M be a topological space and let Q be a set. For each compact subset K of M, equip Q^K with the discrete topology. The topology of discrete convergence on compacta on Q^M is the coarsest topology on Q^M such that, for each compact subset K of M, the projectionπ_KQ^M→ Q^K, π_K, K ⊆ M compact[symbols]piK@π_Kc↦ c_K,is continuous. Note that π_K^-1(b) = 𝔈(K, b), for b ∈ Q^K and K ⊆ M compact. If the set of cells carries the discrete topology, then the topology and uniformity on global configurations reduce to the prodiscrete topology and uniformity.Let M be equipped with the discrete topology. Then, for each subset A of M, the set A is compact if and only if it is finite. Therefore, the topology and uniformity of discrete convergence on compacta on Q^M are the prodiscrete topology and uniformity on Q^M. If the set of cells and the group that acts on it carry the discrete topology, and the set of states is finite, then uniform notions on global configurations reduce to topological ones. Let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space and let Q be a finite set. Equip M with the discrete topology, and equip Q^M with the prodiscrete topology. According to <ref>, the cell space ℛ is tame. Because the topology on GG_0 is discrete, each subset E of GG_0 is compact if and only if it is finite. Because Q^M is compact (see <ref>), each map Δ Q^M → Q^M is uniformly continuous if and only if it is continuous. And, because Q^M is Hausdorff (see <ref>), each map Δ Q^M → Q^M, is a uniform isomorphism if and only if it is continuous and bijective.If the set of states carries the discrete topology, then the well-known uniformity of uniform convergence on compacta is the uniformity of discrete convergence on compacta. Let M be a topological space, let Q be a uniform space, and let 𝒰 be the uniformity of Q. The uniformity on Q^M that has for a subbase the sets𝔈(K, E), for E ∈𝒰 and K ⊆ M compact 𝔈(K, E) = (c,c') ∈ Q^M × Q^Mk ∈ K(c(k),c'(k)) ∈ E, [symbols]EKEfraktur@𝔈(K, E)forE ∈𝒰 andK ⊆ Mcompact,is called uniformity of uniform convergence on compactauniformity of uniform convergence on compacta.If the uniform space Q is (uniformly) discrete, then the uniformity of uniform convergence on compacta on Q^M is the uniformity of discrete convergence on compacta on Q^M. Indeed, because the diagonal D = (q, q)q ∈ Q is an entourage of the uniform space Q, for each compact subset K of M, the set𝔈(K, D) = (c, c') ∈ Q^M × Q^Mk ∈ Kc(k) = c'(k)is an entourage of the uniform space Q^M, which is included in each of the entourages 𝔈(K, E) = (c,c') ∈ Q^M × Q^Mk ∈ K(c(k), c'(k)) ∈ E,forE ∈𝒰.Therefore, the uniformity on Q^M has for a subbase the sets 𝔈(K, D) = 𝔈(K), for K ⊆ M compact.Let Q^M be equipped with the topology of uniform convergence on compacta and let(M, Q) = cM → Qcis continuous⊆ Q^Mbe equipped with the subspace topology. According to theorem 43.7 in <cit.>, this topology on (M, Q) is the well-known compact-open topology.Global transition functions of cellular automata over semi-tame cell spaces with compact sufficient neighbourhoods are characterised by equivariance and continuity.[Uniform Variant; Morton Landers Curtis, Gustav Arnold Hedlund, and Roger Conant Lyndon, 1969]Let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a semi-tame cell space, let Q be a set, let Δ be a map from Q^M to Q^M, let Q^M be equipped with the uniformity of discrete convergence on compacta, and let H be a 𝒦-big subgroup of G. The following two statements are equivalent: *The map Δ is the global transition function of a semi-cellular automaton over ℛ with ∙_H_0-invariant local transition function and compact sufficient neighbourhood.*The map Δ is _H-equivariant and uniformly continuous.First, let Δ be the global transition function of a semi-cellular automaton 𝒞 = ℛ, Q, N, δ with ∙_H_0-invariant local transition function δ and compact sufficient neighbourhood E. Then, according to <ref>, the map Δ is _H-equivariant. Moreover, let K be a compact subset of M. Because ℛ is semi-tame, the set KE is included in a compact subset L of M. For each c ∈ Q^M and each c' ∈ Q^M, if c_KE = c'_KE, then Δ(c)_K = Δ(c')_K, in particular, if c_L = c'_L, then Δ(c)_K = Δ(c')_K. Thus,(Δ×Δ)(𝔈(L)) ⊆𝔈(K).Because the sets 𝔈(K), for K ⊆ M compact, constitute a base of the uniformity on Q^M, the global transition function Δ is uniformly continuous.Secondly, let Δ be _H-equivariant and uniformly continuous. Then, because Δ is uniformly continuous, there is a compact subset E_0 of M such that(Δ×Δ)(𝔈(E_0)) ⊆𝔈(m_0).Therefore, for each c ∈ Q^M, the state Δ(c)(m_0) depends at most on c_E_0. The subset E = (m_0 )^-1(E_0) of GG_0 is compact. Let N be the set G_0 · E. Then, G_0 · N ⊆ N. And, because E_0 = m_0E ⊆ m_0N, for each c ∈ Q^M, the state Δ(c)(m_0) depends at most on c_m_0E, in particular, it depends at most on c_m_0N. Hence, there is a map δ Q^N → Q such thatc ∈ Q^M Δ(c)(m_0) = δ(n ↦ c(m_0n)).The quadruple 𝒞 = ℛ, Q, N, δ is a semi-cellular automaton with compact sufficient neighbourhood E. Conclude with <ref> that δ is ∙_H_0-invariant and that Δ is the global transition function of 𝒞. Note that in the proof semi-tameness of ℛ is used to deduce <ref> from <ref> but not to deduce <ref> from <ref>. Global transition functions of cellular automata with finite sets of states and finite sufficient neighbourhoods are characterised by equivariance and continuity. We already proved this in <ref>, but a slightly less general version also follows from <ref>.Let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space, let Q be a finite set, let Δ be a map from Q^M to Q^M, let Q^M be equipped with the prodiscrete topology, and let H be a 𝒦-big subgroup of G. The following two statements are equivalent: *The map Δ is the global transition function of a semi-cellular automaton over ℛ with ∙_H_0-invariant local transition function and finite sufficient neighbourhood.*The map Δ is _H-equivariant and continuous. With <ref> this follows directly from <ref>.If the set of cells is too small, then the properties concerning locality in the Curtis-Hedlund-Lyndon theorems, namely having a compact or finite sufficient neighbourhood and being uniformly continuous or continuous, are always satisfied.In the situation of <ref> or <ref>, let M be compact or finite respectively. Then, <ref> and <ref> reduce to the statement that the following two statements are equivalent: * The map Δ is the global transition function of a semi-cellular automaton over ℛ with ∙_H_0-invariant local transition function.* The map Δ is _H-equivariant.Indeed, if M is compact, then, for each semi-cellular automaton, there is one with the compact neighbourhood GG_0 ≃ M that has the same global transition function; and, because, according to <ref>, the uniformity on Q^M is the discrete uniformity, the map Δ is uniformly continuous. And, if M is finite, then each semi-cellular automaton has a finite neighbourhood; and, because the topology on Q^M is the discrete topology, the map Δ is continuous. The Curtis-Hedlund-Lyndon theorems presented above are generalisations of known theorems for cellular automata over groups.In the case that M = G andis the group multiplication of G, <ref> is theorem 1.9.1 in <cit.> and <ref> is theorem 1.8.1 in <cit.>. The uniform variant of the Curtis-Hedlund-Lyndon theorem can be applied to some automata that we have already encountered and to generalisations of them.[Plane, Sphere, Left Shift Map] The global transition functions of <ref> are -equivariant and uniformly continuous. [Riemannian Symmetric Space]Extensions of the Laplace–Beltrami operators on Riemannian symmetric spaces of constant curvature, like Euclidean spaces, spheres, and hyperbolic spaces, by the constant 0 like in <ref> are global transition functions of cellular automata and therefore -equivariant and uniformly continuous. Broadly speaking, this is true for all differential operators on such spaces that do not depend on global positions.On cell spaces that are not semi-tame and for subgroups that are not big, the uniform variant of the Curtis-Hedlund-Lyndon theorem does not hold, which is illustrated by[Affine Space]In the situation of <ref> in <ref>, the cell space ℛ = ℳ, 𝒦” is not semi-tame, in particular, neither is its coordinate map continuous nor is it semi-proper. Let Q be the set 0, 1, let N be the set , and let δ be the map Q^N → Q, ℓ↦ℓ(1). Recall that M is identified with GG_0 by ι.The quadruple 𝒞 = ℛ, Q, N, δ is a semi-cellular automaton that has the compact sufficient neighbourhood 1 and whose local transition function is ∙_T_0-invariant, where T is the subgroup α_t, 1 t ∈ of translations of G, which is not 𝒦”-big and T_0 is the stabiliser of m_0 under _T. Note that, because G_0 1 = and we need 1 ∈ N and G_0N ⊆ N, we had to choose N =.The global transition function Δ of 𝒞 is the map Q^M → Q^M, c ↦ [m ↦ c(m1)]. The state Δ()(0) depends on the state of cell 01 = 1 and, for each cell m ∈ M ∖0, the state Δ()(m) depends on the state of m1 = 1/m + m, which diverges to ±∞ as m tends to 0 from above or below. In particular, the restriction Δ()_0, 1 depends on the states of all cells in D = 1∪2, ∞.The set 𝔇 = (c, c') ∈ Q^M × Q^Mc_D = c'_D is the smallest set such that (Δ×Δ)(D) ⊆𝔈(0, 1). However, because D is unbounded, it is neither an entourage nor included in one, hence (Δ×Δ)^-1(𝔈(0, 1)) is not an entourage, and therefore Δ is not uniformly continuous.In conclusion, for cell spaces that are not semi-tame, <ref> of <ref> does not follow from <ref>. However, according to <ref>, even for cell spaces that are not semi-tame, <ref> follows from <ref>. Whether, under the assumption that the set of states is finite, continuity is sufficient in the uniform variant of the Curtis-Hedlund-Lyndon theorem is an Is there a semi-tame cell space ℛ, a finite set Q, a map Δ from Q^M to Q^M, a 𝒦-big subgroup H of G such that Δ is _H-equivariant and continuous with respect to the topology of discrete convergence on compacta on Q^M, but such that Δ is not the global transition function of a semi-cellular automaton over ℛ with ∙_H_0-invariant local transition function and compact sufficient neighbourhood?In example 1.8.2 in <cit.>, for each infinite group G, an -equivariant and continuous map Δ on G^G, equipped with the prodiscrete topology, is constructed that is not the global transition function of a cellular automaton over G with finite neighbourhood. One may try to use the general idea of the construction to construct a -equivariant and continuous map Δ on 0, 1^ that is not the global transition function of a cellular automaton over (, +) with compact neighbourhood.§ INVERTIBILITY OF BIG-CELLULAR AUTOMATASummary. A cellular automaton is invertible if its computations can be made in reverse by another automaton (see <ref>). If we consider only automata with compact sufficient neighbourhoods, it follows from <ref> that a map is the global transition function of an invertible automaton if and only if it is an equivariant uniform isomorphism (see <ref>). Hence, if we consider only those with finite set of states and finite sufficient neighbourhoods, a map is the global transition function of an invertible automaton if and only if it is equivariant, continuous, and bijective (see <ref>). In particular, such an automaton is invertible if and only if its global transition function is bijective (see <ref>). Let 𝒞 = ℛ, Q, N, δ be a semi-cellular automaton. It is called invertibleinvertible if and only if there is a semi-cellular automaton 𝒞', called inverse to 𝒞inverse to 𝒞, such that the global transition functions of 𝒞 and 𝒞' are inverse to each other. Let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a semi-tame cell space, let Q be a set, let Δ be a map from Q^M to Q^M, let Q^M be equipped with the uniformity of discrete convergence on compacta, and let H be a 𝒦-big subgroup of G. The following two statements are equivalent: *The map Δ is the global transition function of an invertible semi-cellular automaton 𝒞 over ℛ that has an inverse 𝒞' such that the local transition functions of 𝒞 and 𝒞' are ∙_H_0-invariant, and 𝒞 and 𝒞' have compact sufficient neighbourhoods.*The map Δ is an _H-equivariant uniform isomorphism. With <ref> this follows directly from main theorem <ref>. Let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space, let Q be a finite set, let Δ be a map from Q^M to Q^M, let Q^M be equipped with the prodiscrete topology, and let H be a 𝒦-big subgroup of G. The following two statements are equivalent: *The map Δ is the global transition function of an invertible semi-cellular automaton 𝒞 over ℛ that has an inverse 𝒞' such that the local transition functions of 𝒞 and 𝒞' are ∙_H_0-invariant, and 𝒞 and 𝒞' have finite sufficient neighbourhoods.*The map Δ is _H-equivariant, continuous, and bijective. With <ref> this follows directly from <ref>.Let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space, let H be a 𝒦-big subgroup of G. Furthermore, let 𝒞 be a semi-cellular automaton over ℛ with finite set of states, finite sufficient neighbourhood, and ∙_H_0-invariant local transition function. The automaton 𝒞 is invertible if and only if its global transition function is bijective.With <ref> and <ref> this follows directly from <ref>.It follows from <ref> that <ref> also holds if Q^M is equipped with the (, L)-prodiscrete topology, where L is a finite subgroup of H.In <ref>, if the assumption that the semi-cellular automaton 𝒞 has a finite set of states does not hold, then it may not be invertible even if its global transition function is bijective, which is shown by the cellular automaton of example 1.10.3 in <cit.>. CHAPTER: RIGHT AMENABILITY AND THE TARSKI-FØLNER THEOREMAbstract. We introduce right amenability, right Følner nets, and right paradoxical decompositions for left-homogeneous spaces and prove the Tarski-Følner theorem for left-homogeneous spaces with finite stabilisers. It states that right amenability, the existence of right Følner nets, and the non-existence of right paradoxical decompositions are equivalent. Remark. Most parts of this chapter appeared in the paper *wacker:amenable:2016<cit.> and they generalise parts of chapter 4 of the monograph *ceccherini-silberstein:coornaert:2010<cit.>. Introduction. The notion of amenability for groups was introduced by John von Neumann in 1929 in the paper *von-neumann:1929<cit.>. It generalises the notion of finiteness. A group G is left or right amenable if there is a finitely additive probability measure on the power set of G that is invariant under left and right multiplication respectively. Groups are left amenable if and only if they are right amenable. A group is amenable if it is left or right amenable.Examples of amenable groups are abound: Finite groups like the Fischer–Griess Monster group, abelian groups like the group of integers, nilpotent groups like the Heisenberg group, solvable groups like the group of invertible upper triangular matrices under multiplication, finitely generated groups of sub-exponential growth like the Grigorchuk group, and many more. Examples of non-amenable groups are the free groups whose rank is greater than 1 and the groups that have a subgroup that is isomorphic to the free group of rank 2. The definitions of left and right amenability generalise to left and right group sets respectively. A left group set M, G, is left amenable if there is a finitely additive probability measure on (M) that is invariant under . There is in general no natural action on the right that is to a left group action what right multiplication is to left group multiplication. Therefore, for a left group set there is no natural notion of right amenability.A transitive left group actionof G on M induces, for each element m_0 ∈ M and each family g_m_0, m_m ∈ M of elements in G such that, for each point m ∈ M, we have g_m_0, m m_0 = m, a right quotient set semi-actionof GG_0 on M with defect G_0 given by mg G_0 = g_m_0, m g g_m_0, m^-1 m, where G_0 is the stabiliser of m_0 under . Each of these right semi-actions is to the left group action what right multiplication is to left group multiplication. They occur in the definition of global transition functions of semi-cellular automata over left-homogeneous spaces. A coordinate system is a choice of m_0 and g_m_0, m_m ∈ M. A left-homogeneous space is right amenable if there is a coordinate system such that there is a finitely additive probability measure on (M) that is semi-invariant under . For example finite left-homogeneous spaces, abelian groups, and finitely right-generated left-homogeneous spaces of sub-exponential growth are right amenable, in particular, quotients of finitely generated groups of sub-exponential growth by finite subgroups acted upon by left multiplication.A net of non-empty and finite subsets of M is a right Følner net if, broadly speaking, these subsets are asymptotically invariant under . A finite subset E of GG_0 and two partitions A_e_e ∈ E and B_e_e ∈ E of M constitute a right paradoxical decomposition if the map e is injective on A_e and B_e, and the family (A_ee)(B_ee)_e ∈ E is a partition of M. The Tarski-Følner theorem states that right amenability, the existence of right Følner nets, and the non-existence of right paradoxical decompositions are equivalent.The Tarski alternative theorem and the theorem of Følner, which constitute the Tarski-Følner theorem, are famous theorems by Alfred Tarski and Erling Følner from 1938 and 1955, see the papers *tarski:1938<cit.> and *folner:1955<cit.>. Contents. In <ref> we introduce finitely additive probability measures and means, and kind of right semi-actions on them. In <ref> we introduce right amenability. In <ref> we introduce right Følner nets and present examples that are used throughout <ref>. In <ref> we introduce right paradoxical decompositions. In <ref> we prove the Tarski alternative theorem and the theorem of Følner. And in <ref> we show under which assumptions left implies right amenability and give two examples of right-amenable left-homogeneous spaces. Preliminary Notions. A left group set is a triple M, G,, where M is a set, G is a group, andis a map from G × M to M, called left group action of G on M, such that G →(M), g ↦ [g ], is a group homomorphism. The actionis transitive if M is non-empty and for each m ∈ M the map m is surjective; and free if for each m ∈ M the map m is injective. For each m ∈ M, the set Gm is the orbit of m, the set G_m = ( m)^-1(m) is the stabiliser of m, and, for each m' ∈ M, the set G_m, m' = ( m)^-1(m') is the transporter of m to m'.A left-homogeneous space is a left group set ℳ = M, G, such thatis transitive. A coordinate system for ℳ is a tuple 𝒦 = m_0, g_m_0, m_m ∈ M, where m_0 ∈ M and, for each m ∈ M, we have g_m_0, m m_0 = m. The stabiliser G_m_0 is denoted by G_0. The tuple ℛ = ℳ, 𝒦 is a cell space. The set g G_0g ∈ G of left cosets of G_0 in G is denoted by GG_0. The map M × GG_0 → M, (m, g G_0) ↦ g_m_0, m gm_0 is a right semi-action of GG_0 on M with defect G_0, which means thatm ∈ MmG_0 = m,andm ∈ Mg ∈ Gg_0 ∈ G_0 𝔤' ∈ GG_0 mg ·𝔤' = (mg G_0)g_0 ·𝔤'.It is transitive, which means that the set M is non-empty and for each m ∈ M the map m is surjective; and free, which means that for each m ∈ M the map m is injective; and semi-commutes with , which means thatm ∈ Mg ∈ Gg_0 ∈ G_0 𝔤' ∈ GG_0 (gm) 𝔤' = g(mg_0 ·𝔤').(See <ref>.)A cell space ℛ is finitely and symmetrically right generated if there is a finite subset S of GG_0 with G_0 · S ⊆ S and S^-1⊆ S, where S^-1 = g^-1 G_0s ∈ S, g ∈ s, such thatm ∈ Mk ∈_0 s_i_i ∈1, 2, …, k inS ∪ S^-1 [][](m_0s_1)s_2… s_k = m.The uncoloured S-Cayley graph of ℛ is the symmetric and 2 S-regular directed graph 𝒢 = M, (m, ms)m ∈ M, s ∈ S, the S-metric on ℛ is the distanceon 𝒢, and the S-length on ℛ is the map = (m_0, ). For each m ∈ M and each ρ∈, the S-ball of radius ρ centred at m is the set (m, ρ) = m' ∈ M (m, m') ≤ρ, the S-sphere of radius ρ centred at m is the set (m, ρ) = m' ∈ M (m, m') = ρ, the ball (m_0, ρ) is denoted by (ρ), and the sphere (m_0, ρ) by (ρ). (See <ref>.)In the present chapter, we assume that the reader is familiar with the basics of the theory of dual spaces and with Hall's harem theorem. A recapitulation of the required basics and of the theory surrounding Hall's theorems is given in <ref>. § FINITELY ADDITIVE PROBABILITY MEASURES AND MEANSIn this section, let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space. Summary. First, we introduce finitely additive probability measures on M (see <ref>); a left group action of G and a kind of right quotient set semi-action of GG_0, both on the set of maps from (M) to 0, 1, which contains all finitely additive probability measures on M (see <ref>); and what it means for a measure to be semi-invariant under the kind of right semi-action (see <ref>). Secondly, we introduce means on M (see <ref>); a left group action of G and a kind of right quotient set semi-action of GG_0, both on the topological dual space of (M), which contains all means on M (see <ref>); and what it means for a mean to be invariant under the kind of right semi-action (see <ref>). Lastly, we give a bijection from the set of means on M to the set of finitely additive probability measures on M (see <ref>). We show in <ref> that this bijection preserves (semi-)invariance (see in particular <ref>). Let μ(M) →0, 1 be a map. It is called * normalisednormalised if and only if μ(M) = 1;* finitely additivefinitely additive if and only if, A ⊆ MB ⊆ M[]A ∩ B = ∅μ(A ∪ B) = μ(A) + μ(B); * finitely additive probability measure on Mfinitely additive probability measure μ on M if and only if it is normalised and finitely additive.The set of all finitely additive probability measures on M is denoted by (M)set (M)[symbols]PMM@(M).Finitely additive measures, which do not need to be normalised, are known as contentcontent.The group G acts on 0, 1^(M) on the left byG ×0, 1^(M) →0, 1^(M), left group actionof G on 0, 1^(M)[symbols]vDash@(g, φ)↦ [A ↦φ(g^-1 A)],such that G (M) ⊆(M). The quotient set GG_0 kind of semi-acts on 0, 1^(M) on the right by[symbols]Dashv@0, 1^(M)× GG_0→0, 1^(M), kind of right quotient set semi-actionof GG_0 on 0, 1^(M)(φ, 𝔤)↦ [A ↦φ(A 𝔤)]. Let μ be a finitely additive probability measure on M and let 𝔤 be an element of GG_0. Then, M 𝔤 need not be equal to M and, for each subset A of M and each subset B of M such that the sets A and B are disjoint, the sets A 𝔤 and B 𝔤 need not be disjoint. Therefore, μ𝔤 need neither be normalised nor finitely additive, in particular, it need not be a finitely additive probability measure on M. Let φ be an element of 0, 1^(M). It is called -semi-invariant-semi-invariantsemi-invariant if and only if, for each element 𝔤∈ GG_0 and each subset A of M such that the map 𝔤 is injective on A, we have (φ𝔤)(A) = φ(A).Let ℛ be the cell space G, G, ·, e_G, g_g ∈ G. Then, G_0 = e_G and = ·. Hence, (φ, g) ↦ [A ↦φ(A · g)]. Except for g not being inverted, this is the right group action of G on (M) as defined in paragraph 4 of section 4.3 in <cit.>. Moreover, for each element g ∈ G, the map g is injective. Hence, being -semi-invariant is the same as being right-invariant as defined in paragraph 2 of section 4.4 in <cit.>. If the set M is finite, then the mapμ(M)→0, 1,A↦A/M,is a -invariant and -semi-invariant finitely additive probability measure on M.The map μ is normalised and finitely additive. Thus, it is a finitely additive probability measure on M. Moreover, for each element g ∈ G and each subset A of M, because the map g^-1 is injective,(g μ)(A) = μ(g^-1 A) = g^-1 A/M = A/M = μ(A).And, for each element 𝔤∈ GG_0 and each subset A of M such that the map 𝔤 is injective on A,(μ𝔤)(A) = μ(A 𝔤) = A 𝔤/M = A/M = μ(A).Hence, the finitely additive probability measure μ is -invariant and -semi-invariant. If the set M is infinite, then, for each finite subset F of M, the mapμ_F (M)→0, 1,A↦A ∩ F/F,is a finitely additive probability measure on M that is neither -invariant nor -semi-invariant. Nevertheless, for certain cell spaces over M, there is a net F_i_i ∈ I of finite subsets of M such that the net μ_F_i_i ∈ I converges in a weak sense to a -invariant or -semi-invariant finitely additive probability measure on M. However, even on , there is no explicit formula for that measure.The vector space of bounded real-valued functions on M with pointwise addition and scalar multiplication is denoted by (M)vector space (M) of bounded real-valued functions on M[symbols]linfinityM@(M),the supremum norm on (M) is denoted by _∞supremum norm _∞ on (M)[symbols]norminfinity@_∞, the topological dual space of (M) is denoted by (M)^*topological dual space (M)^* of (M)[symbols]linfinityMstar@(M)^*, the pointwise partial order on (M) is denoted by ≤pointwise partial order ≤ on (M)[symbols]lessthanorequalto@≤,and the constant function [m ↦ 0] is denoted by zero function = [m ↦ 0][symbols]0 nullity [email protected] A be a subset of M. The map_AM→0,1,[symbols]1doublestrikeA@_Am↦1, if m ∈ A, 0, if m ∉ A,is called indicator function of A on Mindicator function _A of A on M.Let ν(M) → be a map. It is called * normalisednormalised if and only if ν(_M) = 1;* non-negativity preservingnon-negativity preserving if and only iff ∈(M)(f ≥ν(f) ≥ 0); non-negativity preserving* mean on Mmean ν on M if and only if it is linear, normalised, and non-negativity preserving. The set of all means on M is denoted by (M)set (M)[symbols]MSM@(M).Let Ψ be a map from (M) to (M). It is called non-negativity preservingnon-negativity preserving if and only if f ∈(M)(f ≥Ψ(f) ≥ 0).Let G_0 be finite, let A be a finite subset of M, and let 𝔤 be an element of GG_0. Then, (𝔤)^-1(A)≤G_0·A.Let a ∈ A such that (𝔤)^-1(a) ≠∅. There are m and m' ∈ M such that G_m_0,m = 𝔤 and m' 𝔤 = a. For each m”∈ M, we have m”𝔤 = g_m_0, m” m and hence m”𝔤 = a m”𝔤 = m' 𝔤 g_m_0, m'^-1 g_m_0, m” m = m g_m_0, m'^-1 g_m_0, m”∈ G_m g_m_0, m”∈ g_m_0, m' G_m.Moreover, for each m” and each m”' ∈ M with m”≠ m”', we have g_m_0, m”≠ g_m_0, m”'.Thus,(𝔤)^-1(a) =m”∈ Mm”𝔤 = a=m”∈ Mg_m_0, m”∈ g_m_0, m' G_m≤g_m_0, m' G_m=G_m=G_0.Therefore, because (𝔤)^-1(A) = ⋃_a ∈ A (𝔤)^-1(a), we have (𝔤)^-1(A)≤G_0·A.The group G acts on (M) on the left byG ×(M)→(M), left group actionof G on (M)[symbols]Vdash@(g, f)↦ [m ↦ f(g^-1 m)]. Let G_0 be finite. The quotient set GG_0 kind of semi-acts on (M) on the right by[symbols]dashV@(M) × GG_0→(M), kind of right quotient set semi-actionof GG_0 on (M) (f, 𝔤)↦ [m ↦∑_m' ∈ (𝔤)^-1(m) f(m')],such that, for each tuple (f, 𝔤) ∈(M) × GG_0, we have f 𝔤_∞≤G_0·f_∞.Let 𝔤∈ GG_0. Furthermore, let f ∈(M). Moreover, let m ∈ M. Because G_0 is finite, according to <ref>, we have (𝔤)^-1(m)≤G_0 < ∞. Hence, the sum in the definition ofis finite. Furthermore,(f 𝔤)(m) ≤∑_m' ∈ (𝔤)^-1(m)f(m')≤*∑_m' ∈ (𝔤)^-1(m) 1·f_∞=(𝔤)^-1(m)·f_∞≤G_0·f_∞. Therefore, f 𝔤∈(M), f 𝔤_∞≤G_0·f_∞, andis well-defined. In the situation of <ref>, we have (f, g) ↦ [m ↦ f(m · g^-1)]. Hence,is the right group action of G on ^G as defined in paragraph 5 of section 4.3 in <cit.>.Let G_0 be finite and let 𝔤 be an element of GG_0. The map 𝔤 is linear, continuous, and non-negativity preserving.Linearity follows from linearity of summation, continuity follows from linearity and 𝔤_∞≤G_0·_∞, and non-negativity preservation follows from non-negativity preservation of summation. Let ν be a mean on M. Then, ν∈(M)^* and ν_(M)^* = 1. In particular, ν is continuous.Compare proposition 4.1.7 in <cit.>. The group G acts on (M)^* on the left by[symbols]VDash@G ×(M)^*→(M)^*, left group actionof G on (M)^*(g, ψ)↦ [f ↦ψ(g^-1 f)],such that G (M) ⊆(M).Let G_0 be finite. The quotient set GG_0 kind of semi-acts on (M)^* on the right by[symbols]DashV@(M)^* × GG_0→(M)^*, kind of right quotient set semi-actionof GG_0 on (M)^*(ψ, 𝔤)↦ [f ↦ψ(f 𝔤)]. Let ψ∈(M)^* and let 𝔤∈ GG_0. Then, ψ𝔤 = ψ (𝔤). Because ψ and 𝔤 are linear and continuous, so is ψ𝔤.Let G_0 be finite and let ψ be an element of (M)^*. It is called -invariantinvariant-invariant if and only if, for each element 𝔤∈ GG_0 and each function f ∈(M), we have (ψ𝔤)(f) = ψ(f).In the situation of <ref>, we have (ψ, g) ↦ [f ↦ψ(fg]. Except for g not being inverted, this is the right group action of G on (G)^* as defined in paragraph 6 of section 4.3 in <cit.>. Hence, being -invariant is the same as being right-invariant as defined in paragraph 3 of section 4.4 in <cit.>. If the set M is finite, then the map ν(M)→,f↦1/M∑_m ∈ M f(m)(= ∑_r ∈ r ·μ(f^-1(r)),is a -invariant and -invariant mean on M, where μ is the finitely additive probability measure of <ref>. Note that ν is the integral on (M) induced by μ, that, for each function f ∈(M), we have f = ∑_r ∈ r ·_f^-1(r), and that, for each subset A of M, we have μ(A) = ν(_A).The map ν is linear, normalised and non-negativity preserving. Thus, it is a mean on M. Moreover, for each element g ∈ G and each function f ∈(M), because the map g is bijective,(g ν)(f)= ν(g^-1 f)= 1/M∑_m ∈ M (g^-1 f)(m)= 1/M∑_m ∈ M f(gm)= 1/M∑_m ∈ M f(m)= ν(f).And, for each element 𝔤∈ GG_0 and each function f ∈(M), because M = _m ∈ M (𝔤)^-1(m),(ν𝔤)(f)= ν(f 𝔤)= 1/M∑_m ∈ M (f 𝔤)(m)= 1/M∑_m ∈ M∑_m' ∈ (𝔤)^-1(m) f(m')= 1/M∑_m ∈ M f(m)= ν(f).Hence, the mean ν is -invariant and -invariant. Alternatively, we can deduce that ν is -invariant from the -semi-invariance of μ in a way that generalises to the non-finite case with the axiom of choice (compare <ref>). Let 𝔤 be an element of GG_0 and let f be a function of (M). Because (𝔤)^-1(m)≤G_0, for m ∈ M, (in other words, the map 𝔤 is (≤G_0)-to-1), there is a partition M_k_k ∈1, 2, …, G_0 of M such that, for each index k ∈1, 2, …, G_0, the map 𝔤 is injective on M_k. We havef = ∑_r ∈ r ·_f^-1(r) = ∑_r ∈∑_k ∈1, 2, …, G_0 r ·_f^-1(r) ∩ M_k.Therefore, because ν𝔤 is linear,(ν𝔤)(f) = ∑_r ∈∑_k ∈1, 2, …, G_0 r · (ν𝔤)(_f^-1(r) ∩ M_k).Let r be a real number and let k be an index of 1, 2, …, G_0. Then, because 𝔤 is injective on f^-1(r) ∩ M_k, we have _f^-1(r) ∩ M_k𝔤 = _f^-1(r) ∩ M_k 𝔤. Hence, by definition of ν and because μ is -semi-invariant,(ν𝔤)(_f^-1(r) ∩ M_k)= ν(_f^-1(r) ∩ M_k 𝔤)= μ(f^-1(r) ∩ M_k 𝔤)= μ(f^-1(r) ∩ M_k)= ν(_f^-1(r) ∩ M_k).Therefore, with the equation above, because ν is linear,(ν𝔤)(f) = ∑_r ∈∑_k ∈1, 2, …, G_0 r ·ν(_f^-1(r) ∩ M_k) = ν(f). If the set M is infinite, then, for each finite subset F of M, the mapν_F (M)→,f↦1/F∑_m ∈ F f(m)(= ∑_r ∈ r ·μ_F(f^-1(r)),is a mean on M that is neither -invariant nor -invariant, where μ_F is the finitely additive probability measure of <ref>. Nevertheless, for certain cell spaces over M, there is a net F_i_i ∈ I of finite subsets of M such that the net ν_F_i_i ∈ I converges in a weak sense to a -invariant or -semi-invariant mean on M. However, even on , there is no explicit formula for that mean. This construction is used in the subproof <ref> implies <ref> of <ref>. The mapΦ(M)→(M), map Φ from (M) to (M)[symbols]Phi@Φ ν ↦ [A ↦ν(_A)],is bijective. Compare theorem 4.1.8 in <cit.>. The set (M) is a convex and compact subset of (M)^* equipped with the weak-* topology.Compare theorem 4.2.1 in <cit.>.§ RIGHT AMENABILITYContents. In <ref> we introduce the notion of right amenability using finitely additive probability measures. And in <ref> we characterise right amenability of cell spaces with finite stabilisers using means.Let M, G, be a left group set. It is called left amenableleft amenableamenable!left if and only if there is a -invariant finitely additive probability measure on M. Let ℳ = M, G, be a left-homogeneous space. It is called right amenableright-amenable left-homogeneous spaceamenable!right if and only if there is a coordinate system 𝒦 = m_0, g_m_0, m_m ∈ M for ℳ such that there is a -semi-invariant finitely additive probability measure on M, in which case the cell space ℛ = ℳ, 𝒦 is called right amenableright-amenable cell spaceamenable!right.In the situation of <ref>, being right amenable is the same as being amenable as defined in definition 4.4.5 in <cit.>.In the remainder of this section, let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space such that the stabiliser G_0 of m_0 underis finite.The (kind of) right semi-actionsandare compatible in the sense given inLet 𝔤 be an element of GG_0 and let A be a subset of M such that the map 𝔤 is injective on A. Then, _A 𝔤 = _A 𝔤.For each m ∈ M, because 𝔤 is injective on A,_A 𝔤(m)= 1,if m ∈ A 𝔤, 0,otherwise,= m' ∈ Am' 𝔤 = m= ∑_m' ∈ (𝔤)^-1(m)_A(m')= (_A 𝔤)(m).In conclusion, _A 𝔤 = _A 𝔤.Simple functions approximate bounded functions arbitrarily good as stated in The vector space(M) = fM → f(M)is finite (= _AA ⊆ M) (M)[symbols]Ecalligraphic@(M)is dense in the Banach space (M), _∞. Compare lemma 4.1.9 in <cit.>. For a mean, being invariant at certain indicator functions is sufficient for being invariant everywhere, which is shown in Let ψ be an element of (M)^* such that, for each element 𝔤∈ GG_0 and each subset A of M such that the map 𝔤 is injective on A, we have (ψ𝔤)(_A) = ψ(_A). The map ψ is -invariant. Let 𝔤∈ GG_0.First, let A ⊆ M. Moreover, let m ∈ M. According to <ref>, we have k_m = (𝔤)^-1(m)≤G_0. Hence, there are pairwise distinct m_m,1, m_m,2, …, m_m,k_m∈ M such that (𝔤)^-1(m) = m_m,1, m_m,2, …, m_m,k_m. For each i ∈1, 2, …, G_0, putA_i = m_m,i m ∈ M, k_m ≥ i∩ A. Because, for each m ∈ M and each m' ∈ M such that m ≠ m', we have (𝔤)^-1(m) ∩ (𝔤)^-1(m') = ∅, the sets A_1, A_2, …, A_G_0 are pairwise disjoint and the map 𝔤 is injective on each of these sets. Moreover, because ⋃_m ∈ M (𝔤)^-1(m) = M, we have ⋃_i = 1^G_0 A_i = A. Therefore, _A = ∑_i = 1^G_0_A_i. Thus, because ψ𝔤 and ψ are linear,(ψ𝔤)(_A)= (ψ𝔤)*∑_i = 1^G_0_A_i= ∑_i = 1^G_0 (ψ𝔤)(_A_i)= ∑_i = 1^G_0ψ(_A_i)= ψ(_A). Therefore, ψ𝔤 = ψ on the set of indicator functions.Thus, because the indicator functions span (M), and ψ𝔤 and ψ are linear, ψ𝔤 = ψ on (M). Hence, because (M) is dense in (M), and ψ𝔤 and ψ are continuous, ψ𝔤 = ψ on (M).In conclusion, ψ is -invariant. A characterisation of right amenability by means is given inThe cell space ℛ is right amenable if and only if there is a -invariant mean on M. Let Φ be the map in <ref>.First, let ℛ be right amenable. Then, there is -semi-invariant finitely additive probability measure μ on M. Put ν = Φ^-1(μ). Then, for each 𝔤∈ GG_0 and each A ⊆ M such that 𝔤 is injective on A, according to <ref>,(ν𝔤)(_A) = ν(_A 𝔤) = ν(_A 𝔤) = μ(A 𝔤)= μ(A)= ν(_A).Thus, according to <ref>, the mean ν is -invariant.Secondly, let there be a -invariant mean ν on M. Put μ = Φ(ν).Then, for each 𝔤∈ GG_0 and each A ⊆ M such that 𝔤 is injective on A, according to <ref>,(μ𝔤)(A) = μ(A 𝔤) = ν(_A 𝔤) = ν(_A 𝔤) = ν(_A) = μ(A).Hence, μ is -semi-invariant.§ RIGHT FØLNER NETS In this section, let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space. Contents. In <ref> we introduce right Følner nets, and in <ref> we give a necessary and sufficient condition for the existence of such nets. In the case that the stabiliser G_0 is finite, in <ref> we characterise right Følner nets, in <ref> we give another necessary and sufficient condition for the existence of such nets, and in <ref> we show that being a right Følner net does not depend on the choice of coordinate system.Let F_i_i ∈ I be a net in F ⊆ MF ≠∅, Ffinite indexed by (I, ≤). It is called right Følner net in ℛ indexed by (I, ≤)right Følner net F_i_i ∈ I in ℛ indexed by (I, ≤)Følner net in ℛ indexed by (I, ≤)@right Følner net in ℛ indexed by (I, ≤)net!Følner if and only if 𝔤∈ GG_0 lim_i ∈ IF_i ∖ (𝔤)^-1(F_i)/F_i = 0. In the situation of <ref>, for each element g ∈ G and each index i ∈ I, we have ( g)^-1(F_i) = F_i · g^-1. Hence, right Følner nets in ℛ are exactly right Følner nets for G as defined in the first paragraph after definition 4.7.2 in <cit.>. If the set M is finite, then the constant sequence M_k ∈_0 is a right Følner net in ℛ. The necessary and sufficient conditions for the existence of right Følner nets that are given in <ref> follow directly from Let V be a set, let W be a set, and let Ψ be a map from V × W to . There is a net v_i_i ∈ I in V indexed by (I, ≤) such thatw ∈ W lim_i ∈ IΨ(v_i, w) = 0,if and only if, for each finite subset Q of W and each positive real number ε∈_> 0, there is an element v ∈ V such thatq ∈ Q Ψ(v, q) < ε. First, let there be a net v_i_i ∈ I in V indexed by (I, ≤) such that <ref> holds. Furthermore, let Q be a finite subset of W and let ε∈_> 0. Because <ref> holds, for each q ∈ Q, there is an i_q ∈ I such that,i ∈ I(i ≥ i_q Ψ(v_i, q) < ε).Because (I, ≤) is a directed set and Q is finite, there is an i ∈ I such that, for each q ∈ Q, we have i ≥ i_q. Put v = v_i. Then, <ref> holds.Secondly, for each finite Q ⊆ W and each ε∈_> 0, let there be a v ∈ V such that <ref> holds. Furthermore, letI = Q ⊆ WQis finite×_> 0and let ≤ be the preorder on I given by(Q, ε) ∈ I(Q', ε') ∈ I(Q, ε) ≤ (Q', ε')Q ⊆ Q' ε≥ε'.For each (Q, ε) ∈ I and each (Q', ε') ∈ I, the element (Q ∪ Q', min(ε, ε')) of I is an upper bound of (Q, ε) and of (Q', ε'). Hence, (I, ≤) is a directed set.By precondition, for each i = (Q, ε) ∈ I, there is a v_i ∈ V such thatq ∈ Q Ψ(v_i, q) < ε.Let w ∈ W and let ε_0 ∈_> 0. Put i_0 = (w, ε_0). For each i = (Q, ε) ∈ I with i ≥ i_0, we have w ∈ Q and ε≤ε_0. Hence,i ∈ I(i ≥ i_0 Ψ(v_i, w) < ε_0).Therefore, v_i_i ∈ I is a net in V indexed by (I, ≤) such that <ref> holds. There is a right Følner net in ℛ if and only if, for each finite subset E of GG_0 and each positive real number ε∈_> 0, there is a non-empty and finite subset F of M such thate ∈ E F ∖ ( e)^-1(F)/F < ε. This is a direct consequence of <ref> withΨF ⊆ MF ≠∅, Ffinite× GG_0→,(F, 𝔤)↦F ∖ (𝔤)^-1(F)/F.In the proof of <ref>, the upper bound given in <ref> is essential, which itself follows from the upper bound given in lemma <ref> and the inclusion given in <ref>, which in turn follows from the equality given in <ref>. Let m be an element of M, and let 𝔤 be an element of GG_0. There is an element g ∈𝔤 such that𝔤' ∈ GG_0(m 𝔤) 𝔤' = mg ·𝔤',in particular, for said g ∈𝔤, we have (m 𝔤)g^-1 G_0 = m.There is a g ∈ G such that g G_0 = 𝔤. Moreover, becauseis a semi-action with defect G_0, there is a g_0 ∈ G_0 such that𝔤' ∈ GG_0(mg G_0) 𝔤' = mg · (g_0^-1·𝔤').Because g · (g_0^-1·𝔤') = g g_0^-1·𝔤' and g g_0^-1∈𝔤, the statement holds. Let A and A' be two subsets of M, and let 𝔤 and 𝔤' be two elements of GG_0. Then, for each element m ∈ (𝔤)^-1(A) ∖ (𝔤')^-1(A'),m 𝔤∈⋃_g ∈𝔤 A ∖ ( g^-1·𝔤')^-1(A')andm 𝔤' ∈⋃_g' ∈𝔤' ( (g')^-1·𝔤)^-1(A) ∖ A'.Let m ∈ (𝔤)^-1(A) ∖ (𝔤')^-1(A'). Then, m 𝔤∈ A and m 𝔤' ∉ A'. According to <ref>, there is a g ∈𝔤 and a g' ∈𝔤' such that (m 𝔤)g^-1·𝔤' = m 𝔤' ∉ A' and (m 𝔤')(g')^-1·𝔤 = m 𝔤∈ A. Hence, m 𝔤∉ ( g^-1·𝔤')^-1(A') and m 𝔤' ∈ ( (g')^-1·𝔤)^-1(A). Therefore, m 𝔤∈ A ∖ ( g^-1·𝔤')^-1(A') and m 𝔤' ∈ ( (g')^-1·𝔤)^-1(A) ∖ A'. In conclusion, m 𝔤∈⋃_g ∈𝔤 A ∖ ( g^-1·𝔤')^-1(A') and m 𝔤' ∈⋃_g' ∈𝔤' ( (g')^-1·𝔤)^-1(A) ∖ A'. Let G_0 be finite, let F and F' be two finite subsets of M, and let 𝔤 and 𝔤' be two elements of GG_0. Then,(𝔤)^-1(F) ∖ (𝔤')^-1(F') ≤G_0^2 ·max_g ∈𝔤F ∖ ( g^-1·𝔤')^-1(F'), G_0^2 ·max_g' ∈𝔤'( (g')^-1·𝔤)^-1(F) ∖ F'. Put A = (𝔤)^-1(F) ∖ (𝔤')^-1(F'). For each g ∈𝔤, put B_g = F ∖ ( g^-1·𝔤')^-1(F'). For each g' ∈𝔤', put B_g'' = ( (g')^-1·𝔤)^-1(F) ∖ F'.According to <ref>, the restrictions (𝔤)_A →⋃_g ∈𝔤 B_g and (𝔤')_A →⋃_g' ∈𝔤' B_g'' are well-defined. Moreover, for each m ∈ M, according to <ref>, we have (𝔤)^-1(m)≤G_0 and (𝔤')^-1(m)≤G_0. Therefore, because 𝔤 = G_0, A≤G_0·⋃_g ∈𝔤 B_g≤G_0·∑_g ∈𝔤B_g≤G_0^2 ·max_g ∈𝔤B_gand analogouslyA≤G_0^2 ·max_g' ∈𝔤'B_g''.A characterisation of right Følner nets is given in Let G_0 be finite and let F_i_i ∈ I be a net in F ⊆ MF ≠∅, Ffinite indexed by (I, ≤). The net F_i_i ∈ I is a right Følner net in ℛ if and only if𝔤∈ GG_0 lim_i ∈ I(𝔤)^-1(F_i) ∖ F_i/F_i = 0. Let 𝔤∈ GG_0. Furthermore, let i ∈ I. Because F_i = ( G_0)^-1(F_i), according to <ref>,(𝔤)^-1(F_i) ∖ F_i≤G_0^2 ·max_g ∈𝔤F_i ∖ ( g^-1 G_0)^-1(F_i)andF_i ∖ (𝔤)^-1(F_i)≤G_0^2 ·max_g ∈𝔤( g^-1 G_0)^-1(F_i) ∖ F_i.Moreover, 𝔤 = G_0 < ∞. Therefore, if F_i_i ∈ I is a right Følner net in ℛ, thenlim_i ∈ I(𝔤)^-1(F_i) ∖ F_i/F_i = 0;and, if <ref> holds, thenlim_i ∈ IF_i ∖ (𝔤)^-1(F_i)/F_i = 0.In conclusion, F_i_i ∈ I is a right Følner net in ℛ if and only if <ref> holds.Necessary and sufficient conditions for the existence of right Følner nets are given in Let G_0 be finite. There is a right Følner net in ℛ if and only if, for each finite subset E of GG_0 and each positive real number ε∈_> 0, there is a non-empty and finite subset F of M such thate ∈ E ( e)^-1(F) ∖ F/F < ε. This is a direct consequence of <ref> and <ref> withΨF ⊆ MF ≠∅, Ffinite× GG_0→,(F, 𝔤)↦(𝔤)^-1(F) ∖ F/F.That being a right Følner net does not depend on the choice of coordinate system is shown in Let ℳ = M, G, be a left-homogeneous space with finite stabilisers, let 𝒦 = m_0, g_m_0, m_m ∈ M and 𝒦' = m_0', g_m_0', m'_m ∈ M be two coordinate systems for ℳ, and let ℱ = F_i_i ∈ I be a net in F ⊆ MF ≠∅, Ffinite indexed by (I, ≤). The net ℱ is a right Følner net in ℳ, 𝒦 if and only if it is one in ℳ, 𝒦'. First, let ℱ be a right Følner net in ℳ, 𝒦. Furthermore, let g be an element of G such that gm_0 = m_0', let 𝔤' be an element of GG_0' and let i be an index of I. Then, according to <ref>,m ∈ Mg_0 ∈ G_0m ' 𝔤' = mg_0 · (g^-1𝔤').Thus,(' 𝔤')^-1(F_i) ⊆⋃_g_0 ∈ G_0 ( g_0 · (g^-1𝔤'))^-1(F_i),in particular,(' 𝔤')^-1(F_i) ∖ F_i ⊆⋃_g_0 ∈ G_0 ( g_0 · (g^-1𝔤'))^-1(F_i) ∖ F_i.Hence, because the stabiliser G_0 is finite,(' 𝔤')^-1(F_i) ∖ F_i≤∑_g_0 ∈ G_0( g_0 · (g^-1𝔤'))^-1(F_i) ∖ F_i.Therefore, because ℱ is a right Følner net in ℳ, 𝒦,lim_i ∈ I(' 𝔤')^-1(F_i)/F_i = 0.In conclusion, the net ℱ is a right Følner net in ℳ, 𝒦'.Secondly, let ℱ be a right Følner net in ℳ, 𝒦'. It follows as above that ℱ is a right Følner net in ℳ, 𝒦. [Lattice]Let M be the two-dimensional integer lattice ^2, let S_2 be the symmetric group on 1, 2, let V be the multiplicative group -1, 1, let φ be the group homomorphism S_2 →(V^2), π↦ [(v_1, v_2) ↦ (v_π(1), v_π(2))], let V^2 ⋊_φ S_2 be the outer semi-direct product of S_2 acting on V^2 by φ, let ψ be the group homomorphism V^2 ⋊_φ S_2 →(^2), ((v_1, v_2), π) ↦ [(t_1, t_2) ↦ (v_1 · t_π(1), v_2 · t_π(2))], let G = ^2 ⋊_ψ (V^2 ⋊_φ S_2) be the outer semi-direct product of V^2 ⋊_φ S_2 acting on ^2 by ψ, and letbe the transitive left group action of G on M by (((t_1, t_2), ((v_1, v_2), π)), (z_1, z_2)) ↦ (t_1 + v_1 · z_π(1), t_2 + v_2 · z_π(2)). The triple ℳ = M, G, is a left-homogeneous space. The group G encodes the symmetries of the lattice M, more precisely, the component ^2 encodes the translational symmetries and the component V^2 × S_2 the reflectional and rotational ones that stabilise the origin. For example, the element ((4, 2), ((-1, 1), )) encodes the reflection about the x-axis, followed by a translation by (4, 2); the element ((0, 0), ((1, 1), (1 2))) encodes the reflection about the line through the origin of slope 1; and the element ((0, 0), ((-1, 1), (1 2))) encodes the anticlockwise rotation about the origin through 90, where the permutation (1 2), written in cycle notation, is the transposition that swaps 1 and 2. The symmetry group of M is the group gg ∈ G under composition, which is isomorphic to G. Let m_0 be the origin (0, 0) and, for each point m ∈ M, let g_m_0, m be the translation (m, ((1, 1), )). The tuple 𝒦 = m_0, g_m_0, m_m ∈ M is a coordinate system for ℳ. The stabiliser G_0 of m_0 underis the set (0, 0)× (V^2 × S_2), the quotient group GG_0 is isomorphic to the group ^2 by (t, ((v_1, v_2), π)) G_0 ↦ t, and, under this isomorphism, the right quotient group semi-actionof GG_0 on M is the map M ×^2 → M, (z, t) ↦ z + t.The cell space ℛ = ℳ, 𝒦 is finitely and symmetrically right generated by S = (-1, 0), (0, -1), (0, 1), (1, 0). The S-metric , the S-length = (m_0, ), the S-balls (…), and the S-spheres (…) are restrictions of the corresponding notions on ^2 with respect to the taxicab metric on ^2. The balls _^2(…) and spheres _^2(…) induced by the taxicab metric on ^2 are diamonds, that is, filled and unfilled squares with sides oriented at 45 to the coordinate axes. The former notions will be rigorously introduced in <ref>. In the present situation, the S-metric is the graph metric on the S-Cayley graph of ^2, the S-length is the distance to the origin, for each cell m ∈ M and each integer ρ, the S-ball (m, ρ) and S-sphere (m, ρ) of radius ρ centred at m are the sets of points whose distances to m are not greater than and, respectively, equal to ρ, and we write (ρ) and (ρ) for the balls and spheres centred at the origin. Let t be an element of ^2. For each non-negative integer ρ, the preimage ( t)^-1((ρ)) is the ball (-t, ρ) and the complement (ρ) ∖ ( t)^-1((ρ)) is included in the thickened sphere (ρ) ∖(ρ - t) (see <ref> for the case that t = (-1, 0)). Because the size of the latter grows linearly in ρ, the sequence (ρ) ∖ ( t)^-1((ρ))_ρ∈_0 grows linearly in size (in the case that t = (-1, 0), it grows in size like 2 ρ + 1_ρ∈_0). Moreover, the sequence (ρ)_ρ∈_0 grows polynomially in size, more precisely, it grows in size like 2ρ (ρ + 1) + 1_ρ∈_0. Hence, the quotient (ρ) ∖ ( t)^-1((ρ)) / (ρ) converges to 0 as ρ tends to ∞. Therefore, the sequence (ρ)_ρ∈_0 is a right Følner net in ℛ.[Tree]Let M be the vertices of the uncoloured a, b, a^-1, b^-1-Cayley graph of the free group F_2 over a, b, where a ≠ b, let ς be the group automorphism from F_2 to F_2 determined by a ↦ b and b ↦ a^-1, let R be the cyclic group 0, 90, 180, 270 under addition modulo 360, let φ be the group homomorphism R →(F_2), r ↦ς^r / 90, let G = F_2 ⋊_φ R be the outer semi-direct product of R acting on F_2 by φ, and letbe the transitive left group action of G on M by ((f, r), m) ↦ f ·φ(r)(m). The triple ℳ = M, G, is a left-homogeneous space.The group G encodes some graph automorphisms of M, more precisely, the component F_2 encodes the translational automorphisms and the component R the rotational ones that stabilise the origin. For example, the element (a b^-1, 90) encodes the anticlockwise rotation about the origin through 90, followed by a translation by a b^-1, which is the anticlockwise rotation about a through 90; see <ref> for further examples. In general, for each vertex m ∈ M and each angle r ∈ R, the anticlockwise rotation about m through r, is the graph automorphism m ·φ(r)(m^-1·) = m · (φ(r)(m))^-1·φ(r)(), which is encoded by (m, 0) · (e_F_2, r) · (m^-1, 0) = (m · (φ(r)(m))^-1, r). The map g ↦ g embeds the group G into the graph-automorphism group of M.Let m_0 be the neutral element e_F_2 of F_2 and, for each vertex m ∈ M, let g_m_0, m be the translation (m, 0). The tuple 𝒦 = m_0, g_m_0, m_m ∈ M is a coordinate system for ℳ. The stabiliser G_0 of m_0 underis the set e_F_2× R, the quotient group GG_0 is isomorphic to the group F_2 by (f, r) G_0 ↦ f, and, under this isomorphism, the right quotient group semi-actionof GG_0 on M is the map M × F_2 → M, (m, f) ↦ m · f.The cell space ℛ = ℳ, 𝒦 is finitely and symmetrically right generated by S = a, b, a^-1, b^-1. The uncoloured S-Cayley graph of ℛ is equal to the uncoloured S-Cayley graph of F_2. Hence, the S-metricis identical to the S-word metric on F_2 and the S-length = (m_0, ) is identical to the S-word norm on F_2. And, for each element m ∈ M and each integer ρ, the sets (m, ρ) = m' ∈ M (m, m') ≤ρ and (m, ρ) = m' ∈ M (m, m') = ρ are the ball and sphere of radius ρ centred at m, and we denote (m_0, ρ) by (ρ) and (m_0, ρ) by (ρ). The former notions will be rigorously introduced in <ref>.For each non-negative integer ρ, the preimage ( a^-1)^-1((ρ)) is the set (ρ) · a, which is not the ball (a, ρ) in the left a, b, a^-1, b^-1-Cayley graph of F_2, but it is in the right one. The sequence (ρ) ∖ ( a^-1)^-1((ρ))_ρ∈_0 grows exponentially in size, more precisely, it grows in size like 3^ρ_ρ∈_0 (see <ref>). Moreover, the sequence (ρ)_ρ∈_0 also grows exponentially in size, more precisely, it grows in size like ∑_ϱ = 0^ρ(ϱ)_ρ∈_0 = 2 · 3^ρ - 1_ρ∈_0. Hence, the quotient (ρ) ∖ ( a^-1)^-1((ρ)) / (ρ) converges to 1/2 as ρ tends to ∞. Therefore, neither the sequence (ρ)_ρ∈_0 nor any of its subsequences is a right Følner net in ℛ. Actually, as we will see, the cell space ℛ is not right amenable and hence there is no right Følner net in ℛ. § RIGHT PARADOXICAL DECOMPOSITIONS In this section, let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space.Let A and A' be two sets. The set A ∪ A' is denoted by AA'disjoint union AA'[symbols]cupdot@ if and only if the sets A and A' are disjoint. Let E be a finite subset of GG_0, and let A_e_e ∈ E and B_e_e ∈ E be two families of subsets of M indexed by E such that, for each index e ∈ E, the map e is injective on A_e and on B_e, andM = _e ∈ E A_e = _e ∈ E B_e = *_e ∈ E A_ee*_e ∈ E B_ee.The triple E, A_e_e ∈ E, B_e_e ∈ E is called right paradoxical decomposition E, A_e_e ∈ E, B_e_e ∈ E of ℛparadoxical decomposition of ℛ@right paradoxical decomposition of ℛright paradoxical decomposition of ℛ.In the situation of <ref>, for each element g ∈ G, the map g is injective. Hence, right paradoxical decompositions of ℛ are the same as right paradoxical decompositions of G as defined in definition 4.8.1 in <cit.>. [Tree] In the situation of <ref>, let A^+ and A^- be the sets of the elements of M whose reduced form ends with a and a^-1 respectively, let B^+ be the set of the elements of M whose reduced form ends with b or is b^-n for some n ∈_0, and let B^- be the set of the elements of M whose reduced form ends with b^-1 but is not b^-n for any n ∈_+. Furthermore, let E = e_F_2, a, b, let A_e_F_2 = A^-, let A_a = A^+· a^-1, let A_b = ∅, let B_e_F_2 = B^-, let B_a = ∅, and let B_b = B^+· b^-1. The set A_a is the set of the elements of M whose reduced form does not end with a^-1 and the set B_b is the set of the elements of M whose reduced form does not end with b^-1 or is b^-n for some n ∈_0. The triple E, A_e_e ∈ E, B_e_e ∈ E is a right paradoxical decomposition of ℛ (see <ref>; compare example 4.8.2 in <cit.>).A right paradoxical decomposition of M is one of _M in the sense given inLet G_0 be finite and let E, A_e_e ∈ E, B_e_e ∈ E be a right paradoxical decomposition of ℛ. Then,_M = ∑_e ∈ E_A_e = ∑_e ∈ E_B_e = ∑_e ∈ E (_A_e e) + ∑_e ∈ E (_B_e e). This is a direct consequence of <ref> and <ref>.§ TARSKI'S AND FØLNER'S THEOREM In this section, let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space such that the stabiliser G_0 of m_0 underis finite. Contents. In <ref> we show some technical properties that are used in the proof of the Tarski-Følner theorem <ref>, which states for a cell space that right amenability, the existence of a right Følner net, and the non-existence of a right paradoxical decomposition are equivalent. The analogous statement for left-homogeneous spaces is given in <ref>. Let 𝔤 be an element of GG_0. The map 𝔤 is continuous, where (M)^* is equipped with the weak-* topology. For each f ∈(M), let _f be the evaluation map (M)^* →, ψ↦ψ(f). Furthermore, let f ∈(M). Then, for each ψ∈(M)^*,(_f(𝔤))(ψ) = _f(ψ𝔤) = (ψ𝔤)(f) = ψ(f 𝔤) = _f 𝔤(ψ).Thus, _f(𝔤) = _f 𝔤. Hence, because _f 𝔤 is continuous, so is _f(𝔤). Therefore, according to <ref>, the map 𝔤 is continuous.Let ν_i_i ∈ I be a net in (M) such that, for each element 𝔤∈ GG_0, the net (ν_i 𝔤) - ν_i_i ∈ I converges toin (M)^* equipped with the weak-* topology. The cell space ℛ is right amenable. Let 𝔤∈ GG_0. According to <ref>, the set (M) is compact in (M)^* equipped with the weak-* topology. Hence, there is a subnet ν_i_j_j ∈ J of ν_i_i ∈ I that converges to a ν∈(M). Because, according to <ref>, the map 𝔤 is continuous, the net (ν_i_j𝔤) - ν_i_j_j ∈ J converges to (ν𝔤) - ν in (M)^*. Because it is a subnet of (ν_i 𝔤) - ν_i_i ∈ I, it also converges toin (M)^*. Because, according to <ref>, the space (M)^* is Hausdorff, we have (ν𝔤) - ν = and hence ν𝔤 = ν. Altogether, ν is a -invariant mean. In conclusion, according to <ref>, the cell space ℛ is right amenable.Let m be an element of M, and let E and E' be two subsets of GG_0. There is a subset E” of GG_0 such that (mE)E' = mE”; if G_0 ∈ E ∩ E', then G_0 ∈ E”; if E and E' are finite, then E”≤E·E'; and if G_0 · E' ⊆ E', then E” = g · e'e ∈ E, e' ∈ E', g ∈ e.For each e ∈ E, according to <ref>, there is a g_e ∈ e such that 𝔤∈ GG_0(me) 𝔤 = mg_e ·𝔤.Put E” = g_e · e'e ∈ E, e' ∈ E'. Then, (mE)E' = mE”. Moreover, if G_0 ∈ E ∩ E', then G_0 = g_G_0· G_0 ∈ E”; if E and E' are finite, then E”≤E·E'; and if G_0 · E' ⊆ E', then E” is as stated.Let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space such that the stabiliser G_0 of m_0 underis finite. The following statements are equivalent: *The cell space ℛ is not right amenable;*There is no right Følner net in ℛ;*There is a finite subset E of GG_0 such that G_0 ∈ E and, for each finite subset F of M, we have FE≥ 2 F;*There is a 2-to-1 surjective map ϕ M → M and there is a finite subset E of GG_0 such thatm ∈ Me ∈ E ϕ(m)e = m; *There is a right paradoxical decomposition of ℛ.<ref> implies <ref>.Let there be a right Følner net F_i_i ∈ I in ℛ. Furthermore, let i ∈ I. Putν_i (M)→,f↦1/F_i∑_m ∈ F_i f(m).Then, ν_i ∈(M).Moreover, let 𝔤∈ GG_0 and let f ∈(M). Then,(ν_i 𝔤)(f)= ν_i(f 𝔤)= 1/F_i∑_m ∈ F_i (f 𝔤)(m)= 1/F_i∑_m ∈ F_i∑_m' ∈ (𝔤)^-1(m) f(m')= 1/F_i∑_m ∈ (𝔤)^-1(F_i) f(m).Hence,(ν_i 𝔤 - ν_i)(f)= 1/F_i[t] ( ∑_m ∈ (𝔤)^-1(F_i) ∖ F_i f(m)- ∑_m ∈ F_i ∖ (𝔤)^-1(F_i) f(m)).Therefore,(ν_i 𝔤 - ν_i)(f) ≤1/F_i[t] ( ∑_m ∈ (𝔤)^-1(F_i) ∖ F_if(m)+ ∑_m ∈ F_i ∖ (𝔤)^-1(F_i)f(m)) ≤[t] ( (𝔤)^-1(F_i) ∖ F_i/F_i+ F_i ∖ (𝔤)^-1(F_i)/F_i) ·f_∞.According to <ref> and <ref>, the nets (𝔤)^-1(F_i) ∖ F_i / F_i_i ∈ I and F_i ∖ (𝔤)^-1(F_i) / F_i_i ∈ I converge to 0. Hence, so does (ν_i 𝔤 - ν_i)(f)_i ∈ I. Thus, the net ν_i 𝔤 - ν_i_i ∈ I converges toin (M)^* equipped with the weak-* topology. Hence, according to <ref>, the cell space ℛ is right amenable. In conclusion, by contraposition, if ℛ is not right amenable, then there is no right Følner net in ℛ.<ref> implies <ref>. Let there be no right Følner net in ℛ. According to <ref>, there is a finite E_1 ⊆ GG_0 and an ε∈_> 0 such that, for each non-empty and finite F ⊆ M, there is an e_F ∈ E_1 such that F ∖ ( e_F)^-1(F)/F≥ε.Put E_2 = G_0∪ E_1. Let F be a non-empty and finite subset of M. Then, F ⊆ F ∪ (FE_1) = FE_2. Thus,FE_2 - F =(FE_2) ∖ F=(FE_1) ∖ F≥(Fe_F) ∖ F.Moreover, according to <ref>, we have ( e_F)^-1((Fe_F) ∖ F)≤G_0·(Fe_F) ∖ F. Hence,FE_2 - F≥( e_F)^-1((Fe_F) ∖ F)/G_0.Therefore, because F ∖ ( e_F)^-1(F) ⊆ ( e_F)^-1((Fe_F) ∖ F), FE_2 - F ≥F ∖ ( e_F)^-1(F)/G_0≥ε/G_0F.Put ξ = 1 + ε / G_0. Then, FE_2≥ξF. Because ε does not depend on F, neither does ξ. Therefore, for each non-empty and finite F ⊆ M, we have FE_2≥ξF. Let F be a non-empty and finite subset of M. Because ξ > 1, there is an n ∈_+ such that ξ^n ≥ 2. Hence,(((FE_2) …)E_2)E_2_n times ≥ξ((FE_2) …)E_2≥…≥ξ^n F≥ 2 F.Moreover, according to <ref>, there is an E ⊆ GG_0 such that E is finite, G_0 ∈ E, and FE = (((FE_2) …)E_2)E_2. In conclusion, FE≥ 2 F.<ref> implies <ref> (see <ref>). Let there be a finite E ⊆ GG_0 such that, for each finite F ⊆ M, we have FE≥ 2 F. Furthermore, let 𝒢 be the bipartite graph M, M, (m, m') ∈ M × Me ∈ Eme = m'.Moreover, let F be a finite subset of M. The right neighbourhood of F in 𝒢 is𝒩_r(F) = m' ∈ Me ∈ EFe ∋ m' = FEand the left neighbourhood of F in 𝒢 is𝒩_l(F) = m ∈ Me ∈ Eme ∈ F = ⋃_e ∈ E ( e)^-1(F).By precondition 𝒩_r(F) = FE≥ 2 F. Moreover, because G_0 ∈ E,we have F = ( G_0)^-1(F) ⊆𝒩_l(F) and hence 𝒩_l(F)≥F≥ 2^-1F. Therefore, according tothe Hall harem <ref>, there is a perfect (1,2)-matching for 𝒢. In conclusion, there is a 2-to-1 surjective map ϕ M → M such that, for each m ∈ M, the tuple (ϕ(m), m) is an edge in 𝒢, that is, there is an e ∈ E such that ϕ(m)e = m. <ref> implies <ref> (see <ref>). Let there be a 2-to-1 surjective map ϕ M → M and a finite subset E of GG_0 such thatm ∈ Me ∈ E ϕ(m)e = m.By the axiom of choice, there are two injective maps ψ and ψ'M → M such that, for each m ∈ M, we have ϕ^-1(m) = ψ(m), ψ'(m). For each e ∈ E, letA_e = m ∈ Mme = ψ(m)and letB_e = m ∈ Mme = ψ'(m).Let m ∈ M. There is an e ∈ E such that ϕ(ψ(m))e = ψ(m). Because ϕ(ψ(m)) = m, we have m ∈ A_e. And, becauseis free, for each e' ∈ E ∖e, we have me' ≠ me = ψ(m) and thus m ∉ A_e'. Therefore,M = _e ∈ E A_e,and analogouslyM = _e ∈ E B_e.Moreover, ψ(A_e) = A_ee and ψ'(B_e) = B_ee. Hence, because M = ψ(M) ψ'(M), and ψ and ψ' are injective,M= *_e ∈ Eψ(A_e)*_e ∈ Eψ'(B_e)= *_e ∈ E A_ee*_e ∈ E B_ee.Furthermore, because ψ and ψ' are injective, for each e ∈ E, the maps ( e)_A_e = ψ_A_e and ( e)_B_e = ψ'_B_e are injective. In conclusion, E, A_e_e ∈ E, B_e_e ∈ E is a right paradoxical decomposition of ℛ. <ref> implies <ref>. Let there be a right paradoxical decomposition E, A_e_e ∈ E, B_e_e ∈ E of ℛ. According to <ref>,_M = ∑_e ∈ E_A_e = ∑_e ∈ E_B_e = ∑_e ∈ E (_A_e e) + ∑_e ∈ E (_B_e e).Suppose that ℛ is right amenable. Then, according to theorem <ref>, there is a -invariant mean ν on M. Because ν is linear and normalised, 1= ν(_M)= ∑_e ∈ Eν(_A_e e) + ∑_e ∈ Eν(_B_e e)= ∑_e ∈ E (ν e)(_A_e) + ∑_e ∈ E (ν e)(_B_e)= ∑_e ∈ Eν(_A_e) + ∑_e ∈ Eν(_B_e)= ν(_M) + ν(_M)= 1 + 1= 2,which contradicts that 1 ≠ 2. In conclusion, ℛ is not right amenable. Let ℳ be a left-homogeneous space with finite stabilisers. It is right amenable if and only if there is a coordinate system 𝒦 for ℳ such that there is no right paradoxical decomposition of ℳ, 𝒦.Let ℳ be a left-homogeneous space with finite stabilisers. It is right amenable if and only if there is a coordinate system 𝒦 for ℳ such that there is a right Følner net in ℳ, 𝒦. Let ℳ be a left-homogeneous space with finite stabilisers. The following statements are equivalent: * The space ℳ is right amenable;* For each coordinate system 𝒦 for ℳ, there is a -semi-invariant finitely additive probability measure on M;* For each coordinate system 𝒦 for ℳ, there is a -invariant mean on M;* For each coordinate system 𝒦 for ℳ, there is a right Følner net in ℳ, 𝒦;* For each coordinate system 𝒦 for ℳ, there is no right paradoxical decomposition of ℳ, 𝒦.This is a direct consequence of <ref>, <ref>, and <ref>.In the situation of <ref>, <ref> constitute theorem 4.9.1 in <cit.>. [Tree]In <ref> we constructed a right paradoxical decomposition of ℛ. Hence, according to <ref>, the cell space ℛ is not right amenable, there is a subset E of GG_0 as in <ref> of <ref>, there is a map ϕ from M to M as in <ref> of <ref>, and there are maps ψ and ψ' from M to M as in the subproof <ref> implies <ref> of <ref>. For the right paradoxical decomposition we constructed, these sets and maps can be given explicitly and we do so in the following.The mapϕ M→ M,m↦m,if m ∈ A^- (= A_e_F_2), m · a^-1,if m ∈ A^+ (= A_a · a), m,if m ∈ B^- (= B_e_F_2), m · b^-1,if m ∈ B^+ (= B_b · b),is well-defined, because the family A^-, A^+, B^-, B^+ is a partition of M; it is 2-to-1 surjective, because it is bijective from A^- A^+ to M as well as from B^- B^+ to M (see <ref>); and, for each cell m ∈ M, there is an element e ∈ E such that ϕ(m) · e = m, because if m ∈ A^-∪ B^-, then ϕ(m) · e_F_2 = m, if m ∈ A^+, then ϕ(m) · a = m, and if m ∈ B^+, then ϕ(m) · b = m. Hence, the map ϕ satisfies the properties of <ref> of <ref>. The mapsψ M→ M,m↦ m,if m ∈ A^- (= A_e_F_2),m · a,if m ∈ A^+· a^-1 (= A_a),andψ'M→ M,m↦ m,if m ∈ B^- (= B_e_F_2),m · b,if m ∈ B^+· b^-1 (= B_b),are well-defined, because the families A^-, A^+· a^-1 and B^-, B^+· b^-1 are partitions of M; they are injective, because the sets A^- and A^+ as well as the sets B^- and B^+ are disjoint (see <ref>); the family ψ(M), ψ'(M) is a partition of M, because the family A^-, A^+, B^-, B^+ is a partition of M; for each cell m ∈ M, we have ϕ^-1(m) = ψ(m), ψ'(m), as can be verified; and, for each element e ∈ E, we have A_e = m ∈ Mm · e = ψ(m) and B_e = m ∈ Mm · e = ψ'(m), as can be verified. Hence, the maps ψ and ψ' are the ones from the subproof <ref> implies <ref> of <ref> that correspond to ϕ. The set E contains the neutral element e_F_2 and, for each finite subset F of M, we have F · E≥ 2 F, because F = (F ∩ A^-)(F ∩ A^+)(F ∩ B^-)(F ∩ B^+) and hence F · E ⊇ ((F ∩ A^-) · e_F_2)((F ∩ A^-) · b)((F ∩ A^+) · a)((F ∩ A^+) · b)((F ∩ B^-) · e_F_2)((F ∩ B^-) · a)((F ∩ B^+) · a)((F ∩ B^+) · b) (see <ref>). Note that (F ∩ A^-) · e_F_2 = ψ(F ∩ A^-), (F ∩ A^-) · b = ψ'(F ∩ A^-), and so on. According to section 4.5 in <cit.>, amenable groups are closed under taking subspaces, quotients, finite direct products, inductive limits, and arbitrary direct sums. Are right-amenable left-homogeneous spaces closed under analogous notions?§ FROM LEFT TO RIGHT AMENABILITYIntroduction. We begin by stating a general condition for when left amenability implies right amenability, then we take a look at ever more specific situations in which this condition is satisfied, and eventually we present a method to construct right-amenable cell spaces from left-amenable principal left-homogeneous spaces using outer semi-direct products. The cell spaces yielded by this construction though are in the sense degenerated that a subgroup acts freely and transitively. A method to construct non-degenerated right-amenable cell spaces is given in <ref>.Let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space and let H be a subgroup of G such that, for each element 𝔤∈ GG_0, there is an element h ∈ H such that the maps 𝔤 and h are inverse to each other. If M, H, _H × M is left amenable, then ℛ is right amenable.Let μ∈(M). Furthermore, let 𝔤∈ GG_0. There is an h ∈ H such that 𝔤 and h are inverse to each other. Moreover, let A ⊆ M. Because 𝔤 = (h )^-1 = h^-1, we have A 𝔤 = h^-1 A. Therefore,(μ𝔤)(A)= μ(A 𝔤)= μ(h^-1 A)= (h μ)(A).Thus, μ𝔤 = h μ. Hence, if μ is _H ×0, 1^(M)-invariant, then μ is -semi-invariant. In conclusion, if M, H, _H × M is left amenable, then ℛ is right amenable.Let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space and let H be a subgroup of G such that G = G_0 H, for each element 𝔤∈ GG_0, the map 𝔤 is injective, h ∈ Hh G_0 = h ,andh ∈ H 𝔤∈ GG_0( h G_0) 𝔤 =h ·𝔤.If M, H, _H × M is left amenable, then ℛ is right amenable.Let g G_0 ∈ GG_0. Because g^-1∈ G = G_0 H, there is a g_0 ∈ G_0 and there is an h ∈ H such that g^-1 = g_0 h. Thus, h = g_0^-1 g^-1∈ H. Hence, for each m ∈ M,(( g G_0)(h ))(m)= (hm)g G_0= (mh G_0)g G_0= mh g G_0= mg_0^-1 g^-1 g G_0= mG_0= m.Therefore, h is right inverse to g G_0. Hence, g G_0 is surjective and thus, because it is injective by precondition, bijective. Therefore, g G_0 and h are inverse to each other. In conclusion, according to <ref>, if M, H, _H × M is left amenable, then ℛ is right amenable. Let G be a group. The set(G) = z ∈ Gg ∈ Gz g = g zcenter (G) of G[symbols]ZGcentre@(G)is called centre of G.The centre of G is a subgroup of G. Let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space and let H be a subgroup of G such that G is equal to G_0 H, _H × M is free, and g_m_0, m m ∈ M is included in (H) (in particular, _(H) × M is transitive). If M, H, _H × M is left amenable, then ℛ is right amenable.Let g ∈ G. For each m ∈ M,mg G_0 = g_m_0, m gm_0 = g_m_0, m (gm_0).Let m ∈ M. For each m' ∈ M, because _(H) × M is free and g_m_0, m, g_m_0, m'∈(H),m'g G_0 = mg G_0 g_m_0, m' = g_m_0, m m' = m.Therefore, g G_0 is injective. Let m ∈ M and let h ∈ H. Because g_m_0, m∈(G),mh G_0= g_m_0, m hm_0= h g_m_0, m m_0= hm.Put m' = mh G_0. Then,g_m_0, m hm_0= h g_m_0, m m_0= hm= m'.Hence, because g_m_0, m' m_0 = m' also and _H × M is free, g_m_0, m' = g_m_0, m h. Therefore,(mh G_0)g G_0= m'g G_0= g_m_0, m' gm_0= g_m_0, m h gm_0= mh g G_0.In conclusion, according to <ref>, if M, H, _H × M is left amenable, then ℛ is right amenable.Let M = be a field, letG = fM → M, x ↦ a x + ba, b ∈ M, a ≠ 0be the group of affine functions with composition as group multiplication, and letH = fM → M, x ↦ x + bb ∈ Mbe the group of translations also with composition as group multiplication. The group H is an abelian subgroup of G, which in turn is a non-abelian subgroup of the symmetry group of M. Moreover, according to example 4.6.2 and theorem 4.6.3 in <cit.>, the group G is left amenable and hence, according to proposition 4.5.1 in <cit.>, so is its subgroup H. Furthermore, the group G acts transitively on M by function application byand so does H by _H × M, even freely so. Because the groups G and H are left amenable, so are the left group sets M, G, and M, H, _H × M. The stabiliser of m_0 = 0 is the group of dilations G_0 = fM → M, x ↦ a xa ∈ M ∖0.We have G = G_0 H. For each m ∈ M, letg_m_0, m M→ M,x↦ x + m, be the translation by m. Then, g_m_0, m m ∈ M is included in (H) = H. Hence, according to <ref>, the cell space ℛ = M, G, , m_0, g_m_0, m_m ∈ M is right amenable. Let H and N be two groups, let ϕ be a group homomorphism from H to (N), let G be the Cartesian product N × H, and let· G × G→ G,((n, h), (n', h'))↦ (n ϕ(h)(n'), h h').The tuple (G, ·) is a group, called outer semi-direct product N ⋊_ϕ H of H acting on N by ϕouter semi-direct product of H acting on N by ϕsemi-direct product!outer[symbols]NrtimesphiH@N ⋊_ϕ H, and denoted by N ⋊_ϕ H.Let G be a semi-direct product of H acting on N by ϕ. The neutral element of G is (e_N, e_H) and, for each element (n, h) ∈ G, the inverse of (n, h) is (ϕ(h^-1)(n^-1), h^-1). Let M, H, _H be a principal left-homogeneous space. Furthermore, let G_0 be a group, let ϕ be a group homomorphism from G_0 to (H), let m_0 be an element of M, for each element m ∈ M, let h_m_0,m be the unique element of H such that h_m_0,m m_0 = m, and let_G_0 G_0 × M→ M,(g_0, m)↦ϕ(g_0)(h_m_0,m) _H m_0.Moreover, let G be the outer semi-direct product of G_0 acting on H by ϕ, and letG × M→ M,((h, g_0), m)↦ h _H (g_0 _G_0 m).The triple M, G_0, _G_0 is a left group set and the group G_0 is the stabiliser of m_0 under _G_0. Furthermore, the tuple ℛ = M, G, , m_0, (h_m_0, m, e_G_0)_m ∈ M is a cell space and the group e_H× G_0 is the stabiliser of m_0 under . Moreover, under the identification of G_0 with e_H× G_0 and of H with H ×e_G_0, the left group sets M, G_0, _G_0 and M, H, _H are left group subsets of M, G,. Because ϕ(e_G_0) = _(H), for each m ∈ M,e_G_0_G_0 m= ϕ(e_G_0)(h_m_0,m) _H m_0= h_m_0,m_H m_0= m.Let g_0 and g_0' ∈ G_0, and let m ∈ M. Because _H is free and h_m_0, ϕ(g_0')(h_m_0,m) _H m_0_H m_0 = ϕ(g_0')(h_m_0,m) _H m_0, we have h_m_0, ϕ(g_0')(h_m_0,m) _H m_0 = ϕ(g_0')(h_m_0,m). Therefore,g_0 g_0' _G_0 m= ϕ(g_0 g_0')(h_m_0,m) _H m_0= (ϕ(g_0) ϕ(g_0'))(h_m_0,m) _H m_0= ϕ(g_0)(ϕ(g_0')(h_m_0,m)) _H m_0= ϕ(g_0)(h_m_0, ϕ(g_0')(h_m_0,m) _H m_0) _H m_0= g_0 _G_0 (ϕ(g_0')(h_m_0,m) _H m_0)= g_0 _G_0 (g_0' _G_0 m).In conclusion, M, G_0, _G_0 is a left group set.Because h_m_0, m_0 = e_H, for each g_0 ∈ G_0,g_0 _G_0 m_0= ϕ(g_0)(e_H) _H m_0= e_H _H m_0= m_0.In conclusion, G_0 is the stabiliser of m_0 under _G_0.For each m ∈ M,(e_H, e_G_0)m = e_H _H (e_G_0_G_0 m) = m.Let g_0 ∈ G_0, let h ∈ H, and let m ∈ M. Because h h_m_0,m_H m_0 = h _H m, we have h h_m_0,m = h_m_0, h _H m. Hence,ϕ(g_0)(h) _H (g_0 _G_0 m)= ϕ(g_0)(h) _H (ϕ(g_0)(h_m_0,m) _H m_0)= ϕ(g_0)(h) ϕ(g_0)(h_m_0,m) _H m_0= ϕ(g_0)(h h_m_0,m) _H m_0= ϕ(g_0)(h_m_0, h _H m) _H m_0= g_0 _G_0 (h _H m).Therefore, for each g_0 ∈ G_0, each g_0' ∈ G_0, each h ∈ H, each h' ∈ H, and each m ∈ M,(h, g_0) (h', g_0')m= (h ϕ(g_0)(h'), g_0 g_0')m= h ϕ(g_0)(h') _H (g_0 g_0' _G_0 m)= h _H []ϕ(g_0)(h') _H []g_0 _G_0 (g_0' _G_0 m)= h _H []g_0 _G_0[]h' _H (g_0' _G_0 m)= (h, g_0) []h' _H (g_0' _G_0 m)= (h, g_0) [](h', g_0')m.In conclusion, M, G, is a left group action.Because _H is transitive and, for each h ∈ H and each m ∈ M, we have (h, e_G_0)m = hm, the left group actionis transitive and hence ℳ = M, G, is a left-homogeneous space. Moreover, because, for each m ∈ M,(h_m_0, m, e_G_0)m_0= h_m_0, m_H (e_G_0_G_0 m_0)= h_m_0, m_H m_0= m,the tuple 𝒦 = m_0, (h_m_0, m, e_G_0)_m ∈ M is a coordinate system for ℳ. Therefore, ℛ = ℳ, 𝒦 is a cell space.Because G_0 is the stabiliser of m_0 under _G_0, for each (h, g_0) ∈ G, we have (h, g_0)m_0 = h _H (g_0m_0) = hm_0. Because _H is free, e_H× G_0 is the stabiliser of m_0 under .Under the identification of G_0 with e_H× G_0 and of H with H ×e_G_0, we have _G_0 × M = _G_0 and _H × M = _H. In the situation of <ref>, let H be abelian. The cell space ℛ is right amenable. According to theorem 4.6.1 in <cit.>, because H is abelian, it is left amenable. Therefore, M, H, _H is left amenable. Identify G_0 with e_H× G_0 and identify H with H ×e_G_0. Then, H is a subgroup of G, and G = H G_0, and _H × M = _H is free, and, for each m ∈ M, we have (h_m_0, m, e_G_0) ∈ H = (H). Hence, according to <ref>, the cell space ℛ is right amenable.[Euclidean Space (compare <ref>)] Let d be a positive integer; let E be the d-dimensional Euclidean group, that is, the symmetry group of the d-dimensional Euclidean space, in other words, the isometries of ^d with respect to the Euclidean metric with function composition; let T be the d-dimensional translation group; and let O be the d-dimensional orthogonal group. The group T is abelian, a normal subgroup of E, and isomorphic to ^d with addition; the group O is isomorphic to the quotient ET and to the (d × d)-dimensional orthogonal matrices with matrix multiplication; the group E is isomorphic to the outer semi-direct product T ⋊_ι O, where ι O →(^d) is the inclusion map. The groups T, O, and E act on ^d on the left by function application, denoted by _T, _O, andrespectively; under the identification of T with ^d by t ↦ [v ↦ v + t], of O with the orthogonal matrices of ^d × d by A ↦ [v ↦ A v], and of E with T ⋊_ι O by (t, A) ↦ [v ↦ A v + t], we have _TT ×^d→^d,(t, v)↦ v + t,and_OO ×^d→^d,(A, v)↦ A v,andE ×^d→^d,((t, A), v)↦ A v + t,andι O→(^d),A↦ [v ↦ A v].Hence, for each vector v ∈^d, we have v _T 0 = v, therefore, _O = [(A, v) ↦ι(A)(v) _T 0], and thus = [((t, A), v) ↦ t _T (A _O v)]. Moreover, because the group (T, )(^d, +) is abelian, according to theorem 4.6.1 in <cit.>, it is left amenable and so is ^d, ^d, +^d, T,. In conclusion, according to <ref>, the cell space ^d, E, , 0, v_v ∈^d is right amenable.[d-dimensional Lattice] Let d be a positive integer, let G be the symmetry group of ^d, and letbe the left group action of G on ^d by function application. The group G is isomorphic to the outer semi-direct product ^d ⋊_ψ O, where O is the stabiliser of 0 underand ψ is the group homomorphism O ×^d →^d, (f, v) ↦ f(v). As in <ref>, the cell space ^d, G, , 0, (v, _^d)_v ∈^d is right amenable. CHAPTER: THE GARDEN OF EDEN THEOREMAbstract. We prove the Garden of Eden theorem for big-cellular automata with finite set of states and finite neighbourhood over right-amenable left-homogeneous spaces with finite stabilisers. It states that the global transition function of such an automaton is surjective if and only if it is pre-injective. Pre-Injectivity means that two global configurations that differ at most on a finite subset and have the same image under the global transition function must be identical. The theorem is proven by showing that the global transition function of an automaton as above is surjective if and only if its image has maximal entropy and that its image has maximal entropy if and only if it is pre-injective. Entropy of a subset of global configurations measures the asymptotic growth rate of the number of finite patterns with growing domains that occur in the subset. Remark. Most parts of this chapter appeared in the paper *wacker:garden:2016<cit.> and they generalise parts of chapter 5 of the monograph *ceccherini-silberstein:coornaert:2010<cit.>. Summary. For a right-amenable cell space with finite stabilisers we may choose a right Følner net ℱ = F_i_i ∈ I. The entropy of a subset X of Q^M with respect to ℱ, where Q is a finite set, is, broadly speaking, the asymptotic growth rate of the number of finite patterns with domain F_i that occur in X. For subsets E and E' of GG_0, an E, E'-tiling is a subset T of M such that tE_t ∈ T is pairwise disjoint and tE'_t ∈ T is a cover of M. If for each point t ∈ T not all patterns with domain tE occur in a subset of Q^M, then that subset does not have maximal entropy. The global transition function Δ of a big-cellular automaton with finite set of states and finite neighbourhood over a right-amenable cell space with finite stabilisers, as introduced below, is surjective if and only if its image has maximal entropy. Indeed, if Δ is surjective, then its image is equal to the set of all global configurations, which has maximal entropy. And, if Δ is not surjective, then there is a global configuration that is not in its image; thus, because Δ is continuous and Q^M is compact, where Q^M is equipped with the prodiscrete topology, there is a finite pattern that does not occur in the global configurations of Δ(Q^M); and hence, Δ(Q^M) does not have maximal entropy. The image of the global transition function Δ has maximal entropy if and only if it is pre-injective. Indeed, if Δ(Q^M) does not have maximal entropy, that is, the asymptotic growth rate of finite patterns in Δ(Q^M) is less than the one of Q^M, then there are two distinct finite patterns with the same domain that can be identically extended to global configurations with the same image under Δ and thus Δ is not pre-injective. And, if Δ is not pre-injective, then there are two distinct finite patterns p and p' with the same domain that have the same image under a restriction of Δ; thus, Δ(Q^M) is equal to the image of the set Y of all global configurations in which the pattern p does not occur at the cells of a tiling, which is chosen such that its cells are far apart with respect to the domain of p; and hence, because the entropy of Δ(Y) is less than or equal to the one of Y and the entropy of Y is not maximal, the entropy of Δ(Q^M) is not maximal.The previous two paragraphs establish the Garden of Eden theorem, which states that a global transition function as above is surjective if and only if it is pre-injective. This answers a question posed by Sébastien Moriceau at the end of his paper *moriceau:2011<cit.>. The Garden of Eden theorem for cellular automata over ^2 is a famous theorem by Edward Forrest Moore and John R. Myhill from 1962 and 1963, see the papers *moore:1962<cit.> and *myhill:1963<cit.>.Contents. In <ref> we introduce patterns and blocks, and actions on these. In <ref> we introduce E-interiors, E-closures, and E-boundaries of subsets of M, and characterise right Følner nets using boundaries, which motivates the definition of right Erling nets and tractability. In <ref> we introduce E, E'-tilings of cell spaces, show their existence, and relate them, interiors, and right Erling nets combinatorially. In <ref> we introduce entropies of subsets of Q^M, show that applications of global transition functions to subsets of Q^M do not increase entropy, and show that subsets of Q^M that miss a pattern at each cell of a tiling do not have maximal entropy. In <ref> we prove the Garden of Eden theorem by characterising surjectivity and pre-injectivity by maximality of the entropy of the image. And in <ref> we construct non-degenerated right-amenable left-homogeneous spaces.Preliminary Notions. A left group set is a triple M, G,, where M is a set, G is a group, andis a map from G × M to M, called left group action of G on M, such that G →(M), g ↦ [g ], is a group homomorphism. The actionis transitive if M is non-empty and for each m ∈ M the map m is surjective; and free if for each m ∈ M the map m is injective. For each m ∈ M, the set Gm is the orbit of m, the set G_m = ( m)^-1(m) is the stabiliser of m, and, for each m' ∈ M, the set G_m, m' = ( m)^-1(m') is the transporter of m to m'.A left-homogeneous space is a left group set ℳ = M, G, such thatis transitive. A coordinate system for ℳ is a tuple 𝒦 = m_0, g_m_0, m_m ∈ M, where m_0 ∈ M and for each m ∈ M we have g_m_0, m m_0 = m. The stabiliser G_m_0 is denoted by G_0. The tuple ℛ = ℳ, 𝒦 is a cell space. The set g G_0g ∈ G of left cosets of G_0 in G is denoted by GG_0. The map M × GG_0 → M, (m, g G_0) ↦ g_m_0, m g g_m_0, m^-1 m (= g_m_0, m gm_0) is a right semi-action of GG_0 on M with defect G_0, which means thatm ∈ MmG_0 = m,andm ∈ Mg ∈ Gg_0 ∈ G_0 𝔤' ∈ GG_0 mg ·𝔤' = (mg G_0)g_0 ·𝔤'.It is transitive, which means that the set M is non-empty and for each m ∈ M the map m is surjective; and free, which means that for each m ∈ M the map m is injective; and semi-commutes with , which means thatm ∈ Mg ∈ Gg_0 ∈ G_0 𝔤' ∈ GG_0 (gm) 𝔤' = g(mg_0 ·𝔤').The maps ι M → GG_0, m ↦ G_m_0, m, and m_0 are inverse to each other. Under the identification of M with GG_0 by either of these maps, we have (m, 𝔤) ↦ g_m_0, m𝔤. (See <ref>.)A left-homogeneous space ℳ is right amenable if there is a coordinate system 𝒦 for ℳ and there is a finitely additive probability measure μ on M such that 𝔤∈ GG_0A ⊆ M [](𝔤)_Ainjectiveμ(A 𝔤) = μ(A),in which case the cell space ℛ = ℳ, 𝒦 is called right amenable. When the stabiliser G_0 is finite, that is the case if and only if there is a right Følner net in ℛ indexed by (I, ≤), which is a net F_i_i ∈ I in F ⊆ MF ≠∅, Ffinite such that𝔤∈ GG_0 lim_i ∈ IF_i ∖ (𝔤)^-1(F_i)/F_i = 0.If a net is a right Følner net for one coordinate system, then it is a right Følner net for each coordinate system. In particular, a left-homogeneous space ℳ with finite stabilisers is right amenable if and only if, for each coordinate system 𝒦, the cell space ℳ, 𝒦 is right amenable. (See <ref>.)A cell space ℛ is finitely and symmetrically right generated if there is a finite subset S of GG_0 with G_0 · S ⊆ S and S^-1⊆ S, where S^-1 = g^-1 G_0s ∈ S, g ∈ s, such thatm ∈ Mk ∈_0 s_i_i ∈1, 2, …, k inS ∪ S^-1 [][](m_0s_1)s_2… s_k = m.The uncoloured S-Cayley graph of ℛ is the symmetric and 2 S-regular directed graph 𝒢 = M, (m, ms)m ∈ M, s ∈ S, the S-metric on ℛ is the distanceon 𝒢, and the S-length on ℛ is the map = (m_0, ). For each m ∈ M and each ρ∈, the S-ball of radius ρ centred at m is the set (m, ρ) = m' ∈ M (m, m') ≤ρ, the S-sphere of radius ρ centred at m is the set (m, ρ) = m' ∈ M (m, m') = ρ, the ball (m_0, ρ) is denoted by (ρ), and the sphere (m_0, ρ) by (ρ). (See <ref>.)A semi-cellular automaton is a quadruple 𝒞 = ℛ, Q, N, δ, where ℛ is a cell space; Q, called set of states, is a set; N, called neighbourhood, is a subset of GG_0 such that G_0 · N ⊆ N; and δ, called local transition function, is a map from Q^N to Q. A local configuration is a map ℓ∈ Q^N and a global configuration is a map c ∈ Q^M. The stabiliser G_0 acts on Q^N on the left by ∙ G_0 × Q^N → Q^N, (g_0, ℓ) ↦ [n ↦ℓ(g_0^-1· n)], and the group G acts on Q^M on the left by G × Q^M → Q^M, (g, c) ↦ [m ↦ c(g^-1 m)]. The global transition function of 𝒞 is the map Δ Q^M → Q^M, c ↦ [m ↦δ(n ↦ c(mn))].A subgroup H of G is 𝒦-big if the set g_m_0, m m ∈ M is included in H. A big-cellular automaton is a semi-cellular automaton 𝒞 = ℛ, Q, N, δ such that, for some 𝒦-big subgroup H of G, the local transition function δ is ∙_G_0 ∩ H-invariant, which means that, for each h_0 ∈ G_0 ∩ H, we have δ(h_0 ∙) = δ(). Its global transition function is _H-equivariant, which means that, for each h ∈ H, we have Δ(h ) = h Δ(). Note that each 𝒦-big subgroup of G includes the subgroup of G generated by g_m_0, m m ∈ M and that hence a semi-cellular automaton is a big-cellular automaton if and only if its local transition function is ∙_G_0 ∩g_m_0, m m ∈ M-invariant. (See <ref>.) Context. In this chapter, for each subset A of M, let π_A be the restriction map Q^M → Q^A, c ↦ c_A. § PATTERNS In this section, let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space and let Q be a set. Contents. A pattern is a map from a subset of cells to the set of states (see <ref>), its size is the cardinality of its domain (see <ref>), it is empty if its domain is empty (see <ref>), it is finite and called block if its domain is finite (see <ref>), and restricting its domain yields a subpattern (see <ref>). The group of symmetries acts on the set of all patterns on the left (see <ref>) and a pattern centred at the origin can be shifted to a new centre by sort of acting on the new centre (see <ref>). A pattern occurs in a global configuration if a translation of it coincides with a part of the configuration (see <ref>).Let A be a subset of M and let p be a map from A to Q. The map p is called A-patternpattern@A-patternA-pattern p[symbols]pattern@A-pattern and the set (p) = A is called domain of pdomain (p) of p[symbols]domp@(p).Let p be an A-pattern. The cardinal number p = A is called size of psize p of p[symbols]absp@p. The ∅-pattern is called emptyempty pattern ε[symbols]epsilon@ε. The empty pattern is the only one of size 0. Let u be an F-pattern. It is called finitefinite pattern upattern!finite and F-blockblock@F-blockF-block u if and only if its domain F is finite.The set u ∈ Q^FF ⊆ Mfiniteset Q^* of all blocks of all blocks is denoted by Q^*[symbols]Qstar@Q^*.Let p be an A- and let p' be an A'-pattern. The pattern p is called subpattern of p'subpattern p of p'pattern!sub- if and only if A ⊆ A' and p = p'_A. The group G acts on the set of patterns on the left by G ×⋃_A ⊆ M Q^A→⋃_A ⊆ M Q^A, left group actionof G on ⋃_A ⊆ M Q^A[symbols]arrowrightblack@(g, p)↦[g (p)→ Q,m↦ p(g^-1 m). ] The restriction _G × Q^M → Q^M is the left group action on the set of global configurations that was introduced in <ref> and that we so far also denoted by the symbol . Identify M with GG_0 by ι m ↦ G_m_0, m. The mapM ×⋃_A ⊆ M Q^A→⋃_A ⊆ M Q^A, kind of left semi-actionof M by coordinates on ⋃_A ⊆ M Q^A[symbols]arrowleftblackunderscore@(m, p)↦[m (p)→ Q,ma↦ p(a),]broadly speaking, maps a point m and a pattern p that is centred at m_0 to the corresponding pattern centred at m and, as we see in <ref>, it is a kind of left semi-action of the set of cells by coordinates on the set of patterns.Let p be an A-pattern and let m be an element of M. Then, mp = g_m_0, m p and (mp) = m (p) = g_m_0, m(p).Each global transition function Δ of a big-cellular automaton is -equivariantequivariant inducedrightsemiaction@-equivariant-equivariant, which means thatm ∈ Mc ∈ Q^M Δ(mc) = m Δ(c).Note however that -equivariance is only equivalent to _H-equivariance if the subgroup H of G is equal to the subset g_m_0, m m ∈ M of G, which is in general not a subgroup of G.Identify M with GG_0 by ι m ↦ G_m_0, m, let p be an A-pattern, let c be a M-pattern, and let m be an element of M. The pattern p is said to occur at m in cp occurs at m in c and we write p _m cp _m c[symbols]psquaresubseteqmpprime@p _m p' if and only if mp = c_mA.§ INTERIORS, CLOSURES, AND BOUNDARIES; RIGHT FØLNER NETS AND RIGHT ERLING NETS Interiors, Closures, and BoundariesIn this section, let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space. Contents. In <ref> we introduce E-interiors, E-closures, and (internalexternal) E-boundaries of subsets A of M, where the subset E of GG_0 determines thickness and shape of the subtraction from, addition to, or boundary of A. In <ref> we show essential properties of interiors, closures, and boundaries. In <ref> we define surjective restrictions Δ_X, A^- and Δ_A^- of global transition functions to patterns. In <ref> we show that right Følner nets are those nets whose components are asymptotically invariant under taking finite boundaries. In <ref> we introduce right Erling nets and right tractability, which are weak variants of right Følner nets and right amenability. And in <ref> we show that each finitely and symmetrically right-generated cell space is right tractable. Let A be a subset of M and let E be a subset of GG_0. * The set A^-E = m ∈ MmE ⊆ A[]= ⋂_e ∈ E⋃_a ∈ A ( e)^-1(a)E-interior A^-E of Ais called E-interior of Ainterior of A@E-interior of A.* The setA^+E = m ∈ M(mE) ∩ A ≠∅[]= ⋃_e ∈ E⋃_a ∈ A ( e)^-1(a)E-closure A^+E of Ais called E-closure of Aclosure of A@E-closure of A.* The set_E A = A^+E∖ A^-EE-boundary _E A of Ais called E-boundary of Aboundary of A@E-boundary of A.* The set_E^- A = A ∖ A^-Einternal E-boundary _E^- A of Ais called internal E-boundary of Aboundary of A!internal.* The set_E^+ A = A^+E∖ A external E-boundary _E^+ A of Ais called external E-boundary of Aboundary of A!external.Let ℛ be the cell space G, G, ·, e_G, g_g ∈ G, where G is a group and e_G is its neutral element. Then, G_0 = e_G and = ·. Hence, the notions of E-interior, E-closure, and E-boundary are the same as the ones defined in paragraph 2 of section 5.4 in <cit.>. [Lattice] In the situation of <ref>,let A be the ball (2). The (1, 0)-interior of A is the ball A - (1, 0) = ((-1, 0), 2), which is not included in A; the (0, 0), (1, 0)-interior of A is the set A ∩ (A - (1, 0)) = _^2((-1/2, 0), 3/2) ∩ M, which is included in A on the left; the (-1, 0), (0, 0), (1, 0)-interior of A is the ball (A + (1, 0)) ∩ A ∩ (A - (1, 0)) = (1), which is included in A in the middle; and the E-interior of A, where E = (-1, 0), (0, -1), (0, 0), (0, 1), (1, 0) = (1), is also the ball (1) (see <ref>). The (1, 0)-closure of A is the ball A - (1, 0) = ((-1, 0), 2), which does not include A; the (0, 0), (1, 0)-closure of A is the set A ∪ (A - (1, 0)) = (2) ∪((-1, 0), 2), which includes A on the right; the (-1, 0), (0, 0), (1, 0)-closure of A is the set (A + (1, 0)) ∪ A ∪ (A - (1, 0)) = ((1, 0), 2) ∪(2) ∪((-1, 0), 2), which includes A in the middle; and the E-interior of A is the ball (3) (see <ref>). The (1, 0)-boundary of A is the empty set ∅, although (1, 0) and A are non-empty; the (0, 0), (1, 0)-boundary of A is the set (_^2((-1/2, 0), 5/2) ∖_^2((-1/2, 0), 3/2)) ∩ M, which includes A on the right; the (-1, 0), (0, 0), (1, 0)-boundary of A is the set (((1, 0), 2) ∪(2) ∪((-1, 0), 2)) ∖(1), which includes A in the middle; and the E-boundary of A is the thickened sphere (3) ∖(1) (see <ref>).The above calculations suggest that the notions of E-interior, -closure and -boundary behave best if the set E contains (0, 0) and is invariant under G_0, which we also see in the forthcoming <ref>.In general, for each cell m ∈ M and for each pair (ρ, ϱ) ∈_0 ×_0, we have (m, ρ)^-(ϱ) = (m, ρ - ϱ), (m, ρ)^+(ϱ) = (m, ρ + ϱ), and _(ϱ)(m, ρ) = (m, ρ + ϱ) ∖(m, ρ - ϱ).[Tree]In the situation of <ref>, let A be the ball (2). The a-interior of A is the set A a^-1, the a, b-interior of A is the ball A a^-1∩ A b^-1 = (1), and the a, a^-1- and a, b, a^-1, b^-1-interiors of A are also the ball (1). The a-closure of A is the set A a^-1, the a, b-closure of A is the set A a^-1∪ A b^-1, the a, a^-1-closure of A is the set A a^-1∪ A a, and the a, b, a^-1, b^-1-closure of A is the ball (3). The a-boundary of A is the empty set ∅, the a, b-boundary of A is the symmetric difference A a^-1 A b^-1, the a, a^-1-boundary of A is the symmetric difference A a^-1 A a, and the a, b, a^-1, b^-1-boundary of A is the sphere (3) ∖(1) (see <ref>).The above calculations suggest that the notions of E-interior, -closure and -boundary behave best if the set E is invariant under G_0, which we also see in the forthcoming <ref>.In general, for each cell m ∈ M and for each pair (ρ, ϱ) ∈_0 ×_0, we have (m, ρ)^-(ϱ) = (m, ρ - ϱ), (m, ρ)^+(ϱ) = (m, ρ + ϱ), and _(ϱ)(m, ρ) = (m, ρ + ϱ) ∖(m, ρ - ϱ). [Sphere] In the situation of <ref>, let A be a curved circular disk of radius 3 ρ with the north pole m_0 at its centre, let g be the rotation about an axis a in the (x,y)-plane by ρ radians, let E be the set g_0 g G_0g_0 ∈ G_0, and, for each point m ∈ M, let E_m be the set mE. Because G_0 is the set of rotations about the z-axis and m_0E = g_m_0, m_0 G_0 gm_0 = G_0(gm_0), the set E_m_0 is the boundary of a curved circular disk of radius ρ with the north pole m_0 at its centre. And, for each point m ∈ M, because mE = g_m_0, m E_m_0, the set E_m is the boundary of a curved circular disk of radius ρ with m at its centre. The E-interior of A is the curved circular disk of radius 2 ρ with the north pole m_0 at its centre. The E-closure of A is the curved circular disk of radius 4 ρ with the north pole m_0 at its centre. And the E-boundary of A is the annulus bounded by the boundaries of the E-interior and the E-closure of A.Essential properties of and relations between interiors, closures, and boundaries are given inLet A and A' be two subsets of M, let A_i_i ∈ I be a family of subsets of M, let e be an element of GG_0, and let E and E' be two subsets of GG_0. Furthermore, for each element e' ∈ E', let e' · E be the set g' · ee ∈ E, g' ∈ e', let E' · E be the set ⋃_e' ∈ E' e' · E (= g' · ee ∈ E, e' ∈ E', g' ∈ e'), let E^-1 be the set g^-1 G_0e ∈ E, g ∈ e, and let (E')^-1 be the set (g')^-1 G_0e' ∈ E', g' ∈ e'.* A^-G_0 = A, A^+G_0 = A, and _G_0 A = ∅.* A^-G_0, e = A ∩ ( e)^-1(A), A^+G_0, e = A ∪ ( e)^-1(A), and _G_0, e A = A ∖ ( e)^-1(A) ∪ ( e)^-1(A) ∖ A.* (M ∖ A)^-E = M ∖ A^+E and (M ∖ A)^+E = M ∖ A^-E.*⋃_i ∈ I A_i^-E⊆ (⋃_i ∈ I A_i)^-E and ⋃_i ∈ I A_i^+E = (⋃_i ∈ I A_i)^+E.*(⋂_i ∈ I A_i)^-E = ⋂_i ∈ I A_i^-E and (⋂_i ∈ I A_i)^+E⊆⋂_i ∈ I A_i^+E.*Let E ⊆ E'. Then, A^-E⊇ A^-E', A^+E⊆ A^+E', and _E A ⊆_E' A.*Let A ⊆ A'. Then, A^-E⊆ (A')^-E and A^+E⊆ (A')^+E.*Let G_0 ∈ E. Then, A^-E⊆ A ⊆ A^+E. * Let G_0, A, and E be finite. Then, A^-E, A^+E, and _E A are finite. More precisely, A^-E≤G_0·A and A^+E≤G_0·A·E.*Let g ∈ G and let G_0 · E ⊆ E. Then, gA^-E = (gA)^-E, gA^+E = (gA)^+E, and g _E A = _E (gA).*Let m ∈ M, let G_0 · E ⊆ E, and let ι M → GG_0, m ↦ G_m_0, m. Then, m ι(A^-E) = (m ι(A))^-E, m ι(A^+E) = (m ι(A))^+E, and m ι(_E A) = _E (m ι(A)).*Let G_0 · E ⊆ E and let E^-1⊆ E. Then, AE ⊆ A^+E.* Let G_0 · E ⊆ E. Then, (A^-E)^-E' = A^- E' · E and (A^+E)^+E' = A^+ E' · E.*Let G_0 · E ⊆ E. Then, (A^+E)^-E' = ⋂_e' ∈ E' A^+ e' · E and (A^-E)^+E' = ⋃_e' ∈ E' A^- e' · E. And, if (E')^-1⊆ E, then A ⊆ (A^+E)^-E' and (A^-E)^+E'⊆ A. * Because G_0 = _M, this is a direct consequence of <ref>.* Because ( G_0)^-1(A) = A, this is a direct consequence of <ref>.* For each m ∈ M,m ∈ (M ∖ A)^-EmE ⊆ M ∖ A (mE) ∩ A = ∅ m ∈ M ∖ A^+E.Hence, (M ∖ A)^-E = M ∖ A^+E. Therefore,(M ∖ A)^+E = M ∖ (M ∖ (M ∖ A)^+E)= M ∖ (M ∖ (M ∖ A))^-E= M ∖ A^-E. * For each m ∈ M,m ∈⋃_i ∈ I A_i^-Ei ∈ Im ∈ A_i^-E i ∈ ImE ⊆ A_i mE ⊆⋃_i ∈ I A_i m ∈ (⋃_i ∈ I A_i)^-E.Hence, ⋃_i ∈ I A_i^-E⊆ (⋃_i ∈ I A_i)^-E. Moreover, for each m ∈ M,m ∈⋃_i ∈ I A_i^+Ei ∈ Im ∈ A_i^+E i ∈ I(mE) ∩ A_i ≠∅ (mE) ∩ (⋃_i ∈ I A_i) ≠∅ m ∈ (⋃_i ∈ I A_i)^+E.Therefore, ⋃_i ∈ I A_i^+E = (⋃_i ∈ I A_i)^+E. * According to <ref> and <ref>,(⋂_i ∈ I A_i)^-E = M ∖ (M ∖ (⋂_i ∈ I A_i)^-E)= M ∖ (M ∖ (⋂_i ∈ I A_i))^+E= M ∖ (⋃_i ∈ I M ∖ A_i)^+E= M ∖ (⋃_i ∈ I (M ∖ A_i)^+E)= M ∖ (⋃_i ∈ I M ∖ A_i^-E)= M ∖ (M ∖ (⋂_i ∈ I A_i^-E))= ⋂_i ∈ I A_i^-Eand(⋂_i ∈ I A_i)^+E = M ∖ (M ∖ (⋂_i ∈ I A_i)^+E)= M ∖ (⋃_i ∈ I M ∖ A_i)^-E⊆ M ∖ (⋃_i ∈ I M ∖ A_i^+E)= M ∖ (M ∖ (⋂_i ∈ I A_i^+E))= ⋂_i ∈ I A_i^+E. * This is a direct consequence of <ref>.* This is a direct consequence of <ref>.* This is a direct consequence of <ref>.* For each e ∈ E, according to <ref>,⋃_a ∈ A ( e)^-1(a) = ( e)^-1(A)≤G_0·A.Hence, because A^-E = ⋂_e ∈ E⋃_a ∈ A ( e)^-1(a), we have A^-E≤G_0·A < ∞. And, because A^+E = ⋃_e ∈ E⋃_a ∈ A ( e)^-1(a), we have A^+E≤E·G_0·A < ∞.And, because _E A ⊆ A^+E, we also have _E A < ∞.* Let m ∈ M. Becausesemi-commutes with , there is a g_0 ∈ G_0 such that (g^-1 m)E = g^-1 (mg_0 · E). And, because G_0 · E ⊆ E, we have g_0 · E ⊆ E and g_0^-1· E ⊆ E; hence E = g_0 g_0^-1· E = g_0 · (g_0^-1· E) ⊆ g_0 · E; thus g_0 · E = E. Therefore, (g^-1 m)E = g^-1 (mE). Thus, for each m ∈ M,m ∈ gA^-Em' ∈ A^-E gm' = m g^-1 m ∈ A^-E (g^-1 m)E ⊆ A g^-1 (mE) ⊆ A mE ⊆ gA m ∈ (gA)^-E.In conclusion, gA^-E = (gA)^-E. Moreover, for each m ∈ M,m ∈ gA^+Eg^-1 m ∈ A^+E ((g^-1 m)E) ∩ A ≠∅ (g^-1 (mE)) ∩ A ≠∅ (mE) ∩ (gA) ≠∅ m ∈ (gA)^+E.In conclusion, gA^+E = (gA)^+E. Ultimately,g _E A= g(A^+E∖ A^-E)= (gA^+E) ∖ (gA^-E)= (gA)^+E∖ (gA)^-E= _E (gA). * According to<ref>,m ι(A^-E)= g_m_0, m A^-E= (g_m_0, m A)^-E= (m ι(A))^-E,andm ι(A^+E)= g_m_0, m A^+E= (g_m_0, m A)^+E= (m ι(A))^+E,andm ι(_E A)= g_m_0, m_E A= _E (g_m_0, m A)= _E (m ι(A)). * Let m ∈ A and let e ∈ E. Then, there is a g ∈ G such that g G_0 = e. Hence, becauseis a semi-action with defect G_0, there is a g_0 ∈ G_0 such that, for each 𝔤∈ GG_0, we have (me)g_0 ·𝔤 = mg ·𝔤.Put e' = g_0 · g^-1 G_0. Then, because G_0 · E ⊆ E and E^-1⊆ E, we have e' ∈ E. Moreover, (me)e' = mg g^-1 G_0 = m. Therefore, m ∈ (me)E and hence ((me)E) ∩ A ≠∅. Thus, me ∈ A^+E. In conclusion, AE ⊆ A^+E.* For each m ∈ M, according to <ref>, we have (mE')E = mE' · E. Therefore, for each m ∈ M,m ∈ (A^-E)^-E'mE' ⊆ A^-E (mE')E ⊆ A mE' · E ⊆ A m ∈ A^- E' · E.In conclusion, (A^-E)^-E' = A^- E' · E. Moreover, for each m ∈ M,m ∈ (A^+E)^+E'(mE') ∩ A^+E≠∅ e' ∈ E'me' ∈ A^+E e' ∈ E'((me')E) ∩ A ≠∅ ((mE')E) ∩ A ≠∅ (mE' · E) ∩ A ≠∅ m ∈ A^+ E' · E.In conclusion, (A^+E)^+E' = A^+ E' · E.* For each m ∈ M and each e' ∈ E', according to <ref>, we have (me')E = me' · E. Therefore, for each m ∈ M,m ∈ (A^+E)^-E'mE' ⊆ A^+E e' ∈ E'me' ∈ A^+E e' ∈ E'((me')E) ∩ A ≠∅ e' ∈ E'(me' · E) ∩ A ≠∅ e' ∈ E'm ∈ A^+ e' · E m ∈⋂_e' ∈ E' A^+ e' · E.In conclusion, (A^+E)^-E' = ⋂_e' ∈ E' A^+ e' · E. Moreover, for each m ∈ M,m ∈ (A^-E)^+E'(mE') ∩ A^-E≠∅ e' ∈ E'me' ∈ A^-E e' ∈ E'(me')E ⊆ A e' ∈ E'me' · E ⊆ A e' ∈ E'm ∈ A^- e' · E m ∈⋃_e' ∈ E' A^- e' · E.In conclusion, (A^-E)^+E' = ⋃_e' ∈ E' A^- e' · E.From now on, let (E')^-1⊆ E. Then, for each e' ∈ E', we have G_0 ∈ e' · E and hence, according to <ref>, we have A ⊆ A^+ e' · E and A^- e' · E⊆ A. In conclusion, A ⊆ (A^+E)^-E' and (A^-E)^+E'⊆ A. The restriction Δ_X, A^- of Δ given in <ref> is well-defined according to the next lemma, which itself holds due to the locality of Δ. Let 𝒞 = ℛ, Q, N, δ be a semi-cellular automaton, let Δ be the global transition function of 𝒞, let c and c' be two global configurations of 𝒞, and let A be a subset of M. *If c_A = c'_A, then Δ(c)_A^-N = Δ(c')_A^-N.*If c_M ∖ A = c'_M ∖ A, then Δ(c)_M ∖ A^+N = Δ(c')_M ∖ A^+N.* If N^-1⊆ N and c_A^+N = c'_A^+N, then Δ(c)_A = Δ(c')_A. * Let c_A = c'_A. Then, for each m ∈ A^-N, we have mN ⊆ A and hence Δ(c)(m) = Δ(c')(m). In conclusion, Δ(c)_A^-N = Δ(c')_A^-N.* This is a direct consequence of <ref> and <ref> of <ref>.* Let N^-1⊆ N and let c_A^+N = c'_A^+N. Then, for each m ∈ A, according to <ref> of <ref>, we have mN ⊆ A^+N and hence Δ(c)(m) = Δ(c')(m). In conclusion, Δ(c)_A = Δ(c')_A. Let 𝒞 = ℛ, Q, N, δ be a semi-cellular automaton, let Δ be the global transition function of 𝒞, let X be a subset of Q^M, and let A be a subset of M. The mapΔ_X, A^- π_A(X)→π_A^-N(Δ(X)), map Δ_X, A^-[symbols]DeltaXAminus@Δ_X, A^-p↦Δ(c)_A^-N,wherec ∈ Xsuch thatc_A = p,is surjective. The map Δ_Q^M, A^- is denoted by Δ_A^-map Δ_A^-[symbols]DeltaAminus@Δ_A^-. Let p' ∈π_A^-N(Δ(X)). Then, there is a c' ∈Δ(X) such that c'_A^-N = p'. Moreover, there is a c ∈ X such that Δ(c) = c'. Put p = c_A ∈π_A(X). Then, Δ_X, A^-(p) = Δ(c)_A^-N = c'_A^-N = p'. Hence, Δ_X, A^- is surjective. The restrictions Δ_X, A^- of global transition functions Δ of big-cellular automata are _H-equivariant in the sense given inLet H be a 𝒦-big subgroup of G, let 𝒞 = ℛ, Q, N, δ be a semi-cellular automaton such that δ is ∙_H_0-invariant, let Δ be the global transition function of 𝒞, let X be a subset of Q^M, and let A be a subset of M. Then,h ∈ Hp ∈π_A(X) Δ_hX, hA^-(hp) = h Δ_X, A^-(p). Let h ∈ H and let p ∈π_A(X). Then, the domain of hp is hA and we have hp ∈ h π_A(X) = π_hA(hX). Hence, the term Δ_hX, hA^-(hp) is well-defined. Moreover, by definition of p, there is a c ∈ X such that c_A = p, and hence (hc)_hA = hp. Therefore, by definition of Δ_hX, hA^- and of Δ_X, A^-, because Δ is _H-equivariant, we have Δ_hX, hA^-(hp) = Δ(hc)_hA^-N = h Δ(c)_A^-N = h Δ_X, A^-(p). In the situation of <ref>, let M be identified with GG_0 by ι m ↦ G_m_0, m. Then,m ∈ Mp ∈π_A(X) Δ_mX, mA^-(mp) = m Δ_X, A^-(p). This is a direct consequence of <ref>, because g_m_0, m∈ H, m= g_m_0, m, and m= g_m_0, m.A net of non-empty and finite subsets of M is a right Følner net if and only if these subsets are asymptotically invariant under the right semi-action induced by , which broadly speaking means that these subsets are asymptotically invariant under small perturbations. If the stabiliser of the origin underis finite, that is the case if and only if those subsets are asymptotically invariant under taking finite boundaries, in other words, if and only if the finite boundaries of the subsets grow much slower than the subsets themselves. This is shown inLet G_0 be finite and let F_i_i ∈ I be a net in F ⊆ MF ≠∅, Ffinite indexed by (I, ≤). The net F_i_i ∈ I is a right Følner net in ℛ if and only if E ⊆ GG_0finitelim_i ∈ I_E F_i/F_i = 0. First, let F_i_i ∈ I be a right Følner net in ℛ. Furthermore, let E ⊆ GG_0 be finite. Moreover, let i ∈ I. For each e ∈ E and each e' ∈ E, put A_i,e,e' = ( e)^-1(F_i) ∖ ( e')^-1(F_i). For each 𝔤∈ GG_0, put B_i,𝔤 = F_i ∖ (𝔤)^-1(F_i).According to <ref>, _E F_i = []⋃_e ∈ E ( e)^-1(F_i)∖[]⋂_e' ∈ E ( e')^-1(F_i)= ⋃_e, e' ∈ E ( e)^-1(F_i) ∖ ( e')^-1(F_i)= ⋃_e, e' ∈ E A_i,e,e'.Hence, _E F_i≤∑_e, e' ∈ EA_i,e,e'.According to <ref>, we have A_i,e,e'≤G_0^2 ·max_g ∈ eB_i, g^-1· e'. Put E' = g^-1· e'e, e' ∈ E, g ∈ e. Because E is finite, G_0 is finite, and, for each e ∈ E, we have e = G_0, the set E' is finite. Therefore, _E F_i/F_i ≤1/F_i∑_e, e' ∈ EA_i,e,e'≤G_0^2/F_i∑_e, e' ∈ Emax_g ∈ eB_i, g^-1· e'≤G_0^2 ·E^2/F_imax_e' ∈ E'B_i, e'≤G_0^2 ·E^2 ·max_e' ∈ E'F_i ∖ ( e')^-1(F_i)/F_ii ∈ I→ 0.In conclusion, lim_i ∈ I_E F_i/F_i = 0.Secondly, for each finite E ⊆ GG_0, let lim_i ∈ I_E F_i/F_i = 0. Furthermore, let i ∈ I, let e ∈ GG_0, and put E = G_0, e. According to <ref> of <ref>, we have F_i ∖ ( e)^-1(F_i) ⊆_E F_i. Therefore,F_i ∖ ( e)^-1(F_i)/F_i≤_E F_i/F_ii ∈ I→ 0.In conclusion, F_i_i ∈ I is a right Følner net in ℛ. [Lattice] In the situation of <ref>, the sequence (ρ)_ρ∈_+ grows linearly in size, more precisely, it grows like 4 ρ_ρ∈_+; the sequence (ρ)_ρ∈_0 grows polynomially in size, more precisely, it grows like ∑_ϱ = 0^ρ(ϱ)_ρ∈_0 = 2ρ(ρ + 1) + 1_ρ∈_0; the sequence _(1)(ρ)_ρ∈_+ grows linearly in size, more precisely, it grows like (ρ + 1) ∖(ρ - 1)_ρ∈_+ = 4 (2ρ + 1)_ρ∈_+; in general, for each non-negative integer ϱ, we have _(ϱ)(ρ)_ρ∈_+ = 4ϱ (2ρ + 1)_ρ∈_≥ϱ. It follows thatϱ∈_0 lim_ρ→∞_(ϱ)(ρ)/(ρ) = 0.Hence, according to <ref>, the sequence (ρ)_ρ∈_0 is a right Følner net in ℛ, which was also shown in <ref>.[Tree] In the situation of <ref>, the sequence (ρ)_ρ∈_+ grows exponentially in size, more precisely, it grows like 3^ρ + 3^ρ - 1_ρ∈_+; the sequence (ρ)_ρ∈_0 also grows exponentially in size, more precisely, it grows like ∑_ϱ = 0^ρ(ϱ)_ρ∈_0 = 2 · 3^ρ - 1_ρ∈_0; for each non-negative integer ϱ, the sequence _(ϱ)(ρ)_ρ∈_≥ϱ also grows exponentially in size, more precisely, it grows like (ρ + ϱ) ∖(ρ - ϱ)_ρ∈_≥ϱ = 2(3^ρ + ϱ - 3^ρ - ϱ)_ρ∈_≥ϱ. It follows thatϱ∈_0 lim_ρ→∞_(ϱ)(ρ)/(ρ) = 3^ϱ - 3^-ϱ≠ 0.Hence, according to <ref>, each subsequence of (ρ)_ρ∈_0 is not a right Følner net in ℛ, which was also shown in <ref>. Actually, because we deduced in <ref> that the cell space ℛ is not right amenable, there is no right Følner net in ℛ.In <ref> we use a net of non-empty and finite subsets of M to define the entropy of a subset of global configurations. If that net is a right Følner net, then the global transition function of a big-cellular automaton is surjective if and only if the entropy of its image is maximal and the entropy of its image is maximal if and only if it is pre-injective. For the first equivalence it suffices that the net has a weaker property, which we define below; for the second equivalence though that weaker property does not suffice. Let F_i_i ∈ I be a net in F ⊆ MF ≠∅, Ffinite indexed by (I, ≤). It is called right Erling net in ℛ indexed by (I, ≤)right Erling net in ℛ indexed by (I, ≤)Erling net@right Erling netnet!Erling if and only ifE ⊆ GG_0finitelim sup_i ∈ I_E^- F_i/F_i < 1. Regardless of whether F_i_i ∈ I is an Erling net or not, the above limit superior is always less than or equal to 1. The cell space ℛ is called right tractableright tractabletractable right@right tractable if and only if there is a right Erling net in ℛ.Because each right Følner net is a right Erling net, each right-amenable cell space with finite stabilisers is right tractable. The notions of being finitely and symmetrically right generated, of distance, and of balls will be introduced in <ref>. A quick preview was given in the paragraph preliminary notions at the beginning of this chapter. A finitely and symmetrically right-generated cell space is right tractable, which is shown inLet ℛ be finitely and symmetrically right generated. It is right tractable. And, for each symmetric and finite right-generating set S of ℛ such that G_0 ∈ S, the sequence (ρ)_ρ∈_0 is a right Erling net in ℛ. Let S be a symmetric and finite right-generating set of ℛ with G_0 ∈ S, let F_i_i ∈ I be the sequence (ρ)_ρ∈_0, and let E be a finite subset of GG_0. There is a non-negative integer ρ such that m_0E ⊆(ρ) and there is a subset E' of GG_0 such that (ρ) = m_0E'. Note that, becauseis free, we have E ⊆ E'. Let i ∈ I such that i ≥ρ + 1. We have _E^- F_i ⊆_E'^- F_i and _E'^- F_i = F_i ∖ F_i - ρ. Moreover, because G_0 ∈ S, for each element m ∈ F_i ∖ F_i - ρ, there is an element m' ∈ F_i - ρ∖ F_i - ρ - 1 and there is a family s_k_k ∈1, 2, …, ρ in S such that (((m's_1)s_2) …)s_ρ = m. Thus, _E'^- F_i = F_i ∖ F_i - ρ≤S^ρ·F_i - ρ∖ F_i - ρ - 1≤S^ρ·F_i - ρ.Furthermore, F_i = F_i ∖ F_i - ρ + F_i - ρ = _E'^- F_i + F_i - ρ. Hence,F_i/_E^- F_i≥F_i/_E'^- F_i =1 + F_i - ρ/_E'^- F_i≥ 1 + F_i - ρ/S^ρ·F_i - ρ =1 + S^-ρ.Therefore,lim sup_i ∈ I_E^- F_i/F_i≤1/1 + S^-ρ < 1. In conclusion, F_i_i ∈ I is a right Erling net and hence ℛ is right tractable.In the proof of <ref>, we do not need the right-generating set to be symmetric. It is assumed merely for consistency, because we define balls solely for such generating sets. The reason is that only for those is the distance on Cayley graphs a metric and balls behave nicely. [Finitely]Let G be a finitely generated group, let M be the vertices of a Cayley graph of G, and letbe the left group action of G on M by left multiplication. The cell space M, G, , e_G, m_m ∈ M is finitely and symmetrically right generated and hence right tractable.[Lattice]In the situation of <ref>, for each non-negative integer ρ and each non-negative integer ϱ such that ρ≥ϱ, the interior (ϱ)-boundary of (ρ) is equal to (ρ) ∖(ρ - ϱ) and its cardinality is equal to 2ϱ (2ρ - ϱ + 1). Hence,ϱ∈_0 lim sup_ρ→∞_(ϱ)^- (ρ)/(ρ) = 0.Therefore, the sequence (ρ)_ρ∈_0 is a right Erling net in ℛ and hence the cell space ℛ is right tractable.[Tree]In the situation of <ref>, for each non-negative integer ρ and each non-negative integer ϱ such that ρ≥ϱ, the interior (ϱ)-boundary of (ρ) is equal to (ρ) ∖(ρ - ϱ) and its cardinality is equal to 2 · 3^ρ (1 - 3^-ϱ). Hence,ϱ∈_0 lim sup_ρ→∞_(ϱ)^- (ρ)/(ρ) = 1 - 3^-ϱ.Therefore, the sequence (ρ)_ρ∈_0 is a right Erling net in ℛ and hence the cell space ℛ is right tractable. However, as we have seen in <ref>, that sequence is not a right Følner net and that cell space is not right amenable. § TILINGSIn this section, let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space. Contents. In <ref> we introduce the notion of E,E'-tilings. In <ref> we show using Zorn's lemma that, for each non-empty subset E of GG_0, there is an E, E'-tiling. And in lemma <ref> we show that, for each right Erling net F_i_i ∈ I and each E, E'-tiling with finite sets E and E', the net T ∩ F_i^-E_i ∈ I is asymptotically not less than F_i_i ∈ I.Let M_i_i ∈ I be a family of subsets of M. The family M_i_i ∈ I is called * pairwise disjointpairwise disjoint!familypairwise disjoint if and only ifi ∈ Ii' ∈ I(i ≠ i'M_i ∩ M_i' = ∅); * cover of Mcover of M!familycover of M if and only if ⋃_i ∈ I M_i = M;* partition of Mpartition of M!familypartition of M if and only if it is pairwise disjoint and a cover of M.Let T be a subset of M, and let E and E' be two subsets of GG_0. The set T is called E, E'-tiling of ℛtiling of R@E, E'-tiling of ℛE, E'-tiling of ℛ if and only if the family tE_t ∈ T is pairwise disjoint and the family tE'_t ∈ T is a cover of M. The set T is an E, E-tiling of ℛ if and only if the family tE_t ∈ T is a partition of M.Let T be an E, E'-tiling of ℛ. Because M is non-empty and tE'_t ∈ T is a cover of M, the set E' is non-empty.Let T be an E, E'-tiling of ℛ. For each subset F of E and each superset F' of E' with F' ⊆ GG_0, the set T is an F, F'-tiling of ℛ. In particular, the set T is an E, E ∪ E'-tiling of ℛ. In the situation of <ref>, the notion of E, E'-tiling is the same as the one defined in paragraph 2 of section 5.6 in <cit.>. [Lattice] In the situation of <ref>, let E be the ball (1), let E' be the set m - m'm, m' ∈ E, which is the ball (2), and let T be the set (z_1, z_2) ∈ M(z_1, z_2) ∈ 2^2, z_1 + z_2 ∈ 4. The set T is an E, E'-tiling of ℛ (see <ref>). [Tree]In the situation of <ref>, recall thatis the map (m_0, ), let E be the ball (1), let E' be the set e (e')^-1 e, e' ∈ E, which is the ball (2), and let T be the smallest subset of M such that m_0 ∈ T and, for each element t ∈ T, each element x ∈a, b, a^-1, b^-1, and each element y ∈a, b, a^-1, b^-1, we have t x y^2 ∈ T if and only if t x y^2 = t x + 2 and (t x y^2, t) = 3, in other words, if and only if t = m_0 and y ≠ x^-1, or t ≠ m_0, the last symbol of the reduced word that represents t is not x^-1, and y ≠ x^-1, or t ≠ m_0, the last symbol is x^-1, and y ∉x, x^-1. The set T is an E, E-tiling of ℛ, in particular, because E ⊆ E', it is an E, E'-tiling of ℛ (see <ref>).The set m a^3z m ∈ M, z ∈, m a^± 1 = m + 1 is a a^-1, e_F_2, a, a^-1, e_F_2, a-tiling of ℛ (see <ref>). Note that, for each cell m ∈ M, we have m a^± 1 = m + 1 if and only if the last symbol of the reduced word that represents m is neither a nor a^-1 but b or b^-1, and that the set m a^3z z ∈ contains every third element of the set m a^zz ∈, which, in <ref>, is a horizontal bi-infinite path in the a, b, a^-1, b^-1-Cayley graph of F_2. [Sphere]In the situation of <ref>, let E' be the set g (g')^-1 G_0e, e' ∈ E, g ∈ e, g' ∈ e' (= g_0 g g_0' g^-1 G_0g_0, g_0' ∈ G_0) and, for each point m ∈ M, let E'_m = mE'. Because g^-1 is the rotation about the axis a by -ρ radians, the set G_0 g^-1 m_0 is equal to E_m_0 and the set g G_0 g^-1 m_0 is equal to E_gm_0. Because m_0E' = g_m_0, m_0 G_0 g G_0 g^-1 m_0 = G_0(g G_0 g^-1 m_0) = G_0E_gm_0, the set E'_m_0 is the curved circular disk of radius 2 ρ with the north pole m_0 at its centre. And, for each point m ∈ M, because mE' = g_m_0, m E'_m_0, the set E'_m is the curved circular disk of radius 2 ρ with m at its centre (see <ref>).If the radius ρ = π / 2, then the circle E_m_0 is the equator and the curved circular disk E'_m_0 has radius π and is thus the sphere M, and hence the set T = m_0 is an E, E'-tiling of ℛ (see <ref>);if the radius ρ = π / 4, then the curved circular disks E'_m_0 and E'_S, where S is the south pole, have radii π / 2, thus they are hemispheres, and hence the set T = m_0, S is an E, E'-tiling of ℛ (see <ref>);if the radius ρ = π / 8, then the curved circular disks E'_m_0 and E'_S have radii π / 4, and it can be shown with spherical geometry that the set T — consisting of the north pole m_0, the south pole S, four equidistant points m_1, m_2, m_3, and m_4 on the equator, and the circumcentres c_1, c_2, …, c_8 of the 8 smallest spherical triangles with one vertex from m_0, S and two vertices from m_1, m_2, m_3, m_4 — is an E, E'-tiling of ℛ (see <ref>).Tilings exist as shown in Let E be a non-empty subset of GG_0. There is an E, E'-tiling of ℛ, where E' = g (g')^-1 G_0e, e' ∈ E, g ∈ e, g' ∈ e'.Let𝒮 = S ⊆ M sE_s ∈ S is pairwise disjoint.Because m_0∈𝒮, the set 𝒮 is non-empty. Moreover, it is preordered by inclusion.Let 𝒞 be a chain in 𝒮, ⊆. Then, ⋃_S ∈𝒞 S is an element of 𝒮 and an upper bound of 𝒞. According to Zorn's <ref>, there is a maximal element T in 𝒮. By definition of 𝒮, the family tE_t ∈ T is pairwise disjoint. Let m ∈ M. Because T is maximal and mE is non-empty, there is a t ∈ T such that (tE) ∩ (mE) ≠∅. Hence, there are e, e' ∈ E such that te = me'. According to <ref>, there is a g' ∈ e' such that (me')(g')^-1 G_0 = m, and there is a g ∈ e such that (te)(g')^-1 G_0 = tg (g')^-1 G_0. Therefore, m = tg (g')^-1 G_0 (see <ref>). Because g (g')^-1 G_0 ∈ E', we have m ∈ tE'. Thus, tE'_t ∈ T is a cover of M. In conclusion, T is an E, E'-tiling of ℛ. Tilings have asymptotically many points in common with right Erling nets as shown in Let G_0 be finite, let F_i_i ∈ I be a right Erling net in ℛ indexed by (I, ≤), let E and E' be two finite subsets of GG_0, and let T be an E, E'-tiling of ℛ. There is a positive real number ε∈_> 0 and there is an index i_0 ∈ I such that, for each index i ∈ I with i ≥ i_0, we have T ∩ F_i^-E≥εF_i. By the definition of internal boundaries, we have F_i - _E' · E^- F_i≤F_i^- E' · E. And, for nice E and E', we have F_i^- E' · E = (F_i^-E)^-E'. And, because M = TE', we have (F_i^-E)^-E'≤T ∩ F_i^-E·E'. Hence,T ∩ F_i^-E/F_i≥1/E'·*1 - _E' · E^- F_i/F_i.For great enough indices i ∈ I, the right side is bounded below away from zero.According to <ref>, we may suppose, without loss of generality, that E ⊆ E', G_0 · E' ⊆ E', and that (E')^-1⊆ E', where (E')^-1 = (g')^-1 G_0e' ∈ E', g' ∈ e'. Let i ∈ I. PutT_i = T ∩ F_i^-E = t ∈ TtE ⊆ F_iand putT_i' = T ∩ ((F_i^-E)^-E')^+E' = t ∈ T(tE') ∩ (F_i^-E)^-E'≠∅(see <ref>).Because the family tE'_t ∈ T is a cover of M,(F_i^-E)^-E' = M ∩ (F_i^-E)^-E' = ⋃_t ∈ T (tE') ∩ (F_i^-E)^-E'.And, for each element t ∈ T ∖ T_i', we have (tE') ∩ (F_i^-E)^-E' = ∅. Hence,(F_i^-E)^-E' = ⋃_t ∈ T_i' (tE') ∩ (F_i^-E)^-E'.And, for each element t ∈ T_i', we have (tE') ∩ (F_i^-E)^-E'⊆ tE'. Thus,(F_i^-E)^-E'⊆⋃_t ∈ T_i' tE'.And, according to <ref> of <ref>, we have ((F_i^-E)^-E')^+E'⊆ F_i^-E and hence T_i' ⊆ T_i. Therefore,(F_i^-E)^-E'⊆⋃_t ∈ T_i tE'.And, becauseis free, for each t ∈ T_i, we have tE' = E'. Hence,(F_i^-E)^-E'≤T_i·E'.And, according to Items <ref> and <ref> of <ref>, we have (F_i^-E)^-E'⊇ (F_i^- G_0 · E)^-E' = F_i^- E”, where E” = E' · (G_0 · E) = g' · (g_0 · e)e ∈ E, g_0 ∈ G_0, e' ∈ E', g' ∈ e'. And, because _E”^- F_i = F_i ∖ (F_i^-E”∩ F_i),F_i^-E”≥F_i^-E”∩ F_i=F_i - _E”^- F_i.Hence,F_i - _E”^- F_i≤T_i·E'.Therefore,T_i/F_i≥1/E'·*1 - _E”^- F_i/F_i.Because F_i_i ∈ I is a right Erling net, there is a real number ξ∈0, 1 and there is an index i_0 ∈ I such thati ∈ I *i ≥ i_0 _E”^- F_i/F_i≤ξ.Put ε = (1/E”) · (1 - ξ). Then, for each i ∈ I with i ≥ i_0, we have T_i/F_i≥ε. A tiling of a right-amenable and hence right-tractable cell space is given in[Lattice] In the situation of <ref>, for each non-negative integer ρ, one can see that T ∩(ρ) = T ∩(4 ρ/4) = (2 ρ/4 + 1)^2 and, for each positive integer ρ, recall that (ρ)^-(1) = (ρ - 1). Let ρ be a positive integer. The integer ϱ = 4 ρ/4 is the multiple of 4 such that ρ≤ϱ < ρ + 4. Thus, ϱ - 1 is the greatest integer such that (ϱ - 1)/4 = (ρ - 1)/4, in particular, T ∩(ρ - 1) = T ∩(ϱ - 1). Hence, because (ρ) ⊆(ϱ),T ∩(ρ - 1)/(ρ)≥T ∩(ϱ - 1)/(ϱ). Moreover, one can show that the sequence*T ∩(ς - 1)/(ς)_ς∈ 4_+is increasing (and converges to 1/8). Therefore, because ϱ∈ 4_+,T ∩(ϱ - 1)/(ϱ)≥T ∩(4 - 1)/(4) =1/41.In conclusion, T ∩(ρ)^-(1)≥ (1/41) ·(ρ) (actually, for each real number ε∈0, 1/8, there is an index ρ_0 ∈_0 such that, for each index ρ∈_0 with ρ≥ρ_0, we have T ∩(ρ)^-(1)≥ε·(ρ)). A tiling of a right-tractable but not right-amenable cell space is given in[Tree]In the situation of <ref>, by the construction of T, we have m_0 ∈ T; moreover, there are four pairwise distinct elements x_m_0, 1, x_m_0, 2, x_m_0, 3, and x_m_0, 4∈a, b, a^-1, b^-1 and, for each index j ∈1, 2, 3, 4, there are three pairwise distinct elements y_m_0, 1, y_m_0, 2, and y_m_0, 3∈a, b, a^-1, b^-1 such that x_m_0, j y_m_0, k^2 ∈ T and x_m_0, j y_m_0, k^2 = x_t, j + 2 (note that x_t, j + 2 = 3); furthermore, for each element t ∈ T ∖m_0, there are three pairwise distinct elements x_t, 1, x_t, 2, and x_t, 3∈a, b, a^-1, b^-1 such that t x_t, j = t + 1, for j ∈1, 2, 3, and, for each index j ∈1, 2, 3, there are three pairwise distinct elements y_t, 1, y_t, 2, and y_t, 3∈a, b, a^-1, b^-1 such that t x_t, j y_t, k^2 ∈ T and t x_t, j y_t, k^2 = t x_t, j + 2 (note that t x_t, j + 2 = t + 3); and, there is an element x_t, 4∈a, b, a^-1, b^-1∖x_t, 1, x_t, 2, x_t, 3 such that t x_t, 4 = t - 1, and there are two distinct elements y_t, 4 and y_t, 4' ∈a, b, a^-1, b^-1 such that t x_t, 4 y_t, 4^2, t x_t, 4 (y_t, 4')^2 ∈ T and t x_t, 4 y_t, 4^2 as well as t x_t, 4 (y_t, 4')^2 is equal to t x_t, 4 + 2 (note that t x_t, 4 + 2 = t + 1); and, the elements m_0, x_m_0, j y_m_0, k^2, for j ∈1, 2, 3, 4 and k ∈1, 2, 3, t x_t, j y_t, k^2, t x_t, 4 y_t, 4^2, t x_t, 4 (y_t, 4')^2, for t ∈ T ∖m_0 and j, k ∈1, 2, 3, are pairwise distinct and are the only elements of T.Therefore, broadly speaking, the origin m_0 contributes 4 · 3 = 12 elements to T ∩(3); for each integer ρ∈_≥ 4, each element t ∈ T ∩(ρ - 3) contributes 3 · 3 = 9 elements to T ∩(ρ); each element t ∈ T ∩(ρ - 1) contributes 2 elements to T ∩(ρ); and, there are no other elements in T ∩(ρ). And, because (ρ) = (ρ - 1) (ρ), the set T ∩(ρ) contains aside from the elements of T ∩(ρ - 1) only the elements of T ∩(ρ). For each non-negative integer ρ, let σ_ρ = T ∩(ρ) and let β_ρ = T ∩(ρ). Then, σ_0 = 1, σ_1 = 0, σ_2 = 0, σ_3 = 12, and, for each integer ρ∈_≥ 4, we have σ_ρ = 9 σ_ρ - 3 + 2 σ_ρ - 1. Moreover, β_0 = 1 and, for each positive integer ρ∈_+, we have β_ρ = β_ρ - 1 + σ_ρ. One can use mathematical software to determine a closed-form expression for β_ρ and show that the sequence β_ρ - 1/(ρ)_ρ∈_+converges to 1/15 and hence is eventually bounded from below by, say, 1/15 - 1/30 = 1/30. Hence, there is an index ρ_0 ∈_0 such that, for each index ρ∈_0 with ρ≥ρ_0, we have T ∩(ρ)^-(1)≥ (1/30) ·(ρ). § ENTROPIESIn this section, let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space, let 𝒞 = ℛ, Q, N, δ be a semi-cellular automaton, and let Δ be the global transition function of 𝒞 such that the stabiliser G_0 of m_0 under , the set Q of states, and the neighbourhood N are finite, and the set Q is non-empty.Contents. In <ref> we introduce the entropy of a subset X of Q^M with respect to a net F_i_i ∈ I of non-empty and finite subsets of M, which is the asymptotic growth rate of the number of finite patterns with domain F_i that occur in X. In <ref> we show that Q^M has entropy logQ and that entropy is non-decreasing. In <ref> we show that applications of global transition functions of semi-cellular automata on subsets of Q^M do not increase entropy. And in <ref> we show that if for each point t of an E,E'-tiling not all patterns with domain tE occur in a subset of Q^M, then that subset has less entropy than Q^M.Let X be a subset of Q^M and let ℱ = F_i_i ∈ I be a net in F ⊆ MF ≠∅, Ffinite. The non-negative real number or negative infinity _ℱ(X) = lim sup_i ∈ Ilogπ_F_i(X)/F_ientropy _ℱ(X) of X with respect to ℱis called entropy of X with respect to ℱ. In the situation of <ref>, the notion of entropy is the same as the one defined in definition 5.7.1 in <cit.>. As already said, the entropy of X with respect to the net ℱ in ℛ is the asymptotic growth rate of the number of finite patterns with domain F_i that occur in X. In more precise terms but still hand-wavingly, this means that 10^F_i·_ℱ(X)_i ∈ I∼π_F_i(X)_i ∈ I,where ∼ is the binary relation, read asymptotic to, given byr_i_i ∈ Ir_i'_i ∈ I[]r_i_i ∈ I∼r_i'_i ∈ Ilim_i ∈ Ir_i/r_i' = 1. Let ℱ = F_i_i ∈ I be a net in F ⊆ MF ≠∅, Ffinite. Then, *_ℱ(Q^M) = logQ;*X ⊆ Q^MX' ⊆ Q^M []X ⊆ X' _ℱ(X) ≤_ℱ(X');*X ⊆ Q^M _ℱ(X) ≤logQ. * For each i ∈ I, we have π_F_i(Q^M) = Q^F_i and hencelogπ_F_i(Q^M)/F_i= logQ^F_i/F_i = F_i·logQ/F_i = logQ.In conclusion, _ℱ(Q^M) = logQ.* Let X, X' ⊆ Q^M such that X ⊆ X'. For each i ∈ I, we have π_F_i(X) ⊆π_F_i(X') and hence, because log is non-decreasing, logπ_F_i(X)≤logπ_F_i(X'). In conclusion, _ℱ(X) ≤_ℱ(X'). * This is a direct consequence of Items <ref> and <ref>.Due to the locality of the global transition function Δ and the asymptotic invariance of a right Følner net F_i_i ∈ I under taking finite boundaries, the information that flows into F_i from the boundary under an application of Δ is asymptotically negligible and hence applications of Δ to subsets of global configurations do not increase entropy, which is shown in Let ℛ be right amenable, let ℱ = F_i_i ∈ I be a right Følner net in ℛ indexed by (I, ≤), and let X be a subset of Q^M. Then, _ℱ(Δ(X)) ≤_ℱ(X). Suppose, without loss of generality, that G_0 ∈ N.Let i ∈ I. According to <ref>, the map Δ_X, F_i^- π_F_i(X) →π_F_i^-N(Δ(X)) is surjective. Therefore, π_F_i^-N(Δ(X))≤π_F_i(X).Because G_0 ∈ N, according to <ref> of <ref>, we have F_i^-N⊆ F_i. Thus, π_F_i(Δ(X)) ⊆π_F_i^-N(Δ(X)) × Q^F_i ∖ F_i^-N. Hence, logπ_F_i(Δ(X)) ≤logπ_F_i^-N(Δ(X)) + logQ^F_i ∖ F_i^-N≤logπ_F_i(X) + F_i ∖ F_i^-N·logQ. Because G_0 ∈ N, according to <ref> of <ref>, we have F_i ⊆ F_i^+N. Therefore, F_i ∖ F_i^-N⊆ F_i^+N∖ F_i^-N = _N F_i. Because G_0, F_i, and N are finite, according to <ref> of <ref>, the boundary _N F_i is finite. Hence,logπ_F_i(Δ(X))/F_i≤logπ_F_i(X)/F_i + _N F_i/F_ilogQ. Therefore, because N is finite, according to <ref>,_ℱ(Δ(X)) ≤lim sup_i ∈ Ilogπ_F_i(X)/F_i + *lim_i ∈ I_N F_i/F_i·logQ=_ℱ(X).[Tree]In the situation of <ref>, the cell space ℛ is not right amenable and the sequence ℱ = (ρ)_ρ∈_0 is not a right Følner net in ℛ. However, the cell space ℛ is right tractable and the sequence ℱ is a right Erling net in ℛ. Nevertheless, for the majority rule over ℛ, there is a subset X of global configurations whose image has greater entropy than X itself, as we show below. Broadly speaking, such a subset exists because the boundaries of components of ℱ are so big that the information that flows in from them under applications of Δ is asymptotically significant. Let Q be the set 0, 1, let N be the ball (1), let δ be the ∙-invariant map Q^N → Q, ℓ↦ 0, if ∑_n ∈ Nℓ(n) ≤N/2, and ℓ↦ 1, otherwise, which is known as majority rulemajority rulerule!majority, and let 𝒞 be the cellular automaton ℛ, Q, N, δ. Furthermore, for each non-negative integer ρ, let Y_ρ be the set c ∈ Q^M ∖ c_M ∖(ρ)≡ 0, let X_ρ + 1 be the sety ∈ Y_ρ + 1 m' ∈(ρ)q ∈ Qy_(m' · N) ∩(ρ + 1)≡ q, let Y be the set _ρ∈_0 Y_ρ, and let X be the set _ρ∈_0 X_ρ + 1.The global transition function Δ of 𝒞 maps X bijectively onto Y; even more, for each non-negative integer ρ, it maps X_ρ + 1 bijectively onto Y_ρ; more precisely, for each global configuration x ∈ X_ρ + 1, the global configurationy_xM→ Q,m'↦0,if m' ∈ M ∖(ρ), x(m),if m' ∈(ρ), where m ∈ (m' · N) ∩(ρ + 1),is the unique element of Y_ρ that satisfies Δ(x) = y_x, and, for each global configuration y ∈ Y_ρ, the global configurationx_yM→ Q,m↦0,if m ∈ M ∖(ρ + 1), y(m'),if m ∈(ρ + 1),if where m' ∈(ρ) such that m ∈ m' · N,is the unique element of X_ρ + 1 that satisfies Δ(x_y) = y (see <ref>).The entropy of Δ(X) with respect to ℱ is greater than the entropy of X with respect to ℱ. The reason is that, broadly speaking, the cardinality of π_(i)(Δ(X)) is approximately 2^(i), whereas the cardinality of π_(i)(X) is approximately 2^(i - 1), and the cardinality of (i) is approximately (i).First, we prove that Δ(X) = Y. Let y be a global configuration of Y and let m' be a cell of M. Then,Δ(x_y)(m') = δ[]n ↦ x_y(m' · n) =0,if ∑_n ∈ N x_y(m' · n) ≤5/2,1,otherwise.Of the 5 elements of m' · N, the 4 or 3 elements of (m' · N) ∩(m' + 1) have the same state in x_y, namely y(m'). Thus, ∑_m ∈ m' · N x_y(m) ≤5/2 y(m') = 0.Hence,Δ(x_y)(m') = { 0, if y(m') = 0,1, otherwise, } = y(m').Therefore, Δ(x_y) = y. Moreover, for each x ∈ X, if Δ(x) = y, then y_x = y, hence x = x_y_x = x_y, and therefore x_y is unique. Secondly, we prove that _ℱ(Δ(X)) > _ℱ(X). Recall that, for each positive integer i, we have (i) = 3^i + 3^i - 1 and (i) = 2 · 3^i - 1.Let i be an integer such that i ≥ 2. Then,π_(i)(X) = (_ρ = 0^i - 1π_(i)(X_ρ + 1)) = ∑_ρ = 0^i - 1 (2^(ρ) - 1) + 1.Thus, because i ≥ 1,π_(i)(X)≤∑_ρ = 0^i - 1 (2^(i - 1) - 1) + 1 =i · 2^(i - 1) - i + 1 ≤ i · 2^(i - 1).Hence, because i ≥ 2,logπ_(i)(X) ≤log(i) + (i - 1)·log(2)=log(i) + (3^i - 1 + 3^i - 2) ·log(2).Therefore, because (i)≥ 2 · 3^i - 3^i = 3^i,logπ_(i)(X)/(i) ≤log(i) + (3^i - 1 + 3^i - 2) ·log(2)/3^i=log(i)/3^i + 4/9·log(2).Thus, _ℱ(X) ≤ (4/9) ·log(2).Let i be an integer such that i ≥ 1. Then,π_(i)(Y) = _ρ∈_0π_(i)(Y_ρ) = ∑_ρ = 0^i (2^(ρ) - 1) + 1= ∑_ρ = 0^i 2^(ρ) - i.Thus, because ∑_ρ = 0^i - 1 2^(ρ) - i ≥ 0, logπ_(i)(Y)≥log 2^(i) =(i)·log(2) =(3^i + 3^i - 1) ·log(2).Hence, because (i)≤ 2 · 3^i,logπ_(i)(Δ(X))/(i)≥3^i + 3^i - 1/2 · 3^i·log(2)=6/9·log(2).Therefore, _ℱ(Y) ≥ (6/9) ·log(2).In conclusion, _ℱ(Δ(X)) =_ℱ(Y) ≥6/9·log(2) >4/9·log(2) ≥_ℱ(X).For a right-tractable cell space, if for each point t of an E,E'-tiling not all patterns with domain tE occur in a subset of Q^M, then that subset does not have maximal entropy, which is shown in Let ℛ be right tractable, let ℱ = F_i_i ∈ I be a right Erling net in ℛ indexed by (I, ≤), let Q contain at least two elements, let X be a subset of Q^M, let E and E' be two non-empty and finite subsets of GG_0, and let T be an E, E'-tiling of ℛ, such that, for each cell t ∈ T, we have π_tE(X) ⫋ Q^tE. Then, _ℱ(X) < logQ. For each t ∈ T, because π_tE(X) ⫋ Q^tE, Q≥ 2, and tE≥ 1,π_tE(X)≤Q^tE - 1 =Q^tE - 1 ≥ 1.Let i ∈ I. Put T_i = T ∩ F_i^-E and put F_i^* = F_i ∖ (⋃_t ∈ T_i tE) (see <ref>).Because ⋃_t ∈ T_i tE ⊆ F_i and tE_t ∈ T is pairwise disjoint,π_F_i(X) ⊆π_F_i^*(X) ×∏_t ∈ T_iπ_tE(X)⊆ Q^F_i^*×∏_t ∈ T_iπ_tE(X).Therefore,logπ_F_i(X) ≤logQ^F_i^* + ∑_t ∈ T_ilogπ_tE(X)≤logQ^F_i^* + ∑_t ∈ T_ilog[]Q^tE - 1=F_i^*·logQ + ∑_t ∈ T_ilog[]Q^tE (1 - Q^- tE)=[t] F_i^*·logQ + ∑_t ∈ T_itE·logQ+ ∑_t ∈ T_ilog[]1 - Q^- tE.Moreover, for each t ∈ T_i, we have tE ⊆ F_i. Thus,F_i^* = F_i - ∑_t ∈ T_itE.And, becauseis free, we have tE = E. Hence,logπ_F_i(X)≤F_i·logQ + T_i·log[]1 - Q^- E.Put c = - log[]1 - Q^- E. Because Q≥ 2 and E≥ 1, we have Q^- E∈0, 1 and hence c > 0. According to <ref>, there are ε∈_> 0 and i_0 ∈ I such that, for each i ∈ I with i ≥ i_0, we have T_i≥εF_i. Therefore, for each such i, logπ_F_i(X)/F_i≤logQ - c ε.In conclusion,_ℱ(X) =lim sup_i ∈ Ilogπ_F_i(X)/F_i≤logQ - c ε <logQ. For a right-tractable cell space, if for a non-empty and finite subset E of GG_0 not all patterns with domain m_0E occur in a shift-invariant subset of Q^M, then that subset does not have maximal entropy, which is shown inLet ℛ be right tractable, let ℱ = F_i_i ∈ I be a right Erling net in ℛ indexed by (I, ≤), let Q contain at least two elements, let H be a 𝒦-big subgroup of G, let X be a _H-invariant subset of Q^M, and let E be a non-empty and finite subset of GG_0, such that π_m_0E(X) ⫋ Q^m_0E. Then, _ℱ(X) < logQ. According to <ref>, there is a subset E' of GG_0 and an E, E'-tiling T of ℛ. Because G_0 and E are finite, so is E'. Let m ∈ M.Put h = g_m_0, m_0 g_m_0, m^-1. Then, because H is 𝒦-big, we have h ∈ H. And, h(mE) = m_0E. Hence, because X is _H-invariant, π_mE(X)= π_mE(h^-1 X)= h^-1π_h(mE)(X)= h^-1π_m_0E(X).And, because π_m_0E(X) ⫋ Q^m_0E, h^-1π_m_0E(X) ⫋ h^-1 Q^m_0E = Q^h^-1 (m_0E) = Q^mE.Therefore, π_mE(X) ⫋ Q^mE. In conclusion, according to <ref>, we have _ℱ(X) < logQ. This example demonstrates that in <ref> it is necessary that ℱ, E, and T are such that the limit superior of the net T ∩ F_i^-E / F_i_i ∈ I is greater than 0, which, according to <ref>, is the case if ℱ is a right Erling net in ℛ. Let ℱ = F_i_i ∈ I be a net in F ⊆ MF ≠∅, Ffinite indexed by (I, ≤), let Q contain at least two elements, let E and E' be two non-empty and finite subsets of GG_0, and let T be an E, E'-tiling of ℛ, Moreover, let q be an element of Q, let p be the pattern of Q^m_0E such that p ≡ q, and let X be the subset Q^M ∖⋃_t ∈ Tc ∈ Q^Mc_tE = tp of Q^M. Then, for each cell t ∈ T, we have π_tE(X) = Q^tE∖tp⫋ Q^tE. Let i be an index of I. Then, π_F_i(X) = Q^F_i∖⋃_t ∈ T ∩ F_i^-E Y_t, where Y_t = p' ∈ Q^F_i p'_tE = tp, for t ∈ T ∩ F_i^-E. Indeed, we have π_F_i(X) ⊆ Q^F_i∖⋃_t ∈ T ∩ F_i^-E Y_t. To show the other inclusion, let p' ∈ Q^F_i∖⋃_t ∈ T ∩ F_i^-E Y_t. Then, there is a state q' ∈ Q ∖q and there is a global configuration c ∈ Q^M such that c_F_i = p' and c_M ∖ F_i≡ q'. Hence, for each t ∈ T ∩ F_i^-E, we have c_tE = p'_tE≠ tp. And, for each t ∈ T ∖ F_i^-E, there is an e ∈ E such that te ∉ F_i and thus, by definition of c, we have c(te) = q' ≠ q, and hence c_tE≠ tp. Therefore, c ∈ X and hence p' = c_F_i∈π_F_i(X). In conclusion, Q^F_i∖⋃_t ∈ T ∩ F_i^-E Y_t ⊆π_F_i(X). For each subset S of T ∩ F_i^-E, we have ⋂_s ∈ S Y_s = Q^F_i ∖_s ∈ S sE = Q^F_i - S·E, which only depends on the cardinality of S. Hence, according to a special case of the inclusion-exclusion principle and the binomial formula, π_F_i(X) = ∑_k = 0^T ∩ F_i^-E (-1)^k T ∩ F_i^-EkQ^F_i - k E= Q^F_i·∑_k = 0^T ∩ F_i^-ET ∩ F_i^-Ek 1^T ∩ F_i^-E - k (-Q^-E)^k= Q^F_i· (1 - Q^-E)^T ∩ F_i^-E.Thus, logπ_F_i(X)/F_i = logQ + T ∩ F_i^-E/F_i·log(1 - Q^-E).Therefore,_ℱ(X) = logQ + lim sup_i ∈ IT ∩ F_i^-E/F_i·log(1 - Q^-E).Hence, because log(1 - Q^-E) < 0, we have _ℱ(X) < logQ if and only if lim sup_i ∈ IT ∩ F_i^-E / F_i > 0.§ GARDENS OF EDEN In this section, let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space and let 𝒞 = ℛ, Q, N, δ be a semi-cellular automaton such that the stabiliser G_0 of m_0 under , the set Q of states, and the neighbourhood N are finite, and the set Q is non-empty. Furthermore, let Δ be the global transition function of 𝒞. Contents. In <ref> we show that if Δ is not surjective, then the entropy of its image is less than the entropy of Q^M. And the converse of that statement obviously holds. In <ref> we show that if the entropy of the image of Δ is less than the entropy of Q^M, then Δ is not pre-injective. And in <ref> we show the converse of that statement. These four statements establish the Garden of Eden theorem, which is <ref>.Body. We first introduce the difference set of two global configurations and then use it to define pre-injectivity. Let c and c' be two global configurations of Q^M. The set(c, c') = m ∈ Mc(m) ≠ c'(m)difference (c, c') of c and c'is called difference of c and c'. Let c and c' be two global configurations of Q^M. Then,g ∈ G (gc, gc') = g (c, c'). Let g ∈ G. Then, for each m ∈ M,m ∈(gc,c') (gc)(m) ≠ (gc')(m) c(g^-1 m) ≠ c'(g^-1 m) g^-1 m ∈(c, c') m ∈ g (c, c').In conclusion, (gc,c') = g (c, c').The map Δ is called pre-injectivepre-injective if and only if, for each tuple (c, c') ∈ Q^M × Q^M such that (c, c') is finite and Δ(c) = Δ(c'), we have c = c'.Each injective map is pre-injective. And, if M is finite, then each pre-injective map is injective. In the proof of <ref>, the existence of a Garden of Eden pattern, as stated in <ref>, is essential, which itself follows from the existence of a Garden of Eden configuration, the compactness of Q^M, and the continuity of Δ. Garden of Eden configurations and patterns are introduced in * Let cM → Q be a global configuration. It is called Garden of Eden configurationGarden of Eden configuration c of 𝒞 if and only if it is not contained in Δ(Q^M). * Let pA → Q be a pattern. It is called Garden of Eden patternGarden of Eden pattern p of 𝒞 if and only if, for each global configuration c ∈ Q^M, we have Δ(c)_A ≠ p. * The global transition function Δ is surjective if and only if there is no Garden of Eden configuration. * If pA → Q is a Garden of Eden pattern, then each global configuration c ∈ Q^M with c_A = p is a Garden of Eden configuration. * If there is a Garden of Eden pattern, then Δ is not surjective. Let Δ not be surjective. There is a Garden of Eden pattern with non-empty and finite domain.Because Δ is not surjective, there is a Garden of Eden configuration c ∈ Q^M. Equip Q^M with the prodiscrete topology. According to <ref>, the image Δ(Q^M) is closed in Q^M. Hence, its complement Q^M ∖Δ(Q^M) is open. Therefore, because c ∈ Q^M ∖Δ(Q^M), according to <ref>, there is a non-empty and finite subset F of M such that (c, F) = c' ∈ Q^Mc'_F = c_F⊆ Q^M ∖Δ(Q^M).Hence, c_F is a Garden of Eden pattern with non-empty and finite domain. Under which assumptions the entropy of the image of a non-surjective global transition function is not maximal is shown inLet ℛ be right tractable, let ℱ be a right Erling net in ℛ, let H be a 𝒦-big subgroup of G, let δ be ∙_H_0-invariant, let Q contain at least two elements, and let Δ not be surjective. Then, _ℱ(Δ(Q^M)) < logQ. According to <ref>, there is a Garden of Eden pattern pF → Q with non-empty and finite domain. Let E = (m_0 )^-1(F). Then, m_0E = F and, becauseis free, E = F < ∞. Because p is a Garden of Eden pattern, p ∉π_m_0E(Δ(Q^M)). Hence, π_m_0E(Δ(Q^M)) ⫋ Q^m_0E. Moreover, according to <ref>, the map Δ is _H-equivariant. Hence, for each h ∈ H, we have h Δ(Q^M) = Δ(hQ^M) = Δ(Q^M). In other words, Δ(Q^M) is _H-invariant. Thus, according to <ref>, we have _ℱ(Δ(Q^M)) < logQ. This yields the characterisation of surjectivity by entropy that is given in Let ℛ be right tractable, let ℱ be a right Erling net in ℛ, let H be a 𝒦-big subgroup of G, let δ be ∙_H_0-invariant, and let Q contain at least two elements. Then, Δ is surjective if and only if _ℱ(Δ(Q^M)) = logQ.This is a direct consequence of <ref>. In the remainder of this section, let ℛ be right amenable and let ℱ = F_i_i ∈ I be a right Følner net in ℛ indexed by (I, ≤). In the proof of <ref>, the fact that enlarging each element of ℱ does not increase entropy, as stated in the next lemma, is essential.Let X be a subset of Q^M and let E be a finite subset of GG_0 such that G_0 ∈ E. Then, _F_i^+E_i ∈ I(X) ≤_ℱ(X).Let i ∈ I. According to <ref> of <ref>, we have F_i^-E⊆ F_i ⊆ F_i^+E. Hence, π_F_i^+E(X) ⊆π_F_i(X) × Q^F_i^+E∖ F_i and F_i^+E∖ F_i ⊆_E F_i. Thus,logπ_F_i^+E(X) ≤logπ_F_i(X) + F_i^+E∖ F_i·logQ≤logπ_F_i(X) + _E F_i·logQ. Therefore, according to <ref>,_F_i^+E_i ∈ I(X) ≤lim sup_i ∈ Ilogπ_F_i(X)/F_i + *lim_i ∈ I_E F_i/F_i·logQ=_ℱ(X).A global transition function whose image does not have maximal entropy is not pre-injective, which is shown in Let _ℱ(Δ(Q^M)) < logQ. Then, Δ is not pre-injective.The asymptotic growth rate of finite patterns in Δ(Q^M) is less than the one of Q^M. Hence, there are at least two finite patterns that can be identically extended to global configurations that have the same image under Δ. Therefore, Δ is not pre-injective. Suppose, without loss of generality, that G_0 ∈ N. Let X = Δ(Q^M). According to <ref>, we have _F_i^+N_i ∈ I(X) ≤_ℱ(X) < logQ. Hence, there is an i ∈ I such thatlogπ_F_i^+N(X)/F_i < logQ.Thus, π_F_i^+N(X) < Q^F_i. Furthermore, let q ∈ Q and let X' = c ∈ Q^Mc_M ∖ F_i≡ q. Then, Q^F_i = X'. Hence,π_F_i^+N(X) < X'.Moreover, for each (c, c') ∈ X' × X', according to <ref> of <ref>, we have Δ(c)_M ∖ F_i^+N = Δ(c')_M ∖ F_i^+N. Therefore, Δ(X') =π_F_i^+N(Δ(X'))≤π_F_i^+N(Δ(Q^M))=π_F_i^+N(X)<X'.Hence, there are c, c' ∈ X' such that c ≠ c' and Δ(c) = Δ(c'). Thus, because (c, c') ⊆ F_i is finite, the map Δ is not pre-injective. [Muller]In this example we present a cellular automaton on a non-right-amenable but right-tractable cell space that, although the image of its global transition function does not have maximal entropy with respect to a right Erling net, is pre-injective. It is Muller's counterexample to Myhill's theorem, see section 6, page 55, in <cit.>.Let G be the group with presentation x, y, zx^2, y^2, z^2 or, equivalently, the 3-fold free product of the cyclic group of order 2 with itself, let Q be the _2-vector space (_2)^2, where _2 is the finite field of order 2, let N be the set x, y, z, let δ be the map Q^N → Q, ℓ↦ (ℓ(x)_1 + ℓ(x)_2 + ℓ(y)_1 + ℓ(z)_2, 0), where, for each vector v ∈ Q, the first component of v is denoted by v_1 and the second by v_2. The tuple ℛ = G, G, ·, e_G, g_g ∈ G is a cell space and the quadruple 𝒞 = ℛ, Q, N, δ is a cellular automaton. According to <ref>, the cell space ℛ is right tractable and the sequence ℱ = (ρ)_ρ∈_0 is a right Erling net. Moreover, because point evaluation, projection, and addition are linear, the local transition function δ is linear and hence the global transition function Δ of 𝒞 is linear. Furthermore, because the image of Δ is included in (_2 ×0)^G, the global transition function Δ is not surjective. Hence, according to <ref>, we have _ℱ(Δ(Q^G)) < logQ. However, as we show now, the global transition function Δ is pre-injective. Let c and c' be two global configurations of Q^G such that (c, c') is finite and Δ(c) = Δ(c'). Then, because Δ is linear, we have Δ(c - c') ≡ 0. Let c” be the global configuration c - c'. Suppose that c”≢0. Then, because c” has finite support, there is an element g ∈ G such that c”(g) ≠ 0 and c”_M ∖(g)≡ 0. And, there are two distinct elements s and s' ∈x, y, z such that g s, g s' ∈(g + 1) (see <ref>). Hence, because c”_M ∖(g)≡ 0, we have c”(g s) = 0 and c”(g s') = 0. And, for each element s”∈s, s', because g s” s” = g and g s” s”' ∈(g + 2), for s”' ∈x, y, z∖s”, Δ(c”)(g s”)_1= c”(g s” x)_1 + c”(g s” x)_2 + c”(g s” y)_1 + c”(g s” z)_2= c”(g)_1 + c”(g)_2,if s” = x, c”(g)_1,if s” = y, c”(g)_2,if s” = z.Because c”(g) ≠ 0, for each element s”∈s, s', in two of the three cases we have Δ(c”)(g s”)_1 = 1 and hence Δ(c”)(g s)_1 = 1 or Δ(c”)(g s')_1 = 1. Therefore, Δ(c”)(g s) ≠ 0 or Δ(c”)(g s') ≠ 0, which contradicts that Δ(c”) ≡ 0. Thus, contrary to our supposition, we have c”≡ 0 and hence c = c'. In conclusion, the global transition function Δ is pre-injective. In the proof of <ref>, the statement of <ref> is essential, which says that if two distinct patterns have the same image and we replace each occurrence of the first by the second in a global configuration, we get a new configuration in which the first pattern does not occur and that has the same image as the original one.How maps with disjoint domains are glued together, which is used to replace occurrences of patterns in global configurations by other patterns, is introduced in Let I be a set and, for each index i ∈ I, let A_i and B_i be two sets and let f_i be a map from A_i to B_i, such that the sets A_i, for i ∈ I, are pairwise disjoint. The map[symbols]coproductfiiinI@∐_i ∈ I f_i∐_i ∈ I f_i _i ∈ I A_i→_i ∈ I B_i, coproduct ∐_i ∈ I f_i of f_i_i ∈ Ix↦ f_i(x),wherei ∈ Isuch thatx ∈ A_i,is called coproduct of f_i_i ∈ I and, if I is the set 1, 2, …, I, then it is also denoted by f_1 × f_2 ×…× f_If_1 × f_2 ×…× f_I[symbols]crossf1f2@f_1 × f_2 ×…× f_I. In the proof of <ref> we use the technicalLet A be a subset of M, and let E and E' be two subsets of GG_0 such that g^-1· e'e, e' ∈ E, g ∈ e⊆ E'. Then, m ∈ MmE ⊆ M ∖ AormE ⊆ A^+E'.Let m ∈ M such that mE ⊈ M ∖ A. Then, (mE) ∩ A ≠∅. Hence, there is an e' ∈ E such that me' ∈ A. Let e ∈ E. According to <ref>, there is a g ∈ e such that (me)g^-1· e' = me'. Because g^-1· e' ∈ E' and me' ∈ A, we have (me)E' ∩ A ≠∅ (see <ref>). Thus, me ∈ A^+E'. Therefore, mE ⊆ A^+E'.Identify M with GG_0 by ι m ↦ G_m_0, m, let H be a 𝒦-big subgroup of G, let δ be ∙_H_0-invariant, let A be a subset of M, let N' be the subset g^-1· n'n, n' ∈ N, g ∈ n of GG_0, and let p and p' be two maps from A^+N' to Q such that p_A^+N'∖ A = p'_A^+N'∖ A and Δ_A^+N'^-(p) = Δ_A^+N'^-(p'). Furthermore, let c be a map from M to Q and let S be a subset of M, such that the family sA^+N'_s ∈ S is pairwise disjoint and, for each cell s ∈ S, we have p _s c. Putc' = c_M ∖ (⋃_s ∈ S sA^+N')×∐_s ∈ S sp'.Then, for each cell s ∈ S, we have p' _s c', and Δ(c) = Δ(c'). In particular, if p ≠ p', then, for each cell s ∈ S, we have p _s c'. For each s ∈ S, we have (sp) = (sp') = sA^+N'. Hence, c' is well-defined. Moreover, for each s ∈ S, we have (sp)_(sA^+N') ∖ (sA) = (sp')_(sA^+N') ∖ (sA).Let m ∈ M ∖ (⋃_s ∈ S sA). If m ∈ M ∖ (⋃_s ∈ S sA^+N'), then c'(m) = c(m). And, if there is an s ∈ S such that m ∈ sA^+N', then, because m ∉ sA, we have c'(m) = (sp')(m) = (sp)(m) = c(m). Therefore,c' = c_M ∖ (⋃_s ∈ S sA)×∐_s ∈ S s(p'_A). Let m ∈ M. Case 1: mN ⊆ M ∖ (⋃_s ∈ S sA). Then, c'_mN = c_mN. Hence, Δ(c')(m) = Δ(c)(m).Case 2: mN ⊈ M ∖ (⋃_s ∈ S sA). Then, there is an s ∈ S such that mN ⊈ M ∖ (sA). Thus, according to <ref>, we have mN ⊆ (sA)^+N'. Hence, because G_0 · N' ⊆ N', according to <ref> of <ref>, we have mN ⊆ sA^+N' and hence m ∈ (sA^+N')^-N. Therefore, because c_sA^+N' = sp, Δ_A^+N'^-(p) = Δ_A^+N'^-(p'), and c'_sA^+N' = sp', according to <ref>, Δ(c)(m)= Δ_sA^+N'^-(sp)(m)= (s Δ_A^+N'^-(p))(m)= (s Δ_A^+N'^-(p'))(m)= Δ_sA^+N'^-(sp')(m)= Δ(c')(m). In either case, Δ(c)(m) = Δ(c')(m). Therefore, Δ(c) = Δ(c'). Under which assumptions the entropy of the image of a non-pre-injective global transition function is not maximal is shown in Let H be a 𝒦-big subgroup of G, let δ be ∙_H_0-invariant, let Q contain at least two elements, and let Δ not be pre-injective. Then, _ℱ(Δ(Q^M)) < logQ. For N' = N^-1· N, there is a subset A of M and there are two distinct finite patterns p and p' with domain A^+N' that have the same image under Δ_A^+N'^-. The set Y of all global configurations in which p does not occur at the cells of a tiling has the same image under Δ as Q^M, because in a global configuration we may replace occurrences of p by p' without changing the image. It follows that (Δ(Q^M)) = (Δ(Y)) ≤(Y); and, because Y is missing the pattern p at each cell of a tiling, we also have (Y) < (Q^M) = logQ. In conclusion, (Δ(Q^M)) < logQ. Suppose, without loss of generality, that G_0 ∈ N. Identify M with GG_0 by ι m ↦ G_m_0, m.Because Δ is not pre-injective, there are c, c' ∈ Q^M such that (c, c') is finite, Δ(c) = Δ(c'), and c ≠ c'. Put A = (c, c'), put N' = g^-1· n'n, n' ∈ N, g ∈ n, put E = A^+N', and put p = c_E and p' = c'_E. Because Δ(c) = Δ(c'), we have Δ_A^+N'^-(p) = Δ_A^+N'^-(p').Because N is finite and, for each n ∈ N, we have n = G_0 < ∞, the set N' is finite. Moreover, G_0 · N' ⊆ N'. According to <ref> of <ref>, because G_0 ∈ N' and A ≠∅, we have E ⊇ A and hence E is non-empty. According to <ref> of <ref>, because G_0, A, and N' are finite, so is E. Because E is non-empty, according to <ref>, there is a subset E' of GG_0 and an E, E'-tiling T of ℛ. Because G_0 and E are non-empty and finite, so is E'.Let Y = y ∈ Q^Mt ∈ Tp _t y. For each t ∈ T, we have tp ∉π_tE(Y) and therefore π_tE(Y) ⫋ Q^tE. According to <ref>, we have _ℱ(Y) < logQ. Hence, according to <ref>, we have _ℱ(Δ(Y)) < logQ.Let x ∈ Q^M. Put S = t ∈ Tp _t x. According to <ref>, there is an x' ∈ Q^M such that x' ∈ Y and Δ(x) = Δ(x'). Therefore, Δ(Q^M) = Δ(Y). In conclusion, _ℱ(Δ(Q^M)) < logQ. [Tree]In the situation of counterexample <ref>, the global transition function Δ is not pre-injective but the entropy _ℱ(Δ(Q^M)) is maximal as we show now. First, let c and c' be the two global configurations of Q^M such that c ≡ 0, c'(m_0) = 1, and c'_M ∖m_0≡ 0. Then, (c, c') is finite and Δ(c) = Δ(c') but c ≠ c'. Hence, Δ is not pre-injective.Secondly, let c be a global configuration of Q^M. And, let c' be the global configuration of Q^M such that c'(m_0) = 0 and, for each cell m ∈ M and each generator s ∈a, b, a^-1, b^-1 with m s = m + 1, we have c'(m s) = c(m). For each generator s ∈a, b, a^-1, b^-1, we have m_0 s = m_0 + 1; and, for each cell m ∈ M ∖m_0, there are precisely three distinct generators s_1, s_2, and s_3 ∈a, b, a^-1, b^-1 such that m s_k = m + 1, for k ∈1, 2, 3. Hence, Δ(c') = c. Therefore, Δ is surjective. In particular, _ℱ(Δ(Q^M)) = logQ.The three preceding theorems yield(Garden of Eden Theorem; Edward Forrest Moore, 1962; John R. Myhill, 1963)Let ℳ = M, G, be a right-amenable left-homogeneous space with finite stabilisers and let Δ be the global transition function of a big-cellular automaton over ℳ with finite set of states and finite neighbourhood. The map Δ is surjective if and only if it is pre-injective.There is a coordinate system 𝒦 = m_0, g_m_0, m_m ∈ M such that there is a big-cellular automaton 𝒞 = ℛ, Q, N, δ such that Q and N are finite and Δ is its global transition function. Moreover, because 𝒞 is big, there is a 𝒦-big subgroup H of G such that the local transition function δ is ∙_H_0-invariant. And, because G_0 is finite, the cell space ℛ = ℳ, 𝒦 is right amenable.Case 1: Q≤ 1. If Q = 0, then, because M≠ 0, we have Q^M = 0. And, if Q = 1, then Q^M = 1. In either case, Δ is bijective, in particular, surjective and pre-injective.Case 2: Q≥ 2. According to <ref> and <ref> of <ref>, the map Δ is not surjective if and only if _ℱ(Δ(Q^M)) < logQ. And, according to <ref> and <ref>, we have _ℱ(Δ(Q^M)) < logQ if and only if Δ is not pre-injective. Hence, Δ is not surjective if and only if it is not pre-injective. In conclusion, Δ is surjective if and only if it is pre-injective. In the situation of <ref>, <ref> is theorem 5.3.1 in <cit.>. [Tree]The global transition function of the cellular automaton of <ref> is surjective but not pre-injective, which was shown in <ref>.[Muller]The global transition function of the cellular automaton of <ref> is not surjective but pre-injective, which was shown there. [Exclusive Or] The global transition function of the elementary cellular automaton with Wolfram code 90, whose local transition function combines the states of the left and right neighbours by exclusive or, is 4-to-1 surjective and pre-injective but not injective.Laurent Bartholdi constructs in his papers *bartholdi:2010<cit.> and *bartholdi:2016<cit.>, for each non-amenable group G, two finite sets Q and Q', and two cellular automata 𝒞 and 𝒞' over G such that the global transition function Δ Q^G → Q^G of 𝒞 is surjective but not pre-injective and the global transition function Δ'(Q')^G → (Q')^G of 𝒞' is not surjective but pre-injective. Are similar constructions possible for non-right-amenable left-homogeneous spaces with finite stabilisers?§ CONSTRUCTION OF NON-DEGENERATED LEFT HOMOGENEOUS SPACESIntroduction. So far we have only seen examples of right-amenable left-homogeneous spaces with finite stabilisers for which there is a subgroup that acts freely and transitively. The global transition function of a semi-cellular automaton on such a space is essentially the global transition function of a cellular automaton over a group. For those spaces, <ref> states nothing new. A simple construction of right-amenable left-homogeneous spaces with finite stabilisers for which there is no subgroup that acts freely and transitively goes like this: Act with the direct product of the automorphism groups of the coloured Cayley graph of a group and a vertex-transitive, finite, and directed non-Cayley graph on the direct product of the vertices of these graphs. Full details and concrete examples of this construction are given below. Body. The Cartesian product of two cell spaces, and how its right quotient set semi-action and its notions of interior, closure, and boundary relate to the respective constructs and notions of its components is given and shown in Let ℛ = M, G, , m_0, g_m_0, m_m ∈ M and ℛ' = M', G', ', m_0', g_m_0', m''_m' ∈ M' be two cell spaces, and let ℛ” be the cell space M × M', G × G', ×', (m_0, m_0'), (g_m_0, m, g_m_0', m'')_(m, m') ∈ M × M', where×'(G × G') × (M × M')→ M × M', left group action ×' of G × G' on M × M'((g, g'), (m, m'))↦ (gm, g' ' m').Furthermore, let A be a subset of M, let A' be a subset of M', let A” be a subset of M × M' such that m ∈ Mm' ∈ M'(m, m') ∈ A” = A, let E be a subset of GG_0, and let E' be a subset of G'G_0'. Then, *” = ×', where×'(M × M') × ((G × G')(G_0 × G_0')) → M × M',((m, m'), (g, g') (G_0 × G_0')) ↦ (mg G_0, m' ' g' G_0'). *(A × A')^-E × E' = A^-E× (A')^-E', (A × A')^+E × E' = A^+E× (A')^+E', and _E × E' (A × A') = (_E A × (A')^+E') ∪ (A^+E×_E' A'), whereE × E' = (g, g') (G_0 × G_0')g G_0 ∈ E, g' G_0' ∈ E'. *(A”)^-E × (G'G_0')⊆ A^-E× M', (A”)^+E × (G'G_0') = A^+E× M', and _E × (G'G_0') A”⊇_E A × M', whereE × (G'G_0') = (g, g') (G_0 × G_0')g G_0 ∈ E, g' G_0' ∈ G'G_0'. Note that the stabiliser G_0” of (m_0, m_0') under ×' is G_0 × G_0'. * For each (m, m') ∈ M × M' and each (g, g') (G_0 × G_0') ∈ (G × G')(G_0 × G_0'),(m, m') ” (g, g') (G_0 × G_0')= (g_m_0, m, g_m_0', m'') · (g, g') ×' (m_0, m_0')= (g_m_0, m gm_0, g_m_0', m'' g' ' m_0')= (mg G_0, m' ' g' G_0').Therefore, ” = ×'.* Let (m, m') ∈ M × M'. Then, (m, m') ” E × E' = (mE) × (m' ' E'). Hence, if E ≠∅ and E' ≠∅, then (m, m') ” E × E' ⊆ A × A' if and only if mE ⊆ A and m' ' E' ⊆ A'. And, ((m, m') ” E × E') ∩ (A × A') ≠∅ if and only if (mE) ∩ A ≠∅ and (m' ' E') ∩ A' ≠∅. Therefore, (A × A')^-E × E' = A^-E× (A')^-E' and (A × A')^+E × E' = A^+E× (A')^+E'. Note that the first equality holds in the case that E = ∅ or E' = ∅, because in that case (A × A')^-E × E' = ∅, and A^-E = ∅ or (A')^-E' = ∅. Moreover, _E × E' (A × A')= []A^+E× (A')^+E'∖[]A^-E× (A')^-E'= [t] [](A^+E∖ A^-E) × (A')^+E'∪[]A^+E× ((A')^+E'∖ (A')^-E')= []_E A × (A')^+E'∪[]A^+E×_E' A'. * Let (m, m') ∈ M × M'. Then, (m, m') ” E × (G'G_0') = (mE) × M'. Hence, if (m, m') ” E × (G'G_0') ⊆ A”, then mE ⊆ A. And, ((m, m') ” E × (G'G_0')) ∩ A”≠∅ if and only if (mE) ∩ A ≠∅. Therefore, (A”)^-E × (G'G_0')⊆ A^-E× M' and (A”)^+E × (G'G_0') = A^+E× M'. Moreover,_E × (G'G_0') A” ⊇ (A^+E× M') ∖ (A^-E× M')= [t] [](A^+E∖ A^-E) × M'∪[]A^+E× (M' ∖ M')= _E A × M'. The Cartesian product of a left-homogeneous space with a finite one is right amenable if and only if the former one is right amenable, which is shown inLet ℳ = M, G, and ℳ' = M', G', ' be two left-homogeneous spaces with finite stabilisers such that M' is finite, and let ℳ” be the left-homogeneous space M × M', G × G', ×'. The space ℳ” is right amenable if and only if the space ℳ is right amenable. Let 𝒦 = m_0, g_m_0, m_m ∈ M and 𝒦' = m_0', g_m_0', m''_m' ∈ M' be two coordinate systems for ℳ and ℳ' respectively, and let 𝒦” be the coordinate system (m_0, m_0'), (g_m_0, m, g_m_0', m'')_(m, m') ∈ M × M' for ℳ”. Note that, because ℳ and ℳ' have finite stabilisers, so has ℳ”.First, let ℳ” be right amenable. Then, because ℳ” has finite stabilisers, there is a right Følner net F_i”_i ∈ I in ℛ” = ℳ”, 𝒦”. Put F_i = m ∈ Mm' ∈ M'(m, m') ∈ F_i”. Let E be a finite subset of GG_0. Then, according to <ref> of <ref> and because F_i”⊆ F_i × M',_E F_i/F_i≤M'/M'·_E × (G'G_0') F_i”/F_i” = _E × (G'G_0') F_i”/F_i”.Hence, because F_i”_i ∈ I is a right Følner net in ℛ”, the net F_i_i ∈ I is a right Følner net in ℳ, 𝒦. In conclusion, ℳ is amenable.Secondly, let ℳ be amenable. Then, because ℳ has finite stabilisers, there is a right Følner net F_i_i ∈ I in ℛ = ℳ, 𝒦. Let E” be a finite subset of (G × G')(G × G')_0. Put E = g G_0g' ∈ G'(g, g') (G × G')_0 ∈ E” and put E' = G'G'_0. Then, according to <ref> of <ref>, we have _E” (F_i × M') ⊆_E × E' (F_i × M') ⊆ ((_E F_i) × M') ∪ (F_i^+E× (_E' M')) = (_E F_i) × M'. Hence, _E” (F_i × M')/F_i × M'≤_E F_i·M'/F_i·M' =_E F_i/F_i.Therefore, because F_i_i ∈ I is a right Følner net in ℛ, the net F_i × M'_i ∈ I is a right Følner net in ℳ”. In conclusion, ℳ” is right amenable. Vertex-transitivity, the Cartesian product of two graphs, and an action on such a product are introduced in the forthcoming definitions.Let 𝒢 be a directed graph. It is called vertex-transitivevertex-transitive if and only if its automorphism group acts transitively on its vertices by function application. Cayley graphs of groups are vertex-transitive. Let 𝒢 be a directed graph. It is a Cayley graph if and only if a subgroup of its automorphism group acts freely and transitively on its vertices by function application.See proposition 3.1(b) in <cit.>. Let 𝒢 = V, E and 𝒢' = V', E' be two directed graphs. The graph 𝒢𝒢' = V × V', ((v_1, v_1'), (v_2, v_2')) ∈ (V × V') × (V × V')v_1 = v_2(v_1', v_2') ∈ E'orv_1' = v_2'(v_1, v_2) ∈ E is called Cartesian product of 𝒢 and 𝒢'Cartesian product 𝒢𝒢' of 𝒢 and 𝒢'. Let 𝒢𝒢' be the Cartesian product of 𝒢 = V, E and 𝒢' = V', E', and letand ' be the left group actions of (𝒢) on V and of (𝒢') on V' by function application. The direct product (𝒢) ×(𝒢') acts on V × V' on the left by'((𝒢) ×(𝒢')) × (V × V')→ V × V', left group action ' of (𝒢) ×(𝒢') on V × V'((φ, φ'), (v, v'))↦ (φ v, φ' ' v').The map (')(𝒢) ×(𝒢') →(𝒢𝒢') is an injective group homomorphism.If 𝒢 and 𝒢' are vertex-transitive, then the left group action ' is transitive. When a subgroup acts freely and transitively on the Cartesian product of two graphs is characterised inLet 𝒢𝒢' be the Cartesian product of 𝒢 = V, E and 𝒢' = V', E'. There is a subgroup H” of (𝒢) ×(𝒢') that acts freely and transitively on V × V' by ' if and only if there is a subgroup H of (𝒢) and there is a subgroup H' of (𝒢') such that H acts freely and transitively on V byand H' acts freely and transitively on V' by '.First, let H” be a subgroup of (𝒢) ×(𝒢') that acts freely and transitively on V × V' by '. Furthermore, let v' be a vertex of V'. The set F = h”∈ H” h”' V ×v'⊆ V ×v' is a subgroup of H” and the left group action (')_F × (V ×v') → V ×v' is free and transitive. The set H = π_1(F) is a subgroup of (𝒢), which acts freely and transitively on V by , where π_1 is the projection homomorphism from (𝒢) ×(𝒢') onto (𝒢). Analogously, a subgroup H' of (𝒢') can be constructed that acts freely and transitively on V' by '.Secondly, let H be a subgroup of (𝒢) and let H' be a subgroup of (𝒢') such that H acts freely and transitively on V byand H' acts freely and transitively on V' by '. The direct product H” = H × H' acts freely and transitively on V × V' by '. It follows that there is no subgroup that acts freely and transitively on the Cartesian product of a Cayley and a non-Cayley graph, which is stated inLet 𝒢𝒢' be the Cartesian product of 𝒢 = V, E and 𝒢' = V', E', where 𝒢 or 𝒢' is not a Cayley graph. There is no subgroup of (𝒢) ×(𝒢') that acts freely and transitively on V × V' by '.This is a direct consequence of <ref>.Let G be a group, let S be a generating set of G, let 𝒢 = V, E be the coloured S-Cayley graph of G, letbe the left group action of (𝒢) on V by function application, let 𝒢' = V', E' be a vertex-transitive, finite, and directed non-Cayley graph, and let ' be the left group action of (𝒢') on V' by function application, and let ℳ” be the left-homogeneous space V × V', (𝒢) ×(𝒢'), '.Note that V = G, that G ≃(𝒢) by g ↦ g ·, that under this identification the mapis the group multiplication · and the map ' is (G ×(𝒢')) × (G × V') → G × V', ((g, φ'), (v, v')) ↦ (g · v, φ'(v')), and that V, (𝒢), is right amenable if and only if G is amenable.According to <ref>, there is no subgroup of (𝒢) ×(𝒢') that acts freely and transitively on V × V' by '. And, according to <ref> and the note above, the space ℳ” is right amenable if and only if the group G is amenable. For example:* If G is the integer latticeand 𝒢' is the Petersen graph, then ℳ” is right amenable. * If G is the free group F_2 over a, b, where a ≠ b and 𝒢' is the Coxeter graph, then ℳ” is not right amenable.CHAPTER: FINITELY RIGHT GENERATED, GROWTH, AND RIGHT AMENABILITYAbstract. We introduce right-generating sets, left Cayley graphs, growth functions, types and rates, and isoperimetric constants for cell spaces and some of these notions also for left-homogeneous spaces; characterise right-amenable finitely right-generated cell spaces with finite stabilisers as those whose isoperimetric constant is 0; and prove that finitely right-generated left-homogeneous spaces with finite stabilisers of sub-exponential growth are right amenable, in particular, quotient sets of groups of sub-exponential growth by finite subgroups acted upon by left multiplication are right amenable. Remark. Most parts of this chapter appeared in the preprint *wacker:growth:2017<cit.> and they generalise parts of chapter 6 of the monograph *ceccherini-silberstein:coornaert:2010<cit.>. Summary. A cell space ℛ is finitely right generated if there is a finite subset S of GG_0 with G_0 · S ⊆ S such that, for each point m ∈ M, there is a sequence s_i_i ∈1, 2, …, k of elements in S ∪ S^-1 such that m = (((m_0s_1)s_2) …)s_k. The finite right-generating set S induces the left S-Cayley graph structure on M given by: For each point m ∈ M and each generator s ∈ S, there is an edge from m to ms. The length of the shortest path between two points of M yields the S-metric. The ball of radius ρ∈_0 centred at m ∈ M, denoted by _S(m, ρ), is the set of all points whose distance to m is not greater than ρ. The S-growth function is the map γ_S _0 →_0, k ↦_S(m_0, k); the growth type of ℛ, which does not depend on S, is the equivalence class γ_S_, where two growth functions are equivalent if they dominate each other; and the S-growth rate is the limit point of the sequence √(γ_S(k))_k ∈_0.A finitely right-generated cell space ℛ is said to have sub-exponential growth if its growth type is not exp_, which is the case if and only if its growth rates are 1. The S-isoperimetric constant is a real number between 0 and 1 that measures, broadly speaking, the invariance under _M × S that a finite subset of M can have, where 0 means maximally and 1 minimally invariant. In the case that G_0 is finite, this constant is 0 if and only if ℛ is right amenable; and, if ℛ has sub-exponential growth, then it is right amenable; and, if G has sub-exponential growth, then so has ℛ.Cayley graphs were introduced by Arthur Cayley in his paper *cayley:1878<cit.>. The notion of growth was introduced by Vadim Arsenyevich Efremovich and Albert S. Švarc in their papers *efremovich:1952<cit.> and *svarc:1955<cit.>. Mikhail Leonidovich Gromov was the first to study groups through their word metrics, see for example his paper *gromov:1984<cit.>.Contents. In <ref> we introduce right-generating sets. In <ref> we recapitulate directed multigraphs. In <ref> we introduce left Cayley graphs induced by right-generating sets. In <ref> we introduce metrics and lengths induced by left Cayley graphs. In <ref> we consider balls and spheres induced by metrics. In <ref> we consider interiors, closures, and boundaries of any thickness of sets. In <ref> we recapitulate growth functions and types. In <ref> we introduce growth functions and types of cell spaces. In <ref> we introduce growth rates of cell spaces. In <ref> we prove that right amenability and having isoperimetric constant 0 are equivalent, and we characterise right Følner nets. And in <ref> we prove that having sub-exponential growth implies right amenability. Preliminary Notions. A left group set is a triple M, G,, where M is a set, G is a group, andis a map from G × M to M, called left group action of G on M, such that G →(M), g ↦ [g ], is a group homomorphism. The actionis transitive if M is non-empty and for each m ∈ M the map m is surjective; and free if for each m ∈ M the map m is injective. For each m ∈ M, the set Gm is the orbit of m, the set G_m = ( m)^-1(m) is the stabiliser of m, and, for each m' ∈ M, the set G_m, m' = ( m)^-1(m') is the transporter of m to m'.A left-homogeneous space is a left group set ℳ = M, G, such thatis transitive. A coordinate system for ℳ is a tuple 𝒦 = m_0, g_m_0, m_m ∈ M, where m_0 ∈ M and for each m ∈ M we have g_m_0, m m_0 = m. The stabiliser G_m_0 is denoted by G_0. The tuple ℛ = ℳ, 𝒦 is a cell space. The map M × GG_0 → M, (m, g G_0) ↦ g_m_0, m g g_m_0, m^-1 m (= g_m_0, m gm_0) is a right semi-action of GG_0 on M with defect G_0, which means thatm ∈ MmG_0 = m,andm ∈ Mg ∈ Gg_0 ∈ G_0 𝔤' ∈ GG_0 mg ·𝔤' = (mg G_0)g_0 ·𝔤'.It is transitive, which means that the set M is non-empty and for each m ∈ M the map m is surjective; and free, which means that for each m ∈ M the map m is injective; and semi-commutes with , which means thatm ∈ Mg ∈ Gg_0 ∈ G_0 𝔤' ∈ GG_0 (gm) 𝔤' = g(mg_0 ·𝔤').The maps ι M → GG_0, m ↦ G_m_0, m, and m_0 are inverse to each other. Under the identification of M with GG_0 by either of these maps, we have (m, 𝔤) ↦ g_m_0, m𝔤.A left-homogeneous space ℳ is right amenable if there is a coordinate system 𝒦 for ℳ and there is a finitely additive probability measure μ on M such that 𝔤∈ GG_0A ⊆ M [](𝔤)_Ainjectiveμ(A 𝔤) = μ(A),in which case the cell space ℛ = ℳ, 𝒦 is called right amenable. When the stabiliser G_0 is finite, that is the case if and only if there is a right Følner net in ℛ indexed by (I, ≤), which is a net F_i_i ∈ I in F ⊆ MF ≠∅, Ffinite such that𝔤∈ GG_0 lim_i ∈ IF_i ∖ (𝔤)^-1(F_i)/F_i = 0.§ RIGHT GENERATING SETS In this section, let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space. Contents. In <ref> we introduce right-generating sets of ℛ. And in <ref> we show how generating sets of G induce right ones of ℛ.Let S be a subset of GG_0 such that G_0 · S ⊆ S. * The set g^-1 G_0s ∈ S, g ∈ s is denoted by S^-1S^-1[symbols]Sminus1@S^-1.* For each element m ∈ M, each non-negative integer k, and each finite sequence s_i_i ∈1, 2, …, k of elements in S ∪ S^-1, the element(((ms_1)s_2) …)s_kis denoted by m s_i_i ∈1, 2, …, km s_i_i ∈1, 2, …, k[symbols]arrowleftunderscorem@m s_i_i ∈1, 2, …, k, where, in the case that k = 0, the sequence s_i_i ∈1, 2, …, k is the empty sequence () and m s_i_i ∈1, 2, …, k is equal to m.* The set S is said to right generate ℛS right generates ℛgenerate right R@right generate ℛ, called right-generating set of ℛright-generating set S of ℛgenerating set right@right-generating set of ℛ, and each element s ∈ S is called right generatorright generator sgenerator right@right generator if and only ifm ∈ M s_i_i ∈1, 2, …, k inS ∪ S^-1 m_0 s_i_i ∈1, 2, …, k = m. * The set S is called symmetricsymmetric!right-generating setsymmetric if and only if S^-1⊆ S. If S is a right-generating set of ℛ, then S ∪ S^-1 is a symmetric one; and, if S is also finite and G_0 is finite, then S ∪ S^-1 is finite. Let P be an adjective. The cell space ℛ is called Ply right generatedPly right-generated cell space ℛ if and only if there is a right-generating set of ℛ that is P.If ℛ is finitely right generated, then M is countably infinite. For each right-generating set S of ℛ, each coordinate system 𝒦' = m_0', g_m_0', m'_m ∈ M for ℳ, and each element g ∈ G such that gm_0 = m_0', according to <ref>, because G_0 · S ⊆ S, the subset gS of GG_m_0' is a right-generating set of ℳ, 𝒦', which is finite or symmetric if S has the respective property. In particular, being finitely and/or symmetrically right generated does not depend on the coordinate system.The left-homogeneous space ℳ is called finitely and/or symmetrically right-generated left-homogeneous spacefinitely and/or symmetrically right generated if and only if, there is a (for each) coordinate system 𝒦 for ℳ, the cell space ℳ, 𝒦 is finitely and/or symmetrically right generated. Let T be a generating set of G. The set S = g_0 · t G_0g_0 ∈ G_0, t ∈ T is a right-generating set of ℛ. And, if T is symmetric, then so is S. And, if T and G_0 are finite, then so is S.First, let m ∈ M. Then, becauseis transitive, there is a g ∈ G such that m_0g G_0 = m. And, because T generates G, there is a finite sequence t_i_i ∈1, 2, …, k in T ∪ T^-1 such that t_1 t_2 … t_k = g. And, becauseis a semi-action, there is a finite sequence g_i,0_i ∈2,3,…,k in G_0 such thatm_0 t_1 G_0, g_2,0 t_2 G_0, …, g_k,0 t_k G_0 = m_0t_1 t_2 … t_k G_0 = m.In conclusion, because t_1 G_0 ∈ S ∪ S^-1 and g_i,0 t_i G_0 ∈ S ∪ S^-1, for i ∈2,3,…,k, the set S is a right-generating set of ℛ.Secondly, let T be symmetric. Furthermore, let s ∈ S and let g ∈ s. Then, there is a g_0 ∈ G_0, there is a t ∈ T, and there is a g_0' ∈ G_0 such that g_0 · t G_0 = s and g_0 t g_0' = g. Hence, because (g_0')^-1∈ G_0 and t^-1∈ T,g^-1 G_0 = (g_0')^-1 t^-1 g_0^-1 G_0 = (g_0')^-1· t^-1 G_0 ∈ S.Therefore, S^-1⊆ S. In conclusion, S is symmetric.Lastly, let T and G_0 be finite. Then, because S≤G_0·T, the set S is finite.§ DIRECTED MULTIGRAPHS Let V and E be two sets, and letandbe two maps from E to V. The quadruple 𝒢 = V, E, , is called directed multigraph 𝒢directed multigraphmultigraph directed@directed multigraph[symbols]Gcalligraphic@𝒢; each element v ∈ V is called vertexvertex v[symbols]v@v; each element e ∈ E is called edge from (e) to (e)edge e from (e) to (e)[symbols]e@e; for each element e ∈ E, the vertex (e) is called head of ehead (e) of e[symbols]sigmae@(e) and the vertex (e) is called tail of etail (e) of e[symbols]taue@(e). In the case that the map η E → V × V, e ↦ ((e), (e)), is injective, the directed multigraph 𝒢 is a directed graph and is identified with V, η(E). In the remainder of this section, let 𝒢 = V, E, , be a directed multigraph. Let e be an edge of 𝒢. The edge e is called looploop e if and only if (e) = (e).Let v be a vertex of 𝒢. * The cardinal number^+(v) = e ∈ E (e) = vout-degree ^+(v) of v[symbols]degplusv@^+(v)is called out-degree of vdegree!out-.* The cardinal number^-(v) = e ∈ E (e) = vin-degree ^-(v) of v[symbols]degminusv@^-(v)is called in-degree of vdegree!in-.* The cardinal number(v) = ^+(v) + ^-(v) degree (v) of v[symbols]degreev@(v)is called degree of vdegree@degree of v. In the degree of v loops are counted twice.Let v and v' be two vertices of 𝒢. They are called adjacentadjacent vertices v and v' if and only if there is an edge from v to v' or one from v' to v. Let p = e_i_i ∈1, 2, …, k be a finite sequence of edges of 𝒢. It is called path from (e_1) to (e_k)path p from (e_1) to (e_k)[symbols]p@p if and only if, for each index i ∈1, 2, …, k-1, we have (e_i) = (e_i + 1). Let p = e_i_i ∈1, 2, …, k be a path in 𝒢. The non-negative integer p = k is called length of plength p of p[symbols][email protected] directed multigraph 𝒢 is called * symmetricsymmetric directed multigraph if and only if, for each edge e ∈ E, there is an edge e' ∈ E such that (e') = (e) and (e') = (e); * strongly connectedstrongly connected directed multigraph if and only if, for each vertex v ∈ V and each vertex v' ∈ V, there is a path p from v to v'; * regularregular directed multigraph if and only if all vertices of 𝒢 have the same degree and, for each vertex v ∈ V, we have ^-(v) = ^+(v). Let W be a subset of V and let F be a subset of E such that (F) ⊆ W and (F) ⊆ W. The directed multigraph W, F, _F → W, _F → W is called submultigraph of 𝒢submultigraph of 𝒢.Let W be a subset of V, let F be the set e ∈ E (e), (e) ∈ W, let ς be the map _F → W, and let υ be the map _F → W. The submultigraph 𝒢[W] = W, F, ς, υ of 𝒢 is called submultigraph 𝒢[W] of 𝒢 induced by Winduced by W[symbols]GWcalligraphicbrackets@𝒢[W]. Let 𝒢 = V, E, , be symmetric and strongly connected. The mapV × V→_0, distanceon 𝒢[symbols]d@(v, v')↦minpp path from v to v',is a metric on V and called distance on 𝒢.Let Λ be a set and let λ be a map from E to Λ. The map λ is called Λ-edge-labelling of 𝒢edge-labelling of 𝒢@Λ-edge-labelling of 𝒢Λ-edge-labelling λ of 𝒢[symbols]lambda@λ and the multigraph 𝒢 equipped with λ is called Λ-edge-labellededge-labelled@Λ-edge-labelledΛ-edge-labelled directed multigraph. In the case that 𝒢 is a directed graph, the Λ-edge-labelled graph 𝒢 is identified with V, (σ(e), λ(e), τ(e))e ∈ E.§ COLOURED AND UNCOLOURED LEFT CAYLEY GRAPHS In this section, let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space and let S be a right-generating set of ℛ.* The directed graphM, (m, ms)m ∈ M, s ∈ Sis called uncoloured (left) S-Cayley graph of ℛuncoloured left S-Cayley graph of ℛleft S-Cayley graph of ℛ!uncolouredCayley graph of ℛ@left S-Cayley graph of ℛCayley graph of ℛ!uncoloured.* The S-edge-labelled directed graphM, (m, s, ms)m ∈ M, s ∈ Sis called coloured (left) S-Cayley graph of ℛcoloured left S-Cayley graph of ℛleft S-Cayley graph of ℛ!colouredCayley graph of ℛ!coloured. Examples of coloured Cayley graphs of groups that illustrate their dependence on the generating set are given in examples 6.3.2 in <cit.>. Because the semi-actionis free, the uncoloured and coloured S-Cayley graphs are isomorphic (if we ignore labels). Moreover, each automorphism of the coloured S-Cayley graph is an automorphism of the uncoloured one. However, because automorphisms of labelled graphs are label-preserving, the converse is not true in general. According to <ref>, because G_0 · S ⊆ S, the uncoloured S-Cayley graph of ℛ does not depend on the coordinates g_m_0, m_m ∈ M; and, for each element g ∈ G, it is equal to the uncoloured (gS)-Cayley graph of ℳ, gm_0, g_m_0, m g^-1_m ∈ M. In the remainder of this section, let 𝒢 be the uncoloured or coloured S-Cayley graph of ℛ. Because the element G_0 ∈ GG_0 is the only one that acts trivially by , the following three statements are equivalent: * G_0 ∈ S;* At least one vertex of 𝒢 has a loop;* All vertices of 𝒢 have a loop.Becauseis a semi-action and G_0 · S ⊆ S, if S is symmetric, then 𝒢 is symmetric and strongly connected. Let m be a vertex of 𝒢. The map S → mS, s ↦ ms, is a bijection onto the out-neighbourhood of m. It is injective, becauseis free, and it is surjective, by definition. Therefore, if S is symmetric, then the degree of m is 2 S in cardinal arithmetic and the graph 𝒢 is regular. § METRICS AND LENGTHSIn this section, let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space and let S be a symmetric right-generating set of ℛ. Contents. In <ref> we introduce the S-metric _S and the S-length _S on ℛ induced by the uncoloured S-Cayley graph. And in <ref> we show how the S-metric relates to the left group actionand the right quotient set semi-action . The distance on the uncoloured S-Cayley graph of ℛ is called S-metric on ℛmetric on ℛ@S-metric on ℛS-metric _S on ℛ and denoted by _S[symbols]dS@_S.The S-metric on ℛ is the map_SM × M→_0, (m, m')↦min{[t] k ∈_0s_i_i ∈1, 2, …, k inSm s_i_i ∈1, 2, …, k = m'}.According to <ref>, the S-metric on ℛ does not depend on the coordinates g_m_0, m_m ∈ M and is identical to the (gS)-metric on ℳ, gm_0, g_m_0, m g^-1_m ∈ M. Let m and m' be two elements of M, and let s be an element of S. Then, _S(m, m's) ≤_S(m, m') + 1.Let k = _S(m, m'). Then, there is a finite sequence s_i_i ∈1, 2, …, k in S such that m s_i_i ∈1, 2, …, k = m'. Hence, m s_1, s_2, …, s_k, s = m's. Therefore, _S(m, m's) ≤_S(m, m') + 1.Let m and m' be two elements of M, and let g be an element of G. Then, _S(gm, gm') = _S(m, m').Let k = _S(gm, gm'). Then, there is a finite sequence s_i_i ∈1, 2, …, k in S such that (gm) s_i_i ∈1, 2, …, k = gm'. And, becausesemi-commutes with , there is a finite sequence g_i,0_i ∈1, 2, …, k in G_0 such that (gm) s_i_i ∈1, 2, …, k = g(m g_i,0· s_i_i ∈1, 2, …, k). Hence, g(m g_i,0· s_i_i ∈1, 2, …, k) = m'. Therefore, _S(m, m') ≤ k = _S(gm, gm'). Analogously, _S(gm, gm') ≤_S(g^-1 (gm), g^-1 (gm')) = _S(m, m'). In conclusion, _S(gm, gm') = _S(m, m').Let m and m' be two elements of M, let s_i_i ∈1, 2, …, _S(m, m') be a finite sequence in S such that m' = m s_i_i ∈1, 2, …, _S(m, m'), let i be an element of 0, 1, 2, …, _S(m, m'), and let m_i = m s_i_i ∈1, 2, …, i. Then, _S(m, m_i) = i.By definition of m_i, we have _S(m, m_i) ≤ i and _S(m_i, m') ≤_S(m, m') - i. Therefore, because _S(m, m') ≤_S(m, m_i) + _S(m_i, m'),_S(m, m_i)≥_S(m, m') - _S(m_i, m')≥_S(m, m') - []_S(m, m') - i= iIn conclusion, _S(m, m_i) = i. The map_SM→_0, S-length _S on ℛ m↦_S(m_0, m),is called S-length on ℛlength on ℛ@S-length on ℛ. For each element m ∈ M, we have m_S = 0 if and only if m = m_0. § BALLS AND SPHERES In this section, let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space and let S be a symmetric right-generating set of ℛ. Contents. In <ref> we introduce balls _S and spheres _S in the S-metric on ℛ. In <ref> we show how the left group actionand the right quotient set semi-actionact on balls and spheres. And in <ref> we calculate and approximate the distances of balls and spheres in various circumstances.Let m be an element of M and let ρ be an integer. * The set_S(m, ρ) = m' ∈ M _S(m, m') ≤ρS-ball _S(m, ρ) of radius ρ centred at m[symbols]BSmrho@_S(m, ρ)is called S-ball of radius ρ centred at mball of radius ρ centred at m@S-ball of radius ρ centred at m. The ball of radius ρ centred at m_0 is denoted by _S(ρ)[symbols]BSrho@_S(ρ).* The set_S(m, ρ) = m' ∈ M _S(m, m') = ρS-sphere _S(m, ρ) of radius ρ centred at m[symbols]SSmrho@_S(m, ρ)is called S-sphere of radius ρ centred at msphere of radius ρ centred at m@S-sphere of radius ρ centred at m. The sphere of radius ρ centred at m_0 is denoted by _S(ρ)[symbols]SSrho@_S(ρ).According to <ref>, the S-balls and spheres of ℛ do not depend on the coordinates g_m_0, m_m ∈ M and are identical to the (gS)-balls and spheres of ℳ, gm_0, g_m_0, m g^-1_m ∈ M.For each negative integer ρ,_S(m, ρ) = _S(m, ρ) = ∅.And, _S(m, 0) = _S(m, 0) = m. For each integer ρ,_S(m, ρ) = _S(m, ρ) ∖_S(m, ρ - 1).Because the metric _S is symmetric, for each integer ρ, each element m ∈ M, and each element m' ∈ M,m' ∈_S(m, ρ)m ∈_S(m', ρ)andm' ∈_S(m, ρ)m ∈_S(m', ρ). For each integer ρ,_S(ρ) = m ∈ M m_S ≤ρand_S(ρ) = m ∈ M m_S = ρ. Let A_k_k ∈_0 be a sequence of sets. It is called * non-decreasingnon-decreasing sequencedecreasingnon@non-decreasing if and only ifk ∈_0A_k ⊆ A_k + 1. * non-increasingnon-increasing sequenceincreasingnon@non-increasing if and only ifk ∈_0A_k + 1⊆ A_k. Let A_k_k ∈_0 be a sequence of sets. * The setlim inf_k →∞ A_k = ⋃_k ∈_0⋂_j ∈_0 j ≥ k A_j limit inferior lim inf_k →∞ A_k of A_k_k ∈_0[symbols]liminfktoinfinityAk@lim inf_k →∞ A_kis called limit inferior of A_k_k ∈_0.* The setlim sup_k →∞ A_k = ⋂_k ∈_0⋃_j ∈_0 j ≥ k A_j limit superior lim sup_k →∞ A_k of A_k_k ∈_0[symbols]limsupktoinfinityAk@lim sup_k →∞ A_kis called limit superior of A_k_k ∈_0.* Let A be a set. The sequence A_k_k ∈_0 is said to A_k_k ∈_0 converges to Aconverge to A, the set A is called limit set of A_k_k ∈_0limit set lim_k →∞ A_k of A_k_k ∈_0, and A is denoted by lim_k →∞ A_k[symbols]limktoinfinityAk@lim_k →∞ A_k if and only iflim inf_k →∞ A_k = lim sup_k →∞ A_k = A. * The sequence A_k_k ∈_0 is called convergentconvergent sequence A_k_k ∈_0 if and only iflim inf_k →∞ A_k = lim sup_k →∞ A_k. Let A_k_k ∈_0 be a non-decreasing or non-increasing sequence of sets with respect to inclusion. It converges to ⋃_k ∈_0 A_k or ⋂_k ∈_0 A_k respectively. Let m be an element of M. Then, _S(m, 0) = m and the sequence _S(m, ρ)_ρ∈_0 is non-decreasing with respect to inclusion and converges to M. In particular, for each non-negative integer ρ, ⋃_ρ' ∈_0ρ' ≥ρ_S(m, ρ') = M.Indeed, for each m ∈ M, each ρ∈_0, and each m' ∈ M, we have m' ∈_S(m, ρ'), where ρ' = maxρ, _S(m, m'). For each element m ∈ M and each integer ρ, in cardinal arithmetic,_S(m, ρ)≤ (1 + S)^ρ,because the map(G_0∪ S)^ρ →_S(m, ρ), s_i_i ∈1, 2, …, ρ ↦ m s_i_i ∈1, 2, …, ρ,is surjective and G_0∪ S^ρ≤ (1 + S)^ρ.Let A be a finite subset of M and let S' be the set G_0∪ S. Then, there is a non-negative integer k such thatA ⊆m ∈ M s_i'_i ∈1, 2, …, k inS'm_0 s_i'_i ∈1, 2, …, k = m. If A is empty, then any k ∈_0 works. Otherwise, let k = sup_a ∈ A_S(m_0, a). Then, because A is non-empty and finite, we have k ∈_0. And, by the choice of k, we have A ⊆(m_0, k). And, because G_0 ∈ S' and G_0 = _M, we have (m_0, k) = m ∈ M s_i'_i ∈1, 2, …, k inS'm_0 s_i'_i ∈1, 2, …, k = m. In conclusion, the stated inclusion holds. §.§ Acting on Balls and Spheres Let m be an element of M, let ρ be a non-negative integer, and let s be an element of S. Then, _S(m, ρ)s ⊆_S(m, ρ + 1). Let m' ∈_S(m, ρ)s. Then, there is an m”∈_S(m, ρ) such that m” s = m'. Hence, according to <ref>, we have _S(m, m') = _S(m, m” s) ≤_S(m, m”) + 1 ≤ρ + 1. Therefore, m' ∈_S(m, ρ + 1). In conclusion, _S(m, ρ)s ⊆_S(m, ρ + 1). Let m be an element of M, let ρ be a non-negative integer, and let g be an element of G. Then, g _S(m, ρ) = _S(gm, ρ).First, let m' ∈ g _S(m, ρ). Then, g^-1 m' ∈_S(m, ρ) and thus _S(m, g^-1 m') ≤ρ. Hence, according to <ref>,_S(gm, m')= _S(g^-1 (gm), g^-1 m')= _S(m, g^-1 m')≤ρ.Therefore, m' ∈_S(gm, ρ). In conclusion, g _S(m, ρ) ⊆_S(gm, ρ). Secondly, let m' ∈_S(gm, ρ). Then, _S(gm, m') ≤ρ. Thus, according to <ref>,_S(m, g^-1 m')= _S(gm, g(g^-1 m'))= _S(gm, m')≤ρ.Hence, g^-1 m' ∈_S(m, ρ). Therefore, m' ∈ g _S(m, ρ). In conclusion, _S(gm, ρ) ⊆ g _S(m, ρ).Let m be an element of M, let ρ be a non-negative integer, and let g_m be an element of G_m. Then, g_m _S(m, ρ) = _S(m, ρ). In particular, G_m _S(m, ρ) = _S(m, ρ).This is a direct consequence of <ref>, because g_mm = m. Let m and m' be two elements of M, and let ρ be a non-negative integer. Then, _S(m, ρ) = _S(m', ρ).This is a direct consequence of <ref>, because there is a g ∈ G such that gm = m', and g is injective Recall from <ref> that, under the identification of M with GG_0 by ι m ↦ G_m_0, m,g ∈ Gm ∈ Mg · m = gm,andm ∈ Mm' ∈ Mmm' = g_m_0, m m'.Let m, m', and m” be three elements of M and identify M with GG_0 by ι m ↦ G_m_0, m. Then, there is an element g_0 ∈ G_0 such that (mm')m” = m(m'(g_0m”)).Becauseis a right semi-action, there is an element g_0 ∈ G_0 such that mg_m_0, m'· g_0 · G_m_0, m” = (mg_m_0, m' G_0)G_m_0, m”. And, according to <ref>, we have G_m_0, m” = m”, g_m_0, m' G_0 = G_m_0, m' = m', and g_m_0, m'· g_0 · G_m_0, m” = m'(g_0m”). Therefore, m(m'(g_0m”)) = (mm')m”. Let m be an element of M, let ρ be a non-negative integer, and identify M with GG_0 by ι m ↦ G_m_0, m. Then, m _S(ρ) = _S(m, ρ).According to <ref> and <ref>,m _S(ρ)= g_m_0, m_S(ρ)= _S(g_m_0, m m_0, ρ)= _S(m, ρ).Let m be an element of M, let ρ and ρ' be two non-negative integers, and identify M with GG_0 by ι m ↦ G_m_0, m. Then, _S(m, ρ) _S(ρ') = _S(m, ρ + ρ').First, let m' ∈_S(m, ρ) _S(ρ'). Then, there is an m”∈_S(m, ρ) such that m' ∈ m”_S(ρ'). And, according to <ref>, we have m”_S(ρ') = _S(m”, ρ'). Hence, because _S is subadditive, we have _S(m, m') ≤_S(m, m”) + _S(m”, m') ≤ρ + ρ'. Therefore, m' ∈_S(m, ρ + ρ'). In conclusion, _S(m, ρ) _S(ρ') ⊆_S(m, ρ + ρ').Secondly, let m' ∈_S(m, ρ + ρ'). Case 1: m' ∈_S(m, ρ). Then, because m_0 ∈_S(ρ'), we have m' = m'm_0 ∈_S(m, ρ) _S(ρ').Case 2: m' ∉_S(m, ρ). Then, there is a j ∈1, 2, …, ρ' and there is a finite sequence s_i_i ∈1, 2, …, ρ + j in S such that m”s_i_i ∈ρ + 1, ρ + 2, …, ρ + j = m', where m” = m s_i_i ∈1, 2, …, ρ∈_S(m, ρ). Hence, m' ∈_S(m”, j) ⊆_S(m”, ρ') = m”_S(ρ') ⊆_S(m, ρ) _S(ρ').In either case, m' ∈_S(m, ρ) _S(ρ'). In conclusion, _S(m, ρ + ρ') ⊆_S(m, ρ) _S(ρ'). §.§ Distances of Balls and SpheresLet A and A' be two subsets of M. The non-negative number or infinity_S(A, A') = min_S(a, a')a ∈ A, a' ∈ A'distance _S(A, A') of A and A'[symbols]dSAAprime@_S(A, A')is called distance of A and A', where we put min∅ = ∞. In the case that A = a, we write _S(a, A') in place of _S(a, A'); and in the case that A' = a', we write _S(A, a') in place of _S(A, a'). Let m and m' be two elements of M, and let ρ be a non-negative integer such that ρ≤_S(m, m'). Then, _S(_S(m, ρ), m') = _S(m, m') - ρ.Let ρ' = _S(m, m'). Then, there is a finite sequence s_i_i ∈1,2,…,ρ' in S such that m s_i_i ∈1,2,…,ρ' = m'. Let m” = m s_i_i ∈1,2,…,ρ. Then, m”s_i_i ∈ρ + 1, ρ + 2, …, ρ' = m'. And, according to <ref>, we have m”∈_S(m, ρ). Thus, _S(_S(m, ρ), m') ≤_S(m”, m') ≤ρ' - ρ.Suppose that _S(_S(m, ρ), m') < ρ' - ρ. Then, there is an m”∈_S(m, ρ) such that _S(m”, m') < ρ' - ρ. Hence, _S(m, m') ≤_S(m, m”) + _S(m”, m') < ρ + (ρ' - ρ) = ρ', which contradicts _S(m, m') = ρ'. Therefore, _S(_S(m, ρ), m') ≥ρ' - ρ. In conclusion, _S(_S(m, ρ), m') = ρ' - ρ = _S(m, m') - ρ. Let m be an element of M, and let ρ and ρ' be two non-negative integers such that the spheres _S(m, ρ) and _S(m, ρ') are non-empty. Then, _S(_S(m, ρ), _S(m, ρ')) = ρ - ρ'.Without loss of generality, let ρ≤ρ'. Then, for each m' ∈_S(m, ρ'), according to <ref>, we have _S(_S(m, ρ), m') = ρ' - ρ. In conclusion, _S(_S(m, ρ), _S(m, ρ')) = ρ' - ρ = ρ - ρ'. Let m and m' be two elements of M, and let ρ be a non-negative integer. Then, _S(_S(m, ρ), m') ≥_S(m, m') - ρ.If _S(m, ρ) = ∅, then _S(_S(m, ρ), m') = ∞≥_S(m, m') - ρ. Otherwise, let ρ' = _S(m, m'). Then, m' ∈_S(m, ρ') ≠∅. Hence, according to <ref>,_S(_S(m, ρ), m')≥_S(_S(m, ρ), _S(m, ρ'))= ρ - ρ'= _S(m, m') - ρ.Let m and m' be two elements of M, and let ρ and ρ' be two non-negative integers such that ρ + ρ' ≤_S(m, m'). Then, _S(_S(m, ρ), _S(m', ρ')) = _S(m, m') - (ρ + ρ').First, for each m_ρ∈_S(m, ρ) and each m'_ρ'∈_S(m', ρ'), because _S is subadditive,_S(m, m')≤_S(m, m_ρ) + _S(m_ρ, m'_ρ') + _S(m'_ρ', m')≤ρ + _S(m_ρ, m'_ρ') + ρ',and hence _S(m_ρ, m'_ρ') ≥_S(m, m') - (ρ + ρ'). In conclusion,_S(_S(m, ρ), _S(m', ρ')) ≥_S(m, m') - (ρ + ρ'). Secondly, there is a finite sequence s_i_i ∈1,2,…,_S(m, m') in S such that m s_i_i ∈1,2,…,_S(m, m') = m'. Let m_ρ = m s_i_i ∈1,2,…,ρ and let m'_ρ' = m_ρs_i_i ∈ρ + 1, ρ + 2, …, _S(m, m') - ρ'. Then, m_ρ∈_S(m, ρ). And, becausem'_ρ's_i_i ∈_S(m, m') - ρ' + 1, _S(m, m') - ρ' + 2, …, _S(m, m') = m',we have m' ∈_S(m'_ρ', ρ') and hence m'_ρ'∈_S(m', ρ'). And, _S(m_ρ, m'_ρ') ≤_S(m, m') - ρ' - ρ. In conclusion,_S(_S(m, ρ), _S(m', ρ')) ≤_S(m, m') - (ρ + ρ'). § INTERIORS, CLOSURES, AND BOUNDARIES In this section, let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a cell space, let S be a symmetric right-generating set of ℛ, let us omit the subscript S of _S, _S, _S, and _S, and let M be identified with GG_0 by ι m ↦ G_m_0, m. Contents. In <ref> we introduce θ-interiors A^-θ, θ-closures A^+θ, and (internal/external) θ-boundaries _θ A, _θ^- A, or _θ^+ A. And in the lemmata and corollaries of this section we characterise them and show how they and the S-metric relate to each other.Let A be a subset of M, let θ be an integer. * The setA^-θ = A^-(θ)(= m ∈ Mm (θ) ⊆ A) θ-interior A^-θ of A[symbols]A-theta@A^-θis called θ-interior of Ainterior of A theta@θ-interior of A.* The setA^+θ = A^+(θ)(= m ∈ M(m (θ)) ∩ A ≠∅) θ-closure A^+θ of A[symbols]A+theta@A^+θis called θ-closure of Aclosure of A theta@θ-closure of A.* The set_θ A = A^+θ∖ A^-θ(= A^+(θ)∖ A^-(θ) = _(θ) A) θ-boundary _θ A of A[symbols]partialthetaA@_θ Ais called θ-boundary of Aboundary of A theta@θ-boundary of A.* The set_θ^- A = A ∖ A^-θ(= A ∖ A^-(θ) = _(θ)^- A) internal θ-boundary _θ^- A of A[symbols]partialtheta-A@_θ^- Ais called internal θ-boundary of Aboundary of A theta!internal.* The set_θ^+ A = A^+θ∖ A (= A^+(θ)∖ A = _(θ)^+ A) external θ-boundary _θ^+ A of A[symbols]partialtheta+A@_θ^+ Ais called external θ-boundary of Aboundary of A theta!external. According to <ref>, the above notions in ℛ do not depend on the coordinates g_m_0, m_m ∈ M and are identical to the respective notions with respect to the symmetric right-generating set gS in ℳ, gm_0, g_m_0, m g^-1_m ∈ M.For each negative integer θ, we have A^-θ = A, and A^+θ = ∅, and _θ A = _θ^- A = _θ^+ A = ∅. Moreover, A^-0 = A^+0 = A and _0 A = _0^- A = _0^+ A = ∅. Furthermore, for each non-negative integer θ, we have M^-θ = M^+θ = M and _θ M = _θ^- M = _θ^+ M = ∅. Let A be a subset of M and let θ be a non-negative integer. Then, *A^-θ = m ∈ A (m, θ) ⊆ A;*A^+θ = ⋃_m ∈ A(m, θ) = A (θ). * According to <ref> and because m ∈(m, θ), for m ∈ M,A^-θ = m ∈ M (m, θ) ⊆ A= m ∈ A (m, θ) ⊆ A. * According to <ref> and <ref>,A^+θ = m ∈ M (m, θ) ∩ A ≠∅= m ∈ Mm' ∈ Am' ∈(m, θ)= m ∈ Mm' ∈ Am ∈(m', θ)= ⋃_m' ∈ A(m', θ)= ⋃_m' ∈ A m' (θ)= A (θ).Let m be an element of M, let ρ be a non-negative integer, and let θ be a non-negative integer. Then, *(m, ρ)^-θ⊇(m, ρ - θ);*(m, ρ)^+θ = (m, ρ + θ);*_θ(m, ρ) ⊆(m, ρ + θ) ∖(m, ρ - θ). * If ρ < θ, then (m, ρ - θ) = ∅ and hence (m, ρ - θ) ⊆(m, ρ)^-θ. Otherwise, according to <ref>, we have (m, ρ - θ) (θ) ⊆(m, ρ) and hence, according to <ref>, we have (m, ρ - θ) ⊆(m, ρ)^-θ. * According to <ref> of <ref> and <ref>, we have (m, ρ)^+θ = (m, ρ) (θ) = (m, ρ + θ).* This is a direct consequence of <ref>.Let M be finite, let ρ be the least integer such that (m, ρ) = M, which is non-negative because M ≠∅, and let θ be a positive integer. Then, (m, ρ)^-θ = M ≠(m, ρ - θ).Let A be a subset of M, and let θ and θ' be two non-negative integers. The following five statements hold: *(A^-θ)^-θ' = A^-(θ + θ');* _θ'^- A^-θ = A^-θ∖ A^-(θ + θ');*(A^+θ)^+θ' = A^+(θ + θ');*_θ'^+ A^+θ = A^+(θ + θ')∖ A^+θ;*If θ' ≤θ, then A^+(θ - θ')⊆ (A^+θ)^-θ' and (A^-θ)^+θ'⊆ A^-(θ - θ'). * For each m' ∈ A, according to <ref> and <ref> of <ref>, we have m' ∈ A^-θ if and only if m' (θ) = (m', θ) ⊆ A. Hence, according to <ref>,(A^-θ)^-θ' = m' ∈ A (m', θ') ⊆ A^-θ= m' ∈ A (m', θ') (θ) ⊆ A= m' ∈ A (m', θ + θ') ⊆ A= A^-(θ + θ'). * This is a direct consequence of <ref>.* According to <ref> of <ref> and <ref>,(A^+θ)^+θ' = A^+θ(θ')= *⋃_m ∈ A(m, θ)(θ')= ⋃_m ∈ A(m, θ) (θ')= ⋃_m ∈ A(m, θ + θ')= A^+(θ + θ'). * This is a direct consequence of <ref>.* Let θ' ≤θ. Then, according to <ref> of <ref> and <ref>, A^+(θ - θ')(θ')= (A^+(θ - θ'))^+θ'= A^+((θ - θ') + θ')= A^+θ.Therefore, according to <ref>, we have A^+(θ - θ')⊆ (A^+θ)^-θ'.Moreover, according to <ref>, <ref> of <ref>, and <ref>, (A^-θ)^+θ'(θ - θ')= ((A^-θ)^+θ')^+(θ - θ')= (A^-θ)^+θ' + (θ - θ')= (A^-θ)^+θ= A^-θ(θ)⊆ A.Therefore, according to <ref>, we have (A^-θ)^+θ'⊆ A^-(θ - θ'). Let M be finite, let M≥ 2, let A = m_0, let θ be the least integer such that A^+θ = M, which is positive because M≥ 2, and let θ' be a positive integer such that θ' ≤θ. Then, A^+(θ - θ')≠ M = (A^+θ)^-θ'. Let θ be a non-negative integer, and let A and A' be two subsets of M. Then, (A, A' ∖ A^+θ) ≥θ + 1.If A or A' ∖ A^+θ is empty, then (A, A' ∖ A^+θ) = ∞≥θ + 1. Otherwise, let m' ∈ A' ∖ A^+θ. Then, according to <ref> of <ref>, we have A' ∖ A^+θ = (A' ∖ A)^-θ. Hence, according to <ref> of <ref>, we have (m', θ) ⊆ A' ∖ A. Therefore, for each m ∈ A, we have m ∉(m', θ) and hence (m, m') ≥θ + 1. Thus, (A, m') ≥θ + 1. In conclusion, (A, A' ∖ A^+θ) ≥θ + 1. Let θ and θ' be two non-negative integers, and let A be a subset of M. Then, (A, _θ'^+ A^+θ) ≥θ + 1.This is a direct consequence of <ref>, because _θ'^+ A^+θ = (A^+θ)^+θ'∖ A^+θ.§ GROWTH FUNCTIONS AND TYPESIn this section we recapitulate growth functions and types, more or less as presented in section 6.4 in <cit.>. Let γ be a map from _0 to _≥ 0. It is called growth function γgrowth function[symbols]gamma@γ if and only if it is non-decreasingnon-decreasing mapdecreasing non@non-decreasing, that is to say, thatk ∈_0k' ∈_0 (k ≤ k' γ(k) ≤γ(k')).Let γ and γ' be two growth functions. The growth function γ is said to dominateγ dominates γ' γ' and we write γγ'γγ'[symbols]greaterthanorequaltocurly@γγ' if and only ifα∈_+k ∈_+ α·γ(α· k) ≥γ'(k). Let γ and γ' be two growth functions. They are called equivalentγ and γ' are equivalent and we write γγ'γγ'[symbols]tilde@ if and only if γγ' and γ' γ. * The binary relationis reflexive and transitive.* The binary relationis an equivalence relation.*If γ_1 γ_2 and γ_1' γ_2', then γ_1 γ_1' implies γ_2 γ_2'. See proposition 6.4.3 in <cit.>.For each growth function γ, the equivalence class γ_ of γ with respect to equivalence class of γ with respect tois denoted by γ_[symbols]gammabracketstilde@γ_. And, each equivalence class Γ with respect tois called growth typegrowth type Γ[symbols]Gamma@Γ. Let Γ and Γ' be two growth types. The growth type Γ is said to dominateΓ dominates Γ' Γ' and we write ΓΓ'ΓΓ'[symbols]greaterthanorequaltocurly@ΓΓ' if and only if γ∈Γγ' ∈Γ' γγ'.According to <ref> of <ref>, the growth type Γ dominates Γ' if and only ifγ∈Γγ' ∈Γ' γγ'.[<cit.>] *The growth function [k ↦ k] dominatesbut they are not equivalent. For each k ∈_+, we have k ≥ 1 = (k). However, for each α∈_+, there is a k ∈_+, for example k = α + 1, such that α(α k) = α < k. * Let r and s be two non-negative real numbers. Then, [k ↦ k^r][k ↦ k^s] if and only if r ≥ s. And, [k ↦ k^r][k ↦ k^s] if and only if r = s. * Let γ be a growth function such that it is a polynomial function of degree d ∈_0. Then, γ [k ↦ k^d]. *Let r and s be two elements of _> 1. Then, [k ↦ r^k][k ↦ s^k]. In particular, [k ↦ r^k] exp. Without loss of generality, suppose that r ≤ s. Then, for each k ∈_+, we have r^k ≤ s^k. Hence, [k ↦ r^k][k ↦ s^k]. Moreover, let α = log_r s∈_+. Then, for each k ∈_+,s^k = (r^log_r s)^k = r^(log_r s) k≤ r^α k≤α r^α k.Hence, [k ↦ r^k][k ↦ s^k]. In conclusion, [k ↦ r^k][k ↦ s^k]. *Let d be a non-negative integer. Then, exp [k ↦ k^d] and exp [k ↦ k^d]. See examples 6.4.4 (d) in <cit.>.Let γ be a growth function and let d be a non-negative integer such that [k ↦ k^d] γ. Then, expγ and expγ.According to <ref> of <ref>, we have exp [k ↦ k^d] and exp [k ↦ k^d]. Hence, becauseis transitive and [k ↦ k^d] γ, we have expγ and expγ.§ SPACES' GROWTH FUNCTIONS AND TYPES In this section, let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space such that there is a finite and symmetric right-generating set S of ℛ. Contents. In <ref> we introduce the S-growth function γ_S of ℛ. In <ref> and its corollaries we show that γ_S is dominated by exp and that the -equivalence class γ_S_ does not depend on S. In <ref> we introduce the growth type γ(ℳ) of ℳ as that equivalence class. In <ref> and its corollary we relate the inclusion-behaviour of the sequence of balls to the cardinality of M. And in <ref> we say what exponential, sub-exponential, polynomial, and intermediate growth of ℛ mean.The mapγ_S _0→_0, S-growth function γ_S of ℛ[symbols]gammaS@γ_Sk↦_S(k),is called S-growth function of ℛgrowth function of ℛ@S-growth function of ℛ. According to <ref>, the S-growth function of ℛ does not depend on the coordinates g_m_0, m_m ∈ M and is identical to the (gS)-growth function of ℳ, gm_0, g_m_0, m g^-1_m ∈ M. According to <ref>, we have γ_S(0) = 1 and the sequence γ_S(k)_k ∈_0 is non-decreasing with respect to the partial order ≤. Moreover, according to <ref>, for each non-negative integer k, we have γ_S(k) ≤ (1 + S)^k.Let S' be a finite and symmetric right-generating set of ℛ and let α be the non-negative integer mink ∈_0 _S(1) ⊆_S'(k). Then, m ∈ Mm' ∈ M _S'(m, m') ≤α·_S(m, m'),in particular, m ∈ M m_S'≤α·m_S. For each m ∈ M, let α_m = mink ∈_0 _S(m, 1) ⊆_S'(m, k), in particular, α_m_0 = α.First, let m ∈ M. Furthermore, let k ∈_0 and let g ∈ G_m_0, m. Then, because g is bijective, we have _S(1) ⊆_S'(k) if and only if g _S(1) ⊆ g _S'(k). And, according to <ref>, we have g _S(1) = _S(m, 1) and g _S'(k) = _S'(m, k). Hence, _S(1) ⊆_S'(k) if and only if _S(m, 1) ⊆_S'(m, k). Therefore, α_m = α. In conclusion, for each m ∈ M, we have α_m = α.Secondly, we prove by induction on the distance k thatk ∈_0m ∈ Mm' ∈ M[]_S(m, m') = k _S'(m, m') ≤α· k. Base Case Let k = 0. Furthermore, let m and m' ∈ M such that _S(m, m') = k. Then, m = m'. Hence, _S'(m, m') = 0. Therefore, _S'(m, m') ≤α· k.Inductive Step Let k ∈_0 such thatm ∈ Mm' ∈ M []_S(m, m') = k _S'(m, m') ≤α· k.Furthermore, let m and m”∈ M such that _S(m, m”) = k + 1. Then, there is a finite sequence s_i_i ∈1, 2, …, k + 1 in S such that m's_k + 1 = m”, where m' = m s_i_i ∈1, 2, …, k. And, according to <ref>, we have _S(m, m') = k. Therefore, according to the inductive hypothesis, _S'(m, m') ≤α· k. Moreover, by definition of α_m', we have m” = m's_k + 1∈_S(m', 1) ⊆_S'(m', α_m'). Hence, because α_m' = α, we have _S'(m', m”) ≤α_m' = α. In conclusion, because _S' is subadditive, we have _S'(m, m”) ≤_S'(m, m') + _S'(m', m”) ≤α· k + α = α· (k + 1). In the situation of <ref>, for each element m ∈ M and each non-negative integer k, we have _S(m, k) ⊆_S'(m, α· k).This is a direct consequence of <ref>, because for each element m ∈ M, each non-negative integer k, and each element m' ∈ M, if _S(m, m') ≤ k, then _S'(m, m') ≤α· k. In the situation of <ref>, for each non-negative integer k, we have γ_S(k) ≤γ_S'(α· k).This is a direct consequence of <ref>.Let S' be a finite and symmetric right-generating set of ℛ. The metrics _S and _S' are Lipschitz equivalent_S and _S' are Lipschitz equivalentequivalent!Lipschitz, that is to say, that there are two positive real numbers κ and ϰ such that κ·_S'≤_S ≤ϰ·_S', where the scalar multiplication · and the partial order ≤ are pointwise.Let α = mink ∈_0 _S(1) ⊆_S'(k) and let α' = mink ∈_0 _S'(1) ⊆_S(k). If α = 0 or α' = 0, then M = m_0, hence _S == _S', and therefore _S ≤_S'≤_S. Otherwise, according to <ref>, we have 1/α·_S'≤_S ≤α' ·_S'.Let S' be a finite and symmetric right-generating set of ℛ. The S-growth function γ_S of ℛ and the S'-growth function γ_S' of ℛ are equivalent.According to <ref>, there is a α∈_0 such that, for each k ∈_0, we have γ_S(k) ≤γ_S'(α· k). Hence, according to <ref>, for each k ∈_0, we have γ_S(k) ≤ (α + 1) γ_S'((α + 1) · k). Therefore, γ_S is dominated by γ_S'. Switching roles of S and S' yields that γ_S' is dominated by γ_S. In conclusion, γ_S and γ_S' are equivalent. The S-growth function γ_S of ℛ is dominated by exp.According to <ref>, for each k ∈_0, we have γ_S(k) ≤ r^k, where r = 1 + S. Hence, γ_S[k ↦ r^k]. Moreover, according to <ref> of <ref>, we have [k ↦ r^k] exp. In conclusion, γ_S exp. The equivalence class γ(ℳ) = γ_S_ is called growth type of ℳgrowth type γ(ℳ) of ℳ[symbols]gammaMcalligraphic@γ(ℳ).Note that, according to <ref> and corollary <ref>, the equivalence class γ_S_ does neither depend on the right-generating set S nor on the coordinate system 𝒦.[Groups <cit.>] The growth type of the groupof integers, of the direct product of the groupsand 2, and of the infinite dihedral group _∞ is [k ↦ k]_. The growth type of the group ^2 is [k ↦ k^2]_. And, the growth type of each free group of finite rank is [exp]_.Let γ be a growth function such that γ(0) > 0. Then, γ is equivalent toif and only if γ is bounded.See proposition 6.4.6 in <cit.>.The set M is finite if and only if the growth types γ(ℳ) and _ are equal.First, let M be finite. Then, for each k ∈_0, we have γ_S(k) ≤M. Hence, according to <ref>, we have γ_S. In conclusion, γ(ℳ) = _.Secondly, let γ(ℳ) = _. Then, according to <ref>, the growth function γ_S is bounded by some ξ∈_> 0. And, according to <ref>, we have M = ⋃_k ∈_0_S(k) and _S(k)_k ∈_0 is non-decreasing with respect to ⊆. Therefore, because γ_S(k)_k ∈_0 = _S(k)_k ∈_0, we have M≤sup_k ∈_0γ_S(k) ≤ξ. In conclusion, M is finite. Either the sequence _S(k)_k ∈_0 is strictly increasing with respect to ⊆ or eventually constanteventually constant sequence, that is to say, that there is a non-negative integer k such that, for each non-negative integer k' with k' ≥ k, we have _S(k') = _S(k).According to <ref>, the sequence _S(k)_k ∈_0 is non-decreasing with respect to ⊆. If it is strictly increasing with respect to ⊆, it is not eventually constant. Otherwise, there is a k ∈_0 such that _S(k) = _S(k + 1). We prove by induction on k' that, for each k' ∈_0 with k' ≥ k, we have _S(k') = _S(k).Base Case Let k' = k. Then, _S(k') = _S(k).Inductive Step Let k' ∈_0 with k' ≥ k such that _S(k') = _S(k). Furthermore, let m ∈_S(k' + 1). Case 1: m ∈_S(k'). Then, according to the inductive hypothesis, m ∈_S(k).Case 2: m ∉_S(k'). Then, there is a finite sequence s_i_i ∈1, 2, …, k' + 1 in S such that m's_k' + 1 = m, where m' = m_0 s_i_i ∈1, 2, …, k'. Hence, m' ∈_S(k') and thus, according to the inductive hypothesis, m' ∈_S(k). Therefore, according to <ref>, we have m ∈_S(k + 1). Thus, because _S(k + 1) = _S(k), we have m ∈_S(k).In either case, m ∈_S(k). Therefore, _S(k' + 1) ⊆_S(k) ⊆_S(k') ⊆_S(k' + 1). In conclusion, _S(k' + 1) = _S(k). The set M is finite if and only if the sequence _S(k)_k ∈_0 is eventually equal to Meventually equal to M sequence, that is to say, that there is a non-negative integer k such that, for each non-negative integer k' with k' ≥ k, we have _S(k') = M.First, let M be finite. Then, _S(k)_k ∈_0 is not strictly increasing. Hence, according to <ref>, it is eventually constant. And, according to <ref>, it converges to M. In conclusion, it is eventually equal to M.Secondly, let _S(k)_k ∈_0 be eventually equal to M. Then, there is a k ∈_0 such that _S(k) = M. In conclusion, according to <ref>, the set M is finite. The set M is infinite if and only if the sequence _S(k)_k ∈_0 is strictly increasing with respect to ⊆.First, let M be infinite. Then, because _S(k) is finite, for k ∈_0, and _S(k)_k ∈_0 converges to M, the sequence _S(k)_k ∈_0 is not eventually constant. In conclusion, according to <ref>, it is strictly increasing with respect to ⊆.Secondly, let _S(k)_k ∈_0 be strictly increasing with respect to ⊆. Then, because it converges to M, the set M is infinite. The set M is infinite if and only ifρ∈_0 _S(ρ) ≠∅. We have _S(0) = m_0≠∅. And, according to <ref>, for each ρ∈_+, we have _S(ρ) = _S(ρ) ∖_S(ρ - 1). Hence, the sequence _S(k)_k ∈_0 is strictly increasing with respect to ⊆ if and only if <ref> holds. Therefore, according to <ref>, the set M is infinite if and only if <ref> holds. The set M is infinite if and only if the growth type of ℳ dominates k ↦ k_.First, let M be infinite. Then, according to <ref>, the sequence _S(k)_k ∈_0 is strictly increasing with respect to ⊆. Hence, because _S(0) = m_0, for each k ∈_0, we have γ_S(k) = _S(k)≥ k + 1. In conclusion, γ_S dominates [k ↦ k] and hence γ(ℳ) dominates k ↦ k_.Secondly, let M be finite. Then, according to <ref>, we have γ(ℳ) = _. Hence, according to <ref> of <ref>, the growth type γ(ℳ) does not dominate k ↦ k_. Because Cayley graphs of ℛ are in a sense quotients of Cayley graphs of G by the finite subgroup G_0, the distances on these graphs are related by the multiplicative constant G_0, which is implicitly used inLet the group G be finitely generated and let the stabiliser G_0 be finite. The growth types of G and ℳ are equal.Because the group G_0 is finite, there is a finite and symmetric generating set T of G such that G_0 T ⊆ T. And, according to <ref>, the set S = t G_0t ∈ T = g_0 · t G_0g_0 ∈ G_0, t ∈ T is a finite and symmetric right-generating set of ℛ.Let k be a non-negative integer. Furthermore, let m be an element of _S^ℛ(k). Then, there is a non-negative integer j ∈0,1, 2, …, k and there is a finite sequence s_i_i ∈1, 2, …, j of elements in S such thatm = m_0 s_i_i ∈1, 2, …, j.And, by the definition of S, there is a finite sequence t_i_i ∈1, 2, …, j of elements in T such that t_i G_0_i ∈1, 2, …, j = s_i_i ∈1, 2, …, j. And, becauseis a semi-action, there is a finite sequence g_i,0_i ∈1, 2, …, j of elements in G_0 such thatm= m_0 s_i_i ∈1, 2, …, j= m_0t_1 g_2,0 t_2 g_3,0 t_3 … g_j,0 t_j G_0= g_1,0 t_1 g_2,0 t_2 g_3,0 t_3 … g_j,0 t_jm_0,where g_1,0 = e_G. And, because G_0 T ⊆ T, the sequence g_i,0 t_i_i ∈1, 2, …, j is one of elements in T. Hence, the element m is contained in _T^G(j)m_0, which is included in _T^G(k)m_0. Therefore, the set _S^ℛ(k) is included in _T^G(k)m_0. Analogously, one can show that the set _T^G(k)m_0 is included in _S^ℛ(k). It follows that _S^ℛ(k) is equal to _T^G(k)m_0.Hence, _S^ℛ(k)≤_T^G(k). And, because the map m_0 from G to M is G_0-to-1 surjective, we have _T^G(k)≤G_0·_S^ℛ(k)≤G_0·_S^ℛ(G_0· k). Therefore, the growth function γ_T^G dominates γ_S^ℛ and vice versa. In conclusion, the growth types γ(G) and γ(ℳ) are equal.The left-homogeneous space ℳ is said to have* exponential growthexponential growthgrowth!exponential if and only if its growth type is exp_;* sub-exponential growthsub-exponential growthgrowth!sub-exponential if and only if it does not have exponential growth;* polynomial growthpolynomial growthgrowth!polynomial if and only if there is a non-negative integer d ∈_0 such that k ↦ k^d_ dominates γ(ℳ).* intermediate growthintermediate growthgrowth!intermediate if and only if it has sub-exponential growth but not polynomial growth.[Free Groups of Finite Rank] Each free group of finite rank has exponential growth (see examples 6.4.11 (g) in <cit.>). [Virtually Nilpotent Groups]Each virtually nilpotent finitely generated group has polynomial growth (see corollary 6.8.5 in <cit.>. Examples of such groups are abelian groups, like the group of integers under addition, nilpotent but non-abelian groups, like the discrete Heisenberg group, and virtually nilpotent but non-nilpotent groups, like the infinite dihedral group.[Grigorchuk Group]Rostislav Ivanovich Grigorchuk was the first to construct a finitely generated group of intermediate growth in 1984. This group is known as Grigorchuk groupGrigorchuk group. The original construction and proofs were published in the paper *grigorchuk:1985<cit.>; a more accessible exposition can be found in section 6.9 in <cit.>. [Left Homogeneous Spaces] Each quotient set of any group of the previous examples by a finite subgroup acted upon by left multiplication is, according to <ref>, a left-homogeneous space of the respective growth. Let ℳ have polynomial growth. It has sub-exponential growth.There is a d ∈_0 such that k ↦ k^d_γ(ℳ). Hence, [k ↦ k^d] γ_S. Therefore, according to <ref>, we have expγ_S. In conclusion, γ(ℳ) ≠exp_.§ GROWTH RATESIn this section, let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a cell space such that there is a finite and symmetric right-generating set S of ℛ. Contents. In <ref> we introduce the S-growth rate of ℛ. And in <ref> show how that growth rate and exponential growth relate to each other. The sequence √(γ_S(k))_k ∈_0 converges to inf_k ∈_0√(γ_S(k))∈_≥ 1.According to <ref>,γ_S(k + k')=_S(k + k')=_S(k) _S(k')≤_S(k)·_S(k')=γ_S(k) ·γ_S(k').Hence, according to lemma 6.5.1 in <cit.>, the sequence √(γ_S(k))_k ∈_0 converges to inf_k ∈_0√(γ_S(k)). Moreover, because, for each k ∈_0, we have γ_S(k) ≥ 1, that limit point is in _≥ 1.The limit point λ_S = lim_k →∞√(γ_S(k)) is called S-growth rate of ℛgrowth rate of ℛ@S-growth rate of ℛS-growth rate λ_S of ℛ[symbols]lambdaS@λ_S. [Lattice] In the situation of <ref>, the (-1, 0), (0, -1), (0, 1), (1, 0)-growth rate lim_k →∞√((k)) of ℛ is equal to 1. [Tree] In the situation of <ref>, the a, b, a^-1, b^-1-growth rate lim_k →∞√((k)) of ℛ is equal to 3.The S-growth rate of ℛ is greater than 1 if and only if the left-homogeneous space ℳ has exponential growth.First, let λ_S > 1. According to <ref>, for each k ∈_0, we have √(γ_S(k))≥λ_S and hence γ_S(k) ≥λ_S^k. Hence, γ_S dominates λ_S^(). And, because λ_S > 1, according to <ref> of <ref>, the growth function λ_S^() is equivalent to exp. Therefore, γ_S dominates exp. Moreover, according to <ref>, the growth function γ_S is dominated by exp. Altogether, γ_S and exp are equivalent. In conclusion, γ(ℳ) = γ_S_ = exp_.Secondly, let γ(ℳ) = exp_. Then, γ_S and exp are equivalent. In particular, γ_S dominates exp. Hence, there is a α∈_+ such that, for each k ∈_+, we have α·γ_S(α· k) ≥exp(k). Therefore, for each k ∈_+, √(α)·√(γ_S(α· k)) =√(α·γ_S(α· k))≥√(exp(k)) =√().Thus, because √(α)_k ∈_+ converges to 1 and √(γ_S(α· k))_k ∈_+, as subsequence of √(γ_S(k))_k ∈_0, converges to λ_S, we conclude that λ_S ≥√() > 1.The S-growth rate of ℛ is equal to 1 if and only if the left-homogeneous space ℳ has sub-exponential growth.This is a direct consequence of <ref>. Let S' be a finite and symmetric right-generating set of ℛ. The S-growth rate of ℛ is equal to 1 or greater than 1 if and only if the S'-growth rate of ℛ is equal to 1 or greater than 1 respectively. This is a direct consequence of <ref> and theorem <ref>. § AMENABILITY, FØLNER CONDITIONS/NETS, AND ISOPERIMETRIC CONSTANTS In this section, let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a finitely right-generated cell space such that the stabiliser G_0 is finite, and let S be a finite and symmetric right-generating set of ℛ (note that such a set S exists due to the assumptions on ℛ). Contents. In <ref> we introduce the S-isoperimetric constant of ℛ, which measures, broadly speaking, the invariance under _M × S that a finite subset of M can have, where 0 means maximally and 1 minimally invariant. In <ref> we show that ℛ is right amenable if and only if a kind of Følner condition holds, which in turn holds if and only if the S-isoperimetric constant is 0. And in <ref> we characterise right Følner nets using ρ-boundaries.Let E be a subset of GG_0 and let ℱ be the set F ⊆ MF ≠∅, Ffinite. The non-negative real numberι_E(ℛ) = inf_F ∈ℱ⋃_e ∈ E F ∖ ( e)^-1(F)/F∈0, 1E-isoperimetric constant ι_E(ℛ) of ℛ[symbols]iotaERcalligraphic@ι_E(ℛ)is called E-isoperimetric constant of ℛisoperimetric constant of ℛ@E-isoperimetric constant of ℛ. [Lattice]In the situation of <ref>, because the balls (ρ), for ρ∈_0, are non-empty and finite, and the sequence (ρ)_ρ∈_0 is a right Følner net in ℛ, for each finite subset E of ^2, the E-isoperimetric constant of ℛ is equal to 0. Let 𝔤 and 𝔤' be two elements of GG_0, and let A, B, and C be three sets. Then, *((𝔤) 𝔤')^-1(A) = (𝔤)^-1((𝔤')^-1(A));* (𝔤)^-1(A ∖ B) = (𝔤)^-1(A) ∖ (𝔤)^-1(B). Let A, B, and C be three finite sets. Then,A ∖ B≤A ∖ C + C ∖ B.Let A be a subset of M, let 𝔤 and 𝔤' be two elements of GG_0, and identify M with GG_0 by ι m ↦ G_m_0, m. Then,(𝔤)^-1(A) ∖( (𝔤𝔤'))^-1(A) ⊆⋃_g_0 ∈ G_0 (𝔤)^-1(A ∖ ( g_0 ·𝔤')^-1(A)). If (𝔤)^-1(A) ∖ ( (𝔤𝔤'))^-1(A) is empty, then there is nothing to show. Otherwise, let m ∈ (𝔤)^-1(A) ∖ ( (𝔤𝔤'))^-1(A). Then, according to <ref>, there is a g_0 ∈ G_0 such that (m 𝔤)g_0 ·𝔤' = m(𝔤𝔤') ∉ A. Therefore, m ∉ ((𝔤)g_0 ·𝔤')^-1(A) and hence m ∈ (𝔤)^-1(A) ∖ ((𝔤)g_0 ·𝔤')^-1(A).Moreover, according to <ref> of <ref>, we have ((𝔤)g_0 ·𝔤')^-1(A) = (𝔤)^-1(( g_0 ·𝔤')^-1(A)). Hence, according to <ref> of <ref>, we have (𝔤)^-1(A) ∖ ((𝔤)g_0 ·𝔤')^-1(A) = (𝔤)^-1(A ∖ ( g_0 ·𝔤')^-1(A)). Therefore, m ∈⋃_g_0 ∈ G_0 (𝔤)^-1(A ∖ ( g_0 ·𝔤')^-1(A)). In conclusion, the stated inclusion holds.Let ℱ be a subset of F ⊆ MF ≠∅, Ffinite. The following two statements are equivalent: *For each positive real number ε∈_> 0, there is an element F ∈ℱ such thats ∈ S F ∖ ( s)^-1(F)/F < ε; *For each finite subset E of GG_0 and each positive real number ε∈_> 0, there is an element F ∈ℱ such thate ∈ E F ∖ ( e)^-1(F)/F < ε.First, because S is a finite subset of GG_0, <ref> implies <ref>.Secondly, let <ref> hold. Furthermore, let ε' ∈_> 0, let E be a finite subset of GG_0, and identify M with GG_0 by ι m ↦ G_m_0, m. Then, according to <ref>, there is a k ∈_0 such thatE ⊆m ∈ M s_i'_i ∈1, 2, …, k inS'm_0 s_i'_i ∈1, 2, …, k,where S' = G_0∪ S. Let ε = ε' / (G_0^2 · k) and let F ∈ℱ such that <ref> holds. Furthermore, let e ∈ E. Then, there is a finite sequence s_i'_i ∈1, 2, …, k in S' such that m_0 s_i'_i ∈1, 2, …, k = e. For each i ∈0, 1, …, k, let m_i = m_0 s_j'_j ∈1, 2, …, i and let F_i = ( m_i)^-1(F). Note that m_k = e and that F_0 = F. Then, according to <ref>,F ∖ F_k =F_0 ∖ F_k≤F_0 ∖ F_1 + F_1 ∖ F_k≤F_0 ∖ F_1 + F_1 ∖ F_2 + F_2 ∖ F_k≤…≤∑_i = 1^k F_i - 1∖ F_i.Let i ∈1, 2, …, k. Then, because m_i = m_i - 1 s_i', we have F_i - 1∖ F_i = ( m_i - 1)^-1(F) ∖ ( (m_i - 1 s_i'))^-1(F). Hence, according to <ref>, we have F_i - 1∖ F_i ⊆⋃_g_0 ∈ G_0 ( m_i - 1)^-1(F ∖ ( g_0 · s_i')^-1(F)). Therefore,F_i - 1∖ F_i≤∑_g_0 ∈ G_0( m_i - 1)^-1(F ∖ ( g_0 · s_i')^-1(F)).Thus, according to <ref>,F_i - 1∖ F_i≤∑_g_0 ∈ G_0G_0·F ∖ ( g_0 · s_i')^-1(F).Hence, because G_0 · S' ⊆ S', F ∖ ( G_0)^-1(F) = F ∖ F = ∅, and <ref> holds,F_i - 1∖ F_i < ∑_g_0 ∈ G_0G_0·ε·F= G_0^2 ·ε·F= G_0^2 ·ε'/G_0^2 · k·F= ε'/k·F.Therefore,F ∖ F_k≤∑_i = 1^k F_i - 1∖ F_i <k ·ε'/k·F =ε' ·F.Thus, because F_k = ( e)^-1(F),F ∖ ( e)^-1(F)/F < ε'.Hence, <ref> holds. In conclusion, <ref> holds.The following three statements are equivalent: *The cell space ℛ is right amenable;*For each positive real number ε, there is a non-empty and finite subset F of M such thats ∈ S F ∖ ( s)^-1(F)/F < ε; *The isoperimetric constant ι_S(ℛ) is 0. <ref><ref> Let ℛ be right amenable. Then, according to <ref>, there is a right Følner net in ℛ. Hence, according to <ref>, for each ε∈_> 0, there is a non-empty and finite subset F of M such that <ref> holds. <ref><ref> Let <ref> hold. Then, according to <ref> and <ref>, there is a right Følner net in ℛ. In conclusion, according to <ref>, the cell space ℛ is right amenable.<ref><ref> Let <ref> hold. Furthermore, let ε' ∈_> 0 and let ε = ε' / S. Then, there is a non-empty and finite subset F of M such that <ref> holds. Therefore,⋃_s ∈ S F ∖ ( s)^-1(F)/F≤∑_s ∈ SF ∖ ( s)^-1(F)/F <S·ε =ε'.In conclusion, ι_S(ℛ) = 0.<ref><ref> Let ι_S(ℛ) = 0. Furthermore, let ε∈_> 0. Then, because ι_S(ℛ) = 0, there is a non-empty and finite subset F of M such that⋃_s ∈ S F ∖ ( s)^-1(F)/F < ε.Hence, for each s ∈ S, because F ∖ ( s)^-1(F) ⊆⋃_s' ∈ S F ∖ ( s')^-1(F),F ∖ ( s)^-1(F)/F < ε.In conclusion, <ref> holds.In the case that M = G andis the group multiplication of G, <ref> is proposition 6.10.2 in <cit.>.Let F_i_i ∈ I be a net in F ⊆ MF ≠∅, Ffinite indexed by (I, ≤). It is a right Følner net in ℛ if and only ifρ∈_0 lim_i ∈ I_ρ F_i/F_i = 0. First, let F_i_i ∈ I be a right Følner net in ℛ. Furthermore, let ρ be a non-negative integer. Then, for each index i ∈ I, we have _ρ F_i = __S(ρ) F_i. And, according to <ref>, the ball _S(ρ) is finite. Hence, according to <ref>,lim_i ∈ I__S(ρ) F_i/F_i = 0.In conclusion, <ref> holds.Secondly, let <ref> hold. Furthermore, let N be a finite subset of GG_0. Then, according to <ref>, there is a non-negative integer ρ such that N ⊆_S(ρ). Hence, for each index i ∈ I, according to <ref> of <ref>, we have _N F_i ⊆__S(ρ) F_i = _ρ F_i. Therefore,lim_i ∈ I_N F_i/F_i = 0.In conclusion, according to <ref>, the net F_i_i ∈ I is a right Følner net in ℛ.§ SUB-EXPONENTIAL GROWTH AND AMENABILITYIn this section, let ℳ = M, G, be a finitely right-generated left-homogeneous space with finite stabilisers. Note that it is even finitely and symmetrically right generated.Contents. In <ref> we show that if ℳ has sub-exponential growth, then it is right amenable and a sequence of balls is a right Følner net. And in <ref> we show that if G has sub-exponential growth, then so has ℳ.Let r_k_k ∈_0 be a sequence of positive real numbers. Then,lim inf_k →∞r_k + 1/r_k≤lim inf_k →∞√(r_k). See lemma 6.11.1 in <cit.>.Let the space ℳ have sub-exponential growth. Then, it is right amenable. And, for each coordinate system 𝒦 for ℳ and each finite and symmetric right-generating set S of ℛ = ℳ, 𝒦, there is a subsequence of _S(ρ)_ρ∈_0 that is a right Følner net in ℛ.Let 𝒦 be a coordinate system for ℳ and let S be a finite and symmetric right-generating set of ℛ = ℳ, 𝒦. Then, according to <ref> and <ref>,1 ≤lim inf_k →∞γ_S(k + 1)/γ_S(k)≤lim_k →∞√(γ_S(k)) =λ_S =1.Hence, lim inf_k →∞γ_S(k + 1)/γ_S(k) = 1.Let ε∈_> 0. Then, there is a k ∈_+ such that γ_S(k)/γ_S(k - 1) < 1 + ε. Thus, γ_S(k) - γ_S(k - 1) < ε·γ_S(k - 1).Let s ∈ S. Then, according to <ref>, we have _S(k - 1) ⊆ ( s)^-1(_S(k)). Hence, because _S(k - 1) ⊆_S(k) and γ_S(k - 1) ≤γ_S(k),_S(k) ∖ ( s)^-1(_S(k)) ≤_S(k) ∖_S(k - 1)=_S(k) - _S(k - 1)=γ_S(k) - γ_S(k - 1)<ε·γ_S(k - 1)≤ε·γ_S(k)=ε·_S(k).Therefore, for each ε∈_> 0, there is a k ∈_+ such thats ∈ S _S(k) ∖ ( s)^-1(_S(k))/_S(k) < ε.In conclusion, according to <ref>, the cell space ℛ is right amenable and hence so is ℳ. And, by going through the proof of <ref> and the proofs of the lemmata, corollaries, and theorems it uses, one can see that a subsequence of _S(ρ)_ρ∈_0 is a right Følner net in ℛ. As this is rather tedious, we construct such a subsequence in the following.If M is finite, then, according to <ref>, the sequence (ρ)_ρ∈_0 is eventually equal to M. Hence, because𝔤∈ GG_0 M ∖ (𝔤)^-1(M)/M = 0,the sequence (ρ)_ρ∈_0 is a right Følner net in ℛ.From now on, let M be infinite. We showed above that, for each ε∈_> 0, there is a k ∈_+ such that <ref> holds. Hence, according to <ref>, under the identification of M with GG_0, for each j ∈_+ and each ε∈_> 0, there is a k ∈_+ such thate ∈_S(j) _S(k) ∖ ( e)^-1(_S(k))/_S(k) < ε.Therefore, for each n ∈_+, there is a least k_n ∈_+ such thate ∈_S(n) _S(k_n) ∖ ( e)^-1(_S(k_n))/_S(k_n) < 1/n.By the choices of k_n, for n ∈_+, the sequence k_n_n ∈_+ is non-decreasing. If it is not eventually constant, then, by skipping duplicate entries, we get an increasing subsequence k_n_i_i ∈_+.Otherwise, there is an n ∈_+ such that, for each e ∈_S(1), we have _S(k_n) ∖ ( e)^-1(_S(k_n)) = ∅ and thus _S(k_n)e ⊆_S(k_n). Hence, _S(k_n + 1) = _S(k_n) _S(1) ⊆_S(k_n). Therefore, according to <ref>, the set M is finite, which contradicts our assumption that M is infinite. Hence, the case that k_n_n ∈_+ is eventually constant does not occur.In the case that does occur, for each 𝔤∈ GG_0 and each ε∈_> 0, there is an i_0 ∈_+ such that 𝔤∈_S(n_i_0) and 1/n_i_0≤ε, and hence, for each i ∈_+ with i ≥ i_0,_S(k_n_i) ∖ (𝔤)^-1(_S(k_n_i))/_S(k_n_i) < 1/n_i≤1/n_i_0≤ε.Therefore,𝔤∈ GG_0 lim_i →∞_S(k_n_i) ∖ (𝔤)^-1(_S(k_n_i))/_S(k_n_i) = 0.In conclusion, the sequence _S(k_n_i)_i ∈_+ is a right Følner net in ℛ. In the case that M = G andis the group multiplication of G, the first part of <ref> is theorem 6.11.2 in <cit.>. Let the group G be finitely generated and let it have sub-exponential growth. The space ℳ has sub-exponential growth and is right amenable.This is a direct consequence of <ref> and <ref>. Each quotient set of any virtually nilpotent group (see <ref>) or the Grigorchuk group (see <ref>) by a finite subgroup acted upon by left multiplication is a right-amenable left-homogeneous space. CHAPTER: SHIFT SPACES AND THE MOORE AND THE MYHILL PROPERTIESAbstract. We prove the Moore and the Myhill property for strongly irreducible subshifts over right-amenable and finitely right-generated left-homogeneous spaces with finite stabilisers. Both properties together mean that the global transition function of each big-cellular automaton with finite set of states and finite neighbourhood over such a subshift is surjective if and only if it is pre-injective. This statement is known as Garden of Eden theorem. Pre-Injectivity means that two global configurations that differ at most on a finite subset and have the same image under the global transition function must be identical. Remark. This chapter generalises parts of the paper *fiorenzi:2003<cit.>.Summary. A subset X of the phase space Q^M, where Q is a finite set of states, is a shift space of finite type if it is generated by a finite set of forbidden blocks. Such a space X shift-invariant and compact. And it is strongly irreducible if each pair of finite patterns that are allowed in X and at least some fixed positive integer apart, are embedded in a point of X. A map Δ from a shift space X to a shift space Y is local if the state Δ(x)(m) is uniformly and locally determined in m, in other words, if the map Δ is the restriction of the global transition function of a big-cellular automaton with finite neighbourhood to the domain X and the codomain Y. For a right-amenable and finitely right-generated cell space with finite stabilisers we may choose a right Følner net ℱ = F_i_i ∈ I. The entropy of a subset X of Q^M with respect to ℱ, where Q is a finite set, is, broadly speaking, the asymptotic growth rate of the number of finite patterns with domain F_i that occur in X. For non-negative integers θ, κ, and θ', a θ, κ, θ'-tiling is a subset T of M such that (t, θ)_t ∈ T is pairwise at least κ + 1 apart and (t, θ')_t ∈ T is a cover of M. If for each point t ∈ T not all patterns with domain (t, θ) occur in a subset of Q^M, then that subset does not have maximal entropy. A local map from a non-empty strongly irreducible shift space of finite type to a strongly irreducible shift space with the same entropy over a right-amenable and finitely right-generated cell space with finite stabilisers is surjective if and only if its image has maximal entropy and its image has maximal entropy if and only if it is pre-injective. This establishes the Garden of Eden theorem, which states that a local map as above is surjective if and only if it is pre-injective. This answers a question posed by Sébastien Moriceau at the end of his paper *moriceau:2011<cit.>. And it follows that strongly irreducible shift spaces of finite type over right-amenable and finitely right-generated cell spaces have the Moore and the Myhill property.The Garden of Eden theorem for cellular automata over ^2 is a famous theorem by Edward Forrest Moore and John R. Myhill from 1962 and 1963, which was proved in their papers *moore:1962<cit.> and *myhill:1963<cit.>. That theorem also holds for cellular automata over amenable finitely generated groups, which was proved by Tullio Ceccherini-Silberstein, Antonio Machi, and Fabio Scarabotti in their paper *ceccherini-silberstein:machi:scarabotti:1999<cit.>. It even holds for such automata on strongly irreducible shifts of finite type, which was proved by Francesca Fiorenzi in her paper *fiorenzi:2003<cit.>. Contents. In <ref> we introduce full shifts, shift-invariance, shift spaces or subshifts (of finite type), strong irreducibility, bounded propagation, local maps, conjugacies, and the Moore and the Myhill property. In <ref> we introduce tilings, prove their existence, and relate them to entropies. And in <ref> we prove the Garden of Eden theorem, from which we deduce that both the Moore and the Myhill property hold. Preliminary Notions. A left group set is a triple M, G,, where M is a set, G is a group, andis a map from G × M to M, called left group action of G on M, such that G →(M), g ↦ [g ], is a group homomorphism. The actionis transitive if M is non-empty and for each m ∈ M the map m is surjective; and free if for each m ∈ M the map m is injective. For each m ∈ M, the set Gm is the orbit of m, the set G_m = ( m)^-1(m) is the stabiliser of m, and, for each m' ∈ M, the set G_m, m' = ( m)^-1(m') is the transporter of m to m'.A left-homogeneous space is a left group set ℳ = M, G, such thatis transitive. A coordinate system for ℳ is a tuple 𝒦 = m_0, g_m_0, m_m ∈ M, where m_0 ∈ M and for each m ∈ M we have g_m_0, m m_0 = m. The stabiliser G_m_0 is denoted by G_0. The tuple ℛ = ℳ, 𝒦 is a cell space. The map M × GG_0 → M, (m, g G_0) ↦ g_m_0, m g g_m_0, m^-1 m (= g_m_0, m gm_0) is a right semi-action of GG_0 on M with defect G_0, which means thatm ∈ MmG_0 = m,andm ∈ Mg ∈ Gg_0 ∈ G_0 𝔤' ∈ GG_0 mg ·𝔤' = (mg G_0)g_0 ·𝔤'.It is transitive, which means that the set M is non-empty and for each m ∈ M the map m is surjective; and free, which means that for each m ∈ M the map m is injective; and semi-commutes with , which means thatm ∈ Mg ∈ Gg_0 ∈ G_0 𝔤' ∈ GG_0 (gm) 𝔤' = g(mg_0 ·𝔤').The maps ι M → GG_0, m ↦ G_m_0, m, and m_0 are inverse to each other. Under the identification of M with GG_0 by either of these maps, we have (m, 𝔤) ↦ g_m_0, m𝔤. (See <ref>.)A left-homogeneous space ℳ is right amenable if there is a coordinate system 𝒦 for ℳ and there is a finitely additive probability measure μ on M such that 𝔤∈ GG_0A ⊆ M [](𝔤)_Ainjectiveμ(A 𝔤) = μ(A),in which case the cell space ℛ = ℳ, 𝒦 is called right amenable. When the stabiliser G_0 is finite, that is the case if and only if there is a right Følner net in ℛ indexed by (I, ≤), which is a net F_i_i ∈ I in F ⊆ MF ≠∅, Ffinite such thatρ∈_0 lim_i ∈ I_ρ F_i/F_i = 0.If a net is a right Følner net for one coordinate system, then it is a right Følner net for each coordinate system. In particular, a left-homogeneous space ℳ with finite stabilisers is right amenable if and only if, for each coordinate system 𝒦 for ℳ, the cell space ℳ, 𝒦 is right amenable. (See <ref>.)A left-homogeneous space ℳ is finitely right generated if there is a coordinate system 𝒦 for ℳ and there is a finite subset S of GG_0 such that G_0 · S ⊆ S and, for each m ∈ M, there is a k ∈_0 and there is a s_i_i ∈1, 2, …, k in S ∪ S^-1, where S^-1 = g^-1 G_0s ∈ S, g ∈ s, such thatm = [][](m_0s_1)s_2… s_k.in which case the cell space ℛ = ℳ, 𝒦 is called finitely right generated. The left-homogeneous space ℳ is finitely right generated if and only if, for each coordinate system 𝒦 for ℳ, the cell space ℳ, 𝒦 is finitely right generated. The right-generating set S is symmetric if S^-1⊆ S. The S-edge-labelled directed graph M, E, where E = (m, s, ms)m ∈ M, s ∈ S, is the coloured S-Cayley graph. The distance _S on that graph is the S-metric and the map _S = _S(m_0, ) is the S-length. For each m ∈ M and each ρ∈, the sets_S(m, ρ)= m' ∈ M _S(m, m') ≤ρ, _S(m, ρ)= m' ∈ M _S(m, m') = ρare the ball/sphere of radius ρ centred at m, the ball _S(m_0, ρ) is denoted by _S(ρ), and the sphere _S(m_0, ρ) by _S(ρ). For each A ⊆ M, each θ∈_0, the set A^-θ = m ∈ A _S(m, θ) ⊆ A is the θ-interior of A, the set A^+θ = m ∈ M _S(m, θ) ∩ A ≠∅ is the θ-closure of A, the set _θ A = A^+θ∖ A^-θ is the θ-boundary of A, the set _θ^- A = A ∖ A^-θ is the internal θ-boundary of A, and the set _θ^+ A = A^+θ∖ A is the external θ-boundary of A. (See <ref>.)A semi-cellular automaton is a quadruple 𝒞 = ℛ, Q, N, δ, where ℛ is a cell space; Q, called set of states, is a set; N, called neighbourhood, is a subset of GG_0 such that G_0 · N ⊆ N; and δ, called local transition function, is a map from Q^N to Q. A local configuration is a map ℓ∈ Q^N, a global configuration is a map c ∈ Q^M, an A-pattern is a map p ∈ Q^A, where A is a subset of M, the number p = A is the size of p, a finite pattern is a block, and the set of blocks is denoted by Q^*. The stabiliser G_0 acts on Q^N on the left by ∙ G_0 × Q^N → Q^N, (g_0, ℓ) ↦ [n ↦ℓ(g_0^-1· n)], and the group G acts on the set of patterns on the left byG ×⋃_A ⊆ M Q^A→⋃_A ⊆ M Q^A,(g, p)↦[g (p)→ Q,m↦ p(g^-1 m). ]The global transition function of 𝒞 is the map Δ Q^M → Q^M, c ↦ [m ↦δ(n ↦ c(mn))].A subgroup H of G is 𝒦-big if the set g_m_0, m m ∈ M is included in H. A big-cellular automaton is a semi-cellular automaton 𝒞 = ℛ, Q, N, δ such that, for some 𝒦-big subgroup H of G, the local transition function δ is ∙_G_0 ∩ H-invariant, which means that, for each h_0 ∈ G_0 ∩ H, we have δ(h_0 ∙) = δ(). Its global transition function is _H-equivariant, which means that, for each h ∈ H, we have Δ(h ) = h Δ().Under the identification of M with GG_0 by ι m ↦ G_m_0, m, the mapM ×⋃_A ⊆ M Q^A→⋃_A ⊆ M Q^A,(m, p)↦[m (p)→ Q,ma↦ p(a), ]broadly speaking, maps a point m and a pattern p that is centred at m_0 to the corresponding pattern centred at m. For each cell m ∈ M, each subset A of M, and each pattern p ∈ Q^A, we have mp = g_m_0, m p. It follows that the global transition function Δ of a big-cellular automaton is -equivariant, which means that, for each m ∈ M, we have Δ(m ) = m Δ(). (See <ref>.) Context. In this chapter, let ℛ = ℳ, 𝒦 = M, G, , m_0, g_m_0, m_m ∈ M be a finitely right-generated cell space such that the stabiliser G_0 of m_0 underis finite; let S be a finite and symmetric right-generating set of ℛ; let H be a 𝒦-big subgroup of G; let H_0 be the stabiliser of m_0 under _H × M, which is H ∩ G_0; for each cell m ∈ M, let H_m_0, m be the transporter of m_0 to m under _H × M; let Q be a finite set; let Q^M be equipped with the prodiscrete topology; for each subset A of M, let π_A be the restriction map Q^M → Q^A, c ↦ c_A; and identify M with GG_0 by ι m ↦ G_m_0, m. Moreover, we omit the subscript S, in particular, instead of _S we write , instead of _S we write , instead of _S we write , and instead of _S we write . § SHIFT SPACESContents. The full shift is the set of global configurations, the points of the full shift (see <ref>). A subset of the full shift is shift-invariant if it is invariant under a group that contains the coordinates (see <ref>). For example the full shift is shift-invariant. A pattern semi-occurs in another pattern if a rotation of it occurs in the other pattern (see <ref>). It is allowed in a subset of the full shift if it semi-occurs in one of its points and it is forbidden otherwise (see <ref>). The set of points of the full shift in which each block of a given set is forbidden is generated by that set (see <ref>) and it is a shift space (see <ref>), which is shift-invariant (see <ref>), closed (see <ref>), and compact (see <ref>).If there is a finite generating set, then the shift space is of finite type (see <ref>), the radius κ of a ball that includes the domains of the patterns of the generating set is its memory and it itself is called κ-step (see <ref>), and its points are characterised by restrictions to balls with its memory as radius (see <ref>). Finitely many points of shift spaces of finite type can be, in various ways, cut into pieces and glued together to construct new points, as long as the pieces agree on a big enough boundary of the cuts (see <ref>). A shift space is strongly irreducible if allowed finite patterns that are at least a certain distance apart can be embedded in the same point (see <ref>). It has bounded propagation if finite patterns are allowed whenever their restrictions to balls of a certain radius are allowed (see <ref>). Such spaces are strongly irreducible and of finite type (see <ref>).A map from a shift space to another one is local if it is uniformly and locally determined in each cell (see <ref>). Such maps are global transition functions of big-cellular automata with finite neighbourhoods (see <ref>), their domains and codomains can be simultaneously restricted to a subset of cells and its interior (see <ref>), and their images are shift spaces (see <ref>). The difference of two points of the full shift is the set of cells in which they differ (see <ref>) and a local map is pre-injective if it is injective on points with finite support (see <ref>). A local map that has a local inverse is a conjugacy (see <ref>), and its domain and codomain are conjugate (see <ref>). Entropy is invariant under conjugacy (see <ref>). A subshift has the Moore property if each surjective local map is pre-injective, and the Myhill property if the converse holds (see <ref>). Both these properties are invariant under conjugacy (see <ref>).The set Q^M is called full shiftfull shift Q^Mshift!full and each element c ∈ Q^M is called pointpoint c. [<cit.>]Let ℳ be the left-homogeneous space , , +, let 𝒦 be the coordinate system 0, z_z ∈, let ℛ be the cell space ℳ, 𝒦, let S be the set -1, 1, let H be the only 𝒦-big subgroupof , and let Q be the binary set 0, 1. The stabiliser _0 of 0 under + is the singleton set 0; under the identification ofwith _0 by z ↦ z + _0, the right semi-action of _0 onis but +; and the set S is a finite and symmetric right-generating set of ℛ. The full shift Q^ is the usual full 2-shift considered in symbolic dynamics and its points are called bi-infinite binary sequencesbi-infinite binary sequences Q^.There is a bijective map φ from Q to _Q (= 0, 1, …,Q - 1). It induces the bijective mapΦ Q^M→_Q^M,c↦[m ↦φ[]c(m)].Let X be a subset of Q^M. It is called shift-invariantshift-invariant if and only ifh ∈ HhX = X. Shift-invariance is the same as _H-invariance. The set X is shift-invariant if and only ifh ∈ HhX ⊆ X. Let X be a shift-invariant subset of Q^M. Then,m ∈ MmX = X.A pattern occurs in another pattern if a translation of it coincides with a subpattern the other pattern and it semi-occurs in another pattern if a rotation and translation of it coincides with a subpattern of the other pattern, as defined inLet p be an A-pattern and let p' be an A'-pattern.* Let m be an element of M. The pattern p is said to occur at m in p'p occurs at m in p' and we write p _m p'p _m p'[symbols]psquaresubseteqmpprime@p _m p' if and only ifmA ⊆ A'andmp = p'_mA.And it is said to semi-occur at m in p'p semi-occurs at m in p'occur at m in p'!semi- and we write p _m p'p _m p'[symbols]psquaresubseteqmzpprime@p _m p' if and only ifh ∈ H_m_0, m hA ⊆ A'andhp = p'_hA. * The pattern p is said to occur in p'p occurs in p' and we write pp'pp'[symbols]psquaresubseteqpprime@pp' if and only if m ∈ Mp _m p'.And it is said to semi-occur in p'p semi-occurs in p'occur in p'!semi- and we write pp'pp'[symbols]psquaresubseteqzpprime@pp' if and only if m ∈ Mp _m p'.Let ℛ be the cell space G, G, ·, e_G, g_g ∈ G, where G is a group, · is its multiplication, and e_G is its neutral element. Then, G_0 = e_G, = ·, =, and, for each element g ∈ G, we have H_e_G, g = g. Hence, the notions occurs and semi-occurs are identical, and they are the common notion of occurrence as used in <cit.>. Semi-occurrence can be characterised in many ways, each illuminating a different aspect, some of which are given inLet p be an A-pattern, let p' be an A'-pattern, and let m be an element of M. The following statements are equivalent: * p _m p'; * h_0 ∈ H_0h_0p _m p';* h_0 ∈ H_0g_m_0, m h_0A ⊆ A'andg_m_0, m h_0p = p'_g_m_0, m h_0A;* h ∈ H_m, m_0 A ⊆ hA'and(hp')_A = p;* h_0 ∈ H_0A ⊆ h_0 g_m_0, m^-1 A'and(h_0 g_m_0, m^-1 p')_A = p.Let p be an A-pattern and let p' be an A'-pattern. The following statements are equivalent: * pp'; * h_0 ∈ H_0h_0pp';* h ∈ HhA ⊆ A'andhp = p'_hA;* h ∈ HA ⊆ hA'and(hp')_A = p.A pattern is allowed in a subset of the full shift if it semi-occurs in one of its points and forbidden otherwise, as defined inLet X be a subset of Q^M and let p be an A-pattern. The pattern p is called * allowed in Xpattern p allowed in Xpattern!allowed in if and only if x ∈ Xpx; * forbidden in Xpattern p forbidden in Xpattern!forbidden in if and only if x ∈ Xp x.The greatest subset of the full shift with respect to inclusion in which each block of a given set of blocks is forbidden is said to be generated by the set of blocks, as defined inLet 𝔉 be a subset of Q^*. The set𝔉 = c ∈ Q^M 𝔣∈𝔉𝔣cset 𝔉 generated by 𝔉[symbols]angleFanglefraktur@𝔉is said to be generated by 𝔉𝔉 generated by 𝔉.A shift space is a subset of the full shift that is generated by a set of blocks, as defined in Let X be a subset of Q^M. It is called shift spaceshift space Xshift!space and subshift of Q^Msubshift X of Q^Mshift!sub- if and only if there is a subset 𝔉 of Q^* such that 𝔉 = X.[Full Shift] Because ∅ = Q^M, the set Q^M is a shift space. [Empty Shift] If we identify each q ∈ Q with the (m_0)-block [m_0 ↦ q], then Q = ∅ and hence the set ∅ is a shift space. [Golden Mean Shift <cit.>]In the situation of <ref>, the set X of all bi-infinite binary sequences with no two 1's next to each other is the shift space known as golden mean shiftgolden mean shiftshift!golden mean. It is for example generated by the forbidden block 11.[Even Shift <cit.>]In the situation of <ref>, the set X of all bi-infinite binary sequences such that, between any two occurrences of 1's, there are an even number of 0's, is the shift space known as even shifteven shiftshift!even. It is for example generated by the forbidden blocks 1 0^2k + 1 1, for k ∈_0. Each shift space is shift-invariant, which is shown inLet X be a subshift of Q^M. It is shift-invariant. There is a subset 𝔉 of Q^* such that 𝔉 = X. Let x ∈ X and let h ∈ H. Suppose that there is an 𝔣∈𝔉 such that 𝔣 hx. Then, there is an h' ∈ H such that h' 𝔣 = (hx)_h' (𝔣). Thus, because (hx)_h' (𝔣) = h(x_h^-1 h' (𝔣)), we have h^-1 h' 𝔣 = x_h^-1 h' (𝔣). Hence, because h^-1 h' ∈ H, we have 𝔣 x, which contradicts that x ∈𝔉. Therefore, contrary to the supposition, for each 𝔣∈𝔉, we have 𝔣hx. Hence, hx ∈ X. Therefore, hX ⊆ X. In conclusion, according to <ref>, the subshift X is shift-invariant. [Shift-Invariant Non-Shift <cit.>]In the situation of <ref>, the set X of all bi-infinite binary sequences in which the symbol 1 occurs exactly once is shift-invariant, but it is not a shift space. Patterns with the same domain can all be restricted to some subdomain, as done in Let A be a subset of M, let B be a subset of A, and let P be a subset of Q^A. The setP_B = p_Bp ∈ PP_Bis the set of all B-subpatterns of patterns of P. Because subshifts are shift-invariant, restrictions to patterns behave nicely with translations and rotations, as remarked inLet X be a subshift of Q^M, let A be a subset of M, let h be an element of H, and let m be an element of M. For each A-pattern p, we have p ∈ X_A if and only if hp ∈ X_hA, and p ∈ X_A if and only if mp ∈ X_mA. In particular, hX_A = X_hA and mX_A = X_mA. And, if A is finite, then X_A = X_hA and X_A = X_mA.Because shift spaces are generated by forbidden blocks, which have finite domains, a point of the full shift belongs to a shift space if and only if its subpatterns do, which is shown in Let X be a subshift of Q^M and let c be a point of Q^M. Then, c ∈ X if and only if ρ∈_0c_(ρ)∈ X_(ρ). If c ∈ X, then <ref> holds. From now on, let <ref> hold. Because X is a subshift, there is a subset 𝔉 of Q^* such that 𝔉 = X. Let 𝔣∈𝔉. Suppose that 𝔣 semi-occurs in c. Then, because 𝔣 < ∞, according to <ref>, there is a ρ∈_0 such that 𝔣 semi-occurs in c_(ρ). Hence, c_(ρ)∉ X_(ρ), which contradicts <ref>. Therefore, 𝔣 does not semi-occur in c. In conclusion, c ∈ X.Shift spaces are characterised by shift-invariance and compactness, which is shown in Let X be a subset of Q^M. It is a shift space if and only if it is shift-invariant and compact.First, let X be a shift space. Then, according to <ref>, it is shift-invariant. Moreover, let x_k_k ∈_+ be a sequence in X that converges to a point c ∈ Q^M. Then, for each ρ∈_0, there is a k ∈_+ such that c_(ρ) = x_k_(ρ)∈ X_(ρ). Thus, according to <ref>, we have c ∈ X. Hence, X is closed. And, according to the first paragraph of section 1.8 in <cit.>, the set Q^M is compact. Therefore, X is compact. Secondly, let X be shift-invariant and compact. Then, X is closed and Q^M ∖ X is open. Hence, for each c ∈ Q^M ∖ X, there is a ρ_c ∈_0 such that (c_(ρ_c)) ⊆ Q^M ∖ X. Put 𝔉 = c_(ρ_c) c ∈ Q^M ∖ X.Let x ∈ X. Suppose that there is an 𝔣∈𝔉 such that 𝔣 x. Then, according to <ref>, there is an h ∈ H such that (hx)_(𝔣) = 𝔣. Thus, hx ∈(𝔣) ⊆ Q^M ∖ X. And, because X is shift-invariant, we also have hx ∈ X, which contradicts that hx ∈ Q^M ∖ X. Hence, contrary to the supposition, for each 𝔣∈𝔉 we have 𝔣x. Therefore, X ⊆𝔉.Let c ∈𝔉. Suppose that c ∉ X. Then, c_(ρ_c)∈𝔉. Thus c ∉𝔉, which contradicts that c ∈𝔉. Hence, c ∈ X. Therefore, 𝔉⊆ X.Altogether, 𝔉 = X. In conclusion, X is a shift space.Compactness cannot be omitted in the equivalence. For example, the subset of 0, 1^ from <ref> is shift-invariant but not a shift space. A shift space of finite type is one that is generated by finitely many forbidden blocks, as defined in Let X be a subshift of Q^M. It is said to be of finite typeof finite typeshift!of finite typefinite type@of finite type if and only if there is a finite subset 𝔉 of Q^* such that 𝔉 = X. [Of Finite Type or Not]As is apparent from their definition, the full shift (<ref>), the empty shift (<ref>), and the golden mean shift (<ref>) are of finite type. According to example 2.1.5 in <cit.>, the even shift (<ref>) is not of finite type. A κ-step shift space is one that is generated by forbidden (κ)-blocks, as defined in Let X be a subshift of Q^M and let κ be a non-negative integer. The subshift X is called κ-stepstep@κ-stepκ-step and the integer κ is called memory of Xmemory κ of X if and only if there is a subset 𝔉 of Q^(κ) such that 𝔉 = X. Let X be a κ-step subshift of Q^M. Because the set Q^(κ) is finite, the subshift X is of finite type. And, for each non-negative integer κ' such that κ' ≥κ, the subshift X is κ'-step. A shift space of finite type is κ-step, where κ is the radius of a ball that includes all domains of a finite generating set of the shift space, which is shown inLet X be a subshift of Q^M of finite type. There is a non-negative integer κ such that X is κ-step.Because X is of finite type, there is a finite subset 𝔉 of Q^* such that 𝔉 = X. And, because the set 𝔉 is finite, according to <ref>, there is a non-negative integer κ such that, for each 𝔣∈𝔉, we have (𝔣) ⊆(κ). Let 𝔉' be the set p ∈ Q^(κ)𝔣∈𝔉 p_(𝔣) = 𝔣. Then, 𝔉' = 𝔉 = X. In conclusion, X is κ-step.A shift space is κ-step if and only if it contains each point of the full shift whose restrictions to the balls of radius κ are allowed patterns, which is shown in Let X be a subshift of Q^M and let κ be a non-negative integer. The subshift X is κ-step if and only if c ∈ Q^M [][] m ∈ Mc_(m, κ)∈ X_(m, κ) c ∈ X. First, let X be κ-step. Then, there is an 𝔉⊆ Q^(κ) such that 𝔉 = X. Furthermore, let c ∈ Q^M such thatm ∈ Mc_(m, κ)∈ X_(m, κ).Suppose that there is an 𝔣∈𝔉 such that 𝔣 c. Then, there is an m ∈ M such that 𝔣_m c. Thus, there is an h ∈ H_m_0, m such that h 𝔣 = c_h (κ). Moreover, because hm_0 = m, according to <ref>, we have h (κ) = (m, κ). Hence, according to <ref>, we have h 𝔣 = c_(m, κ)∈ X_(m, κ) = X_h (κ). Therefore, there is an x ∈ X such that 𝔣 x, which contradicts that 𝔉 = X. In conclusion, for each 𝔣∈𝔉, we have 𝔣c, and hence c ∈ X.Secondly, let <ref> hold. Furthermore, let 𝔉 = Q^(κ)∖ X_(κ). We show below that X ⊆𝔉 and 𝔉⊆ X. Hence, 𝔉 = X. In conclusion, X is κ-step.Subproof of: X ⊆𝔉 Let c ∈ Q^M ∖𝔉. Then, there is an 𝔣∈𝔉 such that 𝔣 c. Thus, there is an m ∈ M and there is an h ∈ H_m_0, m such that h 𝔣 = c_(m, κ). Therefore, because h is bijective, according to <ref> and <ref>, we have c_(m, κ)∈ h 𝔉 = h(Q^(κ)∖ X_(κ)) = (hQ^(κ)) ∖ (hX_(κ)) = Q^(m, κ)∖ X_(m, κ). Hence, c ∈ Q^M ∖ X. In conclusion, X ⊆𝔉.Subproof of: 𝔉⊆ X Let c ∈ Q^M ∖ X. Then, according to <ref>, there is an m ∈ M such that c_(m, κ)∈ Q^(m, κ)∖ X_(m, κ). Furthermore, let h = g_m_0, m. Then, similar as above, h^-1 c_(m, κ)∈ Q^(κ)∖ X_(κ) = 𝔉. Thus, there is an 𝔣∈𝔉 such that h 𝔣 = c_(m, κ). Hence, because h ∈ H, we have 𝔣 c. Therefore, c ∈ Q^M ∖𝔉. In conclusion, 𝔉⊆ X.If we cut holes in a point of a shift space of finite type that are far enough apart and fill these holes with pieces from other points of the shift space that agree on big enough boundaries of the holes with the holey point, then we still have a point of the shift space, which is shown in Let X be a κ-step subshift of Q^M, let x be a point of X, let A_i_i ∈ I be a family of subsets of M such that the family A_i^+2κ_i ∈ I is pairwise disjoint, and let x_i_i ∈ I be a family of points of X such that, for each index i ∈ I, we have x_i__2κ^+ A_i = x__2κ^+ A_i. The map x_M ∖ (⋃_i ∈ I A_i)×∐_i ∈ I x_i_A_i is identical to x_M ∖ (⋃_i ∈ I A_i^+2κ)×∐_i ∈ I x_i_A_i^+2κ and a point of X. Recall that such coproducts were introduced in <ref>.Because A_i^+2κ_i ∈ I is pairwise disjoint, so is A_i_i ∈ I. Hence, x' = x_M ∖ (⋃_i ∈ I A_i)×∐_i ∈ I x_i_A_i is well-defined. Furthermore, for each i ∈ I, we have x_i_A_i^+2κ∖ A_i = x_A_i^+2κ∖ A_i. Hence, x' = x_M ∖ (⋃_i ∈ I A_i^+2κ)×∐_i ∈ I x_i_A_i^+2κ. Moreover, because X is κ-step, there is an 𝔉⊆ Q^(κ) such that 𝔉 = X.Let u ∈ Q^(κ) semi-occur in x'. Then, there is an m ∈ M and there is an h ∈ H_m_0, m such that hu = x'_h (κ). Moreover, according to <ref> and <ref> of <ref>, we have h (κ) = (m, κ) = m^+κ. Case 1: i ∈ Im ∈ A_i^+κ. Then, according to <ref> of <ref> and <ref> of <ref>, we have m^+κ⊆ (A_i^+κ)^+κ⊆ A_i^+2κ. Hence, h (κ) ⊆ A_i^+2κ. Thus, x'_h (κ) = x_i_h (κ). Therefore, u semi-occurs in x_i. Hence, u ∉𝔉.Case 2: m ∈ M ∖ (⋃_i ∈ I A_i^+κ). Then, according to <ref>, <ref>, and <ref> of <ref> and <ref> of <ref>, we havem^+κ ⊆ (M ∖ (⋃_i ∈ I A_i^+κ))^+κ⊆ M ∖ (⋃_i ∈ I (A_i^+κ)^-κ)⊆ M ∖ (⋃_i ∈ I A_i^+(κ - κ))= M ∖ (⋃_i ∈ I A_i).Hence, h (κ) ⊆ M ∖ (⋃_i ∈ I A_i). Thus, x'_h (κ) = x_h (κ). Therefore, u semi-occurs in x. Hence, u ∉𝔉.In either case, u ∉𝔉. In conclusion, x' ∈ X. If we sew one part and respectively the other part of two sufficiently overlapping points of a shift space of finite type together, then we get a point of the shift space, which is shown in Let X be a κ-step subshift of Q^M, let A be a subset of M, and let x and x' be two points of X such that x__2κ^+ A = x'__2κ^+ A. The map x_A × x'_M ∖ A is identical to x_A^+2κ× x'_M ∖ A^+2κ and a point of X.This is a direct consequence of <ref> with x_i_i ∈ I = x'. If two patterns with the same domain, that are allowed in a shift space of finite type, agree on a big enough boundary, then they can be identically extended to points of the shift space, which is shown in Let X be a κ-step subshift of Q^M, let A be a subset of M, let p and p' be two patterns of X_A^+2κ such that p__2κ^+ A = p'__2κ^+ A. There are two points x and x' of X such that x_A^+2κ = p, x'_A^+2κ = p', and x_M ∖ A = x'_M ∖ A.Because p, p' ∈ X_A^+2κ, there are x”, x' ∈ X such that x”_A^+2κ = p and x'_A^+2κ = p'. Because p__2κ^+ A = p'__2κ^+ A, we have x”__2κ^+ A = x'__2κ^+ A. Hence, according to <ref>, we have x = x”_A × x'_M ∖ A∈ X. Moreover, x_M ∖ A = x'_M ∖ A. And, because x = x”_A^+2κ× x'_M ∖ A^+2κ, we have x_A^+2κ = p. A shift space is strongly irreducible if two allowed finite patterns that are far enough apart are embedded in a point of the shift space, which is defined in Let X be a subshift of Q^M and let κ be a non-negative integer. The subshift X is called κ-strongly irreduciblestrongly irreducible kappa@κ-strongly irreducibleκ-strongly irreducible if and only if, for each tuple (p, p') of finite patterns allowed in X such that ((p), (p')) ≥κ + 1, there is a point x ∈ X such that x_(p) = p and x_(p') = p'. Let X be a κ-strongly irreducible subshift of Q^M. For each non-negative integer κ' such that κ' ≥κ, the subshift X is κ'-strongly irreducible. Let X be a subshift of Q^M. It is called strongly irreduciblestrongly irreducible if and only if there is a non-negative integer κ such that it is κ-strongly irreducible.[Strongly Irreducible] The full shift (<ref>) and the empty shift (<ref>) are 0-strongly irreducible. The golden mean shift is 1-strongly irreducible. The even shift is 2-strongly irreducible. The generalised golden mean shifts (<ref>) and the shift space of <ref> are strongly irreducible. Of these examples, according to <ref>, all but the even shift are of finite type. [Not Strongly Irreducible <cit.>] In the situation of <ref>, the set X of all bi-infinite binary sequences with no two 0's and no two 1's next to each other is a shift space. It is for example generated by the forbidden blocks 00 and 11, in particular, it is of finite type. It consists of the two bi-infinite binary sequences with alternating 0's and 1's. And, it is not strongly irreducible; indeed, for each even and non-negative integer κ, the finite patterns 01 and 10 are allowed in X, the allowed patterns of size κ are of the form (01)^κ/2 or (10)^κ/2, but the patterns 01(01)^κ/210 and 01(10)^κ/210 are not allowed and hence not embedded in a point of X. A shift space has bounded propagation if a finite pattern is allowed whenever all restrictions of it to balls of a fixed radius are allowed, as defined in Let X be a subshift of Q^M and let ρ be a non-negative integer. The subshift X is said to have ρ-bounded propagationbounded propagation@ρ-bounded propagationρ-bounded propagation if and only ifF ⊆ Mfinite p ∈ Q^F[][] f ∈ Fp_(f, ρ) ∩ F∈ X_(f, ρ) ∩ F p ∈ X_F . Let X be a subshift of Q^M with ρ-bounded propagation. For each non-negative integer ρ' such that ρ' ≥ρ, the subshift X has ρ'-bounded propagation. Let X be a subshift of Q^M. It is said to have bounded propagationbounded propagation if and only if there is a non-negative integer ρ such that it has ρ-bounded propagation.The notion of bounded propagation was first introduced by Mikhail Leonidovich Gromov in paragraph 7.A in <cit.>. A subshift with bounded propagation is strongly irreducible and of finite type, which is shown in Let X be a subshift of Q^M with ρ-bounded propagation. It is ρ-strongly irreducible and ρ-step.Let (p, p') be a tuple of finite patterns allowed in X such that ((p), (p')) ≥ρ + 1. Moreover, let F = (p) ∪(p') and let p” = p × p' ∈ Q^F. Furthermore, let f ∈ F. Then, if f ∈(p), then (f, ρ) ∩ F ⊆(p); and, if f ∈(p'), then (f, ρ) ∩ F ⊆(p'). Thus, in either case, p”_(f, ρ) ∩ F∈ X_(f, ρ) ∩ F. Hence, because X has ρ-bounded propagation, we have p”∈ X_F. Therefore, there is an x ∈ X such that x_F = p”, in particular, x_(p) = p and x_(p') = p'. In conclusion, X is ρ-strongly irreducible.Let c ∈ Q^M such that, for each m ∈ M, we have c_(m, ρ)∈ X_(m, ρ). Moreover, let ρ' ∈_0. Then, for each f ∈(ρ'), we have(c_(ρ'))_(f, ρ) ∩(ρ') = (c_(f, ρ))_(ρ') ∩(f, ρ) ∈ (X_(f, ρ))_(ρ') ∩(f, ρ) = X_(f, ρ) ∩(ρ').Thus, because X has ρ-bounded propagation, we have c_(ρ')∈ X_(ρ'). Hence, according to <ref>, we have c ∈ X. In conclusion, according to <ref>, the subshift X is ρ-step. [Has Bounded Propagation or Not]The full shift (<ref>) and the empty shift (<ref>) have 0-bounded propagation. The golden mean shift (<ref>) has 1-bounded propagation. The even shift (<ref>) is, according to example <ref>, not of finite type and hence it is, according to <ref>, does not have bounded propagation. The shift of <ref> is not strongly irreducible and hence it is, according to <ref>, does not have bounded propagation.[Generalised Golden Mean Shifts <cit.>] Let q be a positive integer, let Q be the set 0, 1, …, q, let k be a positive integer, let F_i_i ∈1, 2, …, k be a family of finite subsets of M that contain m_0, let ρ be the least non-negative integer such that ⋃_i ∈1, 2, …, k F_i ⊆(ρ), and let X be the ρ-step subshift p ∈ Q^(ρ) i ∈1, 2, …, k m ∈ F_ip(m) ≠ 0 of Q^M. The subshift X is called generalised golden mean shiftgeneralised golden mean shiftgolden mean shift!generalisedshift!generalised golden mean, it is equal to c ∈ Q^Mi ∈1, 2, …, k h ∈ Hm ∈ hF_ic(m) = 0, and it has ρ-bounded propagation, in particular, according to <ref>, it is ρ-strongly irreducible.In the case that ℛ is the cell space from <ref>, q = 1, k = 1, and F_1 = 0, 1, the non-negative integer ρ is equal to 1 and the generalised golden mean shift X is equal to the golden mean shift from <ref>.Let F be a finite subset of M, let p be a pattern of Q^F such thatf ∈ Fp_(f, ρ) ∩ F∈ X_(f, ρ) ∩ F,and let c be the point of Q^M that is equal to p on F and identically 0 on M ∖ F. Moreover, let i ∈ I and let h ∈ H. If hF_i ⊈ F, then, there is an m ∈ (hF_i) ∖ F, and, by definition of c, we have c(m) = 0. Otherwise, if hF_i ⊆ F, then, because m_0 ∈ F_i, we have hm_0 ∈ F; thus, because p_(hm_0, ρ) ∩ F∈ X_(hm_0, ρ) ∩ F, there is a point x ∈ X such that p_(hm_0, ρ) ∩ F = x_(hm_0, ρ) ∩ F; hence, by the characterisation of X, there is a cell m ∈ hF_i such that x(m) = 0; and therefore, because hF_i ⊆ (h (ρ)) ∩ F = (hm_0, ρ) ∩ F, we have c(m) = p(m) = x(m) = 0. Hence, in either case, there is an m ∈ hF_i such that c(m) = 0. Therefore, by the characterisation of X, we have c ∈ X and hence p = c_F ∈ X_F. In conclusion, X has ρ-bounded propagation. [<cit.>]In the situation of <ref>, the subshift 010, 111 of 0, 1^ is strongly irreducible and of finite type but does not have bounded propagation.A map from a shift space to another shift space is local if the image of a point is uniformly and locally determined in each cell, as defined in Let X and Y be two subshifts of Q^M, let Δ be a map from X to Y, let κ be a non-negative integer, let N be a subset of (κ) such that G_0N ⊆ N, and let ∙_H_0 be the left group action _H_0 × X_N → X_N of H_0 on X_N. The map Δ is called κ-locallocal@κ-localκ-local if and only if there is a ∙_H_0-invariant map δ X_N → Q such that x ∈ Xm ∈ M Δ(x)(m) = δ(n ↦ x(mn)).For each point x ∈ X and each cell m ∈ M, we have Δ(x)(m) = δ((g_m_0, m^-1 x)_N). Let Δ be a κ-local map from X to Y. For each non-negative integer κ' such that κ' ≥κ, the map Δ is κ'-local. Let X and Y be two subshifts of Q^M and let Δ be a map from X to Y. The map Δ is called locallocal if and only if there is a non-negative integer κ such that it is κ-local. Let X and Y be two subshifts of Q^M and let Δ be a map from X to Y. The map Δ is local if and only if it is the restriction to X → Y of the global transition function of a big-cellular automaton over ℛ with set of states Q and finite neighbourhood.[Sliding Block Codes <cit.>] In the situation of <ref>, local maps are but sliding block codes. [From Golden Mean to Even Shift <cit.>]Let X be the golden mean shift (<ref>), let Y be the even shift (<ref>), let δ be the map from X_0, 1 to 0, 1 given by 00 ↦ 1, 01 ↦ 0, and 10 ↦ 0, and let Δ be the map from X to Y given by x ↦ [z ↦δ(n ↦ x(z + n))]. The map Δ is, by definition, local and it is, according to example 1.5.6 in <cit.>, surjective. Domain and codomain of a local map can be restricted simultaneously to a subset of cells and its interior, as is done in Let Δ be a κ-local map from X to Y and let A be a subset of M. The mapΔ_A^-X_A→ Y_A^-κ, restriction Δ_A^- of Δ to A[symbols]DeltaAminus@Δ_A^-p↦Δ(c)_A^-κ,wherec ∈ Xsuch thatc_A = p,is called restriction of Δ to A. The image of a local map is a shift space, which is shown in Let Δ be a local map from X to Y. Its image Δ(X) is a subshift of Q^M. According to <ref>, the shift space X is compact. And, according to <ref> and <ref>, the map Δ is continuous. Therefore, the topological space Δ(X) is compact and hence closed. Moreover, because h Δ(X) = Δ(hX) = Δ(X), the topological space Δ(X) is shift-invariant. Therefore, according to <ref>, the topological space Δ(X) is a subshift of Q^M. The difference of two points of the full shift is the set of cells in which they differ, as defined in Let c and c' be two points of Q^M. The set(c, c') = m ∈ Mc(m) ≠ c'(m)difference (c, c') of c and c'is called difference of c and c'. A local map is pre-injective if it is injective on points with finite support, as defined in Let Δ be a local map from X to Y. It is called pre-injectivepre-injective if and only if, for each tuple (x, x') ∈ X × X such that (x, x') is finite and Δ(x) = Δ(x'), we have x = x'. A bijective local map with local inverse is a conjugacy, and its domain and codomain are conjugate, which is defined in Let Δ be a local map from X to Y. It is called conjugacyconjugacy if and only if it is bijective and its inverse is local.Let X and Y be two shift spaces. They are called conjugateconjugate if and only if there is a conjugacy from X to Y. Entropy of shift spaces is invariant under conjugacy, which is shown inLet ℛ be right amenable, let ℱ be a right Følner net in ℛ, and let X and Y be two conjugate subshifts of Q^M. Then, _ℱ(X) = _ℱ(Y).This is a direct consequence of <ref> and <ref>. Both the Moore and the Myhill property are introduced in Let X be a subshift of Q^M. It is said to have the * Moore propertyMoore propertyproperty!Moore if and only if each surjective local map from X to X is pre-injective;* Myhill propertyMyhill propertyproperty!Myhill if and only if each pre-injective local map from X to X is surjective.Both the Moore and the Myhill property is invariant under conjugacy.§ TILINGS Contents. A θ, κ, θ'-tiling is a subset of cells such that the balls of radius θ about those cells are pairwise at least κ + 1 apart and the balls of radius θ' about those cells cover all cells (see <ref>). If there are infinitely many cells, then, for each θ and κ, there is a θ, κ, 4 θ + 2 κ-tiling (see <ref>). And, a subset of a strongly irreducible shift space has less entropy than that space if about each point of a θ, κ, θ'-tiling the subset has fewer patterns with ball-shaped domains of radius θ than the space (see <ref>); in the proof of that statement we use <ref>. Let A_j_j ∈ J be a family of subsets of M and let κ be a non-negative integer. The family A_j_j ∈ J is called pairwise at least κ + 1 apartpairwise at least κ + 1 apartapart!pairwise at least κ + 1 if and only if j ∈ Jj' ∈ J []j ≠ j' (A_j, A_j') ≥κ + 1.Each pairwise at least κ + 1 apart family is pairwise disjoint. And each pairwise disjoint family is pairwise at least 0 + 1 apart.Let T be a subset of M, and let θ, κ, and θ' be three non-negative integers. The set T is called θ, κ, θ'-tiling of ℛtiling of R@θ, κ, θ'-tiling of ℛθ, κ, θ'-tiling T of ℛtilingthetakappathetaprime@θ, κ, θ'-tiling of ℛ if and only if the family (t, θ)_t ∈ T is pairwise at least κ + 1 apart and the family (t, θ')_t ∈ T is a cover of M.According to <ref>, each θ, κ, θ'-tiling of ℛ is a (θ), (θ')-tiling of ℛ, see <ref>; and each (θ), (θ')-tiling of ℛ is a θ, 0, θ'-tiling of ℛ. [Lattice] The (1), (2)-tiling from <ref> is a 1, 1, 2-tiling of ℛ. [Tree] The (1), (2)-tiling from <ref> is a 1, 0, 2-tiling of ℛ. Greedily picking elements that are pairwise far enough apart yields a tiling, which we show in Let M be infinite, and let θ and κ be two non-negative integers. There is a countably infinite θ, κ, θ'-tiling T of ℛ, where θ' = 4 θ + 2 κ.In the proof, infiniteness of M is used to deduce that spheres of arbitrarily big radii are non-empty.From each of the spheres (i (2 θ + κ + 1)), for i ∈_0, pick elements that are pairwise at least 2 θ + κ + 1 apart and whose (2 θ + κ)-closure covers the sphere — they constitute a set T. The family (t, θ)_t ∈ T is pairwise at least κ + 1 apart and the family (t, 4 θ + 2 κ)_t ∈ T is a cover of M. See <ref> for a schematic representation.Let i ∈_0. Furthermore, let M_i,1 = (i (2 θ + κ + 1)). Then, because M is infinite, according to <ref>, the set M_i,1 is non-empty and finite. For j ∈_+ in increasing order, if M_i,j∖m_i,1, m_i,2, …, m_i,j-1≠∅, then choose m_i,j∈ M_i,j∖m_i,1, m_i,2,…, m_i,j-1 and putM_i,j+1 = m_i,j∪ M_i,j∖(m_i,j, 2 θ + κ)= m ∈ M_i,j(m, m_i,j) = 0or (m, m_i,j) ≥ 2 θ + κ + 1;otherwise stop, put j_i = j and put M_i = M_i, j_i (see <ref>). By construction,m ∈ M_im' ∈ M_i []m ≠ m' (m, m') ≥ 2 θ + κ + 1,andm ∈ M_i,1 m' ∈ M_i (m, m') ≤ 2 θ + κ,and, for each i' ∈_0 with i' ≠ i, because M_i ⊆ M_i,1 and M_i'⊆ M_i',1, and M is infinite, according to <ref>,(M_i, M_i') ≥(M_i,1, M_i',1) ≥ 2 θ + κ + 1. Let T = ⋃_i ∈_0 M_i. Because, for each i ∈_0, the set M_i is finite, the set T is countable. And, because M_i,1_i ∈_0 is pairwise disjoint, so is M_i_i ∈_0 and hence T is infinite.Thus, T is countably infinite.Subproof of: (t, θ)_t ∈ T is pairwise at least κ + 1 apart Let t, t' ∈ T such that t ≠ t'. If there is an i ∈_0 such that t, t' ∈ M_i, then, according to <ref>, we have (t, t') ≥ 2 θ + κ + 1. Otherwise, there are i, i' ∈_0 with i ≠ i' such that t ∈ M_i and t' ∈ M_i', and then, according to <ref>, we have (t, t') ≥(M_i, M_i') ≥ 2 θ + κ + 1. In conclusion, in both cases, according to <ref>, we have ((t, θ), (t', θ)) ≥κ + 1.Subproof of: (t, 4 θ + 2 κ)_t ∈ T is a cover of M (see <ref>) Let m ∈ M. Then, there is an i ∈_0 such that i (2 θ + κ + 1) ≤m < (i + 1) (2 θ + κ + 1). Hence, according to <ref>,(m, M_i,1)= (M_i,1, m)≤(m_0, m) - i (2 θ + κ + 1)< (i + 1) (2 θ + κ + 1) - i (2 θ + κ + 1)= 2 θ + κ + 1.Thus, (m, M_i,1) ≤ 2 θ + κ. Therefore, there is an m' ∈ M_i,1 such that (m, m') ≤ 2 θ + κ. Moreover, according to <ref>, there is an m”∈ M_i such that (m', m”) ≤ 2 θ + κ. Thus,(m, m”) ≤(m, m') + (m', m”) ≤ 4 θ + 2 κ.Hence, m ∈(m”, 4 θ + 2 κ). Therefore, because m”∈ T, we have m ∈⋃_t ∈ T(t, 4 θ + 2 κ). In conclusion, ⋃_t ∈ T(t, 4 θ + 2 κ) = M. One by one banning subpatterns with similar domains that are far enough apart in a set of finite patterns, decreases its size by at least a multiplicative constant between 0 and 1 in each step. In other words, the number of finite patterns with a fixed domain, excluding those in which some subpatterns with similar domains that are far enough apart are embedded, is bounded above by some constant between 0 and 1 raised to the power of the number of forbidden subpatterns times the number of all finite patterns with the fixed domain, which is shown in Let X be a non-empty and κ-strongly irreducible subshift of Q^M, let F be a finite subset of M, let θ be a non-negative integer, let T be a subset of M such that the family (t, θ)_t ∈ T is pairwise at least κ + 1 apart, and, for each element t ∈ T, let p_t be a pattern of X_(t, θ). Furthermore, let ξ be the positive integer X_(θ)^+κ, let S be the finite set T ∩ F^-(θ + κ) (= t ∈ T (t, θ)^+κ⊆ F), and, for each element s ∈ S, let π_s be the map X_F → X_(s, θ), p ↦ p_(s, θ) (see <ref>).Then, X_F ∖⋃_s ∈ Sπ_s^-1(p_s)≤ (1 - ξ^-1)^S·X_F. In the proof, κ-strong irreducibility is used to extend an in X allowed F ∖(s, θ)^+κ-pattern by the (s, θ)-pattern p_s and a _κ^+ (s, θ)-pattern to an in X allowed F-pattern. Let s_j_j ∈1, 2, …, S be an enumeration of S, let Z_0 = X_F, and, for each ϑ∈0, 1, …, S - 1, let Z_ϑ + 1 = Z_ϑ∖π_s_ϑ + 1^-1(p_s_ϑ + 1) ∩ Z_ϑ. Furthermore, let ϑ∈0, 1, …, S - 1. Then, Z_ϑ≤X_(s_ϑ + 1, θ)^+ κ·(Z_ϑ)_F ∖(s_ϑ + 1, θ)^+ κ = ξ·(Z_ϑ)_F ∖(s_ϑ + 1, θ)^+ κ. And, because X is κ-strongly irreducible, each pattern of (Z_ϑ)_F ∖(s_ϑ + 1, θ)^+ κ can be extended by p_s_ϑ + 1 and a pattern with domain (s_ϑ + 1, θ)^+ κ∖(s_ϑ + 1, θ) to a pattern of π_s_ϑ + 1^-1(p_s_ϑ + 1) ∩ Z_ϑ and thus (Z_ϑ)_F ∖(s_ϑ + 1, θ)^+ κ≤π_s_ϑ + 1^-1(p_s_ϑ + 1) ∩ Z_ϑ. Hence, π_s_ϑ + 1^-1(p_s_ϑ + 1) ∩ Z_ϑ≥ξ^-1·Z_ϑ. Therefore,Z_ϑ + 1 =Z_ϑ - π_s_ϑ + 1^-1(p_s_ϑ + 1) ∩ Z_ϑ≤ (1 - ξ^-1) ·Z_ϑ.The statement follows by induction.As claimed, because X ≠∅, the integer ξ is positive; because, according to <ref> and <ref> of <ref>, s (θ + κ) = (s, θ)^+κ, we have S = T ∩ F^-(θ + κ) = t ∈ T (t, θ)^+κ⊆ F; and, because S ⊆⋃_s ∈ S(s, θ)^+κ⊆ F and F is finite, the set S is finite.Let B_s_s ∈ S = (s, θ)_s ∈ S, let s_j_j ∈1, 2, …, S be an enumeration of S, and, for each ϑ∈0, 1, …, S, let Z_ϑ = X_F ∖⋃_j = 1^ϑπ_s_j^-1(p_s_j) ( = p ∈ X_Fj ∈1, 2, …, ϑ p_B_s_j≠ p_s_j).To establish the claim, we prove by induction on ϑ, that, for each ϑ∈0, 1, …, S,Z_ϑ≤ (1 - ξ^-1)^ϑ·X_F. Base Case Let ϑ = 0. Then, because ⋃_j = 1^0 π_s_j^-1(p_s_j) = ∅ and (1 - ξ^-1)^0 = 1, <ref> holds. Note that 0^0 = 1.Inductive Step Let ϑ∈0, 1, …, S - 1 such that <ref>, called inductive hypothesis, holds. Furthermore, let Z = Z_ϑ. Because B_s_ϑ + 1^+κ⊆ F, Z ⊆ Z_B_s_ϑ + 1^+κ× Z_F ∖ B_s_ϑ + 1^+κ⊆ X_B_s_ϑ + 1^+κ× Z_F ∖ B_s_ϑ + 1^+κ.Hence, Z≤X_B_s_ϑ + 1^+κ·Z_F ∖ B_s_ϑ + 1^+κ. Moreover, according to <ref>, we have X_B_s_ϑ + 1^+κ = X_(θ)^+κ = ξ (where we used that B_s_ϑ + 1^+κ = s_ϑ + 1(θ)^+κ, which holds according to <ref> and <ref> of <ref>). Therefore, Z≤ξ·Z_F ∖ B_s_ϑ + 1^+κ. Let p ∈ Z_F ∖ B_s_ϑ + 1^+κ. Then, p ∈ X_F ∖ B_s_ϑ + 1^+κ. Moreover, according to <ref>, we have (B_s_ϑ + 1, F ∖ B_s_ϑ + 1^+κ) ≥κ + 1. Hence, because X is κ-strongly irreducible, there is a p”∈ X_F such that p”_(p) = p and p”_(p_s_ϑ + 1) = p_s_ϑ + 1. Furthermore, because (t, θ)_t ∈ T is pairwise at least κ + 1 apart, for each j ∈1, 2, …, ϑ, we have B_s_j⊆ F ∖ B_s_ϑ + 1^+κ. Therefore, for each j ∈1, 2, …, ϑ, we have p”_B_s_j = p_B_s_j≠ p_s_j and hence p”∉π_s_j^-1(p_s_j). Thus, p”∈ Z. Moreover, p”∈π_s_ϑ + 1^-1(p_s_ϑ + 1). Therefore, Z_F ∖ B_s_ϑ + 1^+κ≤π_s_ϑ + 1^-1(p_s_ϑ + 1) ∩ Z. Together,Z≤ξ·π_s_ϑ + 1^-1(p_s_ϑ + 1) ∩ Z.Because Z_ϑ + 1 = Z ∖π_s_ϑ + 1^-1(p_s_ϑ + 1),Z_ϑ + 1 =Z ∖π_s_ϑ + 1^-1(p_s_ϑ + 1)=Z ∖ (π_s_ϑ + 1^-1(p_s_ϑ + 1) ∩ Z)=Z - π_s_ϑ + 1^-1(p_s_ϑ + 1) ∩ Z≤Z - ξ^-1·Z=(1 - ξ^-1) ·Z.Hence, according to the inductive hypothesis,Z_ϑ + 1 ≤ (1 - ξ^-1) · (1 - ξ^-1)^ϑ·X_F=(1 - ξ^-1)^ϑ + 1·X_F. In conclusion, according to the principle of mathematical induction, for each ϑ∈0, 1, …, S, <ref> holds. The number of elements in a finite set is bounded above by the number of elements its interior shares with a tiling times the number of elements of a big enough ball plus the number of elements of a big enough boundary of the finite set, which is shown in Let F be a finite subset of M, let θ, κ, and θ' be three non-negative integers, let T be a subset of M such that (t, θ')_t ∈ T is a cover of M, and let S be the finite set T ∩ F^-(θ + κ) (= t ∈ T (t, θ)^+κ⊆ F) (see <ref>).Then,F≤S·(θ') + _θ + κ + θ'^- F. For each m ∈ F, if m ∉ S^+ θ', then m ∉ F^-(θ + κ + θ'). Thus, F ⊆ S^+ θ'∪ F ∖ F^-(θ + κ + θ') = ⋃_s ∈ S(s, θ')∪_θ + κ + θ'^- F. Hence, F≤S·(θ') + _θ + κ + θ'^- F.Let m ∈ F ∖⋃_s ∈ S(s, θ'). Because (t, θ')_t ∈ T is a cover of M, there is a t ∈ T such that m ∈(t, θ'). Because m ∉⋃_s ∈ S(s, θ'), we have t ∉ S and hence (t, θ)^+κ⊈ F. Because m ∈(t, θ'), we have (m, t) ≤θ' and hence t ∈(m, θ') = m^+θ'. Suppose that m ∈ F^-(θ + κ + θ'). Then, according to <ref> of <ref>, <ref> of <ref>, and <ref> of <ref>,(t, θ)^+κ = (t^+θ)^+κ⊆[][]m^+θ'^+θ^+κ= m^+(θ + κ + θ')⊆ (F^-(θ + κ + θ'))^+(θ + κ + θ')= F,which contradicts that (t, θ)^+κ⊈ F. Hence, m ∈ F ∖ F^-(θ + κ + θ') = _θ + κ + θ'^- F. Therefore,F ⊆[]⋃_s ∈ S(s, θ')∪_θ + κ + θ'^- F.Moreover, because S ⊆⋃_s ∈ S(s, θ)^+κ⊆ F and F is finite, the set S is finite. And, for each s ∈ S, according to <ref>, we have (s, θ') = (θ'). In conclusion,F ≤∑_s ∈ S(θ') + _θ + κ + θ'^- F=S·(θ') + _θ + κ + θ'^- F.The number of elements that the components of a right Følner net share with a tiling is asymptotically bounded below away from zero, which is shown inLet F_i_i ∈ I be a right Erling net in ℛ (which exists according to <ref>), let θ, κ, and θ' be three non-negative integers, let T be a subset of M such that (t, θ')_t ∈ T is a cover of M. There is a positive real number ε∈_> 0 and there is an index i_0 ∈ I such that, for each index i ∈ I with i ≥ i_0, we have T ∩ F_i^-(θ + κ)≥εF_i.Let i ∈ I and let T_i = T ∩ F_i^-(θ + κ). Then, according to <ref>, we have F_i≤T_i·(θ') + _θ + κ + θ'^- F_i. Hence,T_i/F_i≥1/(θ')·*1 - _θ + κ + θ'^- F_i/F_i.Moreover, because F_i_i ∈ I is a right Erling net, there is a ξ∈0, 1 and there is an i_0 ∈ I such thati ∈ I *i ≥ i_0 _θ + κ + θ'^- F_i/F_i≤ξ.Let ε = (1/(θ')) · (1 - ξ). Then, for each i ∈ I with i ≥ i_0, we have T_i/F_i≥ε. If a shift space has at least two points, then, for each non-empty domain, it has at least two patterns.Let X be a subshift of Q^M such that X≥ 2 and let A be a non-empty subset of M. Then, X_A≥ 2.Because X≥ 2, there are x and x' in X such that x ≠ x'. Thus, there is an m ∈ M such that x(m) ≠ x'(m). And, because A is non-empty, there is an a ∈ A. The element h = g_m_0, a g_m_0, m^-1 is contained in H and satisfies h^-1 a = m. Hence, (hx)(a) ≠ (hx')(a). Therefore, (hx)_A and (hx')_A are distinct and are contained in X_A. In conclusion, X_A≥ 2. A subset of a strongly irreducible shift space has less entropy than that space if about each point of a tiling the subset has fewer patterns of a certain radius than the space, which is shown inLet ℱ = F_i_i ∈ I be a right Erling net in ℛ (which exists according to <ref>), let X be a κ-strongly irreducible subshift of Q^M such that X≥ 2, let Y be a subset of X, and let T be a θ, κ, θ'-tiling of ℛ such that, for each element t ∈ T, we have Y_(t, θ)⫋ X_(t, θ). Then, _ℱ(Y) < _ℱ(X). In the proof, κ-strong irreducibility is used to apply <ref> yielding the inequality X_F_i∖⋃_t ∈ T_iπ_i, t^-1(p_t)≤ (1 - ξ^-1)^T_i·X_F_i. Let p_t ∈ X_(t, θ)∖ Y_(t, θ) and let T_i = T ∩ F_i^-(θ + κ). Then, Y_F_i⊆ X_F_i∖⋃_t ∈ T_iπ_i, t^-1(p_t). Hence, Y_F_i≤ (1 - ξ^-1)^T_i·X_F_i. Therefore, logY_F_i/F_i≤log(1 - ξ^-1) ·T_i/F_i + logX_F_i/F_i. In conclusion, because log(1 - ξ^-1) < 0 and T_i/F_i_i ∈ I is eventually bounded below away from zero, we have _ℱ(Y) < _ℱ(X). For each t ∈ T, because Y_(t, θ)⫋ X_(t, θ), we have X_(t, θ)∖ Y_(t, θ)≠∅. Let p_t_t ∈ T be a transversal of X_(t, θ)∖ Y_(t, θ)_t ∈ T and let ξ = X_(θ)^+κ. Furthermore, let i ∈ I, let T_i = T ∩ F_i^-(θ + κ) (= t ∈ T (t, θ)^+κ⊆ F_i) and, for each t ∈ T_i, let π_i, t X_F_i→ X_(t, θ), p ↦ p_(t, θ). Note that, because X≥ 2, according to <ref>, we have ξ≥ 2 and hence 1 - ξ^-1 > 0. Because (t, θ)_t ∈ T is pairwise at least κ + 1 apart, according to <ref>,X_F_i∖⋃_t ∈ T_iπ_t^-1(p_t)≤ (1 - ξ^-1)^T_i·X_F_i.For each t ∈ T_i, because p_t ∉ Y_(t, θ), we have π_i, t^-1(p_t) ∩ Y_F_i = ∅. Hence, *⋃_t ∈ T_iπ_i, t^-1(p_t)∩ Y_F_i = ∅. Therefore,Y_F_i =Y_F_i∖⋃_t ∈ T_iπ_i, t^-1(p_t)≤X_F_i∖⋃_t ∈ T_iπ_i, t^-1(p_t)≤ (1 - ξ^-1)^T_i·X_F_i.Thus,logY_F_i/F_i≤T_i/F_i·log(1 - ξ^-1)+ logX_F_i/F_i.Because (t, θ')_t ∈ T is a cover of M, according to <ref>, there is an ε∈_> 0 and there is an i_0 ∈ I such thati ∈ I *i ≥ i_0 T_i/F_i≥ε.Hence, because log(1 - ξ^-1) < 0,_ℱ(Y) ≤ε·log(1 - ξ^-1) + _ℱ(X) <_ℱ(X). § THE MOORE AND THE MYHILL PROPERTIES Contents. The image of a local map to a strongly irreducible shift space that is not surjective does not have maximal entropy (see <ref>). And the converse of that statement obviously holds. Moreover, a local map from a strongly irreducible shift space of finite type whose image has less entropy than its domain is not pre-injective (see <ref>). And the converse of that statement also holds (see <ref>). These four statements establish the Garden of Eden theorem (see Main <ref>). It follows that strongly irreducible shift spaces of finite type have the Moore and the Myhill property (see <ref>). Body. Because a local map that is not surjective has a Garden of Eden pattern, the entropy of its image is not maximal, which is shown in Let M be infinite, let X be a non-empty subshift of Q^M, let Y be a strongly irreducible subshift of Q^M, let Δ be a local map from X to Y that is not surjective, and let ℱ be a right Erling net in ℛ (which exists according to <ref>). Then, _ℱ(Δ(X)) < _ℱ(Y).In the proof, infiniteness of M is used to apply <ref> yielding a tiling, locality of Δ is used to apply <ref> yielding that Δ(X) is a subshift of Q^M, and strong irreducibility of Y is used to apply <ref> yielding a strict inequality for entropies. Because Δ is not surjective, there is a Garden of Eden configuration. Thus, because Δ is local, there is a Garden of Eden pattern. Hence, there are too many Garden of Eden configurations for the entropy to be maximal. Because Y is strongly irreducible, there is a κ∈_0 such that Y is κ-strongly irreducible. And, because Δ is not surjective, there is a y ∈ Y ∖Δ(X). Hence, according to <ref>, there is a ρ∈_0 such that y_(ρ)∉ (Δ(X))_(ρ) and thus y_(ρ)∈ Y_(ρ)∖ (Δ(X))_(ρ). And, because M is infinite, according to <ref>, there is a ρ, κ, θ'-tiling T of ℛ.According to <ref>, the set Δ(X) is a subshift of Q^M. And, for each t ∈ T, according to <ref>, we have t (ρ) = (t, ρ). Therefore, for each t ∈ T, because t is bijective and according to <ref>, we have t(y_(ρ)) ∈ Y_(t, ρ)∖ (Δ(X))_(t, ρ) and thus (Δ(X))_(t, ρ)⫋ Y_(t, ρ). Because X is non-empty and Δ is not surjective, we have Y≥ 2. In conclusion, because Y is κ-strongly irreducible, according to <ref>, we have _ℱ(Δ(X)) < _ℱ(Y). If there are less patterns in the codomain of a local map than in its domain, at least two patterns have the same image, which is shown in Let X be a κ-strongly irreducible subshift of Q^M, let Y be a subshift of Q^M, let Δ be a κ-local map from X to Y, and let F be a finite subset of M such that Y_F^+2κ < X_F. There are two patterns p and p' in X_F^+3κ such that p ≠ p', p__2κ^+ F^+κ = p'__2κ^+ F^+κ, and Δ_F^+3κ^-(p) = Δ_F^+3κ^-(p'). In the proof, strong irreducibility of X is used to extend an in X allowed F-pattern by an in X allowed _2κ^+ F^+κ-pattern and an _κ^+ F-pattern to an in X allowed F^+3κ-pattern; and κ-locality of Δ is used to restrict it to a map from X_F^+3κ to Y_F^+2κ.Because Y_F^+2κ < X_F, we have X_F > 0, thus X_F ≠∅, and hence X ≠∅.Therefore, there is a v ∈ X__2κ^+ F^+κ. Let P_v = p ∈ X_F^+3κ p_(v) = v. Note that, according to <ref> of <ref>, we have _2κ^+ F^+κ = F^+3κ∖ F^+κ. Let u ∈ X_F. According to <ref>, we have (F, (v)) ≥κ + 1. Hence, because X is κ-strongly irreducible, there is an x ∈ X such that x_F = u and x_(v) = v. Let p = x_F^+3κ. Then, p_F = u and p ∈ P_v.Therefore, P_v≥X_F > Y_F^+2κ. The restriction ϕ of Δ_F^+3κ^-X_F^+3κ→ Y_F^+2κ to P_v →Δ_F^+3κ^-(P_v) is surjective. Note that, because Δ is κ-local and, according to <ref> of <ref>, we have (F^+3κ)^-κ⊇ F^+2κ, we can choose Y_F^+2κ as the codomain of Δ_F^+3κ^-. If ϕ were injective, then P_v = ϕ(P_v)≤Y_F^+2κ, which contradicts that P_v > Y_F^+2κ. Hence, ϕ is not injective. In conclusion, there are p, p' ∈ P_v such that p ≠ p' and ϕ(p) = ϕ(p'). Because a local map, that has an image whose entropy is less than the entropy of its domain, maps at least two finite patterns to the same pattern, it is not pre-injective, which is shown in Let ℛ be right amenable, let ℱ = F_i_i ∈ I be a right Følner net in ℛ, let X be a strongly irreducible subshift of Q^M of finite type, let Y be a subshift of Q^M, and let Δ be a local map from X to Y such that _ℱ(Δ(X)) < _ℱ(X). The map Δ is not pre-injective.In the proof, strong irreducibility of X and locality of Δ are used to apply <ref> yielding two distinct finite patterns with the same domain, identical boundaries, and identical images; and of finite typeness of X is used to apply <ref> to identically extend these patterns to points of X.Because the entropy of Δ(X) is less than the one of X, the number of finite patterns in Δ(X) grows slower than in X. Hence, there are two distinct finite patterns in X that have the same image and these can be identically extended to two distinct points of X that have the same image. Therefore, the map Δ is not pre-injective.According to <ref>, <ref> and <ref>, and <ref>, there is a κ∈_0 such that X is κ-strongly irreducible, X is κ-step, and Δ is κ-local.Let Y = Δ(X). According to <ref> and <ref> and the precondition _ℱ(Y) < _ℱ(X), we have _F_i^+2κ_i ∈ I(Y) ≤_ℱ(Y) < _ℱ(X). Hence, there is an i ∈ I such thatlogY_F_i^+2κ/F_i < logX_F_i/F_i.Thus, logY_F_i^+2κ < logX_F_i and thus Y_F_i^+2κ < X_F_i.Therefore, because X is κ-strongly irreducible and Δ is κ-local, according to <ref>, there are p and p' in X_F_i^+3κ such that p ≠ p', p__2κ^+ F_i^+κ = p'__2κ^+ F_i^+κ, and Δ_F_i^+3κ^-(p) = Δ_F_i^+3κ^-(p').Hence, because X is κ-step, according to <ref>, there are x and x' in X such that x_(p) = p, x'_(p') = p', and x_M ∖ F_i^+κ = x'_M ∖ F_i^+κ. In particular, because p ≠ p', we have x ≠ x' and, because F_i^+κ is finite, the set (x, x') is finite.Moreover, Δ(x)_F_i^+2κ = Δ_F_i^+3κ^-(p) = Δ_F_i^+3κ^-(p') = Δ(x')_F_i^+2κ. And, according to <ref> and <ref>, we have Δ(x)_M ∖ F_i^+2κ = Δ(x')_M ∖ F_i^+2κ. Therefore, Δ(x) = Δ(x'). In conclusion, Δ is not pre-injective. If in a point of a shift space we replace all occurrences of a pattern by another pattern with the same image that agree on a big enough boundary, we get a new point of the shift space in which the first pattern does not occur that has the same image as the original point, which is shown in Let X be a κ-step subshift of Q^M, let Y be a subshift of Q^M, let Δ be a κ-local map from X to Y, let A be a subset of M, let p and p' be two patterns in X_A^+2κ such that p__2κ^+ A = p'__2κ^+ A and Δ_A^+2κ^-(p) = Δ_A^+2κ^-(p'). Furthermore, let c be a point of X and let T be a subset of M such that the family tA^+2κ_t ∈ T is pairwise disjoint and that, for each element t ∈ T, we have p _t c. Put c' = c_M ∖ (⋃_t ∈ T tA^+2κ)×∐_t ∈ T tp'.Then, for each element t ∈ T, we have p' _t c', and c' ∈ X, and Δ(c) = Δ(c'). In particular, if p ≠ p', then, for each element t ∈ T, we have p _t c'.In the proof, κ-stepness of X is used to apply lemma <ref> to deduce that c' is a point of X; and locality of Δ is used to deduce that Δ(c) = Δ(c'). There are x and x' in X such that x_A^+2κ = p and x'_A^+2κ = p'. Thus, for each t ∈ T, we have (tx')_tA^+2κ = tp'. Hence,c' = c_M ∖ (⋃_t ∈ T tA^+2κ)×∐_t ∈ T (tx')_tA^+2κMoreover, for each t ∈ T, according to <ref>, we have tx' ∈ X. And, by precondition, (tA)^+2κ_t ∈ T is pairwise disjoint (where we used that tA^+2κ = (tA)^+2κ, which holds according to <ref> of <ref>). And, for each t ∈ T, we have (tx')__2κ^+(tA) = (tp')__2κ^+(tA) = (tp)__2κ^+(tA) = c__2κ^+(tA) (where we used that _2κ^+(tA) = t _2κ^+ A, which holds according to <ref> of <ref>). Therefore, because X is κ-step, according to <ref>, we have c' ∈ X.Let m ∈ M. Case 1: t ∈ Tm ∈ tA^+κ. Then, according to <ref> of lemma <ref> and <ref> of <ref>m (κ) ⊆ (tA^+κ) (κ)= (tA^+κ)^+κ=tA^+2κ. Hence, because Δ is κ-local, according to <ref> and <ref>, Δ(c')(m) = Δ_tA^+2κ^-(tp')= t Δ_A^+2κ^-(p')= t Δ_A^+2κ^-(p)= Δ_tA^+2κ^-(tp)= Δ(c)(m). Case 2: t ∈ Tm ∉ tA^+κ. Then, m ∈ M ∖⋃_t ∈ T tA^+κ. Hence, according to <ref> of <ref>, <ref> of <ref>, <ref> of <ref>, and <ref> of <ref>, m (κ) ⊆[]M ∖⋃_t ∈ T tA^+κ(κ)= []M ∖⋃_t ∈ T tA^+κ^+κ=M ∖[]⋃_t ∈ T tA^+κ^-κ⊆M ∖⋃_t ∈ T (tA^+κ)^-κ⊆M ∖⋃_t ∈ T tA^+ 0=M ∖⋃_t ∈ T tA. Therefore, because Δ is κ-local and c'_M ∖⋃_t ∈ T tA = c_M ∖⋃_t ∈ T tA, we have Δ(c')(m) = Δ(c)(m).In either case, Δ(c')(m) = Δ(c)(m). Therefore, Δ(c') = Δ(c). Because a local map that is not pre-injective maps at least two finite patterns to the same pattern, the entropy of its image is less than the entropy of its domain, which is shown in Let M be infinite, let X be a strongly irreducible subshift of Q^M of finite type, let Y be a subshift of Q^M, let Δ be a local map from X to Y that is not pre-injective, and let ℱ be a right Erling net in ℛ (which exists according to <ref>). Then, _ℱ(Δ(X)) < _ℱ(X).In the proof, infiniteness of M is used to apply <ref> yielding a tiling, strong irreducibility of X is used to apply <ref> yielding a strict inequality for entropies, finite typeness of X and locality of Δ is used to apply <ref> yielding that the image of all points of X in which a certain pattern does not occur at points of a tiling is the same as the image of X.Because Δ is not pre-injective, there are two distinct points of X with the same image that differ only in finitely many cells. Thus, there are two distinct finite patterns, say p and p', with the same image. Hence, the image of X is equal to the image of the set Z of all points of X in which the pattern p does not occur. Because there are too many points not in Z, this set does have less entropy than X.According to <ref>, <ref> and <ref>, and <ref>, there is a κ∈_0 such that X is κ-strongly irreducible, X is κ-step, and Δ is κ-local.Because Δ is not pre-injective, there are c and c' in X such that (c, c') is finite, Δ(c) = Δ(c'), and c ≠ c'; in particular, X≥ 2. Hence, there is a ρ∈_0 such that (c, c') ⊆(ρ). Let p = c_(ρ)^+2κ and let p' = c'_(ρ)^+2κ. Then, p ≠ p'; m_0 ∈(ρ); p, p' ∈ X_(ρ)^+2κ; p__2κ^+ (ρ) = p'__2κ^+ (ρ); and, because Δ(c) = Δ(c'), we have Δ_(ρ)^+2κ^-(p) = Δ_(ρ)^+2κ^-(p').Because M is infinite, according to <ref>, there is a ρ + 2 κ, κ, θ'-tiling T of ℛ. LetZ = x ∈ Xt ∈ Tp _t x.For each t ∈ T, according to <ref>, we have tp ∈ X_t (ρ)^+2κ∖ Z_t (ρ)^+2κ and hence Z_t (ρ)^+2κ⫋ X_t (ρ)^+2κ. Moreover, for each t ∈ T, according to <ref> of <ref> and <ref>, we have t (ρ)^+2κ = (t, ρ + 2 κ). Therefore, because X is κ-strongly irreducible and X≥ 2, according to <ref>, we have _ℱ(Z) < _ℱ(X). Hence, according to <ref>, we have _ℱ(Δ(Z)) < _ℱ(X).Let x ∈ X. Put U = t ∈ Tp _t x. Because X is κ-step and Δ is κ-local, according to <ref>, there is an x' ∈ X such that x' ∈ Z and Δ(x) = Δ(x'). Therefore, Δ(X) = Δ(Z). In conclusion, _ℱ(Δ(X)) < _ℱ(X).Because a right Følner net in a finite cell space is eventually equal to the set of cells, the entropy of a subset of the full shift is a function of the cardinality of that set, which is shown in Let ℛ be right amenable, let ℱ = F_i_i ∈ I be a right Følner net in ℛ, let M be finite, and let X be a subset of Q^M. Then,X = exp[]M·_ℱ(X).Let F be a non-empty and finite subset of M such that F ≠ M. Then, becauseis transitive, there is a 𝔤∈ GG_0 such that (F 𝔤) ∩ (M ∖ F) ≠∅. Hence, F 𝔤⊈ F, thus F ⊈ (𝔤)^-1(F), thus F ∖ (𝔤)^-1(F) ≠∅, and therefore F ∖ (𝔤)^-1(F)≠ 0. On the other hand, M ∖ (𝔤)^-1(M) = 0. Moreover, because M is finite, the set F ⊆ MF ≠∅, Ffinite is finite and hence its subset F_ii ∈ I is finite too. Furthermore, because ℱ is a right Følner net,𝔤∈ GG_0 lim_i ∈ IF_i ∖ (𝔤)^-1(F_i)/F_i = 0.Altogether, ℱ is eventually equal to M. Therefore,_ℱ(X) = logπ_M(X)/M = logX/M.In conclusion, X = exp(M·_ℱ(X)). Because surjectivity as well as pre-injectivity of a local map is characterised by maximal entropy of its image with respect to its codomain or domain, if both domains have the same entropy, then a local map is surjective if and only if it is pre-injective, which is shown in[Garden of Eden Theorem; Edward Forrest Moore, 1962; John R. Myhill, 1963; Analogue of <ref>] Let ℛ be right amenable, let ℱ be a right Følner net in ℛ, let X be a non-empty strongly irreducible subshift of Q^M of finite type, let Y be a strongly irreducible subshift of Q^M such that _ℱ(X) = _ℱ(Y), and let Δ be a local map from X to Y. The map Δ is surjective if and only if it is pre-injective.In the proof, non-emptiness of X, strong irreducibility of Y, and locality of Δ are used to apply <ref> yielding a characterisation of surjectivity; and strong irreducibility and finite typeness of X, and locality of Δ are used to apply <ref> yielding a characterisation of pre-injectivity.First, let M be finite. Then, Q^M is finite, and thus X and Y are finite. Hence, because _ℱ(X) = _ℱ(Y), according to <ref>, we have X = Y. Therefore, Δ is surjective if and only if it is injective. Moreover, because M is finite, the map Δ is pre-injective if and only if it is injective. In conclusion, Δ is surjective if and only if it is pre-injective.Secondly, let M be infinite. According to <ref>, the map Δ is not surjective if and only if _ℱ(Δ(X)) < _ℱ(Y). And, according to <ref> and <ref>, because _ℱ(X) = _ℱ(Y), we have _ℱ(Δ(X)) < _ℱ(Y) if and only if Δ is not pre-injective. Hence, Δ is not surjective if and only if it is not pre-injective. In conclusion, Δ is surjective if and only if it is pre-injective. Let ℛ be right amenable. Each strongly irreducible subshift of Q^M of finite type has the Moore and the Myhill property.This is a direct consequence of <ref> and the fact that the empty strongly irreducible subshift of Q^M has the Moore and the Myhill property. Let ℳ = M, G, be a right-amenable and finitely right-generated left-homogeneous space with finite stabilisers and let Q be a finite set. For each coordinate system 𝒦 for ℳ and each 𝒦-big subgroup H of G, each strongly irreducible subshift of Q^M of finite type with respect to ℛ = ℳ, 𝒦 and H has the Moore and the Myhill property with respect to ℛ and H.This is a direct consequence of <ref>.Note that in <ref> we do not have to choose a finite and symmetric right-generating set S, because being a subshift, being strongly irreducible, being of finite type, being local, being surjective, being pre-injective, having the Moore property, and having the Myhill property, do not depend on the choice of a finite and symmetric right-generating set of ℛ; the reason for the properties that depend on the metric induced by such a right-generating set is that those metrics are, according to <ref>, pairwise Lipschitz equivalent. [From Golden Mean to Even Shift] In the situation of <ref>, let ℱ be the right Følner net 1, 2, …, n_n ∈_+. Recall that the golden mean shift X is non-empty (<ref>), strongly irreducible (<ref>), and of finite type (<ref>); and that the even shift Y is strongly irreducible (<ref>) but not of finite type (<ref>). And, according to example 4.1.4 in <cit.> and Example 4.1.6 in <cit.>, the entropy of X with respect to ℱ and the one of Y are both the golden mean (1 + √(5))/2. Therefore, according to <ref>, because the local map Δ from X to Y is surjective (<ref>), it is also pre-injective. However, it is not injective, because the two points of X with alternating 0's and 1's, that is, those of the form … 010101 …, are both mapped to the point of Y with only 0's, that is, the one of the form … 000 ….CHAPTER: A QUASI-SOLUTION OF THE FIRING MOB SYNCHRONISATION PROBLEM Abstract. We construct a time-optimal quasi-solution of the firing mob synchronisation problem over finite, connected, and undirected multigraphs whose maximum degrees are uniformly bounded by a constant. It is only a quasi-solution because its number of states depends on the graph or, from another perspective, does not depend on the graph but is countably infinite. To construct this quasi-solution we introduce signal machines over continuum representations of such multigraphs and construct a signal machine whose discretisation is a cellular automaton that quasi-solves the problem. This automaton uses a time-optimal solution of the firing squad synchronisation problem in dimension one with one general at one end to synchronise edges, and freezes and thaws the synchronisation of edges in such a way that all edges synchronise at the same time. Introduction. The firing squad synchronisation problem in dimension one with one general at one end is to synchronise each finite one-dimensional array of cells starting from one end of the array and the cell at this end is called general. It was proposed by John R. Myhill in 1957, solved by John McCarthy and Marvin Lee Minsky, and published by Edward Forrest Moore in 1962 (see <cit.>). The first time-optimal several-thousand-states solution was found by Eiichi Goto in 1962 (see <cit.>), reduced to 16 states by Abraham Waksman in 1966 (see <cit.>), and reduced to 8 states by Robert Balzer in 1967 (see <cit.>). Hein D. Gerken found another time-optimal 7-states solution in 1987 (see <cit.>) and Jacques Mazoyer found a time-optimal 6-states solution also in 1987 (see <cit.>). It is unknown whether there is a time-optimal 5-states solution but it is known that there is no time-optimal 4-states solution, a result due to Robert Balzer and Peter Sanders (see <cit.>). The firing mob synchronisation problem is to synchronise each finite, connected, and undirected graph whose maximum degree is bounded by a fixed constant starting from any vertex and this vertex is called general. It was solved by P. Rosenstiehl, J. R. Fiksel, and A. Holliger in 1972 (see <cit.>) and also by Francesco Romani in 1976 (see <cit.>), where the latter solution achieves better running times than the former. The problem for specific classes of graphs were for example studied by Kojiro Kobayashi in 1977 and 1978 (see <cit.>) and by Zsuzsanna Róka in 2000 (see <cit.>). Karel Culik II and Simant Dube presented a solution of the general case in 1991 (see <cit.>). It needs 3.5 r-many steps, where r is the maximal distance of the general to a vertex and is called radius of the graph with respect to the general. By using more and more states, the solution can be adjusted such that the number of steps it needs approaches 3 r.It was shown that r + d is a lower bound for the number of steps that solutions of the firing mob synchronisation problem need by John J. Grefenstette in 1983 (see <cit.>), where d is the maximal distance between two vertices of the graph and is called diameter of the graph. Because there are graphs and choices of generals such that the diameter is 2 r, the solutions by Karel Culik II and Simant Dube approach the optimal number of steps, namely 3 r, if r is taken as problem size. However, if r + d is taken as problem size, then their solutions do not approach the optimal number of steps.In the present chapter we construct a time-optimal quasi-solution that needs exactly r + d steps but whose number of states depends on the graph or, from another perspective, does not depend on the graph but is countably infinite (this is why we call it a quasi-solution). It can also be turned into a time-optimal quasi-solution of the firing squad synchronisation problem for any region in any dimension with one general at any position by regarding each region to be synchronised as a graph, where cells in the region are vertices and edges are neighbourhood relationships. However, restricted to specific classes of problems, the quasi-solution may not be time-optimal. For example, restricted to rectangular regions with one general at one corner, the quasi-solution needs 2 (k + ℓ - 2)-many steps whereas (k + ℓ + maxk, ℓ - 3)-many steps is optimal (see for example <cit.>), where k and ℓ are the side lengths of the rectangle. Nevertheless, because the quasi-solution is (trivially) embeddable in the sense of <cit.>, according to theorem 1 in <cit.>, it can be combined with finitely many embeddable time-optimal solutions for specific classes of problems to get one quasi-solution that is also time-optimal for those classes. Examples of solutions for specific classes, like rectangular regions with one general at the upper left corner, are given in sections 5 and 6 in <cit.>.To design, explain, and draw solutions of firing squad/mob synchronisation problems, it is convenient to think about, talk about, and draw continuous space-time diagrams of different kinds of signals that move across the cell space, vanish or give rise to new signals upon reaching boundaries or junctions of the space or upon colliding with each other. This is mostly done in an informal way, but the idea of signals has also been formalised for one-dimensional cellular automata by Jérôme Olivier Durand-Lose in 2005 (see <cit.>).This formalisation however does not handle accumulations of events like collisions and does not allow infinitely many signals of different speeds, which naturally occur and are necessary in descriptions of many solutions of the firing squad synchronisation problem by signals. For example, collisions accumulate at the time synchronisation finishes and infinitely many signals of different speeds may originate from the general. In the time evolutions of the actual cellular automata, the accumulations of collisions disappear due to the discreteness of space and time, and the infinitely many signals are cleverly produced by finitely many states (see for example <cit.>).Because we want to describe our quasi-solution in terms of signals in a formal way, we first introduce continuum representations of finite and connected multigraphs (without self-loops), we secondly introduce signal machines over such representations that allow infinitely many signals of different speeds and seamlessly handle accumulations of events and accumulations of accumulations of events and so forth, and we thirdly construct a signal machine for the continuous firing mob synchronisation problem over such representations and shortly note how to discretise it to get a cellular automaton quasi-solution of the firing mob synchronisation problem. Contents. In <ref> we state the firing squad and the firing mob synchronisation problems. In <ref> we introduce undirected multigraphs (without self-loops) and direction-preserving paths in such graphs, which are paths that do not make U-turns. In <ref> we introduce continuum representations of undirected multigraphs, which are in a sense drawings of graphs in a high-dimensional Euclidean space. In <ref> we introduce signal machines, which can be studied in their own right, but which can also be thought of as high-level views of time evolutions of cellular automata over graphs, like cellular automata over finitely right-generated cell spaces, that are restricted to configurations with a fixed finite support. In <ref> we construct a signal machine whose discretisation is a cellular automaton that quasi-solves the firing mob synchronisation problem in (r + d)-many steps. And in <ref> we sketch a proof for that statement. The impatient may right now have a look at the continuous space-time diagrams of the synchronisations of small trees as performed by the quasi-solution: See <ref> on figure:MazoyerWithOneAndTwoEdges,figure:fsspWithTwoEdgesAndGeneralInBetweenAndAtTheLeft,figure:fsspWithThreeEdgesInARowAndGeneralAtTheSecondVertexFromTheLeft,figure:fsspWithThreeEdgesIncidentToTheSameVertexAndGeneralAtTheVertexAndAtTheLeafOfTheShortestEdge. Preliminary Notions. The affinely extended real numbersaffinely extended real numbers∪-∞, +∞ are denoted by [symbols]Roverline@. For each tuple (r, r') ∈× such that r ≤ r', the closed, open, and the two half-open extended real intervalsextended real intervals r, r', r, r', and [r, r'[ and ]r, r'] with the endpoints r and r' are denoted by r, r', r, r', and [r, r'[ and ]r, r'] respectively. And, for each tuple (z, z') ∈× such that z ≤ z', the closed integer-valued intervalclosed integer-valued interval zz' with the endpoints z and z', namely z, z + 1, …, z' or equivalently z, z'∩, is denoted by zz'.§ THE FIRING SQUAD/MOB SYNCHRONISATION PROBLEMS In this section, let ℛ = M, G, , m_0, g_m_0, m_m ∈ M be a finitely right-generated cell space, let N be a finite right-generating set of ℛ that contains G_0, where G_0 is the stabiliser of m_0 under , let 𝒢 = M, E be the coloured N-Cayley graph of ℛ, let 𝒞 be a semi-cellular automaton over ℛ with state set Q, neighbourhood N, and local transition function δ, and let Δ be the global transition function of 𝒞. To state the problems succinctly we introduce the notions of passive subsets of states, dead states, supports of global configurations with respect to a distinguished dead state, and what it means for a global configuration to be of the form of a pattern in the following four definitions. Let P be a subset of Q. It is called passivepassive set of states if and only if, for each local configuration ℓ∈ Q^N with (ℓ) ⊆ P, we have δ(ℓ) = ℓ(G_0).Let q be a state of Q. It is called deaddead state if and only if, for each local configuration ℓ∈ Q^N with ℓ(G_0) = q, we have δ(ℓ) = q. In the remainder of this section, let Q contain a distinguished dead state named . Let c be a global configuration of Q^M. The set (c) = M ∖ c^-1() is called support of csupport (c) of c[symbols]suppc@(c). Let A be a subset of M, let p be a pattern of Q^A, and let c be a global configuration of Q^M. The global configuration c is said to be of the form pglobal configuration is of the form p if and only if there is an element g ∈ G such that c_gA = gp and c_M ∖ (gA)≡.We state the firing squad synchronisation problem inLet , , , andbe four distinct states, and let Q' be the set that consists of those states. A solution of the firing squad synchronisation problem in dimension one with one general at the left endfiring squad synchronisation problem in dimension one with one general at the left end is a cellular automaton 𝒞 over , , +, 0, z_z ∈ with neighbourhood -1, 0, 1 and finite set of states that includes Q' such that the stateis dead and the set , is passive, and whose global transition function Δ has the following property:For each global configuration c with finite support of the form …, there is a non-negative integer k such that the global configuration Δ^k(c) is of the form … and has the same support as c, and such that the statedoes not occur in any of the global configurations Δ^j(c), for j ∈_0 with j < k.Let 𝒞 be a solution of the above problem, let c be a global configurations of the form …, and let k be the non-negative integer from the problem definition. Then, because the stateis dead and the support of Δ^k(c) is the same as the one of c, for each non-negative integer j with j ≤ k, the support of Δ^j(c) is the same as the one of c. Broadly speaking, in the time evolution of solutions, the support of initial configurations can neither shrink nor grow before synchronisation is finished. Moreover, because the set , is passive, if the support of c consists of at least 3 cells, then Δ(c) cannot be of the form …. Broadly speaking, the problem cannot be solved trivially. As mentioned above, for each global configuration c of the form …, the supports of the global configurations that are observable in the time evolutions that begin in the configuration c of cellular automata that solve the above problem, are included in the support of c. Hence, we can regard such cellular automata as automata over one-dimensional arrays with one dummy neighbour in the stateat each end.The above problem can be generalised in many ways. For example, by allowing the general to be placed anywhere or by allowing more than one general. We state the firing mob synchronisation problem inLet , , , andbe four distinct states, and let Q' be the set that consists of those states. A solution of the firing mob synchronisation problem in ℛ with respect to Sfiring mob synchronisation problem in ℛ with respect to S is a semi-cellular automaton over ℛ with neighbourhood S and finite set of states that includes Q' such that the stateis dead and the set , is passive, and whose global transition function Δ has the following property:For each finite subset A of M such that the subgraph 𝒢[A] of 𝒢 induced by A is connected, each element a ∈ A, each pattern p ∈ Q^A such that p(a) = and p_A ∖a≡, and each global configuration c of the form p, there is a non-negative integer k such that the global configuration Δ^k(c) is of the form A → Q, a ↦, and such that the statedoes not occur in any of the global configurations Δ^j(c), for j ∈_0 with j < k.The firing squad synchronisation problem with one general at an arbitrary position is the firing mob synchronisation problem in , , +, 0, z_z ∈ with respect to -1, 0, 1. Note that the notions of semi-cellular and cellular automata are identical over , , +.Each semi-cellular automaton over ℛ with neighbourhood S is equivalent to a cellular automaton over the coloured S-Cayley graph of ℛ acted upon by its automorphism group, in the sense that, for each of the former kind of automata, there is one of the latter kind with the same global transition function, and vice versa. Note that the stabilisers of coloured S-Cayley graphs of ℛ are trivial, and hence the notions of semi-cellular and cellular automata are identical over such graphs.We can regard semi-cellular automata that solve the above problem as semi-cellular automata over subgraphs of 𝒢 that are induced by finite subsets of M with one dummy neighbour in the stateat each edge that leads out of the subgraph. Note that, because the graph 𝒢 is of bounded degree, the maximum degrees of the subgraphs it induces are uniformly bounded by a constant.Ideally we would like an abstract description of a semi-cellular automaton that does not depend on any specifics of ℛ and S and that yields a solution for each choice of ℛ and S or at least for as huge a class of such choices as possible.§ UNDIRECTED MULTIGRAPHS Undirected multigraphs without self-loops are introduced inLetandbe two disjoint sets, and letbe a map fromto v, v'⊆ v ≠ v'. The triple = , , is called undirected multigraphundirected multigraph = , ,[symbols]Gcalligraphic@[symbols]V@[symbols]E@; each element v ∈ is called vertexvertex v[symbols]v@v; each element e ∈ is called edgeedge e[symbols]e@e; and, for each edge e ∈, each vertex of (e) is called end of eends (e) of e[symbols]epsilonvare@(e).Because each set in the codomain ofconsists of exactly two distinct vertices, there are no self-loops in the undirected multigraph . With minor modifications the theory and the automata presented in this chapter also work if there are self-loops. They were merely excluded to make the presentation a little simpler.In the remainder of this section, let = , , be an undirected multigraph.What being finite means for multigraphs is said in The multigraphis called finitefinite multigraph if and only if the setsandare both finite. Isolated vertices are the ones without incident edges as introduced in Let v be a vertex of . It is called isolatedisolated vertex if and only if, for each edge e ∈, we have v ∉(e). Directed edges are edges with distinguished source and target vertices as introduced inLet e be an edge of , and let v_1 and v_2 be two vertices ofsuch that v_1, v_2 = (e). The triple e = (v_1, e, v_2) is called directed edge from v_1 through e to v_2directed edge e from v_1 through e to v_2edge!directed[symbols]earrowontop@e; the vertex (e) = v_1 is called source of esource (e) of e[symbols]sigmaearrowontop@(e); the edge (e) = e is called bed of ebed (e) of e[symbols]betaearrowontop@(e); and the vertex (e) = v_2 is called target of etarget (e) of e[symbols]tauearrowontop@(e). At each vertex there is an empty path that starts and ends at the vertex, and non-empty paths are concatenations of directed edges with matching source and target vertices as introduced in * Let v be a vertex of . The singleton p = (v) is called empty path in vempty path (v) in vpath!empty[symbols]vleftparenrightparen@(v), the vertex (p) = (p) = v is called sourcesource of p[symbols]sigmap@(p) and target of psource ((v)) and target ((v)) of (v)[symbols]taup@(p), and the non-negative integer p = 0 is called length of plength (v) of (v)[symbols]absolutep@p.* Let n be a positive integer and, for each index i ∈1n, let e_i be a directed edge ofsuch that, if i ≠ 1, then (e_i) = (e_i - 1). The (2n + 1)-tuple p = ((e_1), (e_1), (e_1), …, (e_n), (e_n)) is called path from (e_1) to (e_n)path p from (e_1) to (e_n)[symbols]p@p; the vertex (p) = (e_1) is called source of psource (p) of p[symbols]sigmap@(p); the vertex (p) = (e_n) is called target of ptarget (p) of p[symbols]taup@(p); and the positive integer p = n is called length of plength p of p[symbols]absolutep@p.* The set of paths is denoted by setof paths[symbols]P aths@. Each directed edge is a path of length 1. Subpaths are connected parts of paths as introduced in Let p = (v_0, e_1, v_1, …, e_n, v_n) be a path of , and let k and ℓ be two indices of 0n such that k ≤ℓ. The path (v_k), if k = ℓ, or (v_k, e_k + 1, v_k + 1, …, e_ℓ, v_ℓ), otherwise, is called subpath of psubpath p_kℓ of p and is denoted by p_kℓ[symbols]pklsubscript@p_kℓ. Direction-preserving paths are the ones without U-turns as introduced in Let p = (v_0, e_1, v_1, …, e_n, v_n) be a path of . It is called direction-preservingdirection-preserving if and only if, for each index i ∈1n - 1, we have e_i ≠ e_i + 1. The set _ of direction-preserving pathsset of direction-preserving paths is denoted by _[symbols]Parrowrightshortsubscript@_.Two paths with matching target and source vertices can be concatenated as introduced in Let p = (v_0, e_1, v_1, …, e_n, v_n) and p' = (v_0', e_1', v_1',…, e_n'', v_n'') be two paths ofsuch that v_n = v_0'. The path pp' = (v_0, e_1, v_1, …, e_n, v_n, e_1', v_1', …, e_n'', v_n'') is called concatenation of p and p'concatenation pp' of p and p'[symbols]bullet@∙.Each non-empty path is the concatenation of directed edges. What being connected means for multigraphs is said in The multigraphis called connectedconnected multigraph if and only if, for each tuple (v, v') ∈×, there is a path from v to v'.The multigraphis connected if and only if, for each tuple (v, v') ∈×, there is a direction-preserving path from v to v'. We can assign weights to edges as done inLetbe a map fromto _> 0. It is called edge weightingof edge weighting of [symbols]omega@, and, for each edge e ∈ E, the element (e) is called edge weight of eedge weight (e) of e[symbols]omegae@(e). Edge weights induce weights of paths as introduced in Letbe an edge weighting ofand let p = (v_0, e_1, v_1, …, e_n, v_n) be a path of . The sum (p) = ∑_i = 1^n (e_i) is called weight of pweight (p) of p[symbols]omegap@(p). Each empty path has weight 0.Each directed edge has the same weight as its bed.§ CONTINUUM REPRESENTATION In this section, let = , , be an undirected multigraph without isolated vertices and letbe an edge weighting of . An orientation is a choice of source and target vertices for each edge as introduced in Let σ and τ be two maps fromtosuch that, for each edge e ∈, we have σ(e), τ(e) = (e). The tuple (σ, τ) is called orientation of orientation (σ, τ) of [symbols]sigmatautuple@(σ, τ). Realising weighted edges as disjoint intervals and gluing these intervals together at shared ends yields a continuum representation ofand is done inLet (σ, τ) be an orientation of , let ζ∖0 →σ, τ, map ζ from ∖0 to σ, τ[symbols]zeta@ζr↦σ,if r < 0, τ,if r > 0,let{ϕ →_< 0, [symbols]phi@ϕe↦ - (e)/2, } and {ψ →_> 0, [symbols]psi@ψe↦(e)/2, }maps ϕ and ψ fromto _< 0 and _> 0and let ∼equivalence relation ∼ on ×[symbols]tilde@∼ be the equivalence relation on × such that, for each tuple (r, e) ∈× and each tuple (r', e') ∈×, (r, e) ∼ (r', e')r ∈ϕ(e), ψ(e) r' ∈ϕ(e'), ψ(e')ζ(r)(e) = ζ(r')(e'). The set = (⋃_e ∈ϕ(e), ψ(e)×e) ∼ is called continuum representationof continuum representation of [symbols]Gcalligraphicbar@. Each weighted edge is realised as a closed interval whose length is the edge's weight. These intervals are made disjoint by taking the Cartesian product with the respective edge. And they are glued together at shared ends by taking the set of all these disjoint intervals modulo the equivalence relation ∼. The vertices are implicitly realised as end points or junctions of the glued disjoint intervals.If the graphcontained isolated vertices, then they would not be represented in . With minor modifications the theory presented in this chapter also works if there are isolated vertices. They were merely excluded to make the presentation a little simpler. In the remainder of this section, letbe a continuum representation ofwith respect to an orientation (σ, τ), and let ζ, ϕ, ψ, and ∼ be the maps and the equivalence relation from <ref>.Vertices are canonically embedded intoas is done in The mapv V→, vertex embedding v[symbols]vbar@v (e)↦(ϕ(e), e)_∼, (e)↦(ψ(e), e)_∼,embeds vertices ofinto . Its image is denoted by vertices [symbols]Vfraktur@ and each element 𝔳∈ is called vertexvertex 𝔳[symbols]vfraktur@𝔳. The embedding is well-defined due to the definition of the equivalence relation ∼.Edges are canonically embedded into the power set ofas is done in The mape E→(), edge embedding e[symbols]ebar@ee↦ ([ϕ(e), ψ(e)] ×e) ∼,embeds edges ofinto . Its image is denoted by edges [symbols]Efraktur@ and each element 𝔢∈ is called edgeedge 𝔢[symbols]efraktur@𝔢.At each point ofthere is at least one direction to move: In a vertex of degree k, there are k directions; and on an edge but not in one of its endpoints, there are 2 directions. An inefficient but immediate way to represent these directions is as in The set -1, 1× is denoted by setof directions[symbols]Dir@, each element d = (o, e) ∈ is called direction on edirectiondirection d on e[symbols]direction@d, the element o is called orientation of dorientation o of d[symbols]orientation@o, the involution→, orientation reversing involution [symbols]minus@(o, e)↦ (-o, e),reverses the orientation of directions, and the map→(), mapthat assigns directions[symbols]direction@ (r, e)_∼ ↦{ (-1, e), (1, e),if r ∈ϕ(e), ψ(e),(- (r'), e')(r', e') ∈(r, e)_∼,otherwise,.assigns to each point inthe set of possible directions in which someone standing on that point can move. This representation of directions is inefficient in the following sense: If we stand on an edge but not on one of its endpoints, then the orientation is enough directional information; and if we stand on a vertex, then the edge is enough directional information, because the orientation is implicit in the fact that we can only move onto the edge but not off it since we are in one end of the edge. On a vertex we do not even need the edge itself but only an identifier for the edge that is locally unique; for example, we could colour the edges such that no two edges of the same colour are incident to the same vertex and use this colour instead.Like vertices, paths ofare also canonically embedded intoand each embedding can be unit-speed parametrised by the interval from 0 to the path's weight as is inductively done in The map p →^0, r r ∈_≥ 0, path embedding p[symbols]pbar@p(v)↦[0, 0 →, r↦v]((e), e, (e))↦[0, (e) →, r↦(ϕ(e) + r, e)_∼,]((e), e, (e))↦[0, (e) →, r↦(ψ(e) - r, e)_∼,](v_0, e_1, v_1)p'↦[ 0, ((v_0, e_1, v_1)p')→,r ↦(v_0, e_1, v_1)(r),if r ≤(e_1), p'(r - (e_1)),otherwise,]maps paths ofto unit-speed parametrisations of them in .The base cases of the inductive definition do not overlap because there are no self-loops, and the inductive step is well-defined because ((v_0, e_1, v_1)p') = (e_1) + (p'). The imagesand _ consist broadly speaking of paths and direction-preserving paths infrom vertices to vertices that only change direction at vertices. Restricting the parametrisation intervals of paths in _ to subintervals and doing a reparametrisation such that the new parametrisation starts at 0 yields all direction-preserving paths inand is done in The setp_r, s( + r)p ∈_ andr, s ∈0, (p) withr ≤ s is denoted by _set _ of direction-preserving paths 𝔭[symbols]Parrowrightsubscriptfraktur@_; each element 𝔭∈_ is called direction-preserving path[symbols]pfraktur@𝔭, the length of the interval (𝔭) is called length of 𝔭length (𝔭) of 𝔭 and is denoted by (𝔭)[symbols]omegapfraktur@(𝔭), the point (𝔭) = 𝔭(0) is called source of 𝔭source (𝔭) of 𝔭[symbols]sigmapfraktur@(𝔭), the point (𝔭) = 𝔭((𝔭)) is called target of 𝔭target (𝔭) of 𝔭[symbols]taupfraktur@(𝔭), and the path 𝔭 is called emptyempty path if and only if (𝔭) = 0. Doing the same with the paths indoes not yield all paths inbut only those that change direction at vertices and not on edges. Because we only need direction-preserving paths in what is to come, we do not define what a general path onis.Sources and targets of direction-preserving paths inare in general not vertices. The distance between two points is the length of the shortest path between the points as introduced in The map× →_≥ 0∪∞, distance [symbols]d@(𝔪, 𝔪')↦inf(𝔭) 𝔭∈_ with (𝔭) = 𝔪 and (𝔭) = 𝔪'is called distance, where the infimum of the empty set is infinity.If the graphis finite and connected, then the distance mapis a metric. Otherwise, it may not be a metric. For example, if there are two distinct vertices v, v' ∈ such that, for each n ∈_+, there is an edge e ∈ whose weight is 1/n, then the distance of v and v' is 0 although v≠v'. Or, if the graphis not connected, then there are two points 𝔪, 𝔪' ∈ whose distance is ∞. Each non-zero vector of a vector space is uniquely determined by its magnitude and its direction, and the zero vector is already uniquely determined by its magnitude, which is 0, and can be thought of as pointing in every direction, which can be represented by the set of directions. A generalisation of vector spaces is given inLetbe the set . The set= (0, )∪ (_> 0×) arrow space [symbols]Arr@is called arrow space; each element a ∈ is called arrowarrow a[symbols]a@a; the setis called semi-directionsemi-direction direction!semi-[symbols]vry@; for each element a = (r, d) ∈, the real number a = r is called magnitude of amagnitude a of a[symbols]norma@a, and the (semi-)direction (a) = d is called (semi-)direction of adirection of a semi@(semi-)direction of a(semi-)direction (a) of asemi-direction of a@(semi-)direction of a[symbols]dira@(a). Arrow spaces will be used to represent both velocities, which are directed speeds, and directed distances. Multiplying a velocity by a time yields a directed distance. This scalar multiplication is introduced in The map×_≥ 0 →, scalar multiplication [symbols]dotcentre@[symbols]centredot@((r, d), s)↦(0, ),if s = 0, (rs, d),if s > 0,is called scalar multiplication. When we stand at the beginning of a non-empty direction-preserving path and walk along it until we reach its end, we start our walk on the first edge of the path in a certain direction and we end it on the last edge of the path in a certain direction. These directions are introduced in Let 𝔭 be a direction-preserving path of _. If 𝔭 is empty, let _(𝔭) = and let _(𝔭) =.Otherwise, there are two edges e_, e_∈ E, which may be the same, and there are two positive real numbers ξ_, ξ_∈0, (𝔭) such that 𝔭(0, ξ_) ⊆e_ and 𝔭((𝔭) - ξ_, (𝔭)) ⊆e_. Moreover, there are four real numbers r_, r_', r_, and r_' such that (r_, e_)_∼ = 𝔭(0), (r_', e_)_∼ = 𝔭(ξ_), (r_, e_)_∼ = 𝔭((𝔭) - ξ_), and (r_', e_)_∼ = 𝔭((𝔭)). Let _(𝔭) = ((r_' - r_), e_) and let _(𝔭) = ((r_' - r_), e_).In both cases, the (semi-)direction _(𝔭) is called source direction _(𝔭) of 𝔭source direction of 𝔭[symbols]dirsigmapfraktur@_(𝔭) and the (semi-)direction _(𝔭) is called target direction of 𝔭target direction _(𝔭) of 𝔭[symbols]dirtaupfraktur@_(𝔭). The edge e_ is the first edge of the path 𝔭 and the edge e_ is its last edge. The numbers ξ_ and ξ_ are two positive real numbers such that the first ξ_ length units of the path run on its first edge and the last ξ_ length units of the path run on its last edge. The numbers r_ and r_' are the positions of the path on its first edge at its very beginning and after ξ_ length units, and the numbers r_, and r_' are the positions of the path on its last edge ξ_ length units before its end and at its very end. Therefore, the signum of r_' - r_ is the start direction on the first edge of the path and the signum of r_' - r_ is the end direction on the last edge of the path.For each non-empty path 𝔭∈_, we have _(𝔭) ∈((𝔭)) and _(𝔭) ∈((𝔭)). § SIGNAL MACHINES In this section, let = , , be a non-trivial, finite, and connected undirected multigraph, letbe an edge weighting of , and let M be a continuum representation of . Recall that, according to our definition of undirected multigraphs, there are no self-loops in . To motivate the definitions in this section, we talk as if there were a signal machine in front of us whose time evolution we can observe, although this evolution is not completely defined until the end of this section. If you observe the time evolution of a signal machine on the graph M, you see signals of different kinds and various speeds each carrying some data move along edges. When signals collide, they may be reflected, removed, new signals may be created, and so on. Similarly, when signals reach a vertex, they may be removed, copies of them may be sent onto all incident edges, new signals may be created, and so on. You may also see stationary signals and signals that travel side-by-side at the same speed. What happens when signals collide or reach a vertex is decided by two local transition functions, one that handles such eventseventevent in vertices and one that handles them on edges.The only events on edges are collisions. In each collision on an edge, there are at least two signals involved, the involved signals are either stationary or they move in one of the two possible directions, and at least two of the signals collide head-on or rear-end. Such a collision results in a set of signals that are either stationary or move in one of the two possible directions.A vertex may be reached by just one signal or multiple signals may collide in it. In both cases, there is at least one signal involved, the involved signals are either stationary or they moved towards the vertex just before the event, and at least one signal is moving. Such an event results in a set of signals that are either stationary or move away from the vertex along incident edges. Let set [symbols]Knd@ be a set, let mapbe a map fromto _≥ 0, let _k_k ∈family _k_k ∈ be a family of sets, letset set _e = (k, d, u)k ∈,((k), d) ∈, andu ∈_k, [symbols]Sgnl@let _e = d,d d ∈[symbols]Dire@_e, let(_e) = { S ∈()D ∈_e S≥ 2and set (_e)[symbols]domdeltae@(_e) (k, d, u) ∈ Sd ∈∪ Dand(k, d, u) ∈ S(k', d', u') ∈ S [t]d ≠ d'or (k) ≠(k')},let _emap _e[symbols]deltae@_e be a map from (_e) to () such thatS ∈(_e)D ∈_e(k, d, u) ∈ S ∪_e(S)d ∈∪ D,let _v = ()set _v[symbols]Dirv@_v, let(_v) =(D, S) ∈_v ×() S≥ 1and set (_v)[symbols]domdeltav@(_v) (k, d, u) ∈ Sd ∈∪ Dand(k, d, u) ∈ S (k) > 0,and let _vmap _v[symbols]deltav@_v be a map from (_v) to () such that(D, S) ∈(_v)(k, d, u) ∈_v(D, S)d ∈∪ D. The quadruple 𝒮 = , , _k_k ∈, (_e, _v)signal machine 𝒮[symbols]Scalligraphic@𝒮 is called signal machine; each element k ∈ is called kindkind k[symbols]k@k; for each kind k ∈, the non-negative real number (k) is called speed of kspeed (k) of k[symbols]spdk@(k) and the set _k is called data set of kdata set _k of k[symbols]Dtk@_k; each element s ∈ is called signalsignal s[symbols]s@s; and the maps _e and _v are called local transition function on edges and in verticeslocal transition functions _e and _v on edges and in vertices respectively. The local transition function _e is used to handle events on edges but not in their endpoints. It gets the signals that are involved in the event and returns the resulting signals. Because in each event at least one moving signal is involved, the direction of this signal and the map that reverses orientation can be used by _e to determine the two possible directions the resulting signals may have.The local transition function _v is used to handle events in vertices. It gets the directions signals may take at the respective vertex and the signals that are involved in the event and returns the resulting signals.The local transition functions _e and _v are supposed to regard directions as black boxes that can merely be distinguished and whose orientation can be reversed. They must not determine edges or vertices by deconstructing directions, which is possible with the chosen representation of directions. If they did something like that, the signal machine would not be uniform. At the beginning of this section we fixed a general multigraph. This multigraph should be regarded as the blueprint of a multigraph. A signal machine depends only on that blueprint and not on any specific properties that a concrete choice of a multigraph may have. So, one and the same signal machine can be instantiated for any multigraph, each instantiation results in a machine on a concrete multigraph, and these instantiations are uniform in the chosen multigraphs. In other words, a signal machine is a map from the set of all multigraphs to the set of quadruples that describe instantiations of the machine on concrete multigraphs and this map depends only in a trivial way on its argument. The quadruple that describes signal machines could be made independent of multigraphs by choosing a different representation for directions. They could for example be represented by integers or vectors or colours equipped with an involution to switch the orientation of directions.Even the global transition function, which is introduced below and describes the time evolution of a signal machine, could be made independent of multigraphs by representing them as patterns in high-dimensional Euclidean spaces (think of the drawing of a graph on a piece of paper). The directions are then vectors that are tangential to edges.In classical solutions of the firing squad synchronisation problem the regions to be synchronised are actually represented as patterns in integer lattices: The cells outside the region are in the same state, say 0, and all cells inside the region are not in state 0, more precisely, one cell inside the region is in a state that distinguishes it as the general, say 1, and the other cells inside the region are all in the same state, say 2. In the remainder of this section, let 𝒮 = , , _k_k ∈, (_e, _v) be a signal machine.To describe the time evolution of our signal machine, the following notions are convenient. Let set [symbols]T@ be the set _≥ 0, let set [symbols]Tbar@ be the set ∪∞, let set [symbols]Q@ be the set (), and let set [symbols]Cnf@ be the set Q^M. Each element t ∈ is called timetime t[symbols]t@t and the time ∞ is called improperimproper time[symbols]infinity@∞, each element q ∈ is called statestate q[symbols]q@q, and each element c ∈ is called configurationconfiguration c[symbols]c@c. The components of signals and some compounds of them are named in Let s = (k, d, u) be a signal of . The kind k of s is denoted by (s)kind (s) of s[symbols]knds@(s); the speed (k) of s is denoted by (s)speed (s) of s[symbols]spds@(s); the (semi-)direction d of s is denoted by (s)(semi-)direction (s) of s[symbols]dirs@(s); the velocity ((k), d) of s is denoted by (s)velocity (s) of s[symbols]vels@(s); and the datum u of s is denoted by (s)datum (s) of s[symbols]dts@(s). For each time t ∈, the arrow (s)t is equal to the arrow ((k)t, d), which can be interpreted as a directed distance. Signals of speed 0 are named in Let s be a signal of . It is called stationarystationary signal if and only if (s) = 0. When we stand on a point facing in a direction and from there we walk a fixed distance without making U-turns but otherwise making arbitrary choices at each vertex, then there is a finite number of points that we may reach and we reach them walking in some direction. The set of these points with and without target-directions is introduced in Let m be a point of M and let (ℓ, d) be an arrow. The set of points with target-directions that can be reached from m by a direction-preserving path of length ℓ with source-direction d is_(ℓ, d)^(m) = [t] ((𝔭), _(𝔭))𝔭∈_, (𝔭) = ℓ, _(𝔭) = d , and (𝔭) = m_(ℓ, d)^(m) and without target-directions it is_(ℓ, d)(m) = m'd ∈ (m', d) ∈_(ℓ, d)^(m)._(ℓ, d)(m) An event occurs when a signal reaches a vertex or two signals coming from different points collide. The time of the next event is given a name in Let c be a configuration of . The minimum time until a signal in c reaches a vertex ist' = inf_m ∈ Minf_s ∈ c(m) (s) > 0inft ∈_> 0_(s)t(m) ∩≠∅.The minimum time until at least two signals in c collide ist” = inf_m, m' ∈ Mm ≠ m'inf_s ∈ c(m)s' ∈ c(m')inft ∈_> 0_(s)t(m) ∩_(s')t(m') ≠∅.The minimum time until the next event(s) in c occurs is next event(s) time t_0(c)t_0(c) = mint', t”[symbols]t0c@t_0(c).A stationary signal at a vertex does never reach the vertex (it is already there) and two signals that already are at the same vertex do never collide (they may have collided when they got there but now they just are at the same vertex; if they have a non-zero velocity, then they will leave the vertex without interfering each other, and, if they have the same positive velocity, then they will travel alongside each other).The next event time may be 0, which means that events accumulate at time 0, or ∞, which means that there are no more events in the future; note that inf∅ = ∞. It is for example 0 if there is a sequence of signals moving at the same velocity towards the same vertex each one being already a little closer to the vertex than the previous one. And it is for example ∞ if there are no signals at all or there are only stationary signals. If each signal moves with its velocity and upon reaching a vertex propagates to all incident edges except the one it came from (making copies of itself if necessary) and if collisions of signals are ignored, then the set of points an event occurs in at a time in the future is given a name in Let c be a configuration ofand let t be a time of . The set of vertices that signals in c reach at time t (under the assumptions given in the introduction to the present definition) isM' = ∅,if t = 0, ⋃_m ∈ Ms ∈ c(m) (s) > 0_(s)t(m) ∩,otherwise.The set of points that signals in c collide in at time t isM” = ⋃_m, m' ∈ Mm ≠ m'⋃_s ∈ c(m)s' ∈ c(m')_(s)t(m) ∩_(s')t(m').The set of points that signals in c are involved in an event in at time t is M_t(c) = M' ∪ M”set M_t(c) of points that signals in c are involved in an event in at time t[symbols]Mtc@M_t(c).For times t before and including the time t_0(c) of the next event, the definition of M_t(c) is natural. And for other times, it is plausible with the explanation given before the definition and it is used to handle accumulations of events and accumulations of accumulations of events and so on. As above, if each signal moves with its velocity and upon reaching a vertex propagates to all incident edges except the one it came from (making copies of itself if necessary) and if collisions of signals are ignored, then starting our signal machine in a configuration c and letting it run for a time t without handling propagation of signals in vertices at time t yields a new configuration (t)(c) as defined in → (→), mapfromto →[symbols]boxplus@0↦_, t↦ [c ↦ [m ↦[t]s ∈m' ∈ Ms' ∈ c(m') (m, (s)) ∈_(s')t^(m') , (s) = (s') , and (s) = (s')]]. For each time t ∈ and each configuration c ∈, if there is a signal in c that reaches a vertex in time t (or one of its duplicates does), then in the configuration (t)(c) that signal is at the vertex and its velocity is the one it had just before reaching the vertex. The direction of that velocity points away from the edge the signal came from and it does not point to any edge that is incident to the vertex. It is then up to the local transition function to decide what to do with the signal and, if it is not removed, what its direction shall be.The mapis used to determine future configurations before or right until the next event occurs and needs to be handled, and also to make crude predictions of future configurations beyond the next event time by propagating at vertices and ignoring collisions as explained above. These predictions will be used to handle accumulations of events and accumulations of accumulations of events and so on. Until the first events occur, signals move along edges without colliding. At the time the first events occur, a signal reached a vertex or two signals collided (on a vertex or edge) or multiple such events happened. An event in a vertex is handled by the local transition function _v and on an edge by _e. This global behaviour can be described by a map that maps a configuration to the configuration right after the first events occurred and have been handled. This map is given in [symbols]Deltadot0@_0 _0→, map _0 fromto c↦{ c,if t_0(c) ∈0, ∞,[m ↦{ c'(m), if m ∉ M_t_0(c)(c),_v((m), c'(m)), if m ∈ M_t_0(c)(c) ∩,_e(c'(m)), if m ∈ M_t_0(c)(c) ∖,} ], otherwise,.↦ wherec' = (t_0(c))(c). The map _0 maps a configuration to itself if the next event time is 0, which means that event times accumulated at 0, or ∞, which means that there is no next event. And it maps a configuration to the configuration that is reached after the first events have been handled, by first usingto determine the configuration in which the events occur and then handling all occurring events with _v and _e. If the next event time is 0, which we call singularitysingularity of order -1singularity of order -1singularity!of order -1 (see <ref>), then the machine is sometimes stuck, in the sense that there is no natural way to define what configuration the machine is in at any time in the future, and sometimes the machine can go forward in time, in the sense that there is a natural way to define what configuration the machine is in at least until some time in the future; the latter case is handled later and largely ignored for now. If the next event time is ∞, then the machine does nothing for eternity.Otherwise, the machine can at least proceed until the next event time and handle the occurring events, which we call singularity of order 0singularities of order 0singularity!of order 0, and then the next event time may again be 0, ∞, or something in between. It may happen that the next event times are never 0 or ∞ but accumulate at some time in the future, which we call singularity of order j, for j ∈_+singularity of order 1singularity!of order j, for j ∈_+ (see <ref>). In that case repeated applications of _0 never reach a configuration at that future time or a time beyond. But we can in a sense take the limit of the sequence of configurations that repeated applications of _0 yield. Yet, it may even happen that singularities of order 1 accumulate at some time in the future, which we call singularity of order 2singularity!of order j, for j ∈_+. Again, we can in the same sense as before take the limit of the sequence of configurations at these singularities. It may continue this way ad infinitum. In precise terms this is done in The sequence t_j - 1^n_n ∈_0, where the n in t_j - 1^n is an upper index and does not stand for exponentiation, the map t_j, and the map _j, for j ∈_+, are defined by mutual induction as follows: The maps t_0 and _0 have already been defined and, for each positive integer j, let* t_j - 1^n→, [symbols]tj-1n@t_j - 1^n c↦∑_i = 0^n - 1 t_j - 1(_j - 1^i(c)), _n ∈_0map t_j - 1^n fromto (note that t_j - 1^0 = 0), lett_j→, map t_j fromto [symbols]tjsubscript@t_j c↦lim_n →∞ t_j - 1^n(c),and let[symbols]Deltadotj@_j_j→, map _j fromto c↦{ c,if t_j(c) ∈0, ∞,[m ↦{ c'(m), if m ∉ M_j(c),_v((m), c'(m)), if m ∈ M_j(c) ∩,_e(c'(m)), if m ∈ M_j(c) ∖,} ], otherwise,.↦ wherec' = lim inf_n →∞(t_j(c) - t_j - 1^n(c))(_j - 1^n(c)),↦ andM_j(c) = lim inf_n →∞ M_t_j(c) - t_j - 1^n(c)(_j - 1^n(c)),where the first limit inferior is the pointwise limit inferior of sequences of set-valued maps and the second limit inferior is the limit inferior of sequences of sets. In greater detail, for each sequence c_n_n ∈_0 of set-valued maps from M to Q, the pointwise limit inferior of c_n_n ∈_0 is the map cM → Q, m ↦lim inf_n →∞ c_n(m) and is denoted by lim inf_n →∞ c_n; and, for each sequence A_n_n ∈_0 of subsets of M, the limit inferior of A_n_n ∈_0 is the subset ⋃_n ∈_0⋂_k ≥ n A_k of M and is denoted by lim inf_n →∞ A_n. Let c be a configuration of . Then, for each positive integer j, we have t_j - 1^0(c) = 0 and t_j - 1^1(c) = t_j - 1(c), and the sequence t_j - 1^n(c)_n ∈_0 inis non-decreasing and hence converges in . And, the sequence t_j(c)_j ∈_0 inis non-decreasing and hence converges in . Let the signal machine be in a configuration c at time 0 and let there be no future configuration whose next event time is 0 or ∞. The latter is the case if and only if the sequences t_j - 1^n(c)_n ∈_0, for j ∈_+, and hence also the sequence t_j(c)_j ∈_0, are strictly increasing sequences in .Then, for each positive integer j and each non-negative integer n, at time t_j - 1^n(c) the machine is in configuration _j - 1^n(c). And, the time t_0^n(c) is the n-th time an event occurs (singularity of order 0), the time t_1^n(c) is the n-th time an accumulation of events occurs (singularity of order 1), the time t_2^n(c) is the n-th time an accumulation of accumulation of events occurs (singularity of order 2), and so forth.Moreover, for each non-negative integer j, at time t_j(c) the machine is in configuration _j(c). And, the time t_0(c) is the next time an event occurs, the time t_1(c) is the next time an accumulation of events occurs, the time t_2(c) is the next time an accumulation of accumulations of events occurs, and so forth. Furthermore, for each positive integer j, the map _j maps the configuration c to the configuration that is reached after an accumulation of singularities of order j - 1, which is a singularity of order j. First, it calculates the configurations that accumulate, namely _j - 1^n(c); secondly, for each of these configurations, it usesto determine the configuration that would be reached at the accumulation time if there were no further events, which is a crude prediction of the future that becomes better the greater n is; thirdly, it calculates the pointwise limit inferior of these configurations, which is essentially the configuration that contains the signals that all but finitely many of the configurations have in common (in particular, if for a point m the sequence of signals at m become constant, then the limit at m is that set of signals); lastly, it handles collisions. Let the signal machine be in a configuration c at time 0 and let there be a future configuration whose next event time is 0. Then, there is a least positive integer j such that the sequence t_j - 1^n(c)_n ∈_0 is eventually constant. And, there is a least non-negative integer n such that t_j - 1^n(c) = t_j - 1^n + 1(c). The time t' = t_j - 1^n(c) is the first time at which the signal machine is in a configuration whose next event time is 0 and this configuration is c' = _j - 1^n(c).For each non-negative integer n' such that n' ≥ n, we have t_j - 1^n'(c) = t' and _j - 1^n'(c) = c'. And, for each positive integer j' such that j' ≥ j, the time t_j'(c) is equal to t' and the configuration _j'(c) is equal to c', and the sequences t_j'^n(c)_n ∈_0 and _j'^n(c)_n ∈_0 are the constant sequences t'_n ∈_0 and c'_n ∈_0. In particular, the limit lim_j →∞ t_j(c) is equal to t' and the limit inferior lim inf_j →∞_j(c) is equal to c'.The limit of sequences of configurations and of sets of points does in general not exist. However, the limit inferior and the limit superior always exist. We decided not to use the limit, to avoid case distinctions that would have to be made. Instead, we decided to use the limit inferior; we could as well have decided to use the limit superior. Which of the two has the desired outcome depends on the specific use case.For the signal machine that solves the firing squad synchronisation problem that we construct in the next section and the configurations it is initialised with and the configurations it encounters during its time evolution, the encountered limit inferiors and superiors are actually always the same, which means that the limits exist, and hence the choice of limit inferior or superior is irrelevant in that use case. If the machine never assumes a configuration in which events accumulate at time 0 and if non-negative singularities of ever higher orders ad infinitum do not accumulate, then the machine can be observed for eternity. Otherwise, it can for now only be observed for all times before lim_j →∞ t_j(c), where c is the initial configuration of the machine. This time is given a name inFor each configuration c ∈, the non-negative real number or infinity t_∞(c) = lim_j →∞ t_j(c) is called ∞-existence time of cexistence time infinity@∞-existence time of c∞-existence time t_∞(c) of c[symbols]tinftyc@t_∞(c), and the closed interval 0, t_∞(c) is called ∞-existence interval of cexistence interval infinity@∞-existence interval of c∞-existence interval 0, t_∞(c) of c. Repeated applications of powers of the maps _j, for decreasing j ∈_0, let us jump to and from configurations right after singularities of decreasing orders. The resulting map is given a name in For each non-negative integer j, each non-negative integer k, and each finite sequence n_i_i = j^k of non-negative integers, let_n_i_i = j^k = _,if j > k, _j^n_j_j + 1^n_j + 1…_k^n_k,otherwise,map _n_i_i = j^k fromto [symbols]Deltadotniijksequence@_n_i_i = j^kand lett_n_i_i = j^k = ∑_i = j^k t_i^n_i_n_ℓ_ℓ = i + 1^k. map t_n_i_i = j^k fromto [symbols]tniijksequence@t_n_i_i = j^k Let the signal machine be in a configuration c at time 0 and let there be no future configuration whose next event time is 0 or ∞. The map _n_i_i = j^k applied to c, first applies _k^n_k to jump from c to the configuration right after the n_k-th time a singularity of order k occurs, secondly it applies _k - 1^n_k - 1 to jump from that configuration, namely _n_i_i = k^k(c), to the configuration right after the n_k - 1-th time a singularity of order k - 1 occurs (counting from the time at which the machine is in configuration _n_i_i = k^k(c)), and so forth until it finally applies _j^n_j to jump from the configuration _n_i_i = j + 1^k(c) to the configuration right after the n_j-th time a singularity of order j occurs (counting from the time at which the machine is in configuration _n_i_i = j + 1^k(c)), where in the case that j = 0, a singularity of order 0 is nothing but an event.The time it takes the machine to get from c to the configuration _k^n_k(c) is t_k^n_k(c), the time it takes to get from _k^n_k(c) to _k - 1^n_k - 1(_k^n_k(c)) is t_k - 1^n_k - 1(_k^n_k(c)), and so forth; in total, the time it takes to get from c to _n_i_i = j^k(c) is t_n_i_i = j^k(c). To compute the configuration the machine is in at the ∞-existence time of the initial configuration, we can use the maps _j for increasing j to jump from singularities of non-negative lower orders to singularities of ever higher orders, which in the case there are any singularities of order -1 comes to a halt at the first such singularity at the ∞-existence time and in the other case yields in a sense the limit configuration. And, to compute the configuration at time t before the ∞-existence time, we can use one of the maps _n_i_i = 0^k to jump to the configuration right after the last event before time t and then we can use the mapto jump from there to time t. The resulting map describes the time evolution of the signal machine before or at ∞-existence times and it is given in For each time t ∈, the set of configurations whose ∞-existence interval contains t is_t^∞ = c ∈ t ≤ t_∞(c). set _t^∞[symbols]Cnftinfinity@_t^∞Let→⋃_t ∈ (_t^∞→), mapfromto ⋃_t ∈ (_t^∞→)[symbols]boxminus@ t↦[ _t^∞ →,c↦lim inf_j →∞_j(c),if t = t_∞(c), (t - t_n_i_i = 0^k(c))(_n_i_i = 0^k(c)),otherwise, ↦ for the leastk ∈_0and the n_i_i = 0^k ∈_0^k + 1↦ witht ∈t_n_i_i = 0^k(c), t_(n_0 + 1, n_1, n_2, …, n_k)(c). ] Note that the least index k and the finite sequence n_i_i = 0^k that occur above are uniquely determined by the time t and the configuration c. In the second case in the definition of , we have t ∈t_k^n_k(c), t_k^n_k + 1(c), and t - t_k^n_k(c) ∈t_k - 1^n_k - 1(_k^n_k(c)), t_k - 1^n_k - 1 + 1(_k^n_k(c)), and so forth.For each configuration c ∈, the map ()(c) is defined on the closed interval 0, t_∞(c).Let t be a time ofand let c be a configuration of _t^∞ such that t ≠ t_∞(c). If events do not accumulate for the initial configuration c, then (t)(c) = (t - t_0^n_0(c))(_0^n_0(c)), for the n_0 ∈_0 with t ∈t_0^n_0(c), t_0^n_0 + 1(c); in words, we apply _0 repeatedly, jumping from event to event, until we reach the configuration _0^n_0(c) at time t_0^n_0(c) with the property that the next event (if there even is one) occurs after t, at which point we useto move signals along edges for the remaining time t - t_0^n_0(c).If events do accumulate for c but singularities of order 1 do not accumulate, then we apply _1 repeatedly, jumping from singularity to singularity, until we reach the configuration _1^n_1(c) at time t_1^n_1(c) with the property that the next singularity (if there even is one) occurs after t, at which point we apply _0 repeatedly, jumping from event to event, until we reach the configuration _0^n_0(_1^n_1(c)) at time t_0^n_0(c) + t_1^n_1(c) with the property that the next event (if there even is one) occurs after t, at which point we useto move signals along edges for the remaining time t - t_0^n_0(c) - t_1^n_1(c).And so forth. Let the signal machine be in a configuration c at time 0. If there is a singularity of order -1 in the future, then, according to <ref>, the time t_∞(c) is the first time at which the signal machine is at such a singularity and the corresponding configuration is (t_∞(c))(c). Otherwise, the time t_∞(c), which may be the improper time ∞, is the time just after all singularities and the corresponding configuration is (t_∞(c))(c). In either case, in what is to come, for simplicity, if t_∞(c) is finite, then we talk as if there is a singularity of order -1 at time t_∞(c).At the time of a singularity of order 1, there are infinitely many events that occur just before that time and arbitrarily close to it, and the problem is to define the configuration at the time of the singularity. At the time of a singularity of order -1, there are infinitely many events that occur just after that time and arbitrarily close to it, and the problem is to define the configurations at all times after the singularity. Analogous problems exist for accumulations of singularities of order 1 or -1, and accumulations of accumulations of singularities of order 1 or -1, and so forth. For singularities of positive orders these problems have been solved above but not for singularities of order -1 and its accumulations.At a singularity of order -1 at time 0, to compute the configuration at a small enough time t, first, we make crude predictions of the future withby jumping past the singularity to future times ε ignoring events, secondly, we extrapolate these predictions to the time t withby letting the machine evolve them until the time t, and, lastly, we take the limit inferior of these predictions as ε tends to 0. This does not work for all singularities of order -1 regardless of how small we choose t (for example if at each point in time of a time span after and including 0 an event occurs), but it does work for the singularities of order -1 that occur in our quasi-solution of the firing squad synchronisation problem. If the ∞-existence time of the current configuration is ∞, then the machine can be observed for eternity. Otherwise, it can be observed until the ∞-existence time usingand from there at least until the least time until which all the crude predictions mentioned above can be observed; this time span is named int_-1 →, map t_-1 fromto [symbols]t-1@t_-1c↦∞,if t_∞(c) = ∞, inf_ε∈_> 0((ε + t_∞(c)) + t_∞(c_ε)),otherwise, ↦ wherec_ε = (ε)((t_∞(c))(c)).How the machine evolves until and beyond t_∞ and at most until t_-1 is given in For each time t ∈, let_t^-1 = c ∈ t ≤ t_-1(c)set _t^-1[symbols]Cnft-1@_t^-1and let→⋃_t ∈ (_t^-1→), mapfromto ⋃_t ∈ (_t^-1→)[symbols]boxast@t↦[ _t^-1 →,c↦(t)(c),if t ≤ t_∞(c), lim inf_ε 0(t - (ε + t_∞(c)))(c_ε),otherwise, ↦ wherec_ε = (ε)((t_∞(c))(c)), ]where the limit inferior is the pointwise limit inferior of set-valued maps, in greater detail, for each family c_ε_ε∈_> 0 of set-valued maps from M to Q, the pointwise limit inferior lim inf_ε 0 c_ε is the map cM → Q, m ↦⋃_r ∈_> 0⋂_s ≥ r c_1/s(m).For each configuration c ∈, we have t_∞(c) ≤ t_-1(c); for each time t ∈, we have _t^∞⊆_t^-1; and for each configuration c ∈ and each positive real number ε, we have t_-1(c) - (ε + t_∞(c)) ≤ t_∞(c_ε).Let the signal machine be in a configuration c at time 0. If t_∞((t_∞(c))(c)) = 0, then there is a singularity of order -1 at time t_∞(c). Otherwise, there is not. In the latter case, as one would hope, t_-1(c) = t_∞((t_∞(c))(c)), and, for each time t ∈t_∞(c), t_-1(c), we have (t)(c) = (t - t_∞(c))((t_∞(c))(c)).Unlike for a singularity of a non-negative order, for a singularity of order -1 it is not possible to jump to the time right after the singularity as there is no such time. However, we may jump over the singularity at time t_∞ to the time t_-1. This is done by the map[symbols]Deltadot-1@_-1 _-1 →, map _-1 fromto c↦ c,if t_∞(c) = ∞, (t_-1(c))(c),otherwise.Like for singularities of non-negative orders, such of order -1 may accumulate, which is a singularitysingularity of order -2singularity!of order -j, for j ∈_≥ 2singularity of order -j, for j ∈_≥ 2, such of order -2 my accumulate, which is a singularity of order -3singularity!of order -j, for j ∈_≥ 2, and so forth. The map to jump over singularities of order -1 has already been given and the maps to jump over singularities of smaller orders are introduced in The sequence t_-(j - 1)^n_n ∈_0, where the n in t_-(j - 1)^n is an upper index and does not stand for exponentiation, the map t_-j, and the map _-j, for j ∈_≥ 2, are defined by mutual induction as follows: The maps t_-1 and _-1 have already been defined and, for each integer j such that j ≥ 2, let* t_-(j - 1)^n→, [symbols]tj-1nz@t_-(j - 1)^nc↦∑_i = 0^n - 1 t_-(j - 1)(_-(j - 1)^i(c)), _n ∈_0map t_-(j - 1)^n fromto(note that t_-(j - 1)^0 = 0), lett_-j →, map t_-j fromto [symbols]tjsubscriptz@t_-jc↦lim_n →∞ t_-(j - 1)^n(c),and let[symbols]Deltadotj-@_-j_-j →, map _-j fromto c↦{ c,if t_-j(c) = ∞,lim inf_n →∞(t_-j(c) - t_-(j - 1)^n(c))(_-(j - 1)^n(c)), otherwise..The machine can be observed for all times before lim_j →∞ t_-j(c), where c is the initial configuration of the machine. This time is given a name in For each configuration c ∈, the non-negative real number or infinity t_-∞(c) = lim_j →∞ t_-j(c) is called (-∞)-existence time t_-∞(c) of c(-∞)-existence time of cexistence time infinity minus@(-∞)-existence time of c[symbols]tinftycminus@t_-∞(c), and the closed interval 0, t_-∞(c) is called (-∞)-existence interval 0, t_-∞(c) of c(-∞)-existence interval of cexistence interval infinity minus@(-∞)-existence interval of c. Like for singularities of non-negative orders, repeated applications of powers of the maps _-j, for decreasing j ∈_+, let us jump over singularities of decreasing negative orders down to order -1. The resulting map is given a name in For each positive integer j, each non-negative integer k, and each finite sequence n_i_i = -j^-k of non-negative integers that is indexed from -j down to -k, let_n_i_i = -j^-k = _,if -j < -k, _-j^n_-j_-j - 1^n_-j - 1…_-k^n_-k,otherwise,map _n_i_i = -j^-k fromto [symbols]Deltadotniijksequenceminus@_n_i_i = -j^-kand lett_n_i_i = -j^-k = ∑_i = -j^-k t_i^n_i_n_ℓ_ℓ = i - 1^-k. map t_n_i_i = -j^-k fromto [symbols]tniijksequenceminus@t_n_i_i = -j^-kLike for singularities of non-negative orders, to compute the configuration the machine is in at the (-∞)-existence time of the initial configuration, we can use the maps _-j for increasing j to jump from singularities of negative lower orders to singularities of ever higher orders ad infinitum. And, to compute the configuration at time t before the (-∞)-existence time, we can use one of the maps _n_i_i = -1^-k to jump over all the singularities before time t such that the next jump over a singularity of order -1 would be beyond time t and then we can use the mapto jump from there to time t whereby we may cross a singularity of order -1. The resulting map describes the time evolution of the signal machine before or at (-∞)-existence times, which for the purposes of this treatise is the complete time evolution, and this map is given in For each time t ∈, the set of configurations whose (-∞)-existence interval contains t is_t^-∞ = c ∈ t ≤ t_-∞(c). set _t^-∞[symbols]Cnftinfinityminus@_t^-∞The map→⋃_t ∈ (_t^-∞→), global transition functionfromto ⋃_t ∈ (_t^-∞→)[symbols]boxdot@ t↦[ _t^-∞ →,c↦{ lim inf_j →∞_-j(c),if t = t_-∞(c),(t - t_n_i_i = -1^-k(c))(_n_i_i = -1^-k(c)),otherwise, .↦ for the leastk ∈_+and the n_i_i = -1^-k∈_0^k↦ witht ∈t_n_i_i = -1^-k(c), t_(n_-1 + 1, n_-2, n_-3, …, n_-k)(c), ]is called global transition function. Jérôme Olivier Durand-Lose introduces and studies another notion of signal machines in his paper *durand-lose:2008<cit.>. The notable differences are the following: While our machines are defined over continuum graphs, his machines are defined over the real number line; while our machines may have infinitely many different kinds of signals, his machines may only have signals of a finite number of kinds (which he calls meta-signals); while in a configuration of our machines there may exist infinitely many signals (even at the same point), in configurations of his machines there may only exist finitely many signals; while the time evolution of our machines can be observed beyond singularities of any order, the time evolution of his machines already stops before singularities of order 1, that is, before accumulations of collisions (as there are no vertices, other events do not exist for his machines).§ FIRING SQUAD SIGNAL MACHINES In this section, let = , , be a non-trivial, finite, and connected undirected multigraph, letbe an edge weighting of , let M be a continuum representation of , and letbe a vertex of M, which we call generalgeneral vertex . When we say path, we either mean path inor path in M; when we say longest path, we either mean maximum-weight path inor longest path in M; and, when we say path, we often mean non-empty direction-preserving path from vertex to vertex; in all cases it should be clear from the context what is meant.From a broad perspective, the signal machine we construct in this section performs the following tasks: It cuts the graph such that the graph turns into a virtual tree; it starts synchronisation of edges as soon as possible and freezes it as late as possible; it determines the midpoints of all non-empty direction-preserving paths from vertices to vertices; it determines which midpoints are the ones of the longest paths; starting from the midpoints of the longest paths, it traverses midpoints of shorter and shorter paths and upon reaching midpoints of edges, it thaws synchronisation of the respective edges; all edges finish synchronisation at the same time with the creation of fire signals that lie dense in the graph (see <ref>). A more detailed account is given in* Turn the graph into a tree: Initiate signals of speed 1 spread from the general throughout the graph without making U-turns and vanish at leaves. When they collide on an edge but not in one of its ends, the edge is cut in two at the point of collision by two stationary leaf signals (also called virtual leaves), one for each of the two new ends. When they collide in a vertex coming from all incident edges, all edges are cut off by leaf signals; and when the do not come from all incident edges, all but one of the edges the signals come from are cut off. In the latter case, the initiate signal that comes from the edge that is not cut off spreads onto the edges from which no initiate signals came.In this way all cycles are eventually broken up and the graph is turned into a virtual tree. Because the virtual leaves are created as soon as possible and because they are treated just like normal leaves, we may and will assume in the description of the other tasks that the graph is a tree.* Start and freeze synchronisation of edges (see <ref>): When initiate signals reach a vertex, for each incident edge, synchronisation of the edge is started from the vertex immediately and freezing of this synchronisation is started from the midpoint of the edge after 3/2 times the edge's length time units by sending freeze signals of speed 1 to both ends of the edge and finishes after twice times the edge's length time units.Each edge is synchronised by recursively dividing it into two parts, one having two-third its length and the other having one-third its length. This division procedure becomes finer and finer, in other words, divisions accumulate, the closer the time evolution gets to the time 2 times the edge's length after of the edge's synchronisation started.The division of one part is performed by sending divide signals of speed (2/3)^n / (2 - (2/3)^n), for n ∈_0, from one boundary onto the part, reflecting the divide signal of speed 1 at the other boundary, and, upon collision of the reflected divide signal with a divide signal, letting the reflected divide signal move on, removing the involved divide signal, creating a stationary boundary signal and sending divide signals of the above speeds onto the newly created part the reflected divide signal comes from. * Determine the midpoints of all direction-preserving paths from vertices to vertices (see <ref>): The general and initiate signals that reach vertices send find-midpoint signals of speed 1 and slowed-down find-midpoint signals of speed 1/3 in all directions to determine the midpoints of all paths that contain the general and the reached vertex respectively. These signals spread throughout the graph without making U-turns memorising the paths they take and (fast) find-midpoint signals are additionally reflected. Reflected find-midpoint signals of speed 1 take the paths back they took before being reflected memorising the path to the vertex they were reflected at and the remaining path they have to take. Upon finishing their path at the vertex they originated at, slowed-down find-midpoint signals of speed 1/3 are sent onto all edges except the one the reflected signal comes from, they memorise the path from the origin vertex to the reflection vertex of the reflected find-midpoint signal, and they spread throughout the graph without making U-turns memorizing the paths they take and vanish at leaves.When a reflected find-midpoint signal collides with a slowed-down find-midpoint signal that originated at the same vertex, the point of collision is the midpoint of the concatenation of the paths the two signals took after being reflected, that is, the path from the vertex the reflected find-midpoint signal was reflected at over the point of collision over the vertex both signals originated at to the vertex the reflected find-midpoint signal that spawned the slowed-down find-midpoint signal was reflected at. Each midpoint is designated by a stationary midpoint signal that memorises the path it is the midpoint of along with its position on the path.When a reflected find-midpoint signal collides with a reflected find-midpoint signal that originated at the same vertex and the point of collision is the origin vertex itself, both reflection vertices have the same distance to the origin vertex and this vertex is the midpoint of the concatenation of the paths the two signals took after being reflected. To get a clearer picture of how midpoints are determined, let us focus on only a few signals and let us ignore boundary cases: When an initiate signal reaches a vertex, one find-midpoint signal is sent along one incident edge, and another find-midpoint signal is sent along another incident edge. Both signals travel along edges and upon reaching a vertex they are either reflected and travel back or they take one of the incident edges that leads them further away. At the latest, they are reflected upon reaching a leaf. One of the reflected find-midpoint signals returns first to the vertex it originated at, is slowed down there, and the slowed down signal travels towards the other (reflected) find-midpoint signal. At some time in the future, the slowed-down find-midpoint signal collides with the other, now reflected, find-midpoint signal and the point of collision is the midpoint of the concatenation of the paths the two signals took after being reflected.* Determine the midpoints of the longest direction-preserving paths (see <ref>): The midpoints of the longest paths are eventually found. But this is not sufficient, they also need to be recognised as such. To that end, each reflected find-midpoint signal and each slowed-down find-midpoint signal carries a boolean that indicates whether the path it took from the vertex it was reflected at may be the subpath of a longest path that has the same source (or target), and whether the signal would be the first one to find the longest path's midpoint (recall that a slowed-down find-midpoint signal was either spawned by the general vertex or an initiate signal that reached a vertex, in which case we can think of this vertex as the one the slowed-down find-midpoint signal was reflected at; or it was originally a reflected find-midpoint signal and knows where that signal was reflected at). Let us call signals that carry the booleanmarked and the others not.At each vertex that is not a leaf, a stationary count signal memorises the directions from which marked reflected find-midpoint signals that originated at the vertex or from which marked slowed-down find-midpoint signals with any origin have already returned. Because longest paths always end at leaves, for each leaf, find-midpoint signals are marked when they are the first ones to be reflected at the leaf and not otherwise. When a marked signal reaches a vertex, it stays marked, if, including itself, from all but one direction have marked signals already returned (which is the case if and only if it is the last signal to return from its direction and the penultimate signal to return among such signals from either direction), and it is unmarked, otherwise.This means that a marked reflected find-midpoint signal is unmarked, if there are at least two other signals that started out at the same time at the same vertex but take longer to return (because they have a longer way; the combined paths that two of the other signals take may be a longest path), and it stays marked, if all other signals except for one that all started out at the same time at the same vertex returned earlier (because they had a shorter way).And it means that a marked slowed-down find-midpoint signal is unmarked, if it is unclear which of the incident edges belong to longest paths, and it stays marked, otherwise, which is the case if from precisely one direction no slowest reflected find-midpoint signal that originated at the vertex has returned yet. This is not only pessimistic, meaning that we do not falsely consider midpoints of paths that are not among the longest as such, but also correct, meaning that we still find the midpoints of all longest paths in time (see <ref>).When a marked reflected find-midpoint signal reaches its origin and is the penultimate such signal to do so, it turns into a marked slowed-down find-midpoint signal that travels in the one direction from which no marked signal has returned yet and the point at which this signal collides with the one signal that has not yet returned will be the midpoint of a longest path, if both signals are still marked at the time of collision.* Traverse midpoints, thaw synchronisation of edges, and fire (see <ref>): The midpoints of the longest paths are found at the same time, at which, from each such midpoint, two thaw signals of speed 1 are sent that travel along the midpoint's path towards both of its ends. When a thaw signal collides with the midpoint of a path such that one of the two subpaths from the midpoint to either end of the path coincides with the remaining path the thaw signal travels along, an additional thaw signal is created that travels along the other subpath. On its way from the midpoint of the last edge of the path a thaw signal travels on to the end of the path, the thaw signal thaws all frozen signals it collides with. All thaw signals reach the ends of their paths at the same time, which is also the time all edges finish synchronisation with the creation of stationary fire signals that lie dense in the graph. We introduce a typographic convention in Each word of letters of the Latin alphabet that is written in typewriter font shall denote the word itself and shall for example not be the name of a variable. We introduce a boolean algebra inLet = ,booleans = ,[symbols]Bblackboard@. Each element b ∈ is called boolean[symbols]b@b. The map→, negation [symbols]negate@↦,↦,is called negation. The map× →, conjunction [symbols]wedge@(b, b')↦,if ∈b, b', ,otherwise,is called conjunction. We introduce finite lists of directions in Let ^* be the set w 1n→ n ∈_0set ^*[symbols]Dirstarsuperscript@^*. Each element w ∈^* is called word over word w over [symbols]w@w; for each word w ∈^*, the non-negative integer w = (w) is called length of wlength w of w[symbols]absolutew@w; the word ∅→ is called emptyempty word [symbols]lambda@; the map^* ×^*→^*, concatenation ∙[symbols]bullet@(w, w')↦[1w + w' →, i↦ w(i),if i ≤w,w'(i - w),otherwise, ]is called concatenation.The empty word is the only word of length 0 and it is the neutral element of . The signal machine we construct in this section has infinitely many kinds of signals, which are explained in <ref>, and given names, speeds, and data sets in Let= , , , , , , , , ∪[]⋃_n ∈_0_n, _n∪, , , , setlet→_≥ 0, map ↦ 1,↦ 0,↦ 0,↦ 0,↦ 1,↦ 1,↦1/3,↦ 1,↦ 1, _n↦*2/3^n/2 - *2/3^n,forn ∈_0, _n↦ 0,forn ∈_+,↦ 1,↦ 0,↦ 0,↦ 0,and let= 0, set_ = , family _k_k ∈ _ = , _ = (), _ = w, w'⊆^*w ≠ w', _ = ^*, _ = ^* ×^* ×, _ = ^* ×^* ×, _ = , _ = ^* ×, __n = ,forn ∈_0, __n = ,forn ∈_+, _ = , _ = , _ = , _ = .The kinds together with their speeds and data sets determine the possible signals, which are recalled and given abbreviations in The set of signals is= (k, d, u)k ∈,((k), d) ∈, andu ∈_k. setLetd = (, d, 0),ford ∈, abbreviations of signals like d, w_odw_rb andd = (, , d),ford ∈, D = (, , D),forD ∈_, ww' = (, , w, w'),for w, w'∈_, w_od = (, d, w_o),forw_o ∈_, w_odw_rb = (, d, (w_o, w_r, b)),ford ∈ and(w_o, w_r, b) ∈_, dw_ow_rb = (, d, (w_o, w_r, b)),ford ∈ and(w_o, w_r, b) ∈_, d = (, d, 0),ford ∈, dwb = (, d, (w, b)),ford ∈ and(w, b) ∈_, nd = (_n, d, 0),forn ∈_0andd ∈, nd = (_n, , d),forn ∈_+andd ∈, d = (, d, 0),ford ∈,= (, , 0),= (, , 0),= (, , 0).What signals of various kinds do when they reach a vertex or collide with one another, what the data they carry means, and what we call them is given a glimpse at in * Each signal of kindhas speed 1; at each vertex it reaches it spreads in all directions that lead away from where it comes from, it initiates synchronisation of all incident edges except the one it comes from by sending divide signals onto them, it initiates the search for midpoints of paths that contain the vertex by sending (slowed-down) find-midpoint signals in all directions, and it initiates one component of the search for the longest paths of the graph by marking slowed-down find-midpoint signals if the vertex is a leaf; it carries no data; and it is called initiate signalinitiate signalsignal!initiate. The very first initiate signals spread from the general in all directions.* Each signal of kindis stationary, designates a virtual leaf, carries the direction that leads onto the edge that is incident to the virtual leaf, and is called leaf signalsignal!leafleaf signal. Such signals are created when initiate signals collide in a vertex (or on an edge), which means that there is a cycle in the graph, and this cycle is broken up by virtually terminating the involved edge(s) with leaf signals. Each leaf signal is treated like a leaf in the following way: When signals collide with each other and with leaf signals, for each involved leaf signal, the collision of the signals that move in the opposite direction than the one the leaf signal carries is handled as if those signals collided in a leaf. Because leaf signals are created at points at the same time or before any other signal reaches them, the graph looks like a tree for all other signals.* Each signal of kindis stationary; is positioned at a vertex that is not a leaf; memorises the directions from which find-midpoint signals that originated at the vertex, were reflected, and may be on longest paths and would be the first ones to find their midpoints have already returned, in other words, it memorises the directions from which the slowest find-midpoint signals that originated at the vertex and travelled alongside initiate signals before they were reflected at a leaf have already returned; and is called count signalsignal!countcount signal. When an initiate signal reaches a vertex that is not a leaf, a count signal is created. Note that, for the data set of count signals, instead of the infinite set (), we could have used the finite set (1, 2, …, k), where k is the maximum degree of the graph or the upper bound of the maximum degrees of the graphs to be considered and the numbers represent directions that lead away from vertices. The first choice of k would make the signal machine depend on the graph, which is unconventional, whereas the latter would not and would also fit to the fact that solutions of firing mob synchronisation problems are usually considered for graphs whose maximum degrees are uniformly bounded by a constant.* Each signal of kindis stationary, designates the midpoint of a path, carries the directions that lead from its position to both ends of the path, and is called midpoint signalsignal!midpointmidpoint signal. Such a signal is created when a reflected find-midpoint signal collides with a slowed-down find-midpoint signal that originated at the same vertex, or when two reflected find-midpoint signals that originated at the same vertex collide with each other, which only happens at the origin vertex itself. See <ref>.* Each signal of kindhas speed 1, at each vertex it reaches it spreads in all directions that lead away from where it comes from (in the sense that, in each such direction, a signal of its kind is sent) and it is also reflected (in the sense that a reflected find-midpoint signal is sent in the direction from where it comes from), it carries the directions that lead from its position to the vertex the signal originated at, and it is called find-midpoint signalsignal!find-midpointfind-midpoint signal. When an initiate signal reaches a vertex, for each incident edge, a find-midpoint signal whose origin is the vertex is created that travels onto the edge.* Each signal of kindhas speed 1; is the reflection of a find-midpoint signal at a vertex, travels back along the path this signal took before it was reflected and slows down when it reaches the vertex the find-midpoint signal originated at; carries the directions that lead from its position to the vertex the find-midpoint signal originated at, the directions that lead from its position to the vertex the find-midpoint signal was reflected at, and a boolean that indicates whether the path described by its position and both directions, which leads from the reflection vertex to the origin vertex, may be the subpath of a longest path that has the same source (or target), and the boolean also indicates whether the signal would be the first one to find the longest path's midpoint; and is called reflected find-midpoint signalsignal!reflected find-midpointsignal!reflected find-midpoint!marked(marked) reflected find-midpoint signal and, if the boolean it carries is , it is called marked.As has already been pointed at, when a find-midpoint signal reaches a vertex, a reflected find-midpoint signal is created that travels onto the edge the find-midpoint signal comes from. If the vertex is a leaf and the find-midpoint signal is one of the first signals to reach it, which is precisely the case if the signal reaches the leaf together with an initiate signal, then its reflection is marked, and otherwise, not. The reasons are that both ends of a longest path are leaves and that a find-midpoint signal that is not among the first signals to reach one end of a longest path would not find its midpoint after another one has already found it.When a marked reflected find-midpoint signal reaches a vertex that is not a leaf, the count signal at the vertex memorises the direction the marked signal comes from, and the signal stays marked, if the memory of the count signal contains each but one direction that leads away from the vertex, and it is unmarked, otherwise. Why is that? Each vertex that is not the general is reached precisely once by an initiate signal, at which point find-midpoint signals are sent in all directions; for each direction, the marked reflected find-midpoint signal to return from that direction is memorised, which is the slowest one or, in other words, the last one or the one that had the longest way (note that although only one find-midpoint signal is sent in a direction, multiple reflected find-midpoint signals may return from that direction); the penultimate marked reflected find-midpoint signal to return may come from one edge of a longest path that runs through the vertex and hence it stays marked (note that the other edge of the longest path that is incident to the vertex would be the one from which the marked signal has not yet returned); the signals that return before the penultimate one are too fast to be on a longest path and the last signal to return has already collided with the slowed-down penultimate signal that returned before it (if they do not return at the same time) and hence they are unmarked.* Each signal of kindhas speed 1/3; is the slow-down of a reflected find-midpoint signal at the vertex the find-midpoint signal originated at; at each vertex it reaches it spreads in all directions that lead away from where it comes from; it carries the directions that lead from its position to the vertex the find-midpoint signal originated at, the directions that lead from the origin vertex to the vertex the find-midpoint signal was reflected at, and a boolean that indicates whether the path described by its position and both directions, which leads from the reflection vertex over the origin vertex to its position, may be the subpath of a longest path that has the same source (or target), and the boolean also indicates whether the signal would be the first one to find the longest path's midpoint; and it is called slowed-down find-midpoint signalsignal!slowed-down find-midpointsignal!slowed-down find-midpoint!marked(marked) slowed-down find-midpoint signal and, if the boolean it carries is , it is called marked.As has already been pointed at, when a reflected find-midpoint signal reaches the vertex the find-midpoint signal originated at, for each incident edge except the one the reflected find-midpoint signal comes from, a slowed-down find-midpoint signal is created that travels onto the edge. Additionally, when an initiate signal reaches a vertex, for each incident edge, a slowed-down find-midpoint signal is created that travels onto the edge.The latter case is in the following senses the boundary or limiting case of the former: Imagine that the vertex the initiate signal reaches is in fact two vertices that are infinitesimally close; then a find-midpoint signal is created at one vertex, this signal immediately reaches the infinitesimally close other vertex, there it is reflected, the reflected find-midpoint signal immediately reaches the infinitesimally close other vertex, and there it is slowed down. Or, analogously, imagine the limit of the cascade of the creation of a find-midpoint signal, its reflection, and slow-down for shorter and shorter distances between the vertex the find-midpoint signal originates at and the one it is reflected at; then in the limit the find-midpoint signal and its reflection vanish and only the slowed-down find-midpoint signal remains.When a marked slowed-down find-midpoint signal reaches a vertex that is not a leaf, the count signal at the vertex memorises the direction the marked signal comes from, and the signal stays marked, if the memory of the count signal contains each but one direction that leads away from the vertex, and it is unmarked, otherwise. Why is that? The slowed-down signal reaches the vertex at the same time and from the same direction as the slowest reflected find-midpoint signal from that direction that originated at the vertex. The latter signal is however not marked, because it did not travel alongside initiate signals before it was reflected at a leaf (the reason is that if it had travelled alongside initiate signals, then it would have been reflected at the same time as the find-midpoint signal whose reflection turned into the marked slowed-down find-midpoint signal and hence, because the paths from the reflection leaves to the vertex they reach together have the same lengths, the find-midpoint signal that reaches it slowed down would have taken longer, and therefore the signals would not reach the vertex at the same time).Therefore, the memory of the count signal contains each but one direction if and only if from each but one direction the slowest reflected find-midpoint signals that originated at the vertex have already returned. If this is the case, then the incident edge belonging to that direction may be the edge of a longest path that runs through the vertex and the slowed-down find-midpoint signal may be the one to collide with the not yet returned signal somewhere on or beyond the edge precisely at the midpoint of the longest path. If from more than one direction signals are overdue, then the paths running through each pair of these directions are longer than the paths running through any of these directions and the direction the slowed-down find-midpoint signal comes from. And, if all signals have already returned, then they have already collided with the slowed-down signal and found the midpoints of the longest paths whose determination involves the slowed-down signal if there are any. * Each signal of kindhas speed 1, is created at the midpoint of an edge, moves towards one end of the edge, and freezes synchronisation of the edge, carries no data, and is called freeze signalsignal!freezefreeze signal. When an initiate signal reaches a vertex, for each incident edge, a find-midpoint signal and a slowed-down find-midpoint signal are created that travel onto the edge, the former is reflected at the other end of the edge and collides with the latter at the midpoint of the edge, at which point two freeze signals are created that travel to both ends of the edge. See <ref>* Each signal of kindhas speed 1, is created at the midpoint of a path, travels along the path towards one of end of the path, creates a new signal of its kind when it collides with the midpoint signal that designates the midpoint of a path such that one of the two subpaths from the midpoint to either end of the path (a half-pathhalf-pathpath!half-) coincides with the path it takes itself and the new signal travels along the other half-path, and thaws synchronisation of an edge if it collided with or was created at the midpoint signal that designates the midpoint of the last edge of the path it takes, carries the directions of the path it takes and a boolean that indicates whether it thaws synchronisation of the edge it is on or not, and is called thaw signalsignal!thawthaw signal.The first thaw signals are created simultaneously at the midpoints of longest paths. For each such midpoint, two thaw signals are created, one that travels along the path to one end of the path and the other that travels to the other end of the path. When a thaw signal collides with the midpoint of a path whose one half-path coincides with the remaining path the thaw signal travels along, an additional thaw signal is created that travels along the other half-path. On its way from the midpoint of the last edge of the path a thaw signal travels on to the end of the path, the thaw signal thaws all frozen signals it collides with.In this way, starting at the midpoints of longest paths, thaw signals traverse the midpoints of shorter and shorter paths, reach the ends of their paths at the same time, and thaw synchronisation of edges, which finishes at the same time. See <ref>* Each signal of kind _0 has speed 1, moves from one boundary (which may be one end of an edge or a boundary signal) to the next boundary (which may be the other end of the edge or a boundary signal) and is reflected there, carries no data, and is called divide signal of type 0divide signal of type 0. When an initiate signal reaches a vertex, for each incident edge, a divide signal of type 0 is created that travels onto the edge. And, when a divide signal of any type collides with a reflected divide signal, a divide signal of type 0 is created that travels in the same direction as the (non-reflected) divide signal.* Each signal of kind _n, for n ∈_+, has speed (2/3)^n / (2 - (2/3)^n), moves from one boundary (which may be one end of an edge or a boundary signal) towards the next boundary (which may be the other end of the edge or a boundary signal) but never reaches it and can be frozen, carries no data, and is called divide signal of type ndivide signal of type n. When an initiate signal reaches a vertex, for each incident edge, and for each n ∈_+, a divide signal of type n is created that travels onto the edge. And, when a divide signal of any type collides with a reflected divide signal, for each n ∈_+, a divide signal of type n is created that travels in the same direction as the (non-reflected) divide signal.Note that although _n, for n ∈_+, are different kinds, events that involve signals of these kinds are handled the same way, in other words, signals of these kinds are not differentiated by the two local transition functions of the signal machine. The only reason they are different kinds is because we need them to have different speeds and by definition all signals of the same kind have the same speed.* Each signal of kind _n, for n ∈_+, has speed 0, is a frozen divide signal of type n, carries the direction the non-frozen divide signal had, and is called frozen divide signal of type nfrozen divide signal of type n. When a freeze signal collides with or is created at the same time as a divide signal of type n ∈_+, the divide signal is frozen.* Each signal of kindhas speed 1, is the reflection of a divide signal of type 0, creates a boundary signal when it collides with a divide signal of type n, creates a fire signal when it reaches the end of the edge it traverses, carries no data, and is called reflected divide signalreflected divide signalsignal!reflected divide. When a divide signal of type 0 reaches a boundary (which may be a vertex or a boundary signal), a reflected divide signal is created that travels in the opposite direction.* Each signal of kindis stationary, designates a boundary for the synchronisation of an edge, carries no data, and is called boundary signalboundary signalsignal!boundary. Such signals are created when divide signals collide with reflected divide signals.On each edge, the interplay of divide signals, reflected divide signals, and boundary signals has the following effect: At first a divide signal of type 1 collides with a reflected divide signal that originated at the same end of the edge. This collision results in the creation of a boundary signal that divides the edge into two parts. The length of the part from the origin vertex to the boundary signal is 2/3 times the length of the edge and the length of the part from the boundary signal to the other end of the edge is 1/3 times the length of the edge.In the same manner as the edge itself, the (1/3)-part is recursively divided further and further. In the (2/3)-part, a signal of type 2 collides with the reflected divide signal from before. This collision results in the creation of a boundary signal that divides the (2/3)-part into two subparts. One has (2/3) · (2/3) the length of the edge and the other has (1/3) · (2/3) the length of the edge. The ((1/3) · (2/3))-part is recursively divided further and further. The ((2/3) · (2/3))-part is divided into a ((2/3) · (2/3) · (2/3))-part and a ((1/3) · (2/3) · (2/3))-part and so forth.If there were no freeze and thaw signals, after twice the time the edge is long — which is precisely the time it took the divide signal of type 0 to reach the other end of the edge, to be reflected there, and to return to the end it originated at — the boundary signals together are dense on the edge, which means that each point on the edge is arbitrarily close to a boundary signal, and, at this point in time, each boundary signal collides with a reflected divide signal, which results in the creation of fire signals that designate that synchronisation has finished. However, because the synchronisation of each edge is started at different times and takes different times depending on how far away the edge is from the general and how long the edge is, synchronisation of each edge is frozen at the last possible moment — the freezing starts from the midpoint of the edge 3/2 times the edge's length many time units after synchronisation of the edge was initiated — and it is thawed such that all edges finish synchronisation at the same time — the thawing starts from the midpoint of the edge 1/3 times the edge's length many time units before the total synchronisation finishes, which is the sum of the radius of the graph and its diameter. Recall that the radius is the longest distance from the general to another vertex and that the diameter is the longest distance between two vertices.Note that, for each edge, collisions of divide signals with reflected divide signals and with boundary signals accumulate at the times the two freeze signals and the two thaw signals reach the ends of the edge. These accumulations are singularities of order 1. See <ref>.* Each signal of kindis stationary, designates that synchronisation has finished and can be frozen, carries no data, and is called fire signalsignal!firefire signal. Such signals are created when reflected divide signals collide with boundary signals or reach vertices.* Each signal of kindis stationary, is a frozen fire signal, carries no data, and is called frozen fire signalsignal!frozen firefrozen fire signal. On each edge, one of the two freeze signals reaches an end of the edge at the same time as the reflected divide signal, at which point a frozen fire signal is created; it is thawed at the same time at which all other fire signals are created, which happens on all edges at the same time and the fire signals lie dense in the multigraph. In the forthcoming definitions of maps we make extensive use of pattern matching. To make the exposition concise and readable we introduce some pattern matching conventions in * Inorder matters the case that patterns of multiple rules overlap, the rule that occurs first is the one to use. For example, the map f →, 0 ↦ 1, 1 ↦ 0, z ↦ z, maps 0 to 1 and 1 to 0 and each other integer to itself.* Inwildcard [symbols]underscore@ the case that we do not care to name some part of a matched structure, we writeinstead of a name for the part. For example, the pattern (E, , d) matches each triple whose last component is an initiate signal, gives the first component the name E, does not give a name to the second component, and gives the direction of the initiate signal the name d. And, the pattern w_o matches each find-midpoint signal, gives the directions to the vertex the signal originated at the name w_o, but gives no name to the direction of the signal.* Tosame names express equality express equality of different parts of a matched structure, we give those parts the same name. For example, the pattern d, nd matches each set that consists of a reflected divide signal and a divide signal of any type such that the direction of the reflected divide signal is the opposite of the direction of the divide signal, gives the direction of the divide signal the name d, and gives its type the name n.* To@-notation name both a structure and its parts, we use a Haskell-like @-notation. For example, the pattern s@db matches each reflected find-midpoint signal whose directions to the vertex the signal originated at and to the vertex the signal was reflected at are empty, gives the direction of the signal the name d, gives the boolean that indicates whether the signal may be on a longest path and would be the first to find its midpoint the name b, and gives the signal itself the name s.Some of the maps we define below are actually partial maps. We represent them by (total) maps as specified in Letrepresentations of partial maps using the bottom symbolX and X' be two sets, let Y be a subset of X, letbe an element that is not in X', which we call bottom, and let f be a map from X to X' ∪ such that, for each element x ∈ X, we have f(x) = if and only if x ∉ Y. The map f represents a partial map whose domain of definition is Y, whose domain is X, and whose codomain is X'.In the following, for maps like f, we do not explicitly specify the domain Y of definition (it is the set X ∖ f^-1()) and we implicitly assume thatdoes not occur in the codomain X' of the represented partial map.To define the local transition functions, we begin with definitions for special cases and use those to gradually arrive at definitions for the general case. For trees and without freezing and thawing, the map _v, 1^ handles the event that precisely one signal reaches a vertex, and the maps _v, 2^ and _e, 2^ handle the event that precisely two signals collide in a vertex and an edge respectively (see <ref>). For trees, the maps _v^ and _e^ handle events involving arbitrarily many signals by considering unordered pairs of signals and applying _v, 1^, _v, 2^, and _e, 2^, and by also freezing and thawing signals if needed (see <ref>). For virtual trees, which means that edges of the graph have been virtually cut by leaf signals to remove circles, the maps _v^ and _e^ handle events by partitioning signals at virtual cuts into those belonging to one or the other leaf and applying _v^ and _e^ (see <ref>). For general graphs, the maps _v and _e handle events by virtually cutting the graph, which eventually creates a virtual tree, and applying _v^ and _e^ (see <ref>).Most of the forthcoming definitions and parts of them are annotated with intuitive explanations of what they mean. For example, after each rule that handles a specific kind of event, it is explained what kind of event in the time evolution of the signal machine is handled, how it is handled, and sometimes why.How events for trees and without freezing and thawing, with one or two signals involved are handled is given inThe following map tells whether two words of directions are both empty:^* ×^*→, map (w, w')↦,if w = 0 and w' = 0, ,otherwise.For trees and without freezing and thawing, the case that precisely two signals collide on an edge but not in one of its ends is handled by the following map, which maps colliding unordered pairs of signals for which a collision rule is specified to the resulting signals and all other sets of signals to :_e, 2^()→() ∪, map _e, 2^ 0d,↦ d, ,(If a divide signal of type 0 collides with a boundary signal, then reflect the divide signal.) d, nd ↦∪n'd n' ∈_0, (If a reflected divide signal collides with a divide signal of any type, then create a boundary signal and send divide signals of all types in the direction of the original divide signal.)s@db, s'@ db' ↦ s, s',dd,d, d,if b = or b' =, ∅,otherwise,(If a reflected find-midpoint signal collides with a slowed-down find-midpoint signal that originated at the same end of an edge and only travelled on this edge, then the point of collision is the midpoint of the edge and, if the graph has at least two edges, then let the signals move on, designate the point by a midpoint signal, and send freeze signals to both ends of the edge to freeze synchronisation of the edge, and otherwise, do not create and send any signals.)s@w_odw_rb, s'@ dw_ow_r'b' ↦ s, s', ( d)w_rdw_ow_r',if b = or b' =,dw_r, dw_ow_r',otherwise,(If a reflected find-midpoint signal collides with a slowed-down find-midpoint signal that originated at the same vertex, then the point of collision is the midpoint of the shortest path in the virtual tree from the vertex the reflected find-midpoint signal was reflected at to the vertex the signal that spawned the slowed-down find-midpoint signal was reflected at and, if this midpoint is not the midpoint of a longest path, then let the signals move on and designate the point by a midpoint signal, and otherwise, send thaw signals along that longest path to both its ends.)dw, dwd'w' ↦dw(w, w'), d'w'(w, w'), (If a thaw signal collides with a midpoint signal that designates the midpoint of a path whose one half-path coincides with the path the thaw signal is going to take, then send an additional thaw signal along the other half-path and, if this path is just a directed edge, then make the thaw signals thaw synchronisation of the edge.) ↦. (If none of the above happened, then indicate that by returning bottom.) A signal that reaches a vertex is in a leaf if and only if there is precisely one direction that leads away from the vertex. And a signal that reaches a vertex is the penultimate one to do so if and only if the number of signals that have already returned including the signal itself is one less than the number of directions that lead away from the vertex. The two maps that express this in an abstract way using booleans are()→, map E↦, if E≠ 1, ,otherwise,and() ×_0→, map (E, n)↦, if E - 1 ≠ n, ,otherwise. For trees and without freezing and thawing, the case that precisely one signal reaches a vertex is handled by the following map, which maps quadruples — consisting of first, the set of directions that lead away from the vertex; secondly, the number of the directions from which the slowest reflected find-midpoint signals that originated at the vertex have already returned or have just arrived; thirdly, a boolean that indicates whether the signal is among the first ones to reach the vertex; and lastly, the signal that reaches the vertex — to the resulting signals:_v, 1^() ×_0 ×× →(), map _v, 1^(, , , 0d)↦ d, (If a divide signal of type 0 reaches a vertex, then reflect it.) (, , , d)↦, (If a reflected divide signal reaches a vertex, then create a fire signal.) (E, , , d)↦[t] []⋃_e ∈ E ∖ de∪[]⋃_e ∈ E ∖ dne n ∈_0∪[]⋃_e ∈ Ee, e(E),(If an initiate signal reaches a vertex, then send initiate signals onto all incident edges except the one the original initiate signal comes from, send divide signals of all types onto all incident edges except the one the original initiate signal comes from, and send find-midpoint and slowed-down find-midpoint signals onto all incident edges to find all midpoints of paths that contain the vertex, where, in the case that the vertex is a leaf, the slowed-down find-midpoint signal is marked, where the mark means that it may be on a longest path and would be the first to find its midpoint.) (E, , b, w_od)↦[t] w_o db (E)∪[]⋃_e ∈ E ∖ d( d)w_oe,(If a find-midpoint signal reaches a leaf, then reflect it and, if it is one of the first signals to reach the leaf, then also mark it as a signal that may be on a longest path and would be the first to find its midpoint. And, if a find-midpoint signal reaches a vertex that is not a leaf, then reflect it and send find-midpoint signals onto all incident edges except the one the original signal comes from.) (E, n, , dw_rb)↦⋃_e ∈ E ∖ de( d)w_rb (E, n),(If a reflected find-midpoint signal reaches the vertex it originated at, then send slowed-down find-midpoint signals onto all incident edges except the one the original signal comes from and, if the original signal is marked and is the penultimate marked signal that originated at and has returned to the vertex, then also mark the slowed-down signals as signals that may be on a longest path and would be the first to find their midpoints.) (E, n, , ew_odw_rb)↦w_oe( d)w_rb (E, n), (If a reflected find-midpoint signal reaches a vertex, then it takes the way back it took before it was reflected and, if it is not one of the penultimate marked signals that reaches the vertex, then it is unmarked.) (E, n, , dw_ow_rb)↦⋃_e ∈ E ∖ de( d)w_ow_rb (E, n), (If a slowed-down find-midpoint signal reaches a vertex, then send slowed-down find-midpoint signals onto all incident edges except the one the original signal comes from, and mark these signals, if the original signal is marked — in which case it arrives at the same time and from the same direction as the slowest reflected find-midpoint signal that originated at the vertex, which however is not marked because it arrived too late at the leaf it was reflected at — and from exactly one direction the slowest find-midpoint signal that originated at the vertex has not returned yet.) (, , , d)↦∅, (If a freeze signal reaches the end of the edge it freezes, then it vanishes.) (, , , d)↦∅, (If a thaw signal reaches the end of its path, then it vanishes.) (, , , dew)↦ew, (If a thaw signal reaches a vertex, then it takes the direction that makes it stay on its path.) (E, , , (s, d, y)) ↦{ (s, d, y),if (s) = 0,⋃_e ∈ E ∖ d(s, e, y),otherwise. . (A stationary signal in a vertex stays there, and if a non-stationary signal reaches a vertex, then copies of it are sent onto each edge except the one the signal comes from.) For trees and without freezing and thawing, the case that precisely two signals collide in a vertex is handled by the following map, which maps triples — consisting of first, the set of directions that lead away from the vertex; secondly, the number of the directions from which the slowest reflected find-midpoint signals that originated at the vertex have already returned or have just arrived; and lastly, the set of colliding signals that is supposed to consist of precisely two signals — to the resulting signals, if a collision rule is specified, and to , otherwise:_v, 2^() ×_0 ×()→() ∪, map _v, 2^(, , d, )↦, (If a reflected divide signal collides with a boundary signal, then create a fire signal.) (E, n, s@( d')w_o'dw_rb, s'@d'w_o'w_r'b')↦ . _v, 1^(E, n, , s)∪_v, 1^(E, n, , s')∪( d)w_r( d')w_o'w_r',} if b = or b' =,dw_r,d'w_o'w_r',otherwise,(If a reflected find-midpoint signal collides with a slowed-down find-midpoint signal that originated at the same vertex, then the vertex of collision is the midpoint of the shortest path in the virtual tree from the vertex the reflected find-midpoint signal was reflected at to the vertex the signal that spawned the slowed-down find-midpoint signal was reflected at and, if this midpoint is not the midpoint of a longest path, then treat the original signals as if they reached the vertex alone and designate the point by a midpoint signal, and otherwise, send thaw signals along that longest path to both its ends.) (E, n, s@dw_rb, s'@d'w_r'b')↦ . _v, 1^(E, n, , s)∪_v, 1^(E, n, , s')∪( d)w_r( d')w_r',} if b = or b' = or n ≠E,dw_r,d'w_r',otherwise,(If two reflected find-midpoint signals that originated at the same vertex collide with each other, then the vertex of collision is the vertex the signals originated at and it is the midpoint of the shortest path in the virtual tree between the vertices the signals were reflected at and, if this midpoint is not the midpoint of a longest path, then treat the original signals as if they reached the vertex alone and designate the point by a midpoint signal, and otherwise, send thaw signals along that longest path to both its ends.) (, , dw, w@(d'w')d” w”)↦d'w', d”w”,(If a thaw signal collides with a midpoint signal that designates the midpoint of a path whose one half-path coincides with the path the thaw signal is going to take, then send an additional thaw signal along the other half-path.) ↦. (If none of the above happened, then indicate that by returning bottom.)How events for trees are handled is given inThe maps{ξ →, nd ↦nd,↦,s↦ s, } and {χ →, nd ↦nd,↦,s↦ s, }maps ξ and χfreeze and thaw signals that can be frozen and thawed respectively.The mapν() ×()→(), map ν(S, S')↦ξ(S'),if d∈ S ∪ S'and d∈ S ∪ S', χ(S'),if d∈ S ∪ S'and d∈ S ∪ S', S',otherwise,takes a set of old signals and a set of new signals and freezes the new signals, if the old or new signals contain a freeze signal but not a thaw signal that thaws the synchronisation of an edge; thaws the new signals, if the old or new signals contain a thaw signal that thaws the synchronisation of an edge but not a freeze signal; and does nothing, otherwise. The mapsζ_2 ()→(()), map ζ_2S↦s, s'⊆ Ss ≠ s'and _e, 2^(s, s') ≠,andη_2 () ×()→(()), map η_2(D, S)↦[t] {s, s'⊆ Ss ≠ s'and _v, 2^(D, x, s, s') ≠},both take a set of signals and return the set of unordered pairs of distinct signals from the given set for which a collision rule is specified in _e, 2^ and _v, 2^ respectively.The map_e^()→(), map _e^S↦ν(S,[] S ∖⋃_P ∈ζ_2(S) P ∪[]⋃_P ∈ζ_2(S)_e, 2^(P) ),handles collisions of signals on edges by leaving signals for which no pairwise collision rule with any other signal is specified in _e, 2^ as is, by applying _e, 2^ to each unordered pair of distinct signals for which a collision rule is specified, and by applying ν to freeze or thaw signals if there are or were any freeze or thaw signals.The mapκ()→(), map κS↦[t] ⋃_D∈ S D∪d ∈ w_r ∈^* dw_r∈ S∪d ∈ w_o ∈^*w_r ∈^* dw_ow_r∈ S takes a set of signals that are at a vertex and returns the directions from which the slowest reflected find-midpoint signals that originated at the vertex have already returned or have just arrived. The directions from which signals have already returned is memorised by a count signal, and the other directions are the ones from which marked reflected find-midpoint signals that originated at the vertex or marked slowed-down find-midpoint signals with any origin have just arrived (the slowest ones of the latter kind always arrive at the same time and from the same direction as the slowest but unmarked reflected find-midpoint signal that originated at the vertex arrive from that direction). Note that although we take the union of the memories of all count signals, there is actually gonna be no such signal in leaves (in which case the union is ∅) and precisely one such signal in each vertex that is not a leaf (in which case the union is the memory stored by this signal). The mapϰ()→, map ϰS↦,if d ∈d∈ S, ,otherwise,tells whether a set of signals contains any initiate signals or not. It is used by _v^ to tell whether a find-midpoint signal that reaches a leaf is among the first ones to do so, which is the case if and only if the signal travels alongside an initiate signal.The map_v^ () ×() →(), map _v^(D, S) [t] ↦ν(S,{ ∅, if (D) =,κ(S), otherwise, }∪_v, 1^[]D, κ(S), ϰ(S), S' ∖⋃_P ∈η_2(D, S') P∪⋃_P ∈η_2(D, S')_v, 2^(D, κ(S), P) ), whereS' = S ∖D∈ SD ⊆.handles events in vertices by updating the memory of the directions from which the slowest reflected find-midpoint signals that originated at the vertex have already returned or have just arrived, by applying _v, 1^ to non-count signals for which no pairwise collision rule with any other non-count signal is specified in _v, 2^, by applying _v, 2^ to each unordered pair of distinct non-count signals for which a collision rule is specified, and by applying ν to freeze or thaw signals if there are or were any freeze or thaw signals. How events for virtual trees, where virtual leaves already exist, are handled is given inWhen colliding signals in the graph and the involved directions are partitioned with respect to the virtual tree, some of the components may be degenerated and applying _e^ and _v^ to them may have unwanted effects. Such boundary cases are properly handled by the maps _e^()→(), map _e^S↦∅,if S≤ 1, _e^(S),otherwise,and_v^() ×()→(), map _v^(D, S)↦∅,if D = ∅ or S = ∅, _v^(D, S),otherwise. To handle an event involving the signals S in a virtual tree we do the following: We partition the signals into leaf signals, namely S_, for each leaf signal d∈ S_, the signals coming from the direction d, namely S_d, and all other signals, namely S_o. And we denote the set of directions that no leaf signal has by D'. Intuitively, S_ is the set of virtual leaves, S_d is the set of signals that reach the virtual leaf d, and D' is the set of directions that do not lead away from any virtual leaf. In the configurations we will encounter, in the case of a collision on an edge, there are either no leaf signals, in which case S_o = S, or there are two leaf signals, in which case S_o = ∅ (because there are no stationary signals in virtual leaves besides the leaf signals themselves) and hence either the signals simply collide, or one subset of signals reaches one virtual leaf and the other subset reaches the other virtual leaf. And, in the case of a collision in a vertex, signals coming from some edges may reach a virtual leaf and signals coming from other edges may reach the virtual vertex (we call it virtual because some of its original edges have been cut off; the directions onto the ones that have not been cut off are those in the set D'). Note that some collisions are not collisions in the virtual tree, because the signals came from different directions of virtual cuts. The maps that do what we just explained are _e^()→(), map _e^S↦[t]S_ ∪_e^(S_o)∪[]⋃_d ∈ X_v^(d, S_d),and_v^() ×()→(), map _v^(D, S)↦[t] S_ ∪_v^(D', S_o)∪[]⋃_d ∈ X_v^(d, S_d),where X = d ∈d∈ S, the set of directions that lead away from virtual leaves, S_ = d d ∈ X, the set of virtual leaves, S_d_d ∈ X = s ∈ S (s) =d_d ∈ X, for each direction of a virtual leaf, the set of signals that reach the virtual leaf corresponding to the direction moving towards it, S_o = S ∖ (S_∪ (⋃_d ∈ X S_d)), the signals that are not virtual leaves and that do not reach a virtual leaf, and D' = D ∖ X, the set of directions that do not lead away from virtual leaves.How events for graphs are handled is given inThe mapμ()→(), map μS↦d ∈d∈ S,takes a set of signals and returns the set of the reverses of the directions that initiate signals have.The mapφ_^e ()→(), map φ_^eS↦∅,if μ(S)≤ 1, d d ∈μ(S),otherwise,takes a set of signals and returns the empty set, if there is at most one initiate signal, and the set that consists of a virtual leaf for each initiate signal, otherwise.The mapφ_^v () ×() →(), map φ_^v(D, S) ↦{ ∅,if μ(S)≤ 1,d d ∈ D,if μ(S)≥ 2 and μ(S) = D, and (D) =,d d ∈ D ∖d_x,if μ(S)≥ 2 and μ(S) ≠ D, for some d_x ∈ D ∖μ(S),.takes a set of directions and a set of signals and returns the empty set, if there is at most one initiate signal, or the set that consists of a virtual leaf for each initiate signal, if there are at least two initiate signals and initiate signals reached the vertex from all incident edges, or the set that consists of a virtual leaf for each initiate signal but one, otherwise. Note that right now the choice of d_x is non-deterministic; however, if the finite set of directions carried a total order, then we could deterministically choose for example the smallest direction; or if continuum representations of graphs were embedded in high-dimensional Euclidean spaces and a Cartesian coordinate system was chosen such that the occurring directions are unit vectors, then the lexicographic order is a total order on the set of directions.The map_e (_e)→(), map _eS↦_e^(S ∪φ_^e(S))handles collisions on edges by creating two virtual leaves, if two initiate signals collide, which cuts the edge virtually, and then applying _e^ to the maybe new set of signals.The map_v (_v)→(), map _v(D, S)↦_v^(D, S ∪φ_^v(D, S))handles events in vertices by creating a virtual leaf for each incident edge, if the vertex is not a leaf and initiate signals reached the vertex from all incident edges, or by creating a virtual leaf for each incident edge from which an initiate signal arrived except for one such edge, otherwise, and then applying _v^ to the maybe new set of signals.Intuitively, initiate signals are used to turn the graph into a virtual tree by cutting edges at points where such signals collide. These cuts create virtual leaves which are represented by leaf signals. More precisely: When two initiate signals collide on an edge, it is cut by two leaf signals, one for each of the two directions. And when at least two initiate signals collide in a vertex, there are two cases: If initiate signals arrive from all directions, then each incident edge is cut by a leaf signal; otherwise, the incident edges from which initiate signals arrive are cut except for one such edge — the initiate signal from this excluded edge will spread to all edges that have not been cut. The signal machine signal machine 𝒮𝒮 = , , _k_k ∈, (_e, _v) is a time-optimal quasi-solution of the firing mob synchronisation problem over continuum representations of weighted, non-trivial, finite, and connected undirected multigraphs in the following sense: For each representation M of such a graph, each vertex ∈ M, for the time t = r + d, where r = sup_m ∈ M(, m) is the radius of M with respect toand d = sup_m, m' ∈ M(m, m') is the diameter of M, for the instantiation of 𝒮 for M, for the configuration c ∈ such that c() = ⋃_d ∈()d∪nd n ∈_0∪d, d(()) and c_M ∖≡∅, the points in the configuration (t)(c) at which a fire signal occurs lies dense in M with respect to the metric , and no fire signals occur in any of the configurations (s)(c), for s ∈_≥ 0 with s < t.A proof is sketched in <ref>. For each positive integer k, under the restriction to multigraphs whose maximum degree is bounded by k, for each such multigraph, because directions only need to be locally unique (compare <ref>), the set 1, 2, …, 2 k can be chosen as the set of directions, which makes the data sets of the kindsandfinite and independent of the multigraph, the finite set (1, 2, …, k) can be chosen as the data set of the kind , and, depending on the diameter d of the multigraph, the finite set of words overwith maximum length d can be chosen as the sets of words overthat occur in the data sets of the kinds , , , , and— altogether, the data sets of all kinds can be chosen to be finite but some depend on the multigraph.A discretisation of the signal machine 𝒮 is a time-optimal cellular automaton quasi-solution of the firing mob synchronisation problem over non-trivial, finite, and connected undirected multigraphs.Letbe a non-trivial, finite, and connected undirected multigraph. It is made up of paths whose source and target vertices are not of degree 2 and whose other vertices are of degree 2. Each such path together with its inverse can be regarded as an undirected uber-edge whose weight is the length of the path and whose ends are the source and target vertices of the path or its inverse. We call vertices that are not of degree 2 uber-vertices and vertices that are of degree 2 under-vertices.Signals jump from vertices to vertices along edges. They collide when they jump simultaneously from different vertices onto the same vertex or along the same edge but in different directions, or when signals jump onto vertices on which stationary signals reside. Collisions in uber-vertices are handled as collisions in vertices, collisions in under-vertices are handled as collisions on edges (namely on the uber-edges that contain the under-vertices), and collisions in the midst of edges are handled as collisions on edges (namely on the uber-edges that contain the edges). Signals reach a vertex when they jump onto an uber-vertex, but not when they jump onto an under-vertex (because the latter just means that they travel along an uber-edge).When signals collide in a vertex, be it a uber- or under-vertex, the resulting signals are on the vertex. But when signals collide in the midst of an edge, the resulting signals must be distributed onto both its ends depending on their directions. This last case makes it rather cumbersome to write down the local transition functions explicitly. Vertices must be virtually divided into multiple parts: One part that plays the role of the vertex itself and, for each incident edge, an additional part that plays the role of the midpoint of the edge together with the corresponding part of the other end of the edge. And signals must be cleverly distributed onto these parts depending on their direction and how they came into being, and collisions of signals must also be cleverly handled taking the parts the involved signals came from and are on into account.This discretisation of the signal machine is actually a cellular automaton over the multigraph with appropriate dummy neighbours that are in a dead state (think for example of the multigraph as being embedded in a coloured S-Cayley graph with sufficient maximum degree and of the vertices that do not belong the multigraph as being in a dead state).Jacques Mazoyer showed in 1987 that all infinitely many divide signals of type n, for n ∈_0, that emanate from the same point can be generated by a cellular automaton with only finitely many states (see <cit.>). And, as illustrated in <ref>, under the restriction to multigraphs whose maximum degrees are uniformly bounded by a constant, the data sets of all kinds can be chosen to be finite but some depend on the multigraph. Therefore, depending on the multigraph, the discretisation of 𝒮 is a cellular automaton with a finite number of states.Are there time-optimal signal machine and cellular automaton solutions of the firing mob synchronisation problem over non-trivial, finite, and connected undirected multigraphs whose maximum degrees are uniformly bounded by a constant? Or, more specifically, is it possible to adapt the signal machine 𝒮 (and thereby its discretisation) such that the data sets of all kinds can be chosen to be finite and independent of the multigraph (and thereby making its discretisation have a finite set of states), for example by reducing the number of midpoints that are and need to be determined?§ PROOF SKETCH OF THE MAIN THEOREM In this section, we sketch a proof of <ref>. To that end, let = , , be a non-trivial, finite, and connected undirected multigraph, letbe an edge weighting of , let M be a continuum representation of , identify vertices ofand M and direction-preserving paths from vertices to vertices ofand M, and letbe a vertex of M, which we call generalgeneral vertex . Furthermore, let 𝒮 be the signal machine and let c be the initial configuration of the firing mob synchronisation problem from <ref>, and, whenever we talk about time evolution, we mean the one of 𝒮 that is in the configuration c at time 0, for example, at time t either means in configuration (t)(c) or essentially in configuration (t)(c) but before events have been handled.To proof <ref>, we need to ascertain that the signal machine performs the following tasks: First, it cuts the multigraph such that the multigraph turns into a virtual tree and looks like a tree to all other tasks; secondly, it starts synchronisation of edges and freezes it in time; thirdly, it determines the midpoints of all non-empty direction-preserving paths from vertices to vertices in time; fourthly, it determines which midpoints are the ones of the longest paths; fifthly, starting from the midpoints of the longest paths, it traverses midpoints of shorter and shorter paths and upon reaching midpoints of edges, it thaws synchronisation of the respective edges; sixthly, all edges finish synchronisation at time r + d with the creation of fire signals that lie dense in the graph, where r is the radius of the graph with respect to the general and d is the diameter of the graph.That the first task is performed is evident from the definitions of _e, _v, _e^, and _v^. The only subtlety here is that besides leaf signals there cannot be stationary signals in virtual leaves or, more precisely, at points that are virtually cut by leaf signals, because stationary signals carry the semi-direction , which is insufficient to associate them with one or the other leaf signal as is done for non-stationary signals. This is no problem because the other tasks do not place stationary signals in (virtual) leaves. Therefore, we assume from now on, without loss of generality, that the multigraphis a tree.That the second task is performed can be seen from a careful examination of the definitions of _e^, _v^, _e, 2^, _v, 1^, and _v, 2^, where from the third task it is used that midpoints of edges are found in time to start the freezing process. That the third, fourth, and fifth and sixth parts are performed is proven in <ref>, <ref>, and <ref>. §.§ Midpoints are Determined The midpoint of a path in a multigraph is the midpoint of its embedding in the continuum representation of the multigraph as introduced inLet p be a path in . The point _p = p((p) / 2) is called midpoint of pmidpoint _p of p[symbols]mpfraktur@_p.The midpoint of the empty path in v is the vertex v itself.Let p, q, and q' be three paths such that (p) = (q) = (q'), _pq∈ p, and (q) ≥(q'). Then, _pq'∈ p and(_pq, _pq') = (pq) / 2 - (pq') / 2 = (q) / 2 - (q') / 2. Analogously, let q, q', and p be three paths such that (q) = (q') = (p), _qp∈ p, and (q) ≥(q'). Then, _q'p∈ p and(_qp, _q'p) = (qp) / 2 - (q'p) / 2 = (q) / 2 - (q') / 2. When the midpoints of non-empty direction-preserving paths are found is stated inLet p be a non-empty direction-preserving path in . The midpoint signal that designates the midpoint of p is created at _p at time t_p = max(, (p)),(, (p)) + (p)/2.First, let the generalbe the source or target of p. Then, a find-midpoint signal with originof speed 1 and a slowed-down find-midpoint signal with originof speed 1/3 travel fromtowards the other end of p. The find-midpoint signal is reflected at the other end at time (2/2) ·(p) and this reflection collides with the slowed-down find-midpoint signal at the midpoint of p at time (3/2) ·(p) creating a midpoint signal for p (because both signals have the same origin). Note that the time of collision is equal to t_p (see <ref>).Secondly, let the generallie on p without being its source or target. Then, two find-midpoint signals with origintravel fromto the ends of p; the source of p is reached at time (, (p)) and the target at time (, (p)). When such a signal reaches its end, it is reflected and travels back. If (, (p)) = (, (p)), then this distance is equal to (p)/2, the reflected signals collide at time (p) at the midpoint of p creating a midpoint signal for p; note that the time of collision is equal to t_p. Otherwise, the reflected signal that is nearer toreaches this vertex first, where the signal is slowed down and travels towards the other reflected signal with which it collides at the midpoint of p at time t_p (see <ref>).Lastly, let the generalnot lie on p. Then, an initiate signal travels fromto the nearest vertex v on p, where it creates two find-midpoint signals with origin v that travel to the ends of p, are reflected at these ends and travel back, one is slowed-down upon reaching v, and the slowed-down signal collides with the reflected signal in the midpoint of p at time (, v) + max(v, (p)),(v, (p)) + (p)/2 creating a midpoint signal for p. Note that the time of collision is equal to t_p. When reflected/slowed-down find-midpoint signals with different origins collide, nothing happens, the signals just move on. Hence, no points are falsely found to be midpoints. That the midpoints of maximum-weight direction-preserving paths are identical and found at time r + d is shown inAll maximum-weight direction-preserving paths inhave the same midpointand the midpoint signals that designate the midpoints of such paths are created atat time r + d/2, where r = max_v ∈(, v) is the radius ofwith respect toand d = max_v, v' ∈(v, v') is the diameter of . Note that r is equal to the radius sup_m ∈ M(, m) of M with respect toand d is equal to the diameter sup_m, m' ∈ M(m, m') of M. We prove both statements by contradiction.First, suppose that there are two maximum-weight direction-preserving paths p̂ and p̂' inthat do not have the same midpoint. Then, there is a non-empty direction-preserving path 𝔭_ in M from the midpoint of p̂ to the one of p̂'. And, there is a direction-preserving subpath 𝔭 of p̂ in M from one end of p̂ to its midpoint whose target-direction is not the reverse of the source-direction of p_. And, there is a direction-preserving subpath 𝔭' of p̂' in M from the midpoint of p̂' to one of its ends whose source-direction is not the reverse of the target-direction of p_. The concatenation of 𝔭, 𝔭_, and 𝔭' is a direction-preserving path from vertex to vertex in M whose length is equal to d/2 + (𝔭_) + d/2 > d. It corresponds to a direction-preserving path p inwhose weight is greater than d, which contradicts that d is the diameter of . Therefore, all maximum-weight direction-preserving paths inhave the same midpoint, which we denote by .Secondly, suppose that there is a maximum-weight direction-preserving path p̂ insuch that max(, (p̂)),(, (p̂)) < r. Let v be a vertex ofsuch that (, v) = r, let p_v, p_, and p_ be the direction-preserving paths infrom v, (p̂), and (p̂) to , let v' be the vertex on p_v and p_ or on p_v and p_ that is the furthest from , and let v” be the vertex on p̂ that is the nearest to . Then, the weight of p_v is r, the one of p_ is (, (p̂)) < r, and the one of p_ is (, (p̂)) < r. If v' lies on the subpath of p_ from (p̂) to v” (which is equal to the subpath of p̂ with the same ends), then let p be the direction-preserving path from v over v' over v” to (p̂); if v' lies on the subpath of p_ from (p̂) to v” (which is equal to the subpath of the inverse of p̂ with the same ends), then let p be the direction-preserving path from v over v' over v” to (p̂); and otherwise, let p be the direction-preserving path from v over v' over v” to (p̂) (we could have chosen (p̂) as well). See <ref> for a schematic representation of the three cases. In the first case, because the subpaths of p_v and p_ from v” tocoincide and the weight of p_v is greater than the weight of p_, the weight of the subpath of p_v from v over v' to v” (which is equal to the subpath of p from v over v' to v”) is greater than the weight of the subpath of p_ from (p̂) over v' to v” (which is equal to the subpath of p̂ from (p̂) over v' to v”) and hence, because the subpaths of p and p̂ from v” to (p̂) coincide, the weight of p is greater than the weight of p̂. In the second case, it follows analogously that the weight of p is greater than the weight of p̂. And in the third case, because the subpaths of p_v and p_ from v' tocoincide and the weight of p_v is greater than the weight of p_, the weight of the subpath of p_v from v to v' is greater than the weight of the subpath of p_ from (p̂) over v” to v', hence the weight of the subpath of p from v over v' to v” is greater than the weight of the subpath of p̂ from (p̂) to v”, and therefore, the weight of p is greater than the weight of p̂.In either case, the inequality (p) > (p̂) contradicts that p̂ is a maximum-weight path. Therefore, for each maximum-weight direction-preserving path in , we have max(, (p̂)),(, (p̂)) = r. It follows from <ref> that the midpoint signals that designate the midpoints of maximum-weight direction-preserving paths inare created atat time r + d/2. It follows that the midpoints of non-empty non-maximum-weight direction-preserving paths are found before the ones of maximum-weight direction-preserving paths as shown in Let p be a non-empty direction-preserving path in . The midpoint signal that designates the midpoint of p is created at _p before time r + d/2.This is a direct consequence of <ref>, because (, (p)) ≤ r, (, (p)) ≤ r, and (p) < d. §.§ Midpoints of Maximum-Weight Paths are Recognised as Such That the midpoints of maximum-weight direction-preserving paths are recognised as such is sketched inThe first two reflected find-midpoint signals, or the first two reflected and slowed-down find-midpoint signals to collide that originated at the same vertex and were reflected at the ends of a maximum-weight direction-preserving path are marked at the time of collision and hence recognise that the midpoint of the path they collide at is the one of a maximum-weight direction-preserving path. Let p̂ be a maximum-weight direction-preserving path in(see <ref>). Then, the ends v̂_1 and v̂_2 of p̂ are leaves. And, among the first find-midpoint signals to reach the ends of p̂ are the two that originated at the vertex v̂ on p̂ that is nearest to , and, because they travel alongside initiate signals, their reflections s and s' at the ends of p̂ are marked. When one of them reaches a vertex on its way back it stays marked, because from all directions excluding from the direction it is headed but including the direction it is coming from have the marked reflected find-midpoint signals that originated at the vertex just or already returned, which is memorised by a count signal that is located at the vertex; the reason that such marked signals have already returned is that otherwise there would be a direction-preserving path with more weight than p̂ that would be the concatenation the maximal subpath of p̂ that lies in the direction the signal is headed and a path that begins with one of the edges from which no marked signal has returned yet. Note that the signals s and s' travel back alongside the marked reflected find-midpoint signals that originated at the vertices the signals s and s' passed by before they were reflected and that therefore, when they reach vertices on their way back, the count signals are just updated and hence up-to-date. When s and s' reach the vertex they originated at at the same time, then this vertex is found to be the midpoint of the path p̂ and, because s and s' are marked, it is recognised as the midpoint of a maximum-weight path. When one of the signals, say s, reaches the vertex it originated at, namely v̂, first, then it is slowed down and spreads throughout the graph away from the edge it comes from, in particular, one slowed-down find-midpoint signal, let us denote it by s”, travels towards s'. When s” reaches a vertex on its way towards s' it stays marked, because from all directions excluding from the direction it is headed have the slowest reflected find-midpoint signals that originated at the vertex just or already returned, more precisely, from all directions excluding from the direction it is coming from and the one it is headed have the marked reflected find-midpoint signals that originated at the vertex just or already returned and from the direction it is coming from has the slowest but unmarked reflected find-midpoint signal that originated at the vertex and the first marked slowed-down find-midpoint signal just returned. When s” and s' collide, the midpoint of p̂ is found and, because s” and s' are marked, it is recognised as the midpoint of a maximum-weight path. A detailed proof is given in the remainder of the present subsection. Symbolic notations for the predicates is a vertex of and is an edge of are introduced in Let = W, F be a subtree of , let v be a vertex of , and let e be an edge of . We write v ∈v ∈[symbols]vinTcalligraphic@v ∈ instead of v ∈ W and e ∈e ∈[symbols]einTcalligraphic@e ∈ instead of e ∈ F. The set of greatest subtrees of a vertex v that correspond to its incident edges is named in Let v be a vertex of . * Let _v be the set of edges that are incident to v, and, for each edge e ∈_v, let 𝒯_v, e be the greatest subtree ofthat is rooted at v, contains the edge e, and does not contain any other edge of _v (note that by greatest we mean greatest with respect to the number of vertices). The set 𝒯_v, e e ∈_v is denoted by (v)set (v)[symbols]sbtrsv@(v).* The set(v) = _∈(v)_v set (v) of trees[symbols]mxsbtrsv@(v)is the set of maximum-radius trees of (v). In the case that it is a singleton set, we denote its one and only element by _vtree _v[symbols]Tvcalligraphichat@_v. * The set(v) = _∈(v) ∖(v)_v set (v) of trees[symbols]sbtrsvclosure@(v)is the set of second-maximum-radius trees of (v). For each leaf v of , the set (v) is a singleton and it is equal to (v), and the set (v) is empty. Things associated with a greatest subtree of a vertex are introduced in Let v be a vertex ofand letbe a tree of (v). * Let e be the edge ofthat is incident to v. The direction that leads from v onto e is uniquely determined byand is denoted by _v()direction _v()[symbols]dirvTcalligraphic@_v(). * The non-negative integer _v = max_v' ∈(v, v') is called radius _v ofwith respect to vradius ofwith respect to v[symbols]normvsubscript@_v.* Let _ be the set of direction-preserving paths in . The set_v() = p ∈_(p) = vand (p) = _vset _v() of paths[symbols]mxpthsvTcalligraphic@_v()is the set of maximum-weight direction-preserving paths from v in . * The set_v() = (p)p ∈_v()set _v() of vertices[symbols]mxvrtcsvTcalligraphic@_v()is the set of vertices inthat are furthest away from v. We have (v) = (v) and _v((v)) = (v).Each vertex of _v() is a leaf. The unique direction-preserving path from one vertex to another is named in Let v and v' be two vertices of . The direction-preserving path infrom v to v' is denoted by p_v, v'path p_v, v'[symbols]pvvprime@p_v, v' and the vertices on this path are denoted by _v, v'set _v, v' of vertices[symbols]Vvvprime@_v, v'. When and why count signals, initiate signals, and (maybe-marked slowed-down/reflected) find-midpoint signals are created and how they spread throughout the tree is said in Let the signal machine 𝒮 be in the configuration c_ at time 0. * At time 0, a count signal with empty memory is created at , and initiate signals, find-midpoint signals with origin , and maybe-marked slowed-down find-midpoint signals with originand reflection vertexare sent fromin all directions, where the slowed-down signal is marked if and only ifis a leaf. For each vertex v ∈∖, at time (, v), an initiate signal reaches v from the direction towards , whereupon * a count signal with empty memory is created at v,* initiate signals are sent from v in all directions away from , and* find-midpoint signals with origin v, and maybe-marked slowed-down find-midpoint signals with origin v and reflection vertex v are sent from v in all directions,where the slowed-down signal is marked if and only if v is a leaf.In short, initiate signals spread fromto all leaves, where they vanish, and they initiate the search for midpoints at all vertices. Note that to simplify the exposition of the forthcoming proofs, we also create count signals at leaves. * Let v be a vertex of . As said above, at time (, v), find-midpoint signals with origin v, and maybe-marked slowed-down find-midpoint signals with origin v and reflection vertex v are sent from v in all directions, where the slowed-down signal is marked if and only if v is a leaf. For each vertex v' ∈∖v, * at time (, v) + (v, v'), a find-midpoint signal with origin v reaches v' from the direction towards v, whereupon a maybe-marked reflected find-midpoint signal with origin v and reflection vertex v' is sent from v' towards v and find-midpoint signals with origin v are sent from v' in all directions away from v, where the reflected signal is marked if and only if v is a leaf and an initiate signal just reached v' (the latter is the case if and only if (, v) + (v, v') = (, v')), and,* at time (, v) + 3 ·(v, v'), a maybe-marked slowed-down find-midpoint signal with origin v and reflection vertex v reaches v' from the direction towards v, whereupon maybe-marked slowed-down find-midpoint signals with origin v and reflection vertex v are sent from v' in all directions away from v. Note that even if the slowed-down signal that is sent from v at time (, v) is marked, the slowed-down signal that reaches v' may be unmarked, and even if the latter signal is marked, the slowed-down signals that are sent from v' may be unmarked.In short, find-midpoint signals with origin v and maybe-marked slowed-down find-midpoint signals with origin v and reflection vertex v spread from v to all leaves, and whenever one of the former signals reaches a vertex, it is also reflected. Note that to simplify the exposition of the forthcoming proofs, we talk as if maybe-marked slowed-down find-midpoint signals only vanish at leaves, although when a marked slowed-down find-midpoint signal collides with a marked reflected find-midpoint signal with the same origin, both signals vanish.* Let v and v' be two vertices ofsuch that v ≠ v'. As said above, at time (, v) + (v, v') a maybe-marked reflected find-midpoint signal with origin v and reflection vertex v' is sent from v' towards v. For each vertex w on the direction-preserving path from v' to v except for v', at time (, v) + (v, v') + (v', w), the signal reaches w from the direction towards v', whereupon, if w ≠ v, the signal is sent from w towards v, and otherwise, maybe-marked slowed-down find-midpoint signals with origin v and reflection vertex v' are sent from v in all directions away from v'. And, for each vertex v” ofthat from the viewpoint of v lies in a direction away from v', which means that v”≠ v and there is a tree ∈(v) such that v' ∉ and v”∈, at time (, v) + (v, v') + (v', v) + 3 ·(v, v”), a maybe-marked slowed-down find-midpoint signal with origin v and reflection vertex v' reaches v”, whereupon maybe-marked slowed-down find-midpoint signals with origin v and reflection vertex v' are sent from v” in all directions away from v (or, equivalently, away from v'). In short, each maybe-marked reflected find-midpoint signal travels back to its origin and when it reaches its origin, it is slowed-down and spreads to all leaves away from its reflection vertex. Note that unmarked signals never become marked, but marked signals may become unmarked; precisely when the latter does or does not happen is answered in the present subsection.What is said above is evident from the definition of the signal machine 𝒮 (if it is carefully studied). When and why leaves send marked reflected/slowed-down find-midpoint signals is said inLet v be a leaf of . At time (, v), an initiate signal reaches v, for each vertex w ∈_, v∖v, a find-midpoint signal with origin w reaches v, and no other find-midpoint signal reaches v, whereupon * for each vertex w ∈_, v∖v, a marked reflected find-midpoint signal with origin w and reflection vertex v is sent from v towards w, which is the only possible direction,* a marked slowed-down find-midpoint signal with origin v and reflection vertex v is sent from v towards w, and* no other marked signal is sent from v. And before time (, v) no signals reach and are sent from v, and after time (, v) no marked signals are sent from v (because after this time no initiate signal reaches v).This is a direct consequence of <ref> and the definition of 𝒮. When does a signal that is sent from a vertex towards the general at a special time reach the next vertex is answered inLet v be a vertex of , letbe a tree of (v) ∖(v) such that either v = or ∉, and let v' be the one and only neighbour of v in . Then, * (v') is a singleton set and its only element _v' contains v,* _v() = v',if v' is a leaf, _' ∈(v')_v'('),otherwise,* _v = (v, v') + max_' ∈(v')'_v', and* when a signal of speed 1 is sent from v' towards v at time (, v') + 2 ·max_' ∈(v')'_v', it reaches v at time (, v) + 2 ·_v, where in the case that (v') is empty, we define max_' ∈(v')'_v' as 0. The first item is evident, the second and third follow from it, and the fourth follows from the third with (, v') = (, v) + (v, v') and the fact that the signal needs the time span (v, v') to traverse the edge from v' to v. The set of all non-leaf vertices whose unique maximum-radius tree contains the general is named inThe set of all non-leaf vertices v ofsuch that (v) is a singleton set and its only element _v contains the vertex , is denoted by V_set V_ of vertices[symbols]Vgsubscript@V_.For each vertex v ∈ V_, each tree ∈(v) ∖_v, and each vertex v' of , we have v' ∈ V_. And, for each vertex v ∈ V_, the set (v) is non-empty and, for each tree ∈(v), we have _v = max_∈(v)_v. When and why non-leaf vertices whose unique maximum-radius tree contains the general send marked reflected/slowed-down find-midpoint signals is shown inFor each vertex v ∈ V_, at time (, v) + 2 ·max_∈(v)_v,* the count signal at v, before it is updated, has the memory (v) ∖_v((v) ∪_v),* for each tree ∈(v), each leaf v∈_v(), and each vertex w ∈_, v, a marked reflected find-midpoint signal with origin w and reflection vertex v reaches v from direction _v(), and* no other marked reflected find-midpoint signal reaches v, whereupon * the count signal at v, after it is updated, has the memory (v) ∖_v(_v),* for each tree ∈(v), each leaf v∈_v(), and each vertex w ∈_, v∖v, a marked reflected find-midpoint signal with origin w and reflection vertex v is sent from v in the direction _v(_v) towards w, and no other marked reflected find-midpoint signal with origin w and reflection vertex v is sent from v, and,* for each tree ∈(v), each leaf v∈_v(), and each direction d ∈(v) ∖_v(), a marked slowed-down find-midpoint signal with origin v and reflection vertex v is sent from v in direction d (note that _v(_v) ∈(v) ∖_v()), and, no other marked slowed-down find-midpoint signal with origin v and reflection vertex v is sent from v. And before time (, v) + 2 ·max_∈(v)_v, marked signals may reach v but no marked signal is sent from v, and after that time, no marked signals as above are sent from v. We prove this by induction on n_v = max_(W, F) ∈(v) ∖_vF, for v ∈ V_. For brevity though, we only treat the existence of signals and not their absence. Base Case (see <ref>) Let v ∈ V_ such that n_v = 1. And, let ∈(v) ∖_v. Then,consists of one edge e whose one end is v, whose other end is a leaf v', and whose weight is _v. According to <ref>, at time (, v') = (, v) + _v, for each vertex w ∈_, v'∖v' = _, v, a marked reflected find-midpoint signal with origin w and reflection vertex v' is sent from v' towards v, in particular, one with origin v. These signals reach v at time (, v) + 2 ·_v, whereupon * the count signal at v memorises the direction _v() (because a marked reflected find-midpoint signal with origin v reached v),* for each vertex w ∈_, v∖v, a maybe-marked reflected find-midpoint signal with origin w and reflection vertex v' is sent from v in the direction _v(_v) towards w, and* for each direction d ∈(v) ∖_v(), a maybe-marked slowed-down find-midpoint signal with origin v and reflection vertex v' is sent from v in direction d.On the timeline, for the trees of (v) ∖_v in non-decreasing order with respect to the radius and at the same time for trees with the same radius, the signals reach v and are sent from v. For those trees whose radius is less than the second greatest radius among the trees of (v), which is max_∈(v)_v, the aforementioned maybe-marked signals that are sent from v are unmarked (because the memory of the count signal, after it is updated, does neither contain the directions of _v((v)) nor the direction _v(_v)). And, for the trees of (v), the aforementioned maybe-marked signals that are sent from v are marked (because the count signal, after it is updated, has the memory (v) ∖_v(_v)). In conclusion, at time (, v) + 2 ·max_∈(v)_v, what is to be proven holds. Inductive Step (see <ref>) Let v ∈ V_ such that n_v ≥ 2 and such that what is to be proven holds for each vertex v' ∈ V_ with n_v' < n_v, which is the inductive hypothesis. And, let ∈(v) ∖_v, let e be the edge ofwhose one end is v, and let v' be the other end of e. The vertex v' is either a leaf or an element of V_ with n_v' < n_v. In the first case, according to <ref>, at time (, v') = (, v) + (v, v'), for each vertex w ∈_, v'∖v' = _, v, a marked reflected find-midpoint signal with origin w and reflection vertex v' is sent from v' towards v; note that v' is the one and only element of _v(), the set (v') is empty, and we define max_' ∈(v')'_v' as 0 (see <ref>). In the second case, according to the inductive hypothesis, at time (, v') + 2 ·max_' ∈(v')'_v', for each tree ' ∈(v'), each leaf v' ∈_v'('), and each vertex w ∈_, v'∖v' = _, v, a marked reflected find-midpoint signal with origin w and reflection vertex v' is sent from v' towards v; note that the set of all vertices v' is equal to _v() (see <ref>). In both cases, the marked signals reach v at time (, v) + (v, v') + 2 ·max_' ∈(v')'_v' + (v', v), which is equal to (, v) + 2 ·_v, whereupon* the count signal at v memorises the direction _v() (because at least one marked reflected find-midpoint signal with origin v reached v),* for each leaf v∈_v() and each vertex w ∈_, v∖v, a maybe-marked reflected find-midpoint signal with origin w and reflection vertex v is sent from v in the direction _v(_v) towards w, and,* for each leaf v∈_v() and each direction d ∈(v) ∖_v(), a maybe-marked slowed-down find-midpoint signal with origin v and reflection vertex v is sent from v in direction d.It follows verbatim as in the base case, that what is to be proven holds.The set of all vertices whose unique maximum-radius tree does not contain the general is named in The set of all vertices v ofsuch that (v) is a singleton set and its only element _v does not contain the vertex , is denoted by U_set U_ of vertices[symbols]Ugsubscript@U_.We have ∉ U_. And, for each vertex v ∈ U_, we have _, v∖∈ U_, the vertex v is not a leaf and the set (v) is non-empty.The maximal subtree of a non-general vertex that contains the general is named in Let v be a vertex ofsuch that v ≠. The tree of (v) that does contain the vertexis denoted by _v^tree _v^[symbols]Tvgcalligraphic@_v^. And to avoid case differentiations, we define _^_^ = 0[symbols]Tggcalligraphic@_^ as the number 0. The vertices of the maximal path from a vertex to the general whose vertices excluding its source have the subtree that contains the general as second-maximum-radius subtree is named in Let v be a vertex of U_, and let p be the maximum-weight subpath of p_, v such that (p) = v and, for each vertex w on p with w ≠(p), we have _w^∈(w). The set of the vertices on p is denoted by W_, vset W_, v of vertices[symbols]Wgvsubscript@W_, v. See <ref>. We have v ∈ W_, v∖⊆ U_. And, for each w ∈ W_, v, we have W_, w⊆ W_, v. And, if _v^∈(v), then W_, v = W_, v'∪v, where v' is the neighbour of v on p_, v. When and why vertices whose unique maximum-radius tree does not contain the general send marked reflected/slowed-down find-midpoint signals is shown inFor each vertex v ∈ U_, at time (, v) + 2 ·max_∈(v)_v, * the count signal at v, before it is updated, has the memory (v) ∖_v((v) ∪_v),* for each tree ∈(v) and each leaf v∈_v(), a maybe-marked reflected find-midpoint signal with origin v and reflection vertex v reaches v from direction _v(), where the reflected signal is marked if and only if ≠_v^, and,* for each vertex w ∈ W_, v∖v, each tree ∈(w) ∖_w^, and each leaf w∈_w(), a marked slowed-down find-midpoint signal with origin w and reflection vertex w reaches v from the direction towards w, which is the direction _v(_v^), whereupon * the count signal at v, after it is updated, has the memory (v) ∖_v(_v),* for each tree ∈(v) ∖_v^, each leaf v∈_v(), and each direction d ∈(v) ∖_v(), a marked slowed-down find-midpoint signal with origin v and reflection vertex v is sent from v in direction d (note that _v(_v), _v(_v^)⊆(v) ∖_v()), and,* for each vertex w ∈ W_, v∖v, each tree ∈(w) ∖_w^, each leaf w∈_w(), and each direction d ∈(v) ∖_v(_v^), a marked slowed-down find-midpoint signal with origin w and reflection vertex w is sent from v in direction d (note that _v(_v) ∈(v) ∖_v(_v^) and that, if w =, then (w) ∖_w^ = (w)).We prove this by induction on n_v = _, v, for v ∈ U_.Base Case (compare <ref>)Let v ∈ U_ such that n_v = 1. First, for each tree ∈(v) ∖_v^, _v, according to <ref>, if the neighbour of v inis a leaf, or <ref>, otherwise, and <ref>, at time (, v) + 2 ·_v, for each leaf v∈_v(), a marked reflected find-midpoint signal with origin v and reflection vertex v reaches v from direction _v(), whereupon * the count signal at v memorises the direction _v(),* for each leaf v∈_v() and each direction d ∈(v) ∖_v(), a maybe-marked slowed-down find-midpoint signal with origin v and reflection vertex v is sent from v in direction d. Secondly, according to <ref>, at time (, v) + 2 ·_v^_v, for each leaf v∈_v(_v^), a maybe-marked reflected find-midpoint signal with origin v and reflection vertex v reaches v from direction _v(_v^). This signal is actually unmarked, because it was reflected at v at time (, v) + _v^_v = (, v) + (v, v), which, because v ≠ and v∈_v^, is greater than the only time, namely (, v), at which an initiate signal reaches v.Thirdly, because U_ is non-empty, we have ∈ V_ and v ∈_. According to <ref>, at time 2 ·max_∈()_, for each tree ∈(), and each leaf ∈_(), a marked slowed-down find-midpoint signal with originand reflection vertexis sent fromtowards v; note that the set of all verticesis equal to _v(_v^) (compare <ref>). The marked signals reach v from the direction towards , which is the direction _v(_v^), at time 2 ·max_∈()_ + 3 ·(, v) = (, v) + 2 ·_v^_v, whereupon * the count signal at v memorises the direction _v(_v^) and,* for each leaf ∈_v(_v^) and each direction d ∈(v) ∖_v(_v^), a maybe-marked slowed-down find-midpoint signal with originand reflection vertexis sent from v in direction d.Altogether, on the timeline, for the trees of (v) ∖_v in non-decreasing order with respect to the radius and at the same time for trees with the same radius, the signals reach v and are sent from v. For those trees whose radius is less than the second greatest radius among the trees of (v), which is max_∈(v)_v, the aforementioned maybe-marked signals that are sent from v are unmarked (because the memory of the count signal, after it is updated, does neither contain the directions of _v((v)) nor the direction _v(_v)). And, for the trees of (v), the aforementioned maybe-marked signals that are sent from v are marked (because the count signal, after it is updated, has the memory (v) ∖_v(_v)). Note that, if _v^∈(v), then W_, v∖v =, and otherwise, W_, v∖v = ∅. In conclusion, at time (, v) + 2 ·max_∈(v)_v, what is to be proven holds. Inductive Step (see <ref>) Let v ∈ U_ such that n_v ≥ 2 and such that what is to be proven holds for each vertex v' ∈ U_ with n_v' < n_v, which is the inductive hypothesis. First, for each tree ∈(v) ∖_v^, _v, the same as in the base case happens.Secondly, as in the base case, at time (, v) + 2 ·_v^_v, for each leaf v∈_v(_v^), an unmarked reflected find-midpoint signal with origin v and reflection vertex v reaches v from direction _v(_v^).Thirdly, let v' be the neighbour of v in _v^. Then, v' ∈_, v∖⊆ U_ and n_v' < n_v. According to the inductive hypothesis, at time (, v') + 2 ·max_' ∈(v')'_v', * for each tree ' ∈(v') ∖_v'^ and each leaf v' ∈_v'('), a marked slowed-down find-midpoint signal with origin v' and reflection vertex v' is sent from v' towards v, and,* for each vertex w' ∈ W_, v'∖v', each tree ' ∈(w') ∖_w'^, each leaf w' ∈_w'('), a marked slowed-down find-midpoint signal with origin w' and reflection vertex w' is sent from v' towards v.The marked signals reach v from from direction _v(_v^) at time (, v') + 2 ·max_' ∈(v')'_v' + 3 ·(v', v) = (, v) + 2 ·_v^_v, whereupon * the count signal at v memorises the direction _v(_v^) and,* for each vertex w ∈ W_, v', each tree ∈(w) ∖_w^, each leaf w∈_w(), and each direction d ∈(v) ∖_v(_v^), a maybe-marked slowed-down find-midpoint signal with origin w and reflection vertex w is sent from v in direction d.Note that, if _v^∈(v), then W_, v∖v = W_, v', and otherwise, W_, v∖v = ∅. With that it follows verbatim as in the base case, that what is to be proven holds.That midpoints of maximum-weight direction-preserving paths are recognised as such is shown in Let p̂ be a maximum-weight direction-preserving path in , let v̂ be the vertex on p̂ that is nearest to , let v̂_1 and v̂_2 be the two ends of p̂ such that (v̂, v̂_1) ≤(v̂, v̂_2), and letbe the midpoint of p̂. At time r + d/2, at the midpoint , * if (v̂, v̂_1) = (v̂, v̂_2), two marked reflected find-midpoint signals with origin v̂ and reflection vertices v̂_1 and v̂_2 collide, and* otherwise, a marked slowed-down find-midpoint signal with origin v̂ and reflection vertex v̂_1 collides with a marked reflected find-midpoint signal with origin v̂ and reflection vertex v̂_2. From a broad perspective and ignoring boundary cases the following happens in the given order (see <ref>). At time 0, an initiate signal is sent fromtowards v̂. At time (, v̂), this signal reaches v̂, whereupon find-midpoint signals with origin v̂ are sent from v̂ in all directions. At time (, v̂) + _v̂^_v̂, the slowest but unmarked reflected find-midpoint signal with origin v̂ returns to v̂ from the direction towards . At time (, v̂) + 2 ·(v̂, v̂_1), the slowest and marked reflected find-midpoint signal with origin v̂ returns to v̂ from the direction towards v̂_1, whereupon a marked slowed-down find-midpoint signal with origin v̂ is sent towards v̂_2. At time (, v̂) + 2 ·(v̂, v̂_1) + 3 ·(v̂, ) = (, ) + 2 ·(v̂_1, ) = r + d/2, this signal reaches . And, at the same time, which is equal to (, v̂) + (v̂, v̂_2) + (v̂_2, ), the slowest and marked reflected find-midpoint signal with origin v̂ that is on its way to return to v̂ from the direction towards v̂_2 reaches . The two marked signals collide atrecognising it as the midpoint of a maximum-weight path.The slowest reflected find-midpoint signal with origin v̂ that returns to v̂ from the direction towardsis unmarked, because it reaches the leaf it is reflected at later than the initiate signal and is never marked in the first place. The slowest reflected find-midpoint signal with origin v̂ that returns to v̂ from the direction towards v̂_1 is marked, because it reaches the leaf v̂_1 alongside initiate signals and at each vertex on its way back, it is the penultimate marked signal to return (this is essentially <ref>). For the same reason, the signal from the direction towards v̂_2 is marked when it reaches . And, the slowed-down find-midpoint signal with origin v̂ that reachesfrom the direction towards v̂ is marked, because at each vertex it reaches on its way, it is the penultimate marked signal to do so (this is essentially <ref>). Note that, for each vertex u on the path fromtoexcept for , the direction towardsis added to the memory of the count signal at u, because a marked slowed-down find-midpoint signal whose origin is not u reaches u from that direction, and the other directions are added, because marked reflected find-midpoint signals with origin u reach it from those directions; for all other vertices, the latter is the reason the directions are added.Let v_1 and v_2 be the neighbours ofon p̂ such that (v_1, v̂_1) < (v_2, v̂_1) (or, equivalently, (v_1, v̂_2) > (v_2, v̂_2)). First, let (v̂, v̂_1) = (v̂, v̂_2) (see <ref>). Then, v̂ =, and, for each index i ∈1, 2, we have v_i ≠ (because, iflies on p̂, then = v̂ = ≠ v_i, and otherwise,does not lie on p̂ but v_i does), and, v_i is either a leaf or an element of _𝔤 (because (v_i) consists of the tree of (v_i) that contains v̂_j, where j ∈1, 2∖i, and this tree, namely _v_i, contains ). Hence, according to <ref> or <ref>, for each index i ∈1, 2, at time (, v_i) + 2 ·(v_i, v̂_i), a marked reflected find-midpoint signal with originand reflection vertex v̂_i is sent from v_i towards , and at time (, ) + 2 ·(, v̂_i) = r + d/2, it reaches , where it collides with the other signal. Secondly, let (v̂, v̂_1) ≠(v̂, v̂_2) (see <ref>). Furthermore, let X_ be the union of the set of leaves and the set V_. Then, v̂≠, and either v_1 == v̂∈ X_ or v_1 ∈ U_, and v_2 ∈ X_ (because otherwise, there would be a direction-preserving path with more weight than p̂, which contradicts that p̂ has maximum-weight).According to <ref> or <ref>, at time (, v_2) + 2 ·(v_2, v̂_2), a marked reflected find-midpoint signal with origin v̂ and reflection vertex v̂_2 is sent from v_2 towards , and at time (, v_2) + 2 ·(v_2, v̂_2) + (v_2, ) = r + d/2 it reaches .If v_1 == v̂∈ X_, then, according to <ref> or <ref>, at time 2 ·(v̂, v̂_1) (which is 0 in the case that v_1 is a leaf, because in this case v_1 = v̂ = v̂_1), a marked slowed-down find-midpoint signal with origin v̂ and reflection vertex v̂_1 is sent from v̂ towards , and at time 2 ·(v̂, v̂_1) + (v̂, ) = r + d/2, it reaches .Otherwise, we have v̂∈ W_, v_1∖v_1 (because, for each vertex v ∈_v̂, v_1∖v̂, the set (v) consists of the tree of (v) that contains v̂_2, the tree _v^ contains v̂_1, and due to the maximality of p̂, except for _v, there can be no tree of (v) with a greater radius than _v^), and hence, according to <ref>, at time (, v_1) + 2 ·(v_1, v̂_1), a marked slowed-down find-midpoint signal with origin v̂ and reflection vertex v̂_1 is sent from v_1 towards , and at time (, v_1) + 2 ·(v_1, v̂_1) + (v_1, ) = r + d/2, it reaches .In either case, at time r + d/2, at the midpoint , a marked slowed-down find-midpoint signal with origin v̂ and reflection vertex v̂_1 collides with a marked reflected find-midpoint signal with origin v̂ and reflection vertex v̂_2. On the other hand, for each direction-preserving path that does not have maximum-weight, whenever its midpoint is found, one of the signals is unmarked and hence the midpoint is not falsely thought to be one of a maximum-weight path. The interested reader may prove that for herself. §.§ Thaw Signals Traverse Midpoints and Thaw Synchronisation of Edges Just in Time The inverse of a path is its traversal from target to source as given in Let p = (v_0, e_1, v_1, …, e_n, v_n) be a path in . The path (p) = (v_n, e_n, …, v_1, e_1, v_0) is called inverse of pinverse (p) of p[symbols]invp@(p). The inverse of an empty path is the empty path itself.The weight and the midpoint of the inverse of a path is the same as the weight and the midpoint of the path itself. An undirected path does not know which of its ends is its source and which is its target and it can be represented as in Let ↔equivalence relation ↔ on [symbols]arrowleftright@↔ be the equivalence relation ongiven by p ↔(p). Each equivalence class p_↔∈↔ is called undirected path p_↔undirected pathpath!undirected and the non-negative real number (p_↔) = (p) is called weight of p_↔weight (p_↔) of p_↔[symbols]omegapequivalenceclass@(p_↔).The equivalence class of an empty path is the singleton set that consists of the empty path. For each path p, the set(s) of paths with the same source (or target), less weight, and midpoints on the continuum representation of p are named in For each direction-preserving path p ∈_, letΣ_p = p' ∈_(p') = (p) , (p') < (p) , and _p'∈pset Σ_p[symbols]Sigmap@Σ_pand letT_p = p' ∈_(p') = (p) , (p') < (p) , and _p'∈p. set T_p[symbols]Tp@T_p Note that T_p = (Σ_(p)).Let p be a direction-preserving path of . For each path p' ∈Σ_p ∪ T_p, we have (_p, _p') = (p)/2 - (p')/2. The set of undirected direction-preserving paths can be turned into a graph by having an edge from an undirected direction-preserving path to each such path that has one end in common with the path, has less weight than the path, and has the greatest weight among the paths with the former two properties. The edges can be weighted by the distance of the midpoints of the undirected paths. The graph and its edge-weighting are introduced in Let _ = _↔, let_ = (p_↔, p'_↔) ∈_×_ p' ∈_p”∈Σ_p ∪ T_p(p”), and let__ →_≥ 0,(p_↔, p'_↔)↦(_p, _p').The triple _ = _, _, _edge-weighted directed acyclic graph _[symbols]GTcalligraphictypewriter@_ is an edge-weighted directed acyclic graph.For each maximum-weight direction-preserving path p̂ in , the in-degree of p̂_↔ in _ is 0, because there are only edges to equivalence classes of less-weight paths. For each edge (p_↔, p'_↔) of _, we have _(p_↔,p'_↔) = (p)/2 - (p')/2.There is a bijection between the vertices of _ and the set of all midpoint signals that are created by the signal machine 𝒮. The reason is that each midpoint signal memorises two words of directions in a set, one that leads from its position to one end of the path it designates the midpoint of and the other to the other end; because these words are stored in a set, the midpoint signal does not differentiate between source and target of its path.Each maximum-weight vertex of _ is a maximum-weight undirected direction-preserving path inand is, under suitable identifications and definitions, a longest undirected direction-preserving path in the continuum representation M of . And, each minimum-weight vertex of _ is one of weight 0, is an undirected empty path in , and is, under suitable identifications, an empty path inand in M, and a vertex inand in M.For each path p_ in _ that starts in a maximum-weight vertex and ends in a minimum-weight vertex, in the time evolution of the signal machine 𝒮, there is a thaw signal that traverses the midpoint signals represented by the vertices on the path in the order they occur on the path such that the time the thaw signal takes to get from the vertex v_ on the path to the next vertex v_' on the path is precisely the weight of the edge (v_, v_'), in particular, the time the thaw signal takes to reach the end of its path is the weight of p_.To show that the synchronisation of all edges is thawed and finishes at the same time, we have to show that, for each edge, there is a thaw signal that collides with the midpoint signal of the edge and reaches the end of its path at one end of the edge, and that all thaw signals reach the ends of their paths at the same time. The former is equivalent to showing that, for each edge e ∈ with the two ends v_0 and v_1, there is a maximum-weight direction-preserving path p insuch that there is a path in _ from p_↔ to (v_0, e, v_1)_↔. And the latter is equivalent to showing that the weights of all paths from maximum-weight vertices to minimum-weight vertices in _ are the same. See <ref>.For each non-empty direction-preserving path p in , the midpoint of p is found at time max(, (p)),(, (p)) + (p)/2 (see <ref>) and, for each maximum-weight direction-preserving path p̂ in , the midpoint of p̂ is found at time r + d/2 (see <ref>). If the thaw signals that emanate from the midpoints of maximum-weight direction-preserving paths at the time r + d/2 reach the ends of their paths after the time span d/2, then synchronisation finishes at the optimal time r + d. The condition is equivalent to showing that the weights of all paths from maximum-weight vertices to minimum-weight vertices in _ are equal to d/2. See <ref>. The thaw signals that spread from the midpoint signal of a path p eventually collide with the midpoint signals of all paths that have the same source or target as p, less weight, and whose midpoints lie on p. This is what is essentially shown inLet p be a path inand let p' ∈Σ_p ∪ T_p. There is a path from p_↔ to p'_↔ in _.Case 1: p' ∈Σ_p. If p' ∈_p”∈Σ_p(p”), then (p_↔,p'_↔) ∈_ and the path consisting of this edge is one from p_↔ to p'_↔. Otherwise, there is a q ∈_p”∈Σ_p(p”) such that (p') < (q). Then, (p_↔, q_↔) ∈_. And, because p', q⊆Σ_p, we have (p') = (p) = (q). Thus, because q((q) / 2) = _q ∈p, the paths q and p are direction-preserving, and the multigraphis a tree, we have q_0, (q) / 2 = p_0, (q) / 2 and, analogously, we have p'_0, (p') / 2 = p_0, (p') / 2. Hence, because (p') < (q),_p' = p'((p') / 2) = p((p') / 2) = q((p') / 2) ∈q.Therefore, p' ∈Σ_q. Now, if p' ∈_p”∈Σ_q(p”), then (q_↔,p'_↔) ∈_, and the path consisting of the edges (p_↔, q_↔) and (q_↔, p'_↔) is a path from p_↔ to p'_↔. Otherwise, because _ is finite and (q) > (p'), it follows by induction that there is a path from q_↔ to p'_↔ and therefore one from p_↔ to p'_↔. Case 2: p' ∈ T_p. Then, (p') ∈Σ_(p). Hence, according to the first case, there is a path from (p)_↔ to (p')_↔. Therefore, because p_↔ = (p)_↔ and p'_↔ = (p')_↔, there is a path from p_↔ to p'_↔.Each midpoint signal of a path eventually collides with a matching thaw signal that originated at the midpoint of a maximum-weight direction-preserving path. This is what is essentially shown inLet p be a direction-preserving path in . There is a maximum-weight direction-preserving path p̂ insuch that there is a path from p̂_↔ to p_↔ in _.If p is a maximum-weight path, then the path p̂ = p inand the empty path (p_↔) in _ have the desired properties. From now on, let p not be a maximum-weight path. Furthermore, let p̂ be a maximum-weight path inand let p_d be the minimum-weight path insuch that (p_d) lies on p and (p_d) lies on p̂. Then, there are paths p_1 and p_2 insuch that p_1p_2 = p and (p_1) = (p_d) as well as ((p_2)) = (p_d). And, there are paths p̂_1 and p̂_2 insuch that p̂_1 p̂_2 = p̂ and (p_d) = (p̂_2) as well as (p_d) = ((p̂_1)). Letq_1 = p_1,if (p_1) ≥(p_2), (p_2),otherwise,letq_2 = p_2,if (p_1) ≥(p_2), (p_1),otherwise,and let q = q_1q_2. Furthermore, letq̂_1 = p̂_1,if (p̂_2) ≥(p̂_1), (p̂_2),otherwise,letq̂_2 = p̂_2,if (p̂_2) ≥(p̂_1), (p̂_1),otherwise,and let q̂ = q̂_1 q̂_2. Moreover, let p' = q_1p_d q̂_2. Then, q ∈p, (p) as well as q̂∈p̂, (p̂). And, (q_1) ≥(q_2) as well as (q̂_1) ≤(q̂_2). And, (p') = (q̂) as well as (q) = (p'). And, because (q_1) ≥(q_2), we have _q ∈q_1⊆p'. Case 1: p' is a maximum-weight path. Then, because p is not a maximum-weight path, we have (q) = (p) < (p'). Hence, because (q) = (p') and _q ∈p', we have q ∈Σ_p'. In conclusion, according to <ref>, there is a path from p'_↔ to p_↔ = q_↔.Case 2: p' is not a maximum-weight path. Then, (q_1p_d q̂_2) = (p') < (q̂) = (q̂_1 q̂_2) and thus (q_1p_d) < (q̂_1), in particular, (q_1) < (q̂_1). Hence, because (q_2) ≤(q_1) < (q̂_1) ≤(q̂_2), we have (q) = (q_1) + (q_2) < (q_1) + (q̂_2) ≤(p'). And, because (q_1q_d) < (q̂_1) ≤(q̂_2), we have _p'∈q̂_2⊆q̂. And, recall that _q∈p'. Altogether, because (p') = (q̂) and (q) = (p'), we have p' ∈ T_q̂ and q ∈Σ_p'. Therefore, according to <ref>, there is a path from p̂_↔ = q̂_↔ to p'_↔ and there is a path from p'_↔ to q_↔ = p_↔. In conclusion, there is a path from p̂_↔ to p_↔.The time it takes a thaw signal from the midpoint of a path to collide with the midpoint signal of a matching path is given by half the former path's length minus half the latter path's length. This is what is essentially shown inFor each path p_ in _, the weight of p_ is equal to ((p_))/2 - ((p_))/2. We prove the statement by induction. Base Case Each empty path p_ in _ has weight 0, has the same source and target vertices, and ((p_))/2 - ((p_))/2 is equal to 0 as needed.Inductive Step Let p_ = (p_0_↔, p_1_↔, …, p_n_↔) be a non-empty path in _ such that the weight of the subpath (p_1_↔, …, p_n_↔) is equal to (p_1_↔)/2 - (p_n_↔)/2. Then, according to <ref>, we have _(p_0_↔, p_1_↔) = (_p_0, _p_1) = (p_0)/2 - (p_1)/2 = (p_0_↔)/2 - (p_1_↔)/2. Hence, the weight of the path p_ is equal to (p_0_↔)/2 - (p_1_↔)/2 + (p_1_↔)/2 - (p_n_↔)/2 = (p_0_↔)/2 - (p_n_↔)/2. The time it takes a thaw signal from a maximum-weight direction-preserving path to collide with the midpoint signal of a matching path is essentially given inFor each maximum-weight direction-preserving path p̂ insuch that there is a path from p̂_↔ to p_↔ in _, the weight of this path is d/2 - (p)/2.This is a direct consequence of <ref> with the fact that (p̂) = d. The time it takes a thaw signal from a maximum-weight direction-preserving path to collide with the midpoint signal of an edge it thaws and to reach the ends of the edge is essentially given inLet e be an edge of , and let v_0 and v_1 be the two ends of e. There is a maximum-weight direction-preserving path p̂ such that there is a path from p̂_↔ to (v_0, e, v_1)_↔ and all such paths have weight d / 2 - (e) / 2, and there are also paths from p̂_↔ to (v_0)_↔ and to (v_1)_↔ and all such paths have weight d/2.This is a direct consequence of <ref> and corollary <ref>. CHAPTER: ZORN'S LEMMA Let I be a set and let ≤ be a binary relation on I. The relation ≤ is called preorder on Ipreorder ≤ on I[symbols]lessthanorequalto@≤ and the tuple (I, ≤) is called preordered setpreordered set (I, ≤) if and only if the relation ≤ is reflexive and transitive.Let ≤ be a preorder on I. It is called partial order ≤ on Ipartial order on I and the preordered set (I, ≤) is called partially ordered setpartially ordered set (I, ≤) if and only if the relation ≤ is antisymmetric. Let ≤ be a partial order on I. It is called total order ≤ on Itotal order on I and the partially ordered set (I, ≤) is called totally ordered set (I, ≤)totally ordered set if and only if the relation ≤ is total.Let ≤ be a preorder on I and let i be an element of I. The element i is called maximal in (I, ≤)maximal in (I, ≤) if and only ifi' ∈ I(i' ≥ ii' ≤ i). Let ≤ be a preorder on I and let J be a subset of I. The set J is called chain in (I, ≤)chain in (I, ≤) if and only if the restriction of ≤ to J is a total order on J.Let (I, ≤) be a preordered set such that each chain in (I, ≤) has an upper bound. Then, there is a maximal element in (I, ≤).See section 16 in <cit.> for the proof of the case that ≤ is a partial order. The general case follows from this case, because each preorder ≤ on I induces a partial order on I ∼, where ∼ is the equivalence relation on I given by i ∼ i' if and only if i ≤ i' and i' ≤ i.CHAPTER: TOPOLOGIES Most of the theory of topologies as presented here may be found in more detail in Appendix A in the monograph *ceccherini-silberstein:coornaert:2010<cit.>. § TOPOLOGIES Let X be a set and let 𝒯 be a set of subsets of X. The set 𝒯 is called topology on Xtopology 𝒯 on X[symbols]Tcalligraphic@𝒯 if and only if * ∅, X is a subset of 𝒯,* for each family O_i_i ∈ I of elements in 𝒯, the union ⋃_i ∈ I O_i is an element of 𝒯,* for each finite family O_i_i ∈ I of elements in 𝒯, the intersection ⋂_i ∈ I O_i is an element of 𝒯. Let X be a set, and let 𝒯 and 𝒯' be two topologies on X. The topology 𝒯 is called * coarser than 𝒯'coarser than 𝒯' if and only if 𝒯⊆𝒯';* finer than 𝒯'finer than 𝒯' if and only if 𝒯⊇𝒯'. Let X be a set and let 𝒯 be a topology on X. The tuple (X, 𝒯) is called topological spacetopological space (X, 𝒯), each subset O of X with O ∈𝒯 is called open in Xopen set O in X, each subset A of X with X ∖ A ∈𝒯 is called closed in Xclosed set A in X, and each subset U of X that is both open and closed is called clopen in Xclopen set U in X.The set X is said to be equipped with 𝒯equipped with 𝒯 if and only if it shall be implicitly clear that 𝒯 is the topology on X being considered. The set X is called topological spacetopological space X if and only if it is implicitly clear what topology on X is being considered. Let X be a set. The set (X) is the finest topology on X. Itself as well as the topological space (X, (X)) are called discretediscrete!topologydiscrete topology.Let X be a set, let (X', 𝒯') be a topological space, and let f be a map from X to X'. The set𝒯 = f^-1(O')O' ∈𝒯'is a topology on X and called induced on X by finduced on X by f!topologyinduced topology on X by f. Let (X, 𝒯) be a topological space, let Y be a subset of X, and let ι be the canonical injection from Y to X. The topology 𝒮 on Y induced by ι is called subspacesubspace!topologysubspace topology and the topological space (Y, 𝒮) is called subspace of (X, 𝒯)subspace of (X, 𝒯).Let (X, 𝒯) be a topological space and let Y be a subset of X. The subspace topology on Y is O ∩ YO ∈𝒯.Let (X, 𝒯) be a topological space and let ℬ be a subset of 𝒯. * The set ℬ is called base of 𝒯base ℬ of 𝒯[symbols]Bcalligraphic@ℬ if and only ifO ∈𝒯B_i_i ∈ I in ℬ⋃_i ∈ I B_i = O. * The set ℬ is called subbase of 𝒯subbase ℬ of 𝒯base of 𝒯!sub-[symbols]Bcalligraphic@ℬ if and only if⋂_i = 1^n B_iB_i ∈ℬ, i ∈1,2,…,n, n ∈_+is a base of 𝒯.Let (X, 𝒯) be a topological space, let x be a point of X, and let N be a subset of X. The set N is called neighbourhood N of xneighbourhood of x if and only if there is an open subset O of X such that x ∈ O and O ⊆ N.Let (X, 𝒯) be a topological space, let x be a point of X, and let ℬ_x be a set of neighbourhoods of x. The set ℬ_x is called neighbourhood base of xneighbourhood base ℬ_x of x[symbols]Bxcalligraphic@ℬ_x if and only if, for each neighbourhood N of x, there is a neighbourhood B_x ∈ℬ_x such that B_x ⊆ N.Let (X, 𝒯) be a topological space and let x be a point of X. The set of open neighbourhoods of x is denoted by 𝒯_xset 𝒯_x of open neighbourhoods of x[symbols]Txcalligraphic@𝒯_x.§ NETS Let ≤ be a preorder on I. It is called directeddirected preorder on Idirected preorder ≤ on Ipreorder on I!directed[symbols]lessthanorequalto@≤ and the preordered set (I, ≤) is called directed setdirected set (I, ≤) if and only ifi ∈ Ii' ∈ Ii”∈ Ii ≤ i” i' ≤ i”. Let ≤ be a preorder on I, let J be a subset of I, and let i be an element of I. The element i is called upper bound of J in (I, ≤)upper bound of J in (I, ≤) if and only ifi' ∈ Ji' ≤ i. In the definition of directed sets, the element i” is an upper bound of i, i' in (I, ≤).Let I be a set, let ≤ be a binary relation on I, and let m_i_i ∈ I be a family of elements in M indexed by I. The family m_i_i ∈ I is called net in M indexed by (I, ≤)net m_i_i ∈ I in M indexed by (I, ≤)[symbols]miiinI@m_i_i ∈ I if and only if the tuple (I, ≤) is a directed set. Let m_i_i ∈ I and m_j'_j ∈ J be two nets in M. The net m_j'_j ∈ J is called subnet of m_i_i ∈ Isubnet m_j'_j ∈ J of m_i_i ∈ I if and only if there is a map fJ → I such that m_j'_j ∈ J = m_f(j)_j ∈ J andi ∈ Ij ∈ Jj' ∈ J []j' ≥ jf(j') ≥ i. Let (X, 𝒯) be a topological space, let x_i_i ∈ I be a net in X indexed by (I, ≤), and let x be a point of X. The net x_i_i ∈ I is said to converge to xx_i_i ∈ I converges to x and x is called limit point of x_i_i ∈ Ilimit point x of x_i_i ∈ I if and only ifO ∈𝒯_xi_0 ∈ Ii ∈ I(i ≥ i_0x_i ∈ O). Let (X, 𝒯) be a topological space and let x_i_i ∈ I be a net in X indexed by (I, ≤). The net x_i_i ∈ I is called convergentconvergent net if and only if there is a point x ∈ X such that it converges to x.Let m_i_i ∈ I be a net that converges to x. Each subnet m_j'_j ∈ J of m_i_i ∈ I converges to x.Let (X, 𝒯) be a topological space and let Y be a subset of X. The setY = ⋂_A ⊆ XclosedY ⊆ A A closure Y of Y in X[symbols]Ybar@Yis called closure of Y in X. Let (X, 𝒯) be a topological space, let Y be a subset of X, and let x be an element of X. Then, x ∈Y if and only if there is a net y_i_i ∈ I in Y that converges to x.See proposition A.2.1 in <cit.>. Let (X, 𝒯) be a topological space. It is HausdorffHausdorff if and only if x ∈ Xx' ∈ X ∖x O ∈𝒯_xO' ∈𝒯_x' O ∩ O' = ∅.Let (X, 𝒯) be a topological space. It is Hausdorff if and only if each convergent net in X has exactly one limit point. See proposition A.2.2 in <cit.>.Let (X, 𝒯) be a Hausdorff topological space, let x_i_i ∈ I be a convergent net in X indexed by (I, ≤), and let x be the limit point of x_i_i ∈ I. The point x is denoted by lim_i ∈ I x_ithe limit point lim_i ∈ I x_i of x_i_i ∈ I[symbols]limiinIxi@lim_i ∈ I x_i and we write x_i i ∈ I→ xx_i i ∈ I→ x[symbols]arrow right limit@x_i i ∈ I→ x.Let (X, 𝒯) be a topological space, let x_i_i ∈ I be a net in X indexed by (I, ≤), and let x be an element of X. The point x is called cluster point of x_i_i ∈ Icluster point x of x_i_i ∈ I if and only ifO ∈𝒯_xi ∈ Ii' ∈ I(i' ≥ ix_i'∈ O).Let (X, 𝒯) be a topological space, let x_i_i ∈ I be a net in X indexed by (I, ≤), and let x be an element of X. The point x is a cluster point of x_i_i ∈ I if and only if there is a subnet of x_i_i ∈ I that converges to x.See proposition A.2.3 in <cit.>.Let (X, 𝒯) and (X', 𝒯') be two topological spaces and let f be a map from X to X'. The map f is called continuouscontinuous map if and only ifO' ∈𝒯'f^-1(O') ∈𝒯. Let (X, 𝒯) and (X', 𝒯') be two topological spaces, let f be a continuous map from X to X', let x_i_i ∈ I be a net in X, and let x be an element of X. * If x is a limit point of x_i_i ∈ I, then f(x) is a limit point of f(x_i)_i ∈ I. * If x is a cluster point of x_i_i ∈ I, then f(x) is a cluster point of f(x_i)_i ∈ I. See the last paragraph of section A.2 in <cit.>. Let (X, 𝒯) and (X', 𝒯') be two topological spaces, and let f be a map from X to X'. The map f is continuous if and only if, for each point x ∈ X and each open neighbourhood N' of f(x), the preimage f^-1(N') is an open neighbourhood of x.See proposition 2.4.2 of section 2.4 in <cit.>. Let = ∪-∞,+∞ be the affinely extended real numbers and let r_i_i ∈ I be a net inindexed by (I, ≤).* The limit of the net inf_i' ≥ i r_i'_i ∈ I is called limit inferior lim inf_i ∈ I r_i of r_i_i ∈ Ilimit inferior of r_i_i ∈ I and denoted by lim inf_i ∈ I r_i[symbols]liminfiinIri@lim inf_i ∈ I r_i.* The limit of the net sup_i' ≥ i r_i'_i ∈ I is called limit superior lim sup_i ∈ I r_i of r_i_i ∈ Ilimit superior of r_i_i ∈ I and denoted by lim sup_i ∈ I r_i[symbols]limsupiinIri@lim sup_i ∈ I r_i. § INITIAL AND PRODUCT TOPOLOGIES Let X be a set, let I be a set, and, for each index i ∈ I, let (Y_i, 𝒯_i) be a topological space and let f_i be a map from X to Y_i. The coarsest topology on X such that, for each index i ∈ I, the map f_i is continuous, is called initial with respect to f_i_i ∈ Iinitial!topologyinitial topology with respect to f_i_i ∈ I.The initial topology is explicitly constructed in section A.3 in <cit.>. Let (X, 𝒯) be a topological space, where 𝒯 is the initial topology with respect to f_iX → Y_i_i ∈ I, let (Z, 𝒮) be a topological space, and let g be a map from Z to X. The map g is continuous if and only if, for each index i ∈ I, the map f_ig is continuous.See the last paragraph of section A.3 in <cit.>.Let (X, 𝒯) be a topological space, where 𝒯 is the initial topology with respect to f_iX → Y_i_i ∈ I, let x_i'_i' ∈ I' be a net in X, and let x be a point in X. The point x is a limit point or cluster point of x_i'_i' ∈ I' if and only if, for each index i ∈ I, the point f_i(x) is a limit point or cluster point of f_i(x_i')_i' ∈ I'.See the last paragraph of section A.3 in <cit.>.Let (X_i, 𝒯_i)_i ∈ I be a family of topological spaces, let X be the set ∏_i ∈ I X_i, and, for each index i ∈ I, let π_i be the canonical projection of X onto X_i. The initial topology on X with respect to π_i_i ∈ I is called productproduct!topologyproduct topology and is also known as topology of pointwise convergencetopology of pointwise convergencetopology!of pointwise convergence.The product topology on X has for a base the sets ∏_i ∈ I O_i, where, for each index i ∈ I, the set O_i is an open subset of X_i, and the set i ∈ IO_i ≠ X_i is finite.Let (X_i, 𝒯_i)_i ∈ I be a family of discrete topological spaces and let X be the set ∏_i ∈ I X_i. The product topology on X is called prodiscreteprodiscrete!topologyprodiscrete topology.Let (X_i, 𝒯_i)_i ∈ I be a family of Hausdorff topological spaces. The set ∏_i ∈ I X_i, equipped with the product topology, is Hausdorff.See proposition A.4.1 in <cit.>.Let (X, 𝒯) be a topological space and let D be a subset of X. The set D is called connectedconnected subset if and only ifO ∈𝒯 O' ∈𝒯[t] ( O ∩ D ≠∅ O' ∩ D ≠∅ O ∩ O' ∩ D = ∅ O ∪ O' ⊉ D). Let (X, 𝒯) be a topological space. It is called totally disconnectedtotally disconnected topological space if and only if, for each non-empty and connected subset D of X, we have D = 1. Let (X_i, 𝒯_i)_i ∈ I be a family of totally disconnected topological spaces. The set ∏_i ∈ I X_i, equipped with the product topology, is totally disconnected.See proposition A.4.2 in <cit.>.Let (X_i, 𝒯_i)_i ∈ I be a family of topological spaces and, for each index i ∈ I, let A_i be a closed subset of X_i. The set ∏_i ∈ I A_i is a closed subset of ∏_i ∈ I X_i, equipped with the product topology.See proposition A.4.3 in <cit.>.§ COMPACTNESSLet (X, 𝒯) be a topological space and let O_i_i ∈ I be a family of elements of 𝒯. The family O_i_i ∈ I is called open cover O_i_i ∈ I of Xopen cover of X if and only if ⋃_i ∈ I O_i = X.Let (X, 𝒯) be a topological space. It is called compactcompact topological space if and only if, for each open cover O_i_i ∈ I of X, there is a finite subset J of I such that O_j_j ∈ J is an open cover of X.Let M be a set and let D_i_i ∈ I be a family of subsets of M. The family D_i_i ∈ I is said to have the finite intersection property of D_i_i ∈ Ifinite intersection property if and only if, for each finite subset J of I, we have ⋂_j ∈ J D_j ≠∅.Let (X, 𝒯) be a topological space. It is compact if and only if, for each family A_i_i ∈ I of closed subsets of X that has the finite intersection property, we have ⋂_i ∈ I A_i ≠∅.See the first paragraph of section A.5 in <cit.>. Let (X, 𝒯) be a topological space. The following three statements are equivalent: * The space (X, 𝒯) is compact.* Each net in X has a cluster point with respect to 𝒯.* Each net in X has a convergent subnet with respect to 𝒯. See theorem A.5.1 in <cit.>.Let (X_i, 𝒯_i)_i ∈ I be a family of compact topological spaces. The set ∏_i ∈ I X_i, equipped with the product topology, is compact.See theorem A.5.2 in <cit.>.Let (X_i, 𝒯_i)_i ∈ I be a family of finite topological spaces. The set ∏_i ∈ I X_i, equipped with the product topology, is compact.See the paragraph before corollary A.5.3 in <cit.>. CHAPTER: UNIFORMITIES Most of the theory of uniformities as presented here may be found in more detail in Appendix B in the monograph *ceccherini-silberstein:coornaert:2010<cit.>. Let X be a set. The set Δ_X = (x,x)x ∈ X is called diagonal in X × Xdiagonal Δ_X in X × X[symbols]DeltaX@Δ_X.Let X be a set, let R be a subset of X × X, and let y be an element of X. The set R[y] = x ∈ X(x,y) ∈ R is called y-th column in Rcolumn in R@y-th column in Ry-th column R[y] in R[symbols]Rybrackets@R[y].Let X be a set and let R be a subset of X × X. The set R^-1 = (y, x)(x, y) ∈ R is called inverse of Rinverse R^-1 of R[symbols]Rminus1@R^-1. The set R is called symmetricsymmetric!binary relationsymmetric subset R of X × X if and only if R^-1 = R. Let X be a set, and let R and R' be two subsets of X × X. The set R'R = (x, z)y ∈ X(x, y) ∈ R ∧ (y, z) ∈ R' is called composition of R and R'composition R'R of R and R'[symbols]RcircleRprime@R'R.Let X be a set and let 𝒰 be a set of subsets of X × X. The set 𝒰 is called uniformity on Xuniformity 𝒰 on X[symbols]Ucalligraphic@𝒰 if and only if 𝒰≠∅,E ∈𝒰Δ_X ⊆ E,E ∈𝒰 E' ⊆ X × X(E ⊆ E'E' ∈𝒰),E ∈𝒰 E' ∈𝒰 E ∩ E' ∈𝒰,E ∈𝒰 E^-1∈𝒰,E ∈𝒰 E' ∈𝒰 E'E' ⊆ E.Let X be a set and let 𝒰 be a uniformity on X. The tuple (X, 𝒰) is called uniform spaceuniform space (X, 𝒰) and each element E ∈𝒰 is called entourage of Xentourage E of X.The set X is said to be equipped with 𝒰equipped with 𝒰 if and only if it shall be implicitly clear that 𝒰 is the uniformity on X being considered. The set X is called uniform spaceuniform space X if and only if it is implicitly clear what uniformity on X is being considered.Let X be a set and let 𝒰 be the set of all subsets of X × X that contain Δ_X. Then, 𝒰 is the finest uniformity on X. Itself as well as the uniform space (X, 𝒰) are called discretediscrete!uniformitydiscrete uniformity.Let (X, 𝒰) be a uniform space. The setO ⊆ Xx ∈ OE ∈𝒰 E[x] = Ois a topology on X and called induced by 𝒰topology on X induced by 𝒰.Let (X, 𝒰) be a discrete uniform space. The topology induced by 𝒰 is the discrete topology on X. However, there are non-discrete uniformities whose induced topologies are discrete.Let (X, 𝒰) and (X', 𝒰') be two uniform spaces, let 𝒯 and 𝒯' be the respective topologies on X and X' induced by 𝒰 and 𝒰', and let f be a map from X to X'. The map f is continuous if and only if, for each point x ∈ X and each entourage E' ∈𝒰', there is an entourage E ∈𝒰 such that f(E[x]) ⊆ E'[f(x)].Let (X, 𝒰) be a uniform space and let ℬ be a subset of 𝒰. * The set ℬ is called base of 𝒰base ℬ of 𝒰[symbols]Bcalligraphic@ℬ if and only ifE ∈𝒰 B ∈ℬ B ⊆ E. * The set ℬ is called subbase of 𝒰subbase ℬ of 𝒰base of 𝒰!sub-[symbols]Bcalligraphic@ℬ if and only if the set⋂_i = 1^n B_iB_i ∈ℬ, i ∈1,2,…,n, n ∈_+is a base of 𝒰. Let X be a set and let ℬ be a set of subsets of X × X. * The set ℬ is a base of a uniformity on X if and only ifℬ≠∅,B ∈ℬΔ_X ⊆ B,B ∈ℬ B' ∈ℬ B”∈ℬ B”⊆ B ∩ B',B ∈ℬ B' ∈ℬ B' ⊆ B^-1,B ∈ℬ B' ∈ℬ B'B' ⊆ B. * The set ℬ is a subbase of a uniformity on X if and only ifℬ≠∅,B ∈ℬΔ_X ⊆ B,B ∈ℬ B' ∈ℬ B' ⊆ B^-1,B ∈ℬ B' ∈ℬ B'B' ⊆ B.Let X be a set and let ℬ be a base or subbase of a uniformity on X. The uniformity 𝒰 that has ℬ for a base or subbase respectively is uniquely determined and called generated by ℬuniformity generated by ℬ. Let I be a set and, for each index i ∈ I, let f_i be a map from X_i to X_i'. The map[symbols]productfiiinI@∏_i ∈ I f_i∏_i ∈ I f_i ∏_i ∈ I X_i→∏_i ∈ I X_i', product ∏_i ∈ I f_i of f_i_i ∈ I x_i_i ∈ I ↦f_i(x_i)_i ∈ I,is called product of f_i_i ∈ I and, if I is the set 1, 2, …, I, then it is also denoted by f_1 × f_2 ×…× f_If_1 × f_2 ×…× f_I[symbols]crossf1f2@f_1 × f_2 ×…× f_I.Let X be a set, let (X', 𝒰') be a uniform space, and let f be a map from X to X'. The setℬ = (f × f)^-1(E')E' ∈𝒰'is a base of a uniformity on X and the uniformity on X it generates is called induced on X by finduced on X by f!uniformityuniformity on X induced by f. Let (X, 𝒰) be a uniform space, let Y be a subset of X, and let ι be the canonical injection from Y to X. The uniformity 𝒱 on Y induced by ι is called subspacesubspace!uniformitysubspace uniformity and the uniform space (Y, 𝒱) is called subspace of (X, 𝒰)subspace of (X, 𝒰).Let (X, 𝒰) be a uniform space and let Y be a subset of X. The subspace uniformity on Y is E ∩ (Y × Y)E ∈𝒰.Let (X, 𝒰) and (X', 𝒰') be two uniform spaces and let f be a map from X to X'. The map f is called uniformly continuous mapuniformly continuous if and only ifE' ∈𝒰'E ∈𝒰 (f × f)(E) ⊆ E'. The map f is uniformly continuous if and only ifE' ∈𝒰'(f × f)^-1(E') ∈𝒰.Let (X, 𝒰) and (X', 𝒰') be two uniform spaces, let f be a map from X to X', and let ℬ' be a base or subbase of 𝒰'. The map f is uniformly continuous if and only ifB' ∈ℬ'(f × f)^-1(B') ∈𝒰. Compare the second paragraph of section B.2 in <cit.>.Let (X, 𝒰) and (X', 𝒰') be two uniform spaces, let f be a map from X to X', and let ℬ and ℬ' be two bases or subbases of 𝒰 and 𝒰' respectively. The map f is uniformly continuous if and only ifB' ∈ℬ'B ∈ℬ (f × f)(B) ⊆ B'. Compare the second paragraph of section B.2 in <cit.>. Let (X, 𝒰) and (X', 𝒰') be two uniform spaces and let f be a uniformly continuous map from X to X'. The map f is continuous, where X and X' are equipped with the topologies induced by 𝒰 and 𝒰' respectively.See proposition B.2.2 in <cit.>. Let (X, 𝒰) and (X', 𝒰') be two uniform spaces such that X, equipped with the topology induced by 𝒰, is compact. Each continuous map fX → X' is uniformly continuous.See theorem B.2.3 in <cit.>.Let (X, 𝒰) and (X', 𝒰') be two uniform spaces, and let f be a map from X to X'. The map f is called uniform isomorphismuniform isomorphism if and only if it is bijective, and both f and f^-1 are uniformly continuous.Let (X, 𝒰) and (X', 𝒰') be two uniform spaces, and let f be a map from X to X'. The map f is called uniform embeddinguniform embedding if and only if it is injective and f_X → f(X) is a uniform isomorphism.Let (X, 𝒰) and (X', 𝒰') be two uniform spaces such that X is compact and X' is Hausdorff. Furthermore, let f be a continuous and injective map from X to X'. The map f is a uniform embedding.See proposition B.2.5 in <cit.>. Let X be a set, let I be a set, and, for each index i ∈ I, let (X_i, 𝒰_i) be a uniform space and let f_i be a map from X to X_i. The coarsest uniformity on X such that, for each index i ∈ I, the map f_i is uniformly continuous, is called initial with respect to f_i_i ∈ Iinitial!uniformityinitial uniformity with respect to f_i_i ∈ I. Let (X_i, 𝒰_i)_i ∈ I be a family of uniform spaces, let X be the set ∏_i ∈ I X_i, and, for each index i ∈ I, let π_i be the canonical projection from X onto X_i. The initial uniformity on X with respect to π_i_i ∈ I is called productproduct uniformity.The product uniformity on X has for a base the sets ∏_i ∈ I E_i, where, for each index i ∈ I, the set E_i is an entourage of X_i, and the set i ∈ IE_i ≠ X_i × X_i is finite. Note that∏_i ∈ I E_i ⊆∏_i ∈ I X_i × X_i = (∏_i ∈ I X_i) × (∏_i ∈ I X_i) = X × X.Let (X_i, 𝒰_i)_i ∈ I be a family of discrete uniform spaces and let X be the set ∏_i ∈ I X_i. The product uniformity on X is called prodiscreteprodiscrete!uniformityprodiscrete uniformity. Let M and Q be two sets. The prodiscrete uniformity on Q^M = ∏_m ∈ M Q has for a base the sets(c,c') ∈ Q^M × Q^Mc_F = c'_F,forF ⊆ Mfinite. CHAPTER: DUAL SPACES Most of the theory of dual spaces as presented here may be found in more detail in Appendix F in the monograph *ceccherini-silberstein:coornaert:2010<cit.>. Let X be an -vector space and let X be equipped with a topology. The -vector space X is called topologicaltopological!vector spacetopological -vector space X if and only if the maps +X × X → X, (x, x') ↦ x + x', and ·× X → X, (r, x) ↦ r · x, are continuous, where X × X and × X are equipped with their respective product topology.Let X be an -vector space and let Y be a subset of X. The set Y is called convexconvex subset of X if and only if(y, y') ∈ Y × Yt ∈0, 1 t y + (1 - t) y' ∈ Y.Let X be a topological -vector space. It is called locally convexlocally convex topological -vector spaceconvex!locally if and only if the origin has a neighbourhood base of convex sets.In the remainder of this chapter, let (X, ) be a normed -vector space. The vector spaceX^* = ψ X →ψ is linear and continuoustopological dual space X^* of X[symbols]Xstar@X^*with pointwise addition and scalar multiplication is called topological dual space of X and each map ψ∈ X^* is called continuous linear functional ψcontinuous linear functional.The norm_X^* X^*→, operator norm _X^* on X^*[symbols]normXstar@_X^* ψ ↦sup_x ∈ X ∖0ψ(x)/x,is called operator norm on X^*. And, the topology on X^* induced by _X^* is called strong topology on X^*strong topology on X^*.Let x be an element of X. The map_xX^*→, evaluation map _x at x[symbols]evx@_x ψ ↦ψ(x),is called evaluation map at x. The initial topology on X^* with respect to _x_x ∈ X is called weak-* topology on X^*weak-* topology on X^*.Let ψ_i_i ∈ I be a net in X^*, let ψ be an element of X^*, and let X^* be equipped with the weak-* topology. The net ψ_i_i ∈ I converges to ψ if and only if, for each element x ∈ X, the net ψ_i(x)_i ∈ I converges to ψ(x).This is a direct consequence of <ref>.The weak-* topology on X^* is coarser than the strong topology on X^*.This holds because the evaluation maps are continuous with respect to the strong topology on X^*.Let ψ_i_i ∈ I be a net in X^* that converges to ψ with respect to the strong topology on X^*. The net ψ_i_i ∈ I converges to ψ with respect to the weak-* topology on X^*.This is a direct consequence of <ref>.Let X^* be equipped with the weak-* topology and let ψ be an element of X^*. An open neighbourhood base of ψ is given by the setsB(ψ, F, ε) = ψ' ∈ X^*x ∈ F ψ(x) - ψ'(x) < ε,forF ⊆ Xfinite and ε∈_> 0. B(ψ, F, ε), for ψ∈ X^*, F ⊆ X finite, and ε∈_> 0[symbols]BphiFepsilon@B(ψ, F, ε) Compare the third paragraph of section F.2 in <cit.>. Let X^* be equipped with the weak-* topology. The space X^* is locally convex.See the third paragraph of section F.2 in <cit.>. The space X^*, equipped with the weak-* topology, is Hausdorff.See the last paragraph of section F.2 in <cit.>. Let X^* be equipped with the weak-* topology. The unit ball ψ∈ X^* ψ_X^*≤ 1, equipped with the subspace topology, is compact.See theorem F.3.1 in <cit.>. CHAPTER: HALL'S THEOREMSMost of the theory concerning Hall's theorems as presented here may be found in more detail in Appendix H in the monograph *ceccherini-silberstein:coornaert:2010<cit.>. § BIPARTITE GRAPHS Let X and Y be two sets, and let E be a subset of X × Y. The triple (X, Y, E) is called bipartite graphbipartite graph (X, Y, E), each element x of X is called left vertexleft vertex xvertex!left, each element y of Y is called right vertexright vertex yvertex!right, and each element e of E is called edgeedge e.Let (X, Y, E) and (X', Y', E') be two bipartite graphs. The graph (X, Y, E) is called bipartite subgraph of (X', Y', E')bipartite subgraph (X, Y, E) of (X', Y', E') if and only if X ⊆ X', Y ⊆ Y', and E ⊆ E'.Let (X, Y, E) be a bipartite graph, and let (x, y) and (x', y') be two elements of E. The edges (x, y) and (x', y') are called adjacentadjacent edges if and only if x = x' or y = y'.Let (X, Y, E) be a bipartite graph. * Let x be an element of X. The set𝒩_r(x) = y ∈ Y(x, y) ∈ Eright neighbourhood 𝒩_r(x) of x[symbols]Nrxcalligraphic@𝒩_r(x)is called right neighbourhood of xneighbourhood!right.* Let A be a subset of X. The set 𝒩_r(A) = ⋃_a ∈ A𝒩_r(a) is called right neighbourhood of Aright neighbourhood 𝒩_r(A) of A[symbols]NrAcalligraphic@𝒩_r(A). * Let y be an element of Y. The set𝒩_l(y) = x ∈ X(x, y) ∈ Eleft neighbourhood 𝒩_l(y) of y[symbols]Nlycalligraphic@𝒩_l(y)is called left neighbourhood of yneighbourhood!left.* Let B be a subset of Y. The set 𝒩_l(B) = ⋃_b ∈ B𝒩_l(b) is called left neighbourhood of Bleft neighbourhood 𝒩_l(B) of B[symbols]NlBcalligraphic@𝒩_l(B). Let (X, Y, E) be a bipartite graph. It is called * finitefinite bipartite graphfinite bipartite graph if and only if the sets X and Y are finite.* locally finitelocally finite bipartite graphlocally finite bipartite graphfinite!locally if and only if, for each element x ∈ X, the set 𝒩_r(x) is finite, and for each element y ∈ Y, the set 𝒩_l(y) is finite. Let (X, Y, E) be a locally finite bipartite graph. Then, for each finite subset A of X, the set 𝒩_r(A) is finite; and, for each finite subset B of Y, the set 𝒩_l(B) is finite.§ MATCHINGS Let (X, Y, E) be a bipartite graph and let M be a subset of E. The set M is called matchingmatching M if and only if, for each tuple (e, e') ∈ M × M with e ≠ e', the edges e and e' are not adjacent.The set M is a matching if and only if the maps pM → X, (x, y) ↦ x, and qM → Y, (x, y) ↦ y, are injectiveLet (X, Y, E) be a bipartite graph and let M be a matching. The matching M is called * left-perfectleft-perfect matching if and only ifx ∈ Xy ∈ Y(x, y) ∈ M; * right-perfectright-perfect matching if and only ify ∈ Yx ∈ X(x, y) ∈ M; * perfectperfect matching if and only if it is left-perfect and right-perfect. The matching M is left-perfect if and only if the map pM → X, (x, y) ↦ x, is surjective (and hence bijective); and it is right-perfect if and only if the map qM → Y, (x, y) ↦ y, is surjective (and hence bijective). Let (X, Y, E) be a bipartite graph and let M be a subset of E. The set M is a * left-perfect matching if and only if there is an injective map φ X → Y such that M = (x, φ(x))x ∈ X;* right-perfect matching if and only if there is an injective map ψ Y → X such that M = (ψ(y), y)y ∈ Y;* perfect matching if and only if there is a bijective map φ X → Y such that M = (x, φ(x))x ∈ X. § HALL'S MARRIAGE THEOREM Let (X, Y, E) be a locally finite bipartite graph. It is said to satisfy the* left Hall conditionleft Hall condition if and only ifA ⊆ Xfinite𝒩_r(A)≥A; * right Hall conditionright Hall condition if and only ifB ⊆ Yfinite𝒩_l(B)≥B; * Hall marriage conditionsHall marriage conditions if and only if it satisfies the left and right Hall conditions. Let (X, Y, E) be a locally finite bipartite graph. It satisfies the left or right Hall condition if and only if there is a left- or right-perfect matching respectively.See theorem H.3.2 in <cit.>.Let (X, Y, E) be a bipartite graph such that there is a left-perfect matching and there is a right-perfect matching. Then, there is a perfect matching.See theorem H.3.4 in <cit.>.Let X and Y be two sets such that there is an injective map f from X to Y and there is an injective map g from Y to X. Then, there is a bijective map from X to Y.See corollary H.3.5 in <cit.>.Let (X, Y, E) be a locally finite bipartite graph. It satisfies the Hall marriage conditions if and only if there is a perfect matching.See theorem H.3.6 in <cit.>.§ HALL'S HAREM THEOREM Let X and Y be two sets, let f be a map from X to Y, and let k be a positive integer. The map f is called k-to-1 surjectivesurjectivekto1@k-to-1 surjectivek-to-1 surjective map if and only ify ∈ Y f^-1(y) = k. Let (X, Y, E) be a bipartite graph, let k be a positive integer, and let M be a subset of E. The set M is called perfect (1,k)-matchingperfect (1,k)-matching if and only ifx ∈ X y ∈ Y(x, y) ∈ E = k,andy ∈ Y x ∈ X(x, y) ∈ E = 1. The set M is a perfect (1,k)-matching if and only if there is a k-to-1 surjective map ψ Y → X such that (ψ(y), y)y ∈ Y = M.The set M is a perfect (1,1)-matching if and only if it is a perfect matching.Let (X, Y, E) be a locally finite bipartite graph and let k be a positive integer. The graph (X, Y, E) is said to satisfy the Hall k-harem conditionsHall k-harem conditions if and only if, for each finite subset A of X, we have 𝒩_r(A)≥ k A, and for each finite subset B of Y, we have 𝒩_l(B)≥ k^-1B. Let (X, Y, E) be a locally finite bipartite graph and let k be a positive integer. The graph (X, Y, E) satisfies the Hall k-harem conditions if and only if there is a perfect (1,k)-matching.See theorem H.4.2 in <cit.>.dummy toc tocchapterSymbolsSymbolsdummy toc tocchapterSymbols[symbols]dummy toc tocchapterIndex | http://arxiv.org/abs/1706.08429v1 | {
"authors": [
"Simon Wacker"
],
"categories": [
"math.GR",
"cs.FL"
],
"primary_category": "math.GR",
"published": "20170626151151",
"title": "Cellular Automata on Group Sets"
} |
Email: [email protected] Raymond and Beverly Sackler School of Physics and Astronomy,Tel Aviv University, Tel Aviv 69978, IsraelMassachusetts Institute of Technology, Department of Physics, Cambridge, Massachusetts 02139, USA64.60.F-82.35.Lr05.40.Fb We consider forces acting on objects immersed in, or attached to, long fluctuating polymers. The confinement of the polymer by the obstacles results in polymer-mediated forces that can be repulsive (due to loss of entropy) or attractive (if some or all surfaces are covered by adsorbing layers). The strength and sign of the force in general depends on the detailed shape and adsorption properties of the obstacles, but assumes simple universal forms if characteristic length scales associated with the objects are large. This occurs for scale-free shapes (such as a flat plate, straight wire, or cone), when the polymer is repelled by the obstacles, or is marginally attracted to it (close to the depinning transition where the absorption length is infinite). In such cases, the separation h between obstacles is the only relevant macroscopic length scale, and the polymer mediated force equals Ak_BT/h, where T is temperature. The amplitude A is akin to a critical exponent, depending only on geometry and universality of the polymer system. The value of A, which we compute for simple geometries and ideal polymers, can be positive or negative. Remarkably, we find A=0 for ideal polymers at the adsorption transition point, irrespective of shapes of the obstacles, i.e. at this special point there is no polymer-mediated force between obstacles (scale-free or not).Attractive and repulsive polymer-mediated forces between scale-free surfaces Mehran Kardar December 30, 2023 ============================================================================§ INTRODUCTIONA prototype of soft matter, polymers are long flexible chains that can fluctuate (whether within a cell or in a solution) between a large number of configurations. The presence of hard boundaries or obstacles modifies the number and weight of allowed configurations, in turn resulting in polymer mediated forces between the obstacles. A well known example is the depletion force of polyethylene glycol (PEG) which acts to bundle filaments <cit.>. However, whereas the relevant length scale for depletion force is the overall size of the polymer R, here we focus on polymer-mediated forces on separations h≪ R. The internal structure of a polymer is a self-similar fractal, spanning a wide range of scales from R to a microscopic monomer size a. To compute forces between obstacles embedded in or attached to the polymer, we need to compute modifications to the free energy due to the objects. This is in principle a complex task involving the shapes of the objects, and details of their interactions with the polymer. We demonstrate that this task is considerably eased in certain cases, yielding simple universal expressions for the force.Technological progress in manipulation of single molecules <cit.>, using probes such as atomic force microscopes (AFMs) <cit.>, microneedles <cit.>, optical <cit.> and magnetic <cit.> tweezers, makes it possible to measure forces exerted by polymers with high precision. The central motivation of these experiments is to unravel specific information about shapes, bindings, and interactions of biological molecules from force-displacement curves. For the important class of intrinsically unstructured proteins <cit.>, entropic forces, such as those considered in the paper, are likely to play an important role.In previous work we considered polymers confined by impenetrable obstacles of scale invariant shape, such as a polymer attached to the tip of a conical probe approaching a flat surface <cit.>. The reduction in the number of configurations of the polymer leads to a repulsive entropic force, which we showed to depend on the (tip to surface) separation h and the temperature T as F= Ak_BT/h. The “universal" amplitude A only encodes basic geometrical properties, and gross features of the polymer. For such “repulsive" surfaces, the amplitude A is positive. By considering both repulsive and attractive surfaces (as well as by expanding the types of polymers considered), here we demonstrate that attractive surfaces may indeed lead to polymer-mediated attraction with negative A.The interaction of a polymer with a surface can be changed from repulsive to attractive, e.g. by changing temperature or solvent quality. The competition between energetic attraction and entropic repulsion typically leads to a temperature dependent absorbed layer size, introducing another length into the problem. This length scale diverges at a continuous adsorption transition point introducing a scale-free boundary condition which is distinct from the repulsive surfaces studied previously.In this paper we expand our formalism <cit.> from the treatment of purely repulsive surfaces to adsorbing surfaces, and to mixed repulsive/adsorbing surface combinations. In Section <ref> we beginexaminingseveral polymer types near repulsive or adsorbing flat surfaces, and show that the size and sign of the force between a polymer and a surface depends both on the polymer type and the surface type. In Section <ref> we demonstrate that under certain circumstances the polymer-mediated forces between scale-free surfaces have a universal coefficient, independent ofminute details of the polymers. The calculation of force induced by ideal polymers, taken up in Section <ref>, can be reduced to the solution of a diffusion problem with either absorbing or reflecting boundary conditions. A particularly interesting result is that when all embedded surfaces are at the adsorption transition point, the polymer mediated force is identically zero, independent of shape and geometry. In Sections <ref> and <ref> we consider a number of examples of mixed repulsive/attractive geometries and demonstrate the ability to modify the force amplitude by changing the surface geometries and types. Finally, under Discussion we consider possible generalizations of ideal polymer results to other polymer types. § POLYMERS NEAR ATTRACTIVE OR REPULSIVE FLAT SURFACES Polymers may exist in different phases, with distinct universal characteristics <cit.>. Athigh temperatures in a good solvent polymers expand to maximize the number of available configurations. Ignoring all interactions between monomers, except those imposing its connectivity leads to configurations resembling a random walk; such configurations will be denoted as ideal polymers. However, it is unrealistic to ignore the exclusion of monomers from occupying the same volume in space, and the resulting configurations (which are more swollen than ideal random walks) are designated as self-avoiding polymers. When thequality of a solvent is reduced, the tendency of monomers to aggregate is akin to an effective short-range attraction which eventually collapses the polymer to a globule of finite density. The transition betweengood and bad solvent regimes occurs at the so-called θ-point, with the resulting configurations labeled as θ-polymers.Ideal, self-avoiding and θ polymers are the three polymer types considered in this work, all characterized by (albeit distinct) universal scale-invariant properties. For example, they are characterized by a fractal dimension d_f=1/ν, such that the typical separation between monomers i and j along the chain scales as |i-j|^ν; the overall polymer size R (such as the mean radius of gyration, or the end-to-end distance) grows with the number of monomers N as R=aN^ν, where a is some microscopic length, of the order of monomer size or persistence length. The exponent ν depends only on polymer type, but not on any microscopic details. It ranges from 1/2 to 3/4 depending on space dimension d and the polymer type, as listed in Table <ref>. This universality enables the frequent use of simple lattice models to study real polymers. For example, ideal and self-avoiding polymers can be represented by random walks and self-avoiding walks on lattices, respectively, while θ polymers may be represented as self-avoiding walks on a lattice with added attractive interaction between monomers on adjacent lattice sites. (The attractive interaction must then be tuned to exactly match the boundary betweengood andbad solvents.)The partition function of polymer types described above is in part universal <cit.>. It depends on the number of monomers asZ= b z^NN^γ-1,whereb and z depend on microscopic properties of the polymer, while the power-law exponent γ depends only on geometry and polymer type. Thus, the leading extensive part of the free energy of a single polymerF=-k_BTln Z=-k_BTNln z-k_BT(γ-1)ln N+…is model dependent, while the coefficient of the subleading ln N is universal. Nevertheless, we shall see that this subleading term plays an important role in polymer-mediated forces. In self-avoiding and ideal polymers, the potential energy plays a minor role. In lattice models it is completely absent, and Z coincides with the total number of configurations N, while z is the lattice coordination number for random walks, or the effective coordination number for self-avoiding walks. The free energy is then obtained from the entropy S as F=-T S=-k_BTln N. If one end of a polymer is attached to an infinite impenetrable flat surface in d=3, or to an infinite repulsive line in d=2, then it will be excluded from half of the space. Nevertheless, the metric exponent ν remains unmodified, although the prefactor a in the power law R=aN^ν does change. The number of available configurations, and hence the partition function, is reduced to Z_1=b_1z^NN^γ_1-1. Note that the factor z related to the extensive part of the free energy is unchanged, with the reduction in states captured through the exponent γ_1<γ (see Table <ref>). The change in free energyΔ F_1≡ F_1- F=k_BT(γ-γ_1)ln Nis positive, i.e. the polymer is repelled by the wall, or, a force towards the wall needs to be applied to bring the polymer from infinity to the wall.If the repulsive surface described above is covered by an attractive layer, then a polymer attached by one end to the surface may decrease its energy by frequently visiting the surface. In discrete models we may simply assign an extra (Boltzmann) weight q= e^-V/k_BT, where V<0 is the potential at the attractive layer, for each point visited at the boundary. The reduction in entropy of the polymer in this absorbed state is compensated by a bigger gain in energy. At high temperatures (or for weakened attractive potential) the entropy wins and the polymer depins from the attractive layer. The free energy per monomer in the absorbed state is lower than that of the free polymer due to the gain in absorption energy, and can be cast as-k_BTln z_a(T) with z_a(T)>z. If one end of the polymer is held at some moderate distance h from the surface, then a typical configuration will consist of an “equilibrium bulk" attached to the surface, and a strongly stretched tail going from the surface to the point where it is held (with a force of order of -(k_BT/a)ln [z_a(T)/z]). For some computations, it is more convenient to consider a slightly different situation where the polymer is anchored to the surface and pulled away by application of a force <cit.>. In this situation, the behavior of the polymer is different if we control the distance h versusthe pulling force applied to its end, akin to controlling density versus pressure at a first order liquid-gas transition <cit.>.The transition from adsorbed to desorbed states occurs at a critical (depinning) temperature T_a <cit.>, where z_a(T_a)=z. Exactly at T_a, the partition function of any of the polymer types mentioned <cit.> above again has a simple form Z_a=b_az^NN^γ_a-1. Since almost all monomers are away from the boundary (the fraction of contacts with the boundary increases slower than N), the dominant factor of z remains unchanged. The relation between the exponent γ_a and the free-space γ is not obvious, since the presence of the surface decreases the number of available configuration, which tends to decrease γ, but also decreases the energy, which tends to increase γ.By comparing γ with γ_a in the Table <ref>, we see that for self-avoiding polymers γ_a>γ, while for ideal and θ polymers γ_a=γ. This means thatΔ F_a≡ F_a- F=k_BT(γ-γ_a)ln N ,is either negative, i.e., the polymer is attracted by the wall, or vanishes, which makes the wall “invisible" to the polymer that is brought into its vicinity. When T is not at the adsorption transition point we may expect deviations from the above relations and various crossover effects. However, as long as the correlation length ξ characterizing the transition <cit.> exceeds the polymer size, we may treat the system as if it is at T_a. In the remainder of this article we will always assume that the attractive surfaces are at adsorption transition point without explicitly mentioning this condition.§ POLYMER-MEDIATED FORCES BETWEEN SCALE-FREE SURFACES The results in the previous section relied on the observation that the partition function of a polymer in free space or near a planar surface (either repulsive or at adsorption transition point) hasthe form in Eq. (<ref>). This form is a consequence of the fact that the geometries of free space or infinite plane do not posses a characteristic geometrical length scale, i.e., the relevant space is invariant under the coordinate transformation r→λ r. Similarly, the polymer/surface interactions do not introduce a length scale when they are either repulsive or attractive at adsorption transition point. The same conclusion [hence Eq. (<ref>)] applies to a host of other scale-free shapes such a semi-infinite plane, a sector of a two-dimensional plane in d=3, a semi-infinite line, a cone of any cross section,a wedge, or any combinations of such shapes, such as a cone touching a plane. Scale invariance in most such geometries is with respect to a “center" location, such as the apex of a cone or the terminal point of semi-infinite line. We assume that in such cases an attached polymer is anchored to the “center" point to avoid introducing a new length scale. The partition function of a polymer attached to the central point of any scale-free shape will be described by Eq. (<ref>), with an exponent γ that depends on d, the polymer phase, surface adsorption (repulsive or attractive), and on geometric features characterizing the shape, such as the apex angle Θ of the cone <cit.>, or the tilt angle of the cone touching a plane <cit.>. Furthermore, we can mix surface types, by, say, attaching a cone with attractive cover to a repulsive plane. In fact we can have a scale-free situation when a single surface mixes repulsive and attractive regions: E.g.For example, consider a repulsive plane on which a sector has been covered by an attractive layer as in Fig. <ref> (with the polymer attached to the sector apex). Starting from the polymer partition function in scale-free geometries, we can compute polymer-mediated forces between such surfaces. As an example consider a repulsive cone, with a polymer attachedto its tip, approaching, say, an attractive plane, as depicted in Fig. <ref>. When the distance h between the cone and the plane is significantly shorter that the polymer size R, but larger than the microscopic scale a, h is the only relevant length scale, while k_BT is the only relevant energy scale. In such a case, the force F transmitted by the polymer between the surfaces is constrained to be the only dimensionally correct combinationF= A k_BT/h .The dimensionless prefactor A (the “force amplitude") can be positive or negative corresponding to polymer-mediated repulsion or attraction between the objects. (This also follows from various polymer scaling forms <cit.>.) Note that the form of the force (and independence of R) is a consequence of the objects having a single point of closest approach. Most of the polymer-surface interactions appear in the neighborhood of this point, while the remote tail of the polymer is not much influenced by the constriction. Equation (<ref>) fails in truly confining geometries; e.g., if confined between parallel planes a distance H apart, the polymer has nowhere to escape and the total polymer-mediated force can be viewed as a sum of forces exerted by Pincus–de Gennes blobs <cit.> whose size depends on H, while their number is proportional to N <cit.>, leading to a force proportional N.Equation (<ref>) is valid only for h ranging between a and R. When h decreases and approaches the microscopic size a, the force saturates at order Ak_BT/a, while for h exceeding R it rapidly drops to 0. Thus, the work that the external force needs to perform to bring the surfaces from far awayto a microscopic distance isW=∫_a^R dh Ak_BT/h = Ak_BTlnR/a= A ν k_BTln N.(The slight uncertainty in the integration limits is not important since it only affects an additive constant to a term that diverges as ln N.) The same work can also be computed from the change in free energies between the final and initial states. Both the initial and final states are scale-free as depicted in Fig. <ref>: Far away only the cone needs to be taken into account, while at the point where the cone touches the plane, we again have a scale-free situation. Therefore, the partition functions in these two extremes will have the form of Eq. (<ref>), but with exponents γ_ far and γ_ near corresponding to the two limiting geometries, with appropriate polymer and surface types in dimension d. As in Eqs. (<ref>) and (<ref>) the free energy difference isΔ F≡ F_ near- F_ far =k_BT(γ_ far-γ_ near)ln N.By equating this Δ F with the work W in Eq. (<ref>) we findA=γ_ far-γ_ near/ν=η_ near-η_ far.In the final step we employed the exponent identityγ=(2-η)ν ,to relate the exponent γ to the exponent η characterizing the anomalous decay of density correlations (as 1/r^d-2+η). Equation (<ref>) indicates that the force amplitude A is a universal quantity akin to critical exponents. In the trivial case, when the polymer, held by a very small probe (point), is moved towards a plane, γ_ far coincides with γ of the free space, while γ_ near is γ_1 or γ_a for repulsive or attractive surfaces, respectively. For example, for a self-avoiding polymer in d=3 approaching an attracting surface, Eq. (<ref>) with the exponents from Table <ref> leads toA≈ -0.25.The case of purely repulsive boundaries was considered previously for both ideal and self-avoiding polymers.The latter required either numerical simulations or resorting to expansions in ϵ=4-d <cit.> to compute the relevant exponents, while ideal polymers could be treated analytically for simple (highly symmetric) geometries <cit.>, only requiring simple numerical solutions of diffusion equations for less symmetric scale-free geometries <cit.>. § IDEAL POLYMERS NEAR REPULSIVE AND ATTRACTIVE SURFACESThe absence of interactions between non-adjacent monomers of an ideal polymer significantly simplifies its treatment. For an N-step polymer on a regular lattice with lattice constant a, such as the square lattice depicted in Fig. <ref> with coordination number μ=4,the partition function (beginning at point r in free space) is simply the number of configurations Z( r,N)=N( r,N)=μ^N. We will define a reduced partition function Z̃≡ Z/μ^N, which in general should scale as Z̃∼ N^γ-1. In free spaceZ̃=1, and therefore γ=1. The partition function of a polymer of (N+1) stepsthat begins at r and ends at r' can be calculated recursively asZ( r, r',N+1) =∑_ r”nnofr' Z( r, r”,N),with the initial condition Z( r, r',0)=δ_ r, r'. Similarly, the reduced partition function sarisfiesZ̃( r, r',N+1)-Z̃( r, r',N) =1/μ∑_ r” nnofr'[Z̃( r, r”,N)-Z̃( r, r',N)].For slowly varying functions, we can employ a continuum formulation in which the left hand side is replaced with a first derivative, while the right hand side represents a second derivative (discrete Laplacian). Regarding the continuous version of N as a time-like variable t, the continuum equation for Z̃ becomes the diffusion equation for the probability density P( r, r',t) of a diffusingparticle that starts its motion at r and ends up at r' in time t, which satifies∂ P( r, r',t)/∂ t=D∇'^2 P( r, r',t) ,with initial condition P( r, r',t=0)=δ^d( r- r'). The prime sign on the Laplacian indicates spatial derivatives with respect to r'. The diffusion constant D is chosen such that in free space the mean squared distance coincides with the random walk value of ⟨( r- r')^2⟩=a^2N=2dDt in d space dimensions, and thus D=a^2/2d. The probability density P is related to the discrete probability Z̃ by Z̃=a^dP.In free space all configurations have identical weight. However, in the presence of a repulsive wall,such as depicted by the lower (red) horizontal line in Fig. <ref>, walks that touch or cross that line,such as walk “a" in the figure, must be eliminated from consideration. This can be achieved by applying Eq. (<ref>) only to the points r above the repulsive line, while setting Z̃( r, r',N)=0, whenever r' is on the repulsive boundary. The continuum limit for P will then correspond to the solution of Eq. (<ref>) with absorbing boundary conditions.Since the statistical weight of every path is independent of its direction,Z̃( r, r',N)=Z̃( r', r,N),i.e. there is symmetry with respect to interchange of the start and end points of the chain. Consequently, in the diffusion equation (<ref>), the prime can be removed from the Laplacian. (This is the usual reciprocity relation of the diffusion problem <cit.>.) After such a change, both sides of the modified Eq. (<ref>) can be integrated over r', the resulting survival probability S( r,t)≡∫ P( r, r',t) d^d r' evolving as∂ S( r,t)/∂ t=D∇^2 S( r,t),with absorbing boundary conditions.The initial condition for survival probability is S( r,t=0)=1, everywhere inside the space where the particle can diffuse, and S( r,t)=0 on the absorbing boundaries. This quantity coincides with the total reduced partition function Z̃( r,N). Our previous works considered a variety of cases with scale-free repulsive boundaries <cit.>, while in this work we are mostly interested in attractive boundaries or in mixtures of the two. When a repulsive surface is covered by an attractive (adsorbing) layer (blue top horizontal line in Fig. <ref>), every time a polymer visits that layer its statistical weight is increased by a factor q= e^-β V, where V<0 is the energy gain. (For q=1 the layer has no effect, while q<1 corresponds to a repulsive potential.) The partition functioncan now be calculated fromZ( r, r',N+1) =x( r)∑_ r”nnofr' Z( r, r”,N),where x( r)=q, for r in the adsorbing layer, and x( r)=1, otherwise. This equation must be supplemented with the initial condition Z( r, r',0)=x( r)δ_ r, r' to ensure reciprocity. The usual diffusion equation, still applicable outside the absorbing layer, is thus modified by the potential near thelayer. It is important to note that the region below the absorbing layer is still impenetrable, and thus the partition function is strictly zero below the surface.The phenomenology of polymer absorption is as follows: For zero temperature (q→∞) all monomers are on the absorbing layer, and the partition function is dominated by the energy contribution. At small, but finite, temperatures parts of the polymer detach from the surface gaining entropy. (We can assume that one end of the polymer is always attached to the surface to avoid discussion of the center of mass entropy.) The average number of visits to the absorbing layer will be proportional to N (n=cN, with c depending on temperature). Despite the loss of entropy, the energy gain from such visits leads to a partition function Z_a≃ [z_a(T)]^N≫μ^N in theadsorbed phase. As temperature increases and q is reduced, there is apoint where the decrease in the number of configurations due to the impenetrable boundary is exactly compensated by the extra weight provided by the adsorbing layer to configurations that touch the adsorbing layer. From the perspective of a random walker, the reduction in the number of possible paths by the boundary is exactly made up by the extra weight of the walks that arrive at the attractive potential (blue line) but do not touch the absorbing boundary (red line).The adsorption of a discrete ideal polymer was studied by Rubin <cit.>. He determined the transition point q_c, and demonstrated that for a planar attractive surface Z=μ^N, i.e. γ_a=1.Comparing the behavior of an ideal polymer at T_a, to a random walk (diffusion) with reflecting boundary conditions, Rubin concluded that for large N these two problems coincide, although subtle differences remain for small N. Clearly, in the presence of reflecting boundaries the survival probability of a diffusing particle is always S( r,t)=1, which corresponds to a polymer at theadsorption transition point with Z̃( r,N)=1.The universal aspects of Rubin's results can be captured in the continuum limit, taking advantage of the mapping between configurations of the ideal random walks (path integral), and quantum mechanics of a particle in a potential <cit.>. In particular the (ideal) polymer adsorption problem is mapped to a quantum particle in a one-dimensional potential of an attractive well adjacent to an impenetrable barrier. Depending on the strength of attraction, such a potential may or may not admit a bound state. The bound state (corresponding to the absorbed polymer) has a wave-function decaying as ψ(z)∼ e^-λ z away from the potential; its energy (∝ -λ^2) designating the gain in polymer free energy on adsorption. As the potential is weakened, λ vanishes (linearly in T_a-T) indicative of the adsorption transition point. At coarse-grained level, the combination of barrier and potential can be expressed as the mixed (Robin) boundary condition ψ'+λψ=0. Under further coarse-graining, at scales larger than λ^-1 (irrespective of its sign), this requirement becomes equivalent to the Dirichlet boundary condition ψ=0, while for λ=0(an unstable fixed point under coarse-graining), it is the Neumann boundary condition ψ'=0. From the perspective of random walks, ψ=0 corresponds to absorbing boundaries, and ψ'=0 to reflecting boundaries; both limits are scale invariant (i.e., such boundaries do not introduce a new length scale to the polymer problem.) The above considerations lead to the following interesting result: If all the confining boundaries and inclusions immersed in a long ideal polymer are at adsorption transition point, and thus in the corresponding diffusion problem all barriers are reflective, then the trivial solution of Eq. (<ref>) is S( r,t)=1 for any t. As this does not depend on the positions of the various obstacles, there can be no polymer-mediated force between them! Note that this is true for arbitrary shapes, and the boundaries do not need to be scale-free. (In the particular case of scale-free surfaces, we note the result γ_a=1 for future reference.)Analytical solutions of Eqs. (<ref> and <ref>) are available for a number of simple shapes <cit.>. For scale-free shapes it is convenient to choose a coordinate system centered on the center of symmetry (such as the tip of a cone). The dimensionless survival probability S can only depend on the dimensionless vector w= r/√(Dt). Thus, S( r,t)=H( w), and Eq. (<ref>) reduces to∇_𝐰^2 H+1/2 w·∇⃗_𝐰H=0,where the subscript w indicates derivatives with respect to components of w. In terms of these dimensionless variables, either the function H or its normal derivative vanish on the absorbing or reflecting surfaces respectively.For some geometries, the solution to Eq. (<ref>) can be expressed in terms of a radial distance w, and a combination of angular variables, such as the polar angle θ and (d-2) azimuthal angles (ϕ, ψ, ⋯). For w≪ 1, i.e. for long times t,the distance dependence is expected to be a simple power law w^ηΨ(θ,ϕ,…). In this limit, the second term in Eq. (<ref>) becomes negligible, and the problem reduces to solving the Laplace equation∇_𝐰^2 (w^ηΨ)=0.For small fixed r we have S∼ t^-η/2, and comparing it with the expected Z̃∼ N^γ-1, we find that η=2(1-γ), i.e. it is the same exponent η that appears in Eq. (<ref>) for any polymer type. Thus obtaining the exponent γ, and the related force amplitude, is reduced to finding η in the solution of Eq. (<ref>) with appropriate boundary conditions. If all boundaries are attractive, then we already know the solution, corresponding to γ_a=1. If allsurfaces are repulsive,thenthe solution to absorbing conditions of the diffusion equation will need to vanish on the boundaries. Problems of this type have been solved for a variety of geometries in the past <cit.>. In the following Sections we consider several examples of mixed boundaries.§ TWO-DIMENSIONAL IDEAL POLYMERS NEAR MIXED BOUNDARIES In a d=2 scale-free geometry the (Laplace) Eq. (<ref>) simplifiesto Ψ”+η^2Ψ=0, where prime denotes the derivative of Ψ with respect to the angle θ. This equation is solved bylinear combinations of sin(ηθ) and cos(ηθ). For repulsive or attractive boundaries of the polymer problem, we must use boundary conditions of vanishing Ψ or vanishing derivative Ψ', respectively. This should be viewed as an eigenvalue equation, and the primary goal is finding the correct value of η. This equation (complemented by the boundary conditions) has many eigensolutions. Since Ψ determines the probability or partition function, its sign cannot change, and consequently we are interested in the “ground state" that corresponds to the lowest value of η.As an example, consider an ideal polymer anchored close to the central angle 2α of a two dimensional wedge with mixed (repulsive/attractive) boundaries as depictedin Fig. <ref>. If the angle θ' is measured from the lower boundary, then the appropriate solution is Ψ=cos(πθ'/4α), i.e.,η=π/4α orγ=1-π/8α,Fortwo repulsive boundaries the eigenfunction is Ψ=sin(πθ'/2α), corresponding toη=π/2α orγ=1-π/4α.Figure <ref> depicts the α-dependence of η both for mixed and for purely repulsive boundaries. Note, that in both cases the exponent divergesin the limit of α→0, capturing the vanishing of the available states. Equation (<ref>) with 2α=π, is the same as Eq. (<ref>) for 2α=2π, both representing a single repulsive semi-infinite line with η=1/2. This means that in d=2 the presence of such obstruction does limit the behavior of an ideal polymer. (An obstructed line only has marginal effects in d=3.) It is interesting to note that while the mixed wedge is equivalent to a repulsive wedge of twice the angle, it can explore configurations not accessible to the repulsive case with 2α>2π, i.e., beyond what would be possible in d=2.The above results for d=2 are also applicable to wedges in any dimension d, since the function Ψis independent of coordinates parallel to the edge. In particular, in d=3 the 2α=π case of Eq. (<ref>) corresponds to a plane half of which is repulsive while the other half is attractive, with γ=3/4. The above expressions for η enable us to compute the force amplitude A in the situation depicted in Fig. <ref> in d=2, when a wedge with a polymer attached to it approaches an excluded half space. We simply need to find the exponents in the two extreme situations, when the wedge is either far away, or is touching the line. Anticipating the generalization from the wedge in d=2 to the cone of opening angle Θ in d=3 in the next section, we shall use the notation depicted in Fig. <ref>. The isolated cone in Fig. <ref>(a) has purely repulsive boundaries and is solved by Eq. (<ref>) giving η_ far=π/[2(π-Θ)], while the touching of cone and plate in Fig. <ref>(b) is described by mixed boundary situation and Eq. (<ref>) results in η_ near=π/(π-2Θ). (We have set α→π-Θ for the repulsive wedge, and α→π/4-Θ/2 in the mixed case.) This results in the force amplitudeA=π^2/2(π-2Θ)(π-Θ).Note that A is always positive, even in the limit of a “needlelike" wedgewith Θ→0. In the reversed situation where the cone is attractive, while the plane is repulsive, η=0 in the remote configurations but retains the same value as before when the plane and cone are in contact, leading toA=π/π-2Θ.This amplitude is larger than in the previous example, and in the Θ→0 limit is the same as a polymer that is brought to the vicinity of a repulsive surface while held at an endpointwithout a wedge.§ THREE-DIMENSIONAL IDEAL POLYMERS NEAR MIXED BOUNDARIES In the presence of azimuthal symmetry in three dimensions, aswith cones of circular cross section, or such a cone touching a plane and perpendicular to it, the Laplace Eq. (<ref>) simplifies. With Ψ depending only on the polar angle θ as illustrated in Fig. <ref>, it takes the form(1-u^2)d^2Ψ/du^2-2μ dΨ/du +η(η+1)Ψ=0,where u≡cosθ. We seek a regular eigensolution Ψ(θ) that vanishes on repulsive boundaries or has a vanishing derivative on attractive boundaries. The general solution to this equation is given by regular (rather than associated) Legendre functionsΨ(θ)=a_1P_η(u)+a_2Q_η(u).Note that P_η(1)=1 for any η, while P_η(-1) diverges for noninteger η. Similarly, Q_η(±1) is divergent. The linear combination in Eq. (<ref>) can be made regular at -1 by a proper choice of a_1/a_2. In Refs. <cit.> we described the analytical solutions of this equation for purely repulsive cones, or such cones touching a repulsive plane. (Geometries without azimuthal symmetry can be easily handled numerically <cit.>.)In many cases, the regularization procedure can be avoided by a convenient choice of functions. For the geometry depicted in Fig. <ref>,the solution must be regular for Θ≤θ≤π. Instead of using combinations of P_η and Q_η, we can simply use P_η(-cosθ), which will be regular at cosθ=-1. The value of η for a repulsive cone is then determined by requiringP_η(-cosΘ)=0.Since Ψ cannot change sign in the physically permitted region, the smallest possible η must be chosen. This procedure is described in detail in Refs. <cit.>.An attractive boundary requires ∂Ψ/∂θ to vanish at θ=Θ. This, as in allcases of purely attractive boundaries, is trivially achieved by Ψ= constant, and η=0.Calculation of η for the situation when the cone touches a plane (as in Fig. <ref>b) is simplified by noting that for non-integer η, P_η(u) and P_η(-u) are linearly independent and both solve Eq. (<ref>) <cit.>. Thus for non-integer η, Eq. (<ref>) can be replaced byΨ(θ)=P_η(cosθ)± P_η(-cosθ).The - and + signs enforce vanishing of Ψ or its derivative at θ=π/2, respectively, corresponding to a repulsive or attractive plane. The remaining boundary condition at θ=Θ can be implemented by proper choice of the exponent η(Θ), which can be obtained numerically. New results pertain to the mixed boundary setups depicted in Fig. <ref>. (The case of all attractive boundaries is trivial, while that of repulsive boundaries was considered in Refs. <cit.>.) From knowledge of the values of η in situations depicted in Fig. <ref>, we can use Eq. (<ref>) to determine the force amplitudes. Figure <ref> depicts the force amplitude for the two cases as a function of cone apex angle Θ. Similarly, to analogous solution in d=2 the force amplitude for repulsive plane and attractive cone is larger than that for the reversed situation.For a less symmetric setup in d=3, the Laplace equation Eq. (<ref>) will depend on two angular variables, and implementation of boundary conditions may prove difficult. Fortunately, a direct numerical implementation of Eq. (<ref>), typically only requires a few thousand iterations for a good estimate of γ from the dependence of Z̃ on N. As an example we considered a repulsive plane decorated by an attractive sector of opening angle 2α as depicted in Fig. <ref>, and results from a numerical estimation are shown in Fig. <ref>. For α=0 we naturally recover γ=1/2, as for a purely repulsive plane, while γ=1 for α=π, corresponds to a purely attractive plane. For intermediate angles, the critical exponent simply interpolates between these two limiting values. At α=π/2 the plane is divided into repulsive and attractive halves, with γ=3/4 as we found in an equivalent geometry in d=2. § DISCUSSIONIn this work we considered several cases of polymer-mediated interactions between repulsive or attractive surfaces. While the loss of entropy leads to a repulsive force between impenetrable obstacles, the gain in energy may cause an attractive force for absorbing surfaces. If the confining objects do not introduce a length scale, which is the case for impenetrable obstacles, and surfaces at the adsorption transition point, the entropic force is dimensionally constrained to the form A k_BT/h where h is a characteristic distance between the scale-free surfaces. The amplitude A depends on geometry and universality class of the polymer system (ideal, self-avoiding, or θ polymer).A hypothetical setup, such as in Fig. <ref>, involves a long polymer (or several polymers) of length N a attached to the tip of a cone at a separation h from a plane. For an adsorbing surface before the desorption transition, a typical configuration will consist of a nearly straight segment of the polymer stretched from the cone tip to the surface, followed by a much longer segment absorbed to the surface. This situation will hold as long as Na≫ h and Na ≫ξ, where ξ is a characteristic segment size that diverges close to adsorption transition point as ξ∝ (T-T_a)^-1/ϕ <cit.>. The free energy difference between adsorbed and free polymers vanishes at the adsorption transition point as Δ F_a∼ -Nk_BT(a/ξ). The adsorbed polymer also fluctuates away from the surface, forming a layer of thickness ℓ∝ξ^ν that also diverges at the adsorption transition point. Equation (<ref>) should apply only in the separation range ξ≫ h≫ℓ, where the short and long-scale cutoffs are immaterial. On shorter scales, the force should saturate, presumably to order of k_BT/ℓ, while at large scales, the polymer should be stretched, with the force reduced to k_BT/ξ, corresponding to the loss of free energy per unit length. We thus expect the following sequence of crossovers for the forceF= -k_BT/ξif h≫ξAk_BT/ξif ξ≫ h≫ℓ ∼±k_BT/ℓif h≪ℓ .(The amplitude A, and the sign of the force, is determined by surface and polymer types.) Closer still to the transition, such that ξ∼ Na, additional crossovers are expected that are not discussed here. For separations of order of 0.1 μm at room temperature, the entropic force is of order 0.1pN. Forces of such magnitude are now measurable by a host of single molecule manipulation techniques <cit.>, e.g. by atomic force microscopes (AFMs) <cit.>, microneedles <cit.>, and optical <cit.> and magnetic <cit.> tweezers. With a good AFM tip, distances can be measured to accuracy of a few nanometers <cit.>, with forces of order of 1 pN measured in nearly biological conditions <cit.>. We note that these accuracies fall within the range of entropic forces for fluctuating, featureless polymers described above.While we considered here the case of a single polymer, related fluctuation-induced forces are also expected in the case of dense melts of long polymers. Such forces have been proposed (dubbed anti-Casimir forces) for dense polymer melts between parallel plates <cit.>. For the scale free geometries that we propose, these forces should have the general forms proposed in this paper, albeit with different universal amplitudes.Finally, we note the interesting observation about the lack ofpolymer mediated forces for any number of objects (scale-free or not) immersed in ideal polymers, as long as all surfaces are at the special adsorption transition point. Compensation ofthe loss of entropy by marginally attractive energies renders such obstacles invisible to ideal polymers, in a situation similar to index matching of colloids by a fluid of the same dielectric constant. (The van der Waals interaction vanishes in such a case.) It is tempting to imagine that such a situation can also occur for objects in a self-avoiding polymer. However, the results (γ_a>γ_1) in Table I indicate that A<0 for a self-avoiding polymer near a plane at adsorption transition point. Nonetheless, by appropriate coatings of the surfaces (as in Fig. <ref>) it should be possible to reducethe force prefactor to A=0. Thus there is indeed hope for engineering (at least scale-free) obstacles that are force-free in a self-avoiding polymer solution. This work was supported by the National Science Foundation under Grant No. DMR-1206323 (M.K.), and the Israel Science Foundation Grant No. 453/17 (Y.K.).49 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Dogic(2016)]Dogic16 author author Z. Dogic, 10.3389/fmicb.2016.01013 journal journal Front. Microbiol. volume 7,pages 1013 (year 2016)NoStop [Bustamante et al.(2003)Bustamante, Bryant, and Smith]bustamante03 author author C. Bustamante, author Z. Bryant,and author S. B. 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Department of Mathematical Sciences, Durham University, United Kingdom mailto:[email protected]@durham.ac.uk Mathematics Subject Classification subjclassname@1991=subjclassname@2000=57M25, 57M27, 57N70This paper subsumes the (now withdrawn) arXiv submission https://arxiv.org/abs/1603.02893On the virtual Rasmussen invariant. A virtual knot is an equivalence class of embeddings of S^1 into thickened (closed oriented) surfaces, up to self-diffeomorphism of the surface and certain handle stabilisations. The slice genus of a virtual knot is defined diagrammatically, in direct analogy to that of a classical knot. However, it may be defined, equivalently, as follows: a representative of a virtual knot is an embedding of S^1 into a thickened surface Σ_g × I; what is the minimal genus of oriented surfaces S ↪ M × I with the embedded S^1 as boundary, where M is an oriented 3-manifold with ∂ M = Σ_g?We compute and estimate the slice genus of all virtual knots of 4 classical crossings or less. We also compute or estimate the slice genus of 46 virtual knots of 5 and 6 classical crossings whose slice status is not determined in the work of Boden, Chrisman, and Gaudreau. The computations are made using two distinct virtual extensions of the Rasmussen invariant, one due to Dye, Kaestner, and Kauffman, the other due to the author. Specifically, the computations are made using bounds on the two extensions of the Ramussen invariant which we construct and investigate. The bounds are themselves generalisations of those on the classical Rasmussen invariant due, independently, to Kawamura and Lobb. The bounds allow for the computation of the extensions of the Rasmussen invariant in particular cases. As asides we identify a class of virtual knots for which the two extensions of the Rasmussen invariant agree, and show that the extension due to Dye, Kaestner, and Kauffman is additive with respect to the connect sum. Computations of the slice genus of virtual knots William Rushworth December 30, 2023 ================================================§ INTRODUCTION §.§ Statement of resultsA virtual knot is an equivalence class of embeddings of S^1 into thickened (closed oriented) surfaces, up to self-diffeomorphism of the surface and handle stabilisations whose attaching spheres do not intersect the embedded S^1; virtual links are defined analogously <cit.>. They are represented diagrammatically using knot diagrams with an extra crossing decoration, the virtual crossing -2.5pt < g r a p h i c s > , up to the virtual Reidemeister moves; see <Ref> for such a diagram.The slice genus of a virtual knot is defined in direct analogy to that of classical knots (see <Ref>); it is less well-studied than that of classical knots, but obstructions to sliceness of virtual knots have been developed by a number of authors. They include the index polynomial of Heinrich <cit.> and the graded genus of Turaev <cit.>. Boden, Chrisman, and Gaudreau <cit.> have used these invariants and others to compute or estimate the slice genus of a very large number of the 92800 virtual knots of 6 crossing or less (as given in Green's table <cit.>).In another direction, Manturov and Fedoseev have produced slice obstructions for free knots <cit.>. A free knot is an equivalence class of 4-valent graphs, and a Gauss code representing a virtual knot may be projected to a code representing a free knot by forgetting the signs and directions of its chords. Given a free knot Γ, obstructing the sliceness of Γ necessarily obstructs the sliceness of every virtual knot which projects to it.We shall focus on the Rasmussen invariant. It has been extended to virtual knots in two different ways, producing two distinct Rasmussen-like invariants: the virtual Rasmussen invariant due to Dye, Kaestner, and Kauffman <cit.>, and the doubled Rasmussen invariant due to the author <cit.>. Both of these extensions provide obstructions to the sliceness of virtual knots (again see <Ref>).In this paper we employ these extensions of the Rasmussen invariant to compute or estimate the slice genus of virtual knots. The extensions themselves are derived from two distinct generalisations of Khovanov homology to virtual links, reviewed in <Ref>. The results of the computations are given in two tables which begin on Tab:slicegenera and Tab:slicegaps respectively, and are outlined in <Ref>.Let K be a virtual knot; postponing their definition until <Ref>, let s( K ) ∈ 2 ℤ and s ( K ) = ( s_1 ( K ), s_2 ( K ) ) ∈ℤ×ℤ denote, respectively, the virtual Rasmussen invariant, and the doubled Rasmussen invariant. Like the classical Rasmussen invariant the quantities s ( K ) and s_1 ( K ) are difficult to compute, in general (in constrast s_2 ( K ) can be computed by hand, as described below). In <Ref> four integer quantities are associated to a diagram D of K - U_v ( D ), U_d ( D ), Δ_v ( D ), and Δ_d ( D ) - which allow for the estimation of s ( K ) and s_1 ( K ).[<Ref> of <Ref>] Let D be a diagram of a virtual knot K. Then U_v ( D ) - 2 Δ_v ( D ) ≤ s ( K ) ≤ U_v ( D ) and U_d ( D ) - Δ_d ( D ) ≤ s_1 ( K ) ≤ U_d ( D ).The bounds U_v ( D ) and U_d ( D ) are generalisations of the slice-Bennequin bounds due, independently, to Kawamura <cit.> and <cit.> (see <Ref>). They are easy to compute for any diagram of K. Further, there are classes of diagrams for which the quantities Δ_v ( D ) and Δ_d ( D ) simplify. In fact, in <Ref>, we characterise exactly the class of diagrams D for which Δ_v ( D ) = 0 so that U_v ( D ) = s ( K ).As an aside, we show that although the extensions s and s_1 are distinct in general there is a class of virtual knots on which they agree. A classical crossing within a virtual knot diagram D is even if it is resolved into its oriented resolution in the alternately colourable smoothing of D; otherwise it is odd. A virtual knot diagram is known as even if all of its classical crossings are even. A virtual knot is even if it possesses an even diagram. This definition of odd and even crossings is shown to be equivalent to the standard definition involving Gauss codes in <cit.>. Classically, the oriented smoothing is necessarily alternately colourable (so that every classical knot is even). Virtually, this is no longer the case; consider the diagram given in <Ref> (both of its classical crossings are odd).[<Ref> of <Ref>] Let K be an even virtual knot. Then s ( K ) = s_1 ( K ). As a final aside, we show that the virtual Rasmussen invariant is additive with respect to connect sum. By an abuse of notation K_1 # K_2 denotes any of the knots which can be obtained as a connect sum between K_1 and K_2.[<Ref> of <Ref>] For virtual knots K_1 and K_2 s ( K_1 # K_2 ) = s ( K_1 ) + s ( K_2 ).§.§.§ Results of the computation of U_v and U_d<Ref> contains two tables which give the results of the computation of the bounds U_v and U_d, along with the results of the computation or estimation of the slice genus which follows (see <Ref>). The first table, beginning on Tab:slicegenera, contains the results for all virtual knots of 4 classical crossings or less, as given in Green's table <cit.>. The second table, beginning on Tab:slicegaps, contains the results for 46 of the 248 virtual knots of 6 classical crossings or less whose slice status is not determined in <cit.>.Many of the calculations and estimations of the virtual and doubled Rasmussen invariants are made by identifying that the knot in question is a connect sum, and applying the additivity of both invariants under that operation. §.§ Virtual cobordismIn direct analogue to those of the classical case we make the following definitions (see <cit.> and <cit.>). Two virtual knot diagrams K_1 and K_2 are cobordant if one can be obtained from the other by a finite sequence of births and deaths of circles, oriented saddles, and virtual Reidemeister moves. Such a sequence describes a compact, oriented surface, S, such that ∂ S = K_1 ⊔ K_2. If g ( S ) = 0 we say that K_1 and K_2 are concordant. If K_2 is the unknot, and K_1 is concordant to K_2 we say that K_1 is slice. In general, we define the slice genus of a virtual knot K, denoted g^∗ ( K ), asg^∗ ( K ) = min{ g ( S ) | S a compact oriented connected surface with ∂ S = K }(here we have simply capped off the unknot in ∂ S with a disc). It is natural to ask whether or not the slice genus of a classical knot may be lowered by treating it as a virtual knot. That is, given a classical knot, does the addition of virtual Redeimeister moves allow one to construct a surface bounding it of lower genus than its classical slice genus? This has been answered in the negative by Boden and Nagel <cit.>, a concordance analogue to the result of Goussarov, Polyak, and Viro that classical links are left unaltered if one views them as virtual links <cit.>.Behind the scenes, the cobordism surface S is embedded in a 4-manifold of the form M × I, where M is a compact, oriented 3-manifold with ∂ M = Σ_k ⊔Σ_l, where Σ_i denotes a closed oriented surface of genus i. The 3-manifold M is described in the standard way in terms of codimension 1 submanifolds and critical points: starting from ∂ M = Σ_k, codimension 1 submanifolds are Σ_k until we pass a critical point, after which they are Σ_k ± 1. Critical points of M correspond to handle stabilisation. A finite number of handle stabilisations are needed to reach Σ_l.As mentioned in the abstract the slice genus of a virtual knot may be defined in a more natural manner. Let K be a virtual knot and (by an abuse of notation) let K ↪Σ_g × I be representative of K. Theng^∗ ( K ) = min{ g ( S ) | S ↪ M × I an oriented connected surface with ∂ S = K M an oriented 3-manifold with ∂ M = Σ_g . } .That this second definition is equivalent to the first follows from the observation that given two representatives of K in Σ_g × I and Σ_g'× I with g ≠ g', there exists a cylinder (embedded in a thickened oriented 3-manifold) which cobounds them. Further, this definition highlights the higher-dimensional topology at play when one considers the slice genus of virtual knots. In constrast to the classical case, in which the slice genus of a knot depends only on how surfaces bounding that knot may be embedded into B^4, the slice genus of a virtual knot depends on the surface S and on the 3-manifold M. §.§ The slice-Bennequin boundsThe Rasmussen invariant of a classical knot extracts geometric information from Khovanov homology, yielding a lower bound on the slice genus <cit.>. Given a classical knot K it is, in principle, difficult to compute its Rasmussen invariant, denoted s ( K ), as it is equivalent to the maximal filtration grading of all elements homologous to a certain generator of the Lee homology of K.Kawamura <cit.> and Lobb <cit.> independently defined diagram-dependent upper bounds on s (K), denoted U ( D ) (for D a diagram of K), which are easily computable by hand, along with an error term, Δ ( D ), the vanishing of which implies that s ( K ) = U ( D ), in fact. More precisely,U ( D ) - 2 Δ ( D ) ≤ s ( K ) ≤ U ( D ).The bounds U ( D ) are henceforth referred to as the strong slice-Bennequin bounds; in <Ref> we construct analogous bounds on the virtual and doubled Rasmussen invariants. §.§ Estimating the slice genusThis paper is concerned with the computation of the slice genus of virtual knots. These computations are achieved using the obstructions to sliceness offered by the two extensions of the Rasmussen invariant mentioned above. As stated, the virtual Rasmussen invariant, and one component of the doubled Rasmussen invariant are difficult to compute (this necessitates the construction of the bounds as mentioned in <Ref>). The other component of the doubled Rasmussen invariant is, however, readily computable. Precisely, the quantity s_2 ( K ) can be computed from quickly from any diagram of K, as it is equal to the odd writhe of K. That is:[[Proposition 4.11 of <cit.>] Let D be a diagram of a virtual knot K. Let J ( D ) denote the sum of the signs of the odd crossings of D. This is a knot invariant, known as the odd writhe of K, and denoted J ( K ) <cit.>. Then s_2 ( K ) = J ( K ). [Theorem 5.8 of <cit.>] Let K be a virtual knot such that s_2 ( K ) ≠ 0. Then K is not slice. Whilst it is more difficult to compute, the other component of the doubled Rasmussen invariant also obstructs sliceness.[Corollary 5.5 of <cit.>] Let K be a virtual knot with s_2 ( K ) = 0. If s_1 ( K ) ≠ 0 then K is not slice. The virtual Rasmussen invariant provides a lower bound on the slice genus of a virtual knot.[Theorem 5.6 of <cit.>] Let K be a virtual knot. Then | s ( K ) | ≤ 2 g^∗ ( K ). The computations and estimations of the slice genus are made as follows. Let D be the diagram of a virtual knot K given in Green's table <cit.>, then: * Compute U_v ( D ), U_d ( D ), Δ_v ( D ), Δ_d ( D ), and s_2 ( K ) for D, in order to estimate or compute s ( K ) and s_1 ( K ). * Take the greatest of the upper bounds on g^∗ ( K ) provided by the estimations or computations of s ( K ), s_1 ( K ), and s_2 ( K ). * Attempt to find a cobordism from D to the unknot of genus equal to the greatest upper bound on g^∗ ( K ), thus computing g^∗ ( K ). * Failing that, find a cobordism of higher genus so that a region in which g^∗ ( K ) lies is identified.§.§ Plan of the paperFirst, in <Ref>, we outline the issues faced when extending Khovanov homology to virtual links, and review two distinct ways of overcoming them i.e. two extensions of Khovanov homology to virtual links. Further, we review the extensions of the Rasmussen invariant produced from each of the homology theories. We also identify in <Ref> a class of virtual knots for which the two extensions of the Rasmussen invariant are equal.Next, in <Ref>, we produce canonical chain-level generators of one of the relevant homology theories. This is done by simplifying the decorated diagrammatic generators defined in <cit.>, so that elements of the algebraic chain complex may be read off from them.These canonical generators are required in <Ref>, in which we construct the strong slice-Bennequin bounds on both the virtual and the doubled Rasmussen invariant. In this we follow much the same path as Lobb <cit.>; in fact, in the case of the virtual Rasmussen invariant, we recover formulae identical to his. In the case of the doubled Rasmussen invariant, however, the formulae arrived at are substantially different, a consequence of the structural differences between doubled Khovanov homology and its classical predecessor.Finally, in <Ref>, we use the tools we have developed to compute or estimate the slice genus of a large portion of the knots given in Green's table <cit.>.Acknowledgements. We thank A Referee for very helpful comments on an earlier version of this paper, and Hans Boden, Micah Chrisman, and Robin Gaudreau for sharing and discussing their work.§ REVIEWWe review the two homology theories used throughout this work. In an attempt to avoid confusion we shall refer to the theory due to Manuturov and reforumulated by Dye, Kaestner, and Kauffman as MDKK homology, and denote it by . We denote the other theory in question, doubled Khovanov homology, by . Classical Khovanov homology, where required, is denoted by . The perturbed versions of the theories are denoted by ', ', and '.The review of MDKK homology contained in <Ref> is substantially more detailed than the review of doubled Khovanov homology (contained in <Ref>). This is because the methods used in <Ref> require chain-level generators of the complexes ' and '. We are already in possession of such generators in case of ' but not '. (In <Ref> we construct these generators.)Before outlining the homology theories we describe the complications one encounters when attempting to extend Khovanov homology to virtual links. §.§ Extending Khovanov homologyManturov first defined Khovanov homology for virtual links <cit.>. His theory was reformulated by Dye, Kaestner, and Kauffman in order to define a virtual Rasmussen invariant <cit.>. An alternative extension of Khovanov homology to virtual links is doubled Khovanov homology, which provides the doubled Rasmussen invariant <cit.>. Here we briefly outline the problems encountered in attempting to extend Khovanov homology to virtual links, and the paths taken in <cit.> and <cit.> to overcome them.The fundamental obstruction to transferring Khovanov homology to the virtual setting is the existence of the single-cycle smoothing depicted in <Ref> (otherwise known as a one-to-one bifurcation). If the module assigned to a circle within a smoothing is the same as that assigned by classical Khovanov homology the map associated to this smoothing, denoted η, must be identically zero, in order to preserve the quantum grading. This, in turn, causes the face depicted in <Ref> to fail to commute. Notice that the differential along the top and right-hand edges is η∘η = 0, but along the left-hand and bottom edges it is m ∘Δ≠ 0 so that d^2 ≠ 0.Thus classical Khovanov homology must be augmented in order to detect this face, if one wishes to assign η the zero map. This is the approach taken by Manutrov and subsequently Dye et al, and outlined in <Ref>. In <cit.> another approach is taken: the module assigned to a circle within a smoothing is altered, allowing for η to be assigned a non-zero map while being grading preserving. The resulting theory is outlined in <Ref>. Tubbenhauer <cit.> has constructed a virtual Khovanov homology theory in the manner of Bar-Natan <cit.> using non-orientable cobordisms, but there are compatibility issues with the theory presented in <cit.>.§.§ Review of MDKK homologyWe review the construction of MDKK homology and the virtual Rasmussen invariant.§.§.§ The complex Let 𝒜 = ℛ[X]/(X^2-t) for ℛ a commutative ring and t ∈ℛ. In order to detect the problem face a symmetry present in 𝒜 (which corresponds to the two possible orientations of S^1) is exploited using the following automorphism: The barring operator is the mapX : 𝒜→𝒜, X ↦ -X.Applying the barring operator is referred to as conjugation. Note that if ℛ = ℝ and t = -1 then 𝒜 = ℂ and the barring operator is just standard complex conjugation. How the barring operator is applied within the Khovanov complex is determined using an extra decoration on link diagrams, the source-sink decoration as depicted in <Ref>. A new diagram is formed by replacing the classical crossings with the source-sink decoration, which induces an orientation on the incident arcs of a crossing. Arcs of the diagram on which the induced orientations due to separate crossings disagree are marked by a cut locus. We refer the reader to <cit.>.§.§.§ The virtual Rasmussen invariantThere is a degeneration of Khovanov homology due to Lee <cit.>. There is such a degeneration of MDKK homology also. Dye, Kaestner, and Kauffman use the methods of Bar-Natan and Morrison <cit.> to show this. Specifically, they employ the Karoubi envelope of a category and the interpretation of virtual links as abstract links <cit.>, and define the virtual Rasmussen invariant.As such diagrams are used extensively below, we describe the process given in <cit.> to obtain a (representative of an) abstract link from a (representative of a) virtual link (examples are given in<Ref>). Let D be a diagram of a virtual link, as in <Ref>, then * About the classical crossings place a disc as shown in<Ref>. * About the virtual crossings place two discs as shown in <Ref>. * Join up these discs with collars about the arcs of the diagram.The result is a knot diagram on a surface which deformation retracts onto the underlying curve of the diagram. We will denote abstract link diagrams by ( F, D) for D a knot diagram and F a compact, oriented surface (which deformation retracts on to the underlying curve of D). We treat such diagrams up to stable equivalence, defined below. Let ( F_1, D_1 ) and ( F_2, D_2 ) be abstract link diagrams. We say that ( F_1, D_1 ) and ( F_2, D_2 ) are equivalent, denoted ( F_1, D_1 ) ↭( F_2, D_2 ), if there exists a closed, connected, oriented surface F_3 and orientation-preserving embeddings f_1 : F_1 → F_3, f_2 : F_2 → F_3 such that f_1 ( D_1 ) and f_2 ( D_2 ) are related by Reidemeister moves on F_3. We say that two abstract link diagrams ( F, D ) and ( F^', D^') are stably equivalent if there is a chain of equivalences( F, D ) = ( F_0, D_0 ) ↭( F_1, D_1 ) ↭…↭( F_n, D_n ) = ( F^', D^')for n ∈ℕ.Stable equivalence classes of abstract link diagrams are in bijective correspondence to equivalence classes of virtual link diagrams <cit.>. A smoothing of an abstract link diagram ( F , D ) is a diagram formed by smoothing the crossings of D into either their 0- or 1-resolution on F. The result is a collection of disjoint copies of S^1 on the surface F. A copy of S^1 is called a cycle.The diagram-level canonical generators of the Lee complex given in <cit.> are smoothings of abstract link diagrams with extra information added. This extra information keeps track of the source-sink structure of the virtual knot. The information is in the form of cross cuts which are added in the following way: before beginning the procedure described above mark the virtual knot diagram with cut loci as inherited from the source-sink orientation and preserve them on the abstract link diagram. Replace each cut locus with a cross cut which bisects the surface as shown in <Ref>. Henceforth by abstract link diagram we mean an abstract link diagram with cross cuts.Using the source-sink decoration we add yet more information to abstract link diagrams in the form of a checkerboard colouring: From an abstract link diagram ( F , D ) form its associated checkerboard coloured abstract link diagram from the surface and curve pair ( F , S ( D ) ) (where S ( D ) denotes the source-sink diagram formed by replacing each crossing by the source-sink decoration) by colouring the surface F using the recipe given in <Ref> and <Ref>. Notice that <Ref> allows us to induce a checkerboard colouring of smoothings of abstract link diagrams by simply joining the shaded or unshaded areas produced by smoothing the crossing.From checkerboard coloured smoothings of abstract link diagrams we are able to produce the tools used by Dye, Kaestner, and Kauffman to prove theorems analogous to those in <cit.>. Henceforth we set ℛ = ℚ and t = -1. Let { r, g } be the basis for 𝒜 where “red" = r = 1 + X2 “green" = g = 1 - X2. On the level of diagrams, arcs of a smoothing are coloured red or green to denote which generator they are labelled with.The properties of r and g are listed in Lemma 4.1 of <cit.>. The most important for our purposes is that r and g are conjugates with respect to the barring operator. That isr = g and g = r. An alternately coloured smoothing of an abstract link diagram is a smoothing for which the arcs have been coloured either red or green such that the arcs passing through each crossing neighbourhood are coloured different colours. At a cut locus the colouring of an arc switches.Using alternately coloured smoothings the following theorems are stated and proved: Within the Karoubi envelope the Lee complex of a virtual link K is homotopy equivalent to a complex with one generator for each alternately coloured smoothing of K on an abstract link diagram with cross cuts and with vanishing differentials. A virtual link K with | K | components has exactly 2^|K| alternately coloured smoothings on an abstract link diagram with cross cuts. These smoothings are in bijective correspondence with the 2^|K| orientations of K. In <Ref> we describe the bijective correspondence of <Ref>, but we conclude this section by stating the definition of the virtual Rasmussen invariant and its properties. Let K be a virtual knot diagram, ' ( K ) and ' ( K ) the associated Lee complex and Lee homology, respectively. Let s be the grading on ' ( K ) induced by j on ' ( K ). Define s_min ( K )= min{ s ( x ) | x ∈ ' ( K ), x ≠ 0 } s_max ( K )= max{ s ( x ) | x ∈ ' ( K ), x ≠ 0 }. The virtual Rasmussen invariant of K is s ( K ) = 1/2( s_max + s_min). The virtual Rasmussen invariant satisfies the following * s ( K ) = s_max - 1 = s_min + 1. * s ( K ) = - s ( K ), for K the mirror image of K: the diagram formed by switching all positive classical crossings to negative classical crossings and vice versa. * | s ( K ) | ≤ 2 g^∗ ( K ), where g^∗ ( K ) denotes the slice genus of K. Notice that the virtual Rasmussen invariant lacks the out-of-the-box additivity of its classical counterpart (a consequence of the ill-defined nature of the connect sum operation on virtual knots). In <Ref> we show, however, that the virtual s invariant is indeed additive. §.§ Doubled Khovanov homologyWe review doubled Khovanov homology and the doubled Rasmussen invariant.§.§.§ ConstructionDoubled Khovanov homology provides an alternative extension of Khovanov homology to virtual links <cit.>. The problem face is dealt with by “doubling up” the module assigned to a smoothing; this allows the map assigned to the single-cycle smoothing to be non-zero.A schematic picture of this “doubling up” process is given in <Ref>; the left hand complex depicts the situation when the module 𝒜 is assigned to a cycle within a smoothing. One sees immediately that the η map must be zero if it is to be degree-preserving. This is path followed by Manturov and Dye et al, and outlined in the previous section. The right hand complex, however, depicts the situation arrived at if one assigns the module 𝒜⊕𝒜{ -1 } to a cyle, where 𝒜 = ⟨, ⟩ and 𝒜{ -1 } = ⟨, ⟩ (the superscripts are u for “upper” and l for “lower”). This allows for η to be non-zero and degree preserving.Given a virtual link diagram, D, the complex ( D ) is formed in the usual way: form the cube of resolutions of D, then assign modules to the vertices and maps to the edges. The module assigned to a smoothing of j cycles is 𝒜^⊗ j⊕𝒜^⊗ j{ -1 }. The maps constituting the differential are as follows. The m and Δ maps are effectively unchanged: m( v^u/l_+ ⊗ v^u/l_+ ) = v^u/l_+Δ ( v^u/l_+ ) = v^u/l_+ ⊗ v^u/l_-+ v^u/l_- ⊗ v^u/l_+ m( v^u/l_+ ⊗ v^u/l_- )= m( v^u/l_- ⊗ v^u/l_+ ) = v^u/l_-Δ( v^u/l_- ) = v^u/l_- ⊗ v^u/l_- m(v^u/l_- ⊗ v^u/l_- ) = 0(notice that they do not map between the upper and lower summands). The η map associated to the single cycle smoothing as in <Ref> is given by η ( v^u_+ ) = v^l_+η ( v^l_+ ) = 2 v^u_-η ( v^u_- ) = v^l_-η ( v^l_- ) = 0.We denote by ( L ) the homology of the complex ( D ), where L is the link represented by D. We refer the reader to <cit.>.§.§.§ The doubled Rasmussen invariantAs in classical Khovanov and MDKK theories there is a perturbation of doubled Khovanov homology produced by adding a term of degree +4 to the differential. As in the other cases, this perturbation allows the definition of a concordance invariant. In this section we give the essentials we require for <Ref>, for full details we refer the reader to <cit.>.Given a virtual link diagram, D, let ' ( D ) denote the complex with the chain spaces of ( D ) but with altered differential. The homology of ' ( D ) is an invariant of the link represented by D, and is denoted ' ( L ) (where L is the link represented by D). The complex ' ( D ) is refered to as the doubled Lee complex, and the homology as the doubled Lee homology.The rank of doubled Lee homology of a link depends on the number of alternately coloured smoothings the link possesses - here we mean the usual notion of alternately coloured smoothing, rather than the augmented notion of alternately coloured smoothings on abstract link diagrams used in <Ref>. Unlike classical links, virtual links may posesses no alternately coloured smoothings. (In fact, one of the purposes of the extra decoration applied to diagrams in the construction of MDKK homology is to ensure that the oriented smoothing of the augmented diagrams is always alternately colourable.) Given a virtual link L rank( ' ( L ) ) = 2 | {alternately coloured smoothings of L } |.Further, given a diagram D of a virtual link L, each alternately coloured smoothing, 𝒮, (if any exist) defines two generators of ' ( L ), denoted ^u and ^l and known as an alternately coloured generators.A virtual knot has two alternately coloured smoothings <cit.> so that its doubled Lee homology is of rank 4. The four generators of the homology lie in a single homological degree, and the quantum grading of any one of them determines that of the others <cit.>. Thus, for a virtual knot, K, the information contained in ' ( K ) is equivalent to a pair of integers. For a virtual knot K the doubled Rasmussen invariant is denoted s ( K ) = ( s_1 ( K ), s_2 ( K ) ) ∈ℤ×ℤ, where s_1 ( K ) is equivalent to the highest non-trivial quantum degree of ' ( K ), and s_2 ( K ) is the single non-trivial homological degree of ' ( K ). The component s_2 ( K ) is easy to compute from any diagram, D, of K: it is the height of the alternately coloured smoothings of D. It is also equal to the odd writhe of K (see <cit.>). §.§ Even knotsTo conclude this section we give a class of virtual knots for which the two extensions of the Rasmussen invariant are equal.Recall the definition of an even virtual knot given in <Ref>; here prove a fact about the cube of resolutions associated to even virtual knot diagrams. Let D be an even virtual knot diagram. Then vCKh ( D ) and CDKh ( D ) contain no η maps. As D is even it possesses a global source-sink orientation i.e. applying the source-sink decoration does not yield any cut loci. (In fact, possessing a global source-sink structure is equivalent to being even, but here we only need one direction.) To see this orient D with either of it's orientations (the usual notion of orientation, not source sink), and consider leaving a classical crossing of D and returning to the arc proscribed by the usual orientation. One sees from <Ref> that passing through a classical crossing reverses the source-sink orientation. As all classical crossings of D are even, one passes through an even number of crossings between leaving and returning at the proscribed arc. Thus the source-sink orientation has been reversed an even number of times, yielding no overall change. This argument can be applied at every crossing to show that D has a global source-sink orientation. Next, notice that every smoothing of D inherits an orientation from the global source-sink orientation of D: looking again at <Ref> one sees that both resolutions of the classical crossing inherit an orientation from the source-sink decoration. That the orientation inherited is consistent between distinct classical crossings of D follows from that fact that D has no cut loci. Finally, we notice that if every smoothing of D inherits a coherent orientation from the global source-sink orientation of D then every cycle within a smoothing must look as in the left or center of <Ref>, as the configuration on the right is prohibited for reasons of (source-sink) orientation. But we see that the configurations on the left and center correspond to either a merge or a split, while the configuration on the right corresponds to the single-cycle smoothing. Thus no single-cycle smoothings can occur in the cube of resolutions of D and we arrive at the desired result. Let K be an even virtual knot. Then DKh ( K ) = vKh ( K ) ⊕ vKh ( K ) { -1 } so that s ( K ) = s_1 ( K ). Let D be an even diagram of K. Then both vCKh ( D ) and CDKh ( D ) contain no η maps by <Ref>. As m and Δ do not map between the shifted and unshifted summands of CDKh ( D ), the complex splits as the direct sum CDKh ( D ) = vCKh ( D ) ⊕ vCKh ( D ) { -1 }. § CHAIN-LEVEL GENERATORS OF VKH 'In <cit.> canonical generators are produced at a diagrammatic level i.e. they are alternately coloured smoothings of (checkerboard-coloured) abstract link diagrams. These generators are sufficient to prove Theorems <ref> and <ref>. Below, we give a method to produce the corresponding chain-level generators of ' ( K ). Before doing so, however, it is instructive to recall the bijection of <Ref> between orientations of a virtual link and alternately coloured smoothings of the associated abstract link diagram as given in <cit.>. We use <Ref> as an example. * Given a virtual link diagram D construct the checkerboard coloured abstract link diagram as in <Ref>. Note that for a virtual knot the checkerboard colouring is independent of the orientation, a consequence of the invariance of the source-sink decoration under 180^∘ rotations. See <Ref>. * For a given orientation o of D form the corresponding oriented smoothing on the checkerboard coloured abstract link diagram as in <Ref>. See <Ref>. *Place a clockwise orientation on the shaded regions of the oriented smoothing, which in turn induces a new orientation on the arcs of the smoothing. On each arc compare this orientation to that induced by o. If these two orientations agree colour the arc red, if they disagree colour the arc green (as in <Ref>). See <Ref>.At this stage we have produced alternately coloured smoothings on abstract link diagrams as in <Ref>. We need a way of reading off from these diagrams elements of '_0 ( K ) (as the oriented resolution is always at height 0), which will be the chain-level canonical generators of ' ( K ). We are unable to do so at this point as the cycles of the alternately coloured smoothings possess more than one colour. We now describe a process by which single coloured smoothings can be produced, and hence chain-level generators of ' ( K ).Firstly, we utilise the stable equivalence relation given in <Ref> to work with alternately coloured smoothings of abstract link diagrams for which the surface deformation retracts onto the curve of the smoothing, for example the abstract link diagrams given in <Ref>. We can always do this as the curve of the smoothing is simply a disjoint union of copies of S^1. Note that the resulting smoothing (of a checkerboard coloured abstract link diagram) may not be connected.Next, we interpret the cross cuts as half-twists with the parity of the twist ignored. That is-22.5pt < g r a p h i c s >= -26.5pt < g r a p h i c s > or equivalently -26.5pt < g r a p h i c s > .The author learnt of this interpretation in the talks of Dye and of Kaestner during Special Session 35, “Low Dimensional Topology and Its Relationships with Physics”, of the 2015 AMS/EMS/SPM Joint Meeting.Replacing cross cuts with appropriate half-twists we are able to view the surface of the smoothing (of a checkerboard coloured abstract link diagram) as a two-sided surface such that the curve of the smoothing appears on both sides. That cross cuts always come in pairs ensures that the surface has two sides. Importantly, on each side of the surface the curve of the smoothing is coloured exactly one colour. This is because passing a cross cut causes the arc to change to change colour (c.f. <Ref>), and to pass a cut locus is to pass onto the other side of the surface. (From this one can see that passing a cut locus, or equivalently moving on to the other side of the surface, is replicated in𝒜 by applying the barring operator.)In summary, we view alternately coloured smoothings (of checkerboard coloured abstract link diagrams) such as those in <Ref> as two sided surfaces such that the curve of the smoothing is coloured exactly one colour on each side. At this point it is clear that in order to read off generators of '_0 ( K ) from such alternately coloured smoothings we must make a choice of side (or sides, if the surface of the smoothing is disconnected) of the surface to read. Further, we must also ensure that this choice is the same for both the alternately coloured smoothings associated to o and o. We must have this as they are both coloured versions of the same smoothing of an abstract link diagram (the oriented smoothing) c.f. the left hand smoothing of <Ref> with <Ref>. In effect we are making the choice on this uncoloured smoothing, which the alternately coloured smoothings then inherit.With all this in mind, let us make a choice: given a virtual knot diagram K with orientations o and o, let A denote the oriented smoothing of the checkerboard coloured abstract link diagram associated to K. On A cancel an arbitrary pair of adjacent cross cuts against one another so that the strand they bound is removed. An example is given in <Ref>. This cancellation of cross cuts is simply `flipping' the segment of the surface they bound so that the other side of the surface is shown. Continue cancelling available arbitrary pairs of cross cuts until all have been removed. In our interpretation, that the smoothing has no cross cuts means that we are looking at exactly one side of surface. Now return to part (iii) of the process given on page part3, and colour the cycles of the oriented smoothings associated to o and o as dictated there. Denote by A_o and A_o the resulting alternately coloured abstract link diagrams associated to o and o, respectively. That the cycles of A_o and A_o are coloured with opposite colours follows from the fact that their orientations are opposite but the checkerboard colouring of A_o and A_o is the same.Examples of such single coloured smoothings are given in <Ref> and <Ref>. In this case a choice of top and bottom is equivalent to picking either the two smoothings on the left of the Figures, or the two on the right. After all that we are left with smoothings of abstract link diagrams the cycles of which are coloured with exactly one colour, either red or green. We form the canonical generators of ' ( K ), denoted _o for o an orientation of K, by taking the appropriate tensor product of r and g as dictated by the colours of the cycles. In this way we obtain two distinct algebraic generators.We conclude by remarking that the s invariant is independent of this choice of which side of the surface to read. Making another choice results in an application of the barring operator to one or more tensor factors of _o and _o, because if a cycle is coloured green on one side of the surface it must be coloured red on the other. But conjugation does not interact with the filtration, that isj(r) = j (g) and j(g) = j (r). To conclude this section we prove a Lemma analogous to Lemma 3.5 of Rasmussen <cit.> which we will use in both the following sections. Let n be the number of components of K. There is a direct sum decomposition '(K) ≅'_o(K) ⊕'_e(K), where '_o (K) is generated by all states with q-grading conguent to 2+n 4, and '_e (K) is generated by all states with q-grading congruent to n 4. If o is an orientation on K, then _o + _o is contained in one of the two summands, and _o - _o is contained in the other. The first statement follows exactly as in the classical case. Regarding the second statement, following <cit.> let ι :' ( K ) → ' ( K ) be the map which acts by the identity on '_e(K) and multiplication by -1 on '_o(K). We claim that ι ( _o ) = ±_o. To show this we define a new grading on 𝒜 with respect to which X has grading 2 and 1 has grading 4. We have that X = - X and 1 = 1 so that r = g and g = r, and the map X^⊗ n : 𝒜^⊗ n→𝒜^⊗ n (which applies the barring operator to all tensor factors) acts as the identity on elements with new grading congruent to 0 4 and multiplication by - 1 on elements with new grading congruent to 2 4. The new grading differs from the q-grading by an overall shift so that ι ( _o ) = ±_o^⊗ n = ±_o as in the classical case. A direct corollary of <Ref> is that _o is not of top filtered degree, that is:s ( _o ) = s ( _o ) = s_min ( K ).§.§ Additivity of the virtual Rasmussen invariant We can use the chain-level generators of ' ( K ) to show that the virtual Rasmussen invariant is additive with respect to connect sum, confirming that the virtual invariant behaves in the same way as its classical counterpart in this respect.The connect sum operation on virtual knots is ill-defined. That is, the result of the operation depends on both the diagrams used and the site at which the sum is conducted. As a result there exist multiple inequivalent virtual knots which can be obtained as connect sums of a fixed pair of virtual knots. By an abuse of notation we shall denote by K_1 # K_2 any of the knots obtained as a connect sum of virtual knots K_1 and K_2. For virtual knots K_1 and K_2 s ( K_1 # K_2 ) = s ( K_1 ) + s ( K_2 ). With the chain-level generators in place, along with <Ref>, the proof follows much the same path as that in <cit.>. For all connect sums K_1 # K_2 there exists the map ' ( K_1 # K_2 ) Δ '⟶ ' ( K_1 ⊔ K_2 ) ≅ ' ( K_1 ) ⊗ ' ( K_2 ). It sends a canonical generator _o of ' ( K_1 # K_2 ) to a canonical generator of ' ( K_1 ) ⊗ ' ( K_2 ) of the form ^1 ⊗^2 where ^i is a generator of ' ( K_i ) for i = 1, 2. As in the classical case, the map is of filtered degree -1 and we obtain s ( _o ) - 1≤ s ( ^1 ⊗^2 ) = s ( ^1 ) + s ( ^2 ) s_min ( K_1 # K_2 )≤ s_min ( K_1 ) + s_min ( K_2 ), by <Ref>. From this point the proof proceeds as in that of the analogous statement in <cit.>: utilising the fact that s_min ( K ) = - s_max (K ) we are able to obtain from <Ref> that s_min ( K_1 # K_2 )= s_min ( K_1 ) + s_min ( K_2 ) + 1 s_max ( K_1 # K_2 )= s_max ( K_1 ) + s_max ( K_2 ) - 1 as required.In light of <Ref> we see that the Rasmussen invariant cannot distinguish between connect sums of a fixed pair of virtual knots. In general it is not known, for K_1 and K_2 both (possibly inequivalent) connect sums of a fixed pair of virtual knots, if K_1 is concordant to K_2. It is known, however, that neither the Jones polynomial <cit.> nor the Rasmussen invariant can distinguish them. This leads one to posit whether Khovanov homology can; in the case of connect sums of trivial diagrams it is shown in <cit.> that doubled Khovanov homology cannot.§ COMPUTABLE BOUNDSIn this section we extend the strong slice-Bennequin bounds to the virtual and doubled Rasmussen invariants. The bounds are constructed, and cases in which they vanish partly or wholly are described. §.§ The virtual Rasmussen invariant§.§.§ Formulation Given a virtual link diagram D denote by O ( D ) the oriented smoothing of D. Denote by T_O ( D ) the signed graph with a vertex for each cycle of O ( D ) and an edge for each classical crossing of D, decorated with the sign of the crossing. The edge associated to a crossing is between the vertex or vertices associated to the cycles involved in the smoothing site of that crossing. The subgraph of T_O ( D ) formed by removing all the edges labelled with + (respectively -) is denoted T^-_O ( D ) (respectively T^+_O ( D )).The graph T_O ( D ) is often called the Seifert graph of D, but in order to avoid confusion with a graph defined in <Ref> we shall not use that term. Given a virtual knot diagram D the quantities U_v ( D ), Δ_v ( D ) ∈ℤ are given by U_v ( D )= # vertices ( T_O ( D ) ) - 2 # components ( T^-_O ( D ) ) + wr ( D ) + 1Δ_v ( D )= # vertices ( T_O ( D ) ) - # components ( T^+_O ( D ) ) - # components ( T^-_O ( D ) ) + 1. The quantities U_v ( D ) and Δ_v ( D ) are dependent on the diagram D and are not invariants of the virtual knot. For D a diagram of a virtual knot K s ( K ) ≤ U_v ( D ). Notice that the left hand side is a knot invariant whereas the right is diagram-dependent. To prove this we require <Ref>, as we have canonical generators in terms of r and g instead of a = 2r and b = -2g and the proof given in <cit.> relies on the sign of a and b. (of <Ref>) The proof is practically identical to that of the classical case given in <cit.>. Form the diagram K^- from K by smoothing all the positive classical crossings of K to their oriented resolution, and suppose that K^- is the disjoint union of l virtual link diagrams. Label these diagrams K^-_1, K^-_2, …, K^-_l. Then the canonical generator _o splits as a tensor product of canonical generators of ' ( K^-_r ) as _o = _1 ⊗_2 ⊗…⊗_l. Classically, _r can either be _o^' or _o^' where o^' denotes the induced orientation on K^-_r, as we are possibly altering the number of cycles separating others from infinity. In the virtual case, however, _r = _o^' by construction as we use abstract link diagrams to produce the canonical generators rather than the method due to Lee. Where the proof given in <cit.> invokes Theorem 3.5 of <cit.> we invoke <Ref> as given above. If Δ_v ( D ) = 0 then s ( K ) = U_v ( D ), where K is the virtual knot represented by D. In fact U_v ( D ) - 2 Δ_v ( D ) ≤ s ( K ) ≤ U_v ( D ).The proof of <Ref> is identical to that of the classical case, owing to the identical behaviour of the virtual and classical Rasmussen invariants with respect to the mirror image.§.§.§ The case Δ_v ( D ) = 0 Cromwell defined homogeneous knots <cit.>. Here we recap his definition, which works equally well for virtual knots. A cut vertex of a graph G is a vertex such that the graph obtained by removing the vertex along with its boundary edges has more connected components than G. A block of a graph G is a maximal connected subgraph of G containing no cut vertices. A signed graph G is homogeneous if every block B of G is such that all edges contained in B are decorated with the same sign. A virtual link diagram K is homogeneous if T_O ( K ) is homogeneous. A virtual link is homogeneous if there exists a diagram of it which is homogeneous.Positive and negative virtual knots are homogeneous trivially (as T_O ( D ) possesses only one kind of decoration). In the classical case alternating knots are also homogeneous <cit.>. In the virtual case, however, this no longer holds. For example, the virtual knot diagram given in <Ref> is alternating but not homogeneous.Abe showed that for a classical knot diagram D Δ_v ( D ) = 0 if and only if D is homogeneous <cit.>. However, Abe's proof relies on T_O ( D ) containing no loops (an edge which begins and ends at the same vertex); classically, this is always the case as the oriented resolution is the alternately coloured resolution, so that T_O ( D ) is bipartite. Virtually, however, there are knots whose oriented resolution is not the alternately coloured resolution; this is explained fully below. An example is given in <Ref>. For now, it suffices to recall that the quantity Δ_v can be expressed as the first Betti number of the graph, G_O, defined as follows. Let T_O ( D ) be associated to the virtual knot diagram D. Form the graph G_O in the following way: * For each connected component of T^+_O ( D ) place a vertex, and a vertex for each connected component of T^-_O ( D ). * Each vertex of T_O ( D ) lies in exactly one connected component of T^+_O ( D ), and exactly one connected component of T^-_O ( D ). For each vertex of T_O ( D ) place an edge linking the vertices of G_Δ corresponding to the connected components in which it lies. Let T_O ( D ) be associated to the virtual knot diagram D, and T_O ( D ) be a graph obtained from T_O ( D ) by adding or removing a loop (of arbitrary sign). Further, let G_O be the graph formed from T_O ( D ) following the method given in <Ref>, where T^+_O ( D ) and T^-_O ( D ) are formed in the obvious way. Then G_O = G_O. It is clear that #components ( T^+/-_O ( D ) ) = #components ( T^+/-_O ( D ) ) (we have only added or removed a loop) so that #vertices ( G_O ) = #vertices ( G_O ). Further, as loops do not connect distinct vertices, two vertices are linked in G_O if and only if they are linked in G_O. In light of <Ref> it is clear that we need only consider homogeneity of T_O ( D ) up to the addition and removal of loops. Let G be a signed graph and let G be the graph formed by removing all loops of G. Then G is l-homogenous if G is homogenous. A virtual knot diagram is l-homogenous if T_O ( D ) is, and a virtual knot is l-homogenous if it has an l-homogenous diagram. A virtual knot diagram D is l-homogeneous if and only if Δ_v ( D ) = 0. Hence, for an l-homogeneous diagram D of a virtual knot K U ( D ) = s ( K ). Abe's original proof yields the following statement: if D is such that T_O ( D ) is loopless, then D is homogenous if and only if Δ_v ( D ) = 0. By <Ref> we may remove any loops from T_O ( D ), leaving the associated G_O unchanged. Recalling that Δ_v ( D ) = b_1 ( G_O ), the first Betti number of G_O, we obtain the desired result.§.§ The doubled Rasmussen invariant §.§.§ Formulation In formulating the bounds on the doubled Rasmussen invariant we follow much the same path as in <Ref>. The formulae arrived at in this section exhibit important differences between those of <Ref>, however, owing to the structural differences between MDKK homology and doubled Lee homology.We begin by making some preliminary definitions. Let D be a diagram of a virtual knot and G ( D ) its Gauss diagram. A classical crossing of D, associated to the chord labelled c in G ( D ), is known as odd if the number of chord endpoints appearing between the two endpoints of c is odd. Otherwise it is known as even. The odd writhe of D is defined J ( D ) = ∑_odd crossings of Dsign of the crossing. Let D be a virtual knot diagram of K. The odd writhe is an invariant of K and we define J ( K )J ( D ). Let D be a virtual knot diagram. The alternately coloured resolution of a classical crossing of D is the resolution it is smoothed into in the alternately colourable smoothing of D. A classical crossing of a virtual knot diagram is odd if and only if it's alternately coloured resolution is the unoriented resolution. In the construction of MDKK homology source-sink decorations are used to ensure that the oriented resolution of a virtual knot is, in fact, alternately colourable; doubled Khovanov homology does not do so. In the definition below, therefore, we consider the graph associated to the alternately coloured smoothing of a virtual knot. Given a virtual link diagram D denote by 𝒮 ( D ) the alternately coloured smoothing of D. Denote by T_𝒮 ( D ) the graph with a vertex for each cycle of 𝒮 ( D ) and an edge for each classical crossing of D, decorated with the sign and parity of the crossing: every edge is decorated with an element of { (e,+),(e,-),(o,+),(o,-) }, where (e,+) denotes an even positive crossing, (o,+) an odd positive crossing, and so on. The edge associated to a crossing is between the vertex or vertices associated to the cycles involved in the smoothing site of that crossing. The subgraph of T_𝒮 ( D ) formed by removing all the edges labelled with either (e,+) or (o,-) is denoted T^_𝒮 ( D ). The subgraph of T_𝒮 ( D ) formed by removing all the edges labelled with either (e,-) or (o,+) is denoted T^_𝒮 ( D ). Let D be a virtual knot diagram with n^o_+ (n^o_-) odd positive (negative) classical crossings. Define the quantities U_d ( D )= #vertices ( T_𝒮 ( D ) ) - 2 #comp ( T^_𝒮 ( D ) ) + wr ( D ) + J ( D ) + n^o_+ +1Δ_d ( D )= 2 ( #vertices ( T_𝒮 ( D ) ) - #comp ( T^_𝒮 ( D ) ) - #comp ( T^_𝒮 ( D ) ) + 1 )+ n^o_+ + n^o_- where #comp denotes the number of components of a graph.In direct analogy to <Ref> we have the following. Let D be a diagram of a virtual knot K. Then U_d ( D ) - Δ_d ( D ) ≤ s_1 ( K ) ≤ U_d ( D ). We shall go through the proof of <Ref> in more detail than that of it's counterpart <Ref>, owing to the aforementioned differences between the theories vKh ' and DKh '. The gist of the proof is unchanged, however: as computation of s_1 ( K ) only requires knowledge of the partial chain complex [roundnode/.style=] [roundnode] (s0)at (-5,0)_s_2(K)-1 ( D ) '; [roundnode] (s1)at (0,0)_s_2(K) ( D ) '; [->,thick] (s0)–(s1) node[above,pos=0.5]d_s_2(K)-1; we ignore (by resolving them) classical crossings whose alternately coloured resolution is the 0-resolution; such crossings are associated to outgoing maps from the alternately coloured resolution of D and do not contribute to d_s_2(K)-1. This comes at the price, of course: we lose a large amount of the information contained in '_s_2(K) ( D ). Nevertheless, the trade is a worthwhile one, as we are able to use what's left to obtain bounds on s_1 ( K ). Let D be a diagram of a virtual knot K, with n_+ (n_-) positive (negative) classical crossings. Further, let n_+ = n^e_+ + n^o_+ and n_- = n^e_- + n^o_-, where a superscript e (o) denotes the number of even (odd) crossings. Form a virtual link diagram, D, by resolving all even positive crossings and all odd negative crossings of D into their alternately coloured resolutions. (One readily observes that such crossings are those with alternately coloured resolution the 0-resolution, as mentioned above.) We can write D = D_1 ⊔D_2 ⊔…⊔D_r where D_i is a virtual link diagram with n^i_+ positive and n^i_- negative classical crossings (the parity of positive (negative) crossings is necessarily odd (even), of course). Further, for 𝒮 the alternately colourable smoothing of D, we have 𝒮 = 𝒮_1 ⊔𝒮_2 ⊔…⊔𝒮_r where 𝒮_i is the unique alternately colourable smoothing of D_i formed by resolving all crossings into the resolution they are resolved into in 𝒮. Notice that while ' ( D ) does not split as a tensor product of the ' ( D_i )'s, the alternately coloured generators of ' ( K ) do. That is, if ^u is associated to 𝒮, then ^u = ^u_1 ⊗^u_2 ⊗⋯⊗^u_r where ^u_i is the alternately coloured generator defined by 𝒮_i. We have J ( D_i ) = n^i_+ (as all negative crossings of D_i are even), so that the highest non-trivial quantum grading of '_n^i_+ ( D_i ) containing [ ^u_i ] is e_i + n^i_+ + n^i_+ - n^i_-, where e_i denotes the number of cycles of 𝒮_i. Further, as a corollary to Lemma 4.2 of <cit.>, we determine that [ ^u_i ] is not of top degree, and that e_i + n^i_+ + n^i_+ - n^i_- - 2 is the highest non-trivial degree of '_n^i_+ ( D_i ) containing it. By <Ref> and an argument directly analogous to Lobb's <cit.> we obtain s^u_min ( K )≤ n^e_+ - n^o_- + ∑_i=1^r( e_i + n^i_+ + n^i_+ - n^i_- - 2 ) = wr ( D ) + J ( D ) + n^o_+ + #vertices ( T_𝒮 ( D ) ) - 2 #comp ( T^_𝒮 ( D ) ). Recalling that s^u_min ( K ) = s_1 ( K ) + 1, we arrive at s_1 ( K ) ≤ U_d ( D ). To see that U_d ( D ) - Δ_d ( D ) ≤ s_1 ( K ) repeat the proof of <Ref>, which we are free to do as the doubled Rasmussen invariant replicates the behaviour of its classical counterpart with respect to the mirror image. §.§.§ Simplifying Δ_d ( D ) Much of the analysis used in the <Ref> may be repeated in order to characterise a case in which the Δ_d formula simplifies. However, we do not recover the vanishing result as in the case of Δ_v. Let D be a virtual knot diagram and T_𝒮 ( D ) the graph associated to it. Recall that each edge of T_𝒮 ( D ) is decorated with exactly one element of { (e,+),(e,-),(o,+),(o,-) }. Let = { (e,-),(o,+) } and = { (e,+),(o,-) }. The graph T_𝒮 ( D ) is d-homgenous if every block is decorated with elements of eitheror , but not both. The diagram D is d-homogenous if T_𝒮 ( D ) is d-homogenous. A virtual knot is d-homogenous if it has a d-homogenous diagram. Let D be a virtual link diagram and T_𝒮 ( D ) the graph associated to it. Then D is d-homogenous if and only if #vertices ( T_𝒮 ( D ) ) - #comp ( T^_𝒮 ( D ) ) - #comp ( T^_𝒮 ( D ) ) + 1 = 0. Let G_𝒮 denote the graph formed from T_𝒮 ( D ) in direct analogy to G_O, as given in <Ref>, with T^_𝒮 ( D ) and T^_𝒮 ( D ) taking the place of T^+_O ( D ) and T^-_O ( D ). The graph T_𝒮 ( D ) is bipartite as 𝒮 ( D ) is alternately coloured. Thus it is loopless and Abe's proof may be employed to show that T_𝒮 ( D ) is homogenous if and only if b_1 ( G_𝒮 ) = 0. We conclude by noticing that b_1 ( G_𝒮 ) = #vertices ( T_𝒮 ( D ) ) - #comp ( T^_𝒮 ( D ) ) - #comp ( T^_𝒮 ( D ) ) + 1, which follows exactly as in the case of Δ_v and G_O. Let D be diagram of a virtual knot K. If D is d-homogenous then U_d ( D ) - n^o_+ - n^o_- ≤ s_1 ( K ) ≤ U_d ( D ) where n^o_+ (n^o_-) denotes the number of odd positive (negative) classical crossings of D. § COMPUTATION AND ESTIMATION OF THE SLICE GENUS In this section we use the bounds U_v and U_d to compute or estimate the slice genus of a number of virtual knots. The computations are made by finding a surface of appropriate genus between the given knot and the unknot.The following table contains the results of the analysis for the virtual knots of 4 crossing or less in Green's table <cit.>. A blank entry denotes an unknown, and most computations of s, s_1, and s_2 (or the interval in which they lie) are made by computing U_v/d, Δ_v/d, and J for the diagram given in the table. The exceptions to this are s_1 ( 3.3 ), which the author computed by hand from ' ( 3.3 ), and leftmost knots, for which the definition and the method of computation of s_1 are given in <cit.>. Further, many computations of s, s_2, and s_2 are made by spotting that the knot in question is a connect sum of two other knots, and employing the additivity of the invariants along with their invariance under flanking <cit.>. (As observed in <Ref>, s and s_1 coincide for even knots, so that the invariants are buy one get one free in this case.)Exact values of g^∗ are obtained by constructing a cobordism which attains a lower bound given by s, s_1, or s_2. Upper bounds on g^∗ are obtained by constructing a cobordism of the given genus, and employing the fact that half the crossing number bounds the slice genus of a knot from above (as in the classical case) <cit.>. Shortly after posting a previous version of this paper to the arXiv the author learned of the work of Boden, Chrisman, and Gaudreau in which they compute or estimate the slice genus of a very large number of the virtual knots of 6 crossings or less <cit.>. In the table below we do not include the values of g^∗ they arrive at in order to demonstrate the infomation that can be obtained using the bounds U_v, U_d, and the properties of the virtual and doubled Rasmussen invariants. [head to column names,longtable=|c|c|c|c|c|c|c|c|,table head=Knot l-hom. d-hom. s s_1 s_2 g^∗,table foot=]vknotinfo_v2.csv 1=,5=,6=,7=,8=,9=,10= From the table we are able to make some observations regarding the two extensions of the Rasmussen invariant. We see that only s_1 is able to distinguish between 2.1 and 3.3. Further, there are a number of knots for which the easy to compute s_2 obstructs sliceness while the harder to compute s does not. The virtual and doubled Rasmussen invariants are also able to distinguish many pairs of knots which have the same positive slice genus, showing that they are not concordant to one another.We also give presentations of the surfaces of genus 0, 1, and 2 used to determine the slice genus of the knots 4.8, 3.5, and 4.15 respectively; they are contained in <Ref>. Unlabeled arrows denote virtual Reidemeister moves, while those which denote 1-handle additions are so labelled. Red arcs between strands denote the locations of such handle additions within individual diagrams.To conclude we list the results of similar analysis as that used to produce the previous table, this time on the virtual knots for which Boden, Chrisman, and Gaudreau's methods are unable to obstruct sliceness but the virtual and doubled Rasmussen invariants can. The upper bounds on g^∗ are those given by Boden, Chrisman, and Gaudreau <cit.>. As in the case of knots of 4 or less crossings many of the computations are made by spotting connect sums. [head to column names,longtable=|c|c|c|c|c|c|c|,table head=Knot l-hom. d-hom. s s_1 s_2 g^∗,table foot=]vknotinfo_gaps.csv 1=,5=,6=,7=,8=,9=,10= plain | http://arxiv.org/abs/1706.08279v2 | {
"authors": [
"William Rushworth"
],
"categories": [
"math.GT",
"57M25, 57M27, 57N70"
],
"primary_category": "math.GT",
"published": "20170626084537",
"title": "Computations of the slice genus of virtual knots"
} |
Processing of ices rich in SO_2 Astronomical Institute of Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovakia Centre de Recherche sur les Ions, les Matériaux et la Photonique, Normandie Univ, ENSICAEN, UNICAEN, CEA, CNRS, CIMAP, 14000 Caen, France INAF-Osservatorio Astrofisico di Catania, via Santa Sofia 78, I-95123 Catania, ItalyZ. Kaňuchová,[email protected] is an abundant element in the cosmos and it is thus an important contributor toastrochemistry intheinterstellar medium and in the Solar System. Astronomical observationsof the gas and of the solid phases in the dense interstellar/circumstellar regions have evidenced that sulfur is underabundant. The hypothesis to explain such a circumstance is that it is incorporated in some speciesin the solid phase (i.e. as frozen gases and/or refractory solids) and/or in the gas phase, which for different reasons have not been observed so far.Here we wish to give a contribution to the field by studying the chemistry induced by thermal and energetic processing of frozen mixtures of sulfur dioxide (one of the most abundant sulfur-bearing molecules observed so far) and water. We present the results of a series of laboratory experimentsconcerning thermal processing of differentH_2O:SO_2 mixtures and ion bombardment (30 keV He^+) of the same mixtures. We usedin situ Fourier transform infrared(FTIR) spectroscopy to investigate the induced effects.The results indicate that ionic species such as HSO_3^-, HSO_4^-, and S_2O_5^2- areeasily produced. Energetic processing also producesSO_3 polymers and a sulfurous refractory residue. The produced ionic species exhibit spectral features in a region that, in astronomical spectra of dense molecular clouds, is dominated bystrong silicate absorption. However, such a dominant feature isassociated with some spectral features, some of which havenot yet been identified. We suggestadding the sulfur-bearing ionic species to the list of candidates to helpexplain some of those features. In addition, we suggest that once expelled in the gas phase by sublimation, due to the temperature increase, and/or by non-thermal erosion those species would constitute a class of molecular ions not detected so far.We also suggest that molecular sulfur-bearing ions could be present on the surfaces and/or in the atmospheres of several objects in the Solar System, for example icy satellites of the giant planets and comets. Kaňuchová & al Thermal and energetic processing of SO_2 rich ices Thermal and energetic processing of astrophysical ice analogues rich in SO_2 Z. Kaňuchová corresponding author1,3 Ph. Boduch 2 A. Domaracka 2 M.E. Palumbo3 H. Rothard2G. Strazzulla 3Received / Accepted========================================================================================================================================================================================================================================== § INTRODUCTION Atomic sulfur is an abundant element in the cosmos. Its abundance relative to hydrogen is about 1.32 × 10^-5 <cit.> andis thus an important contributor tochemical evolution in the galaxies and in the Solar System. However, many astronomical observations (both of the gas and of the solid phases) in the dense interstellar medium (ISM) and in star forming regions have evidenced that sulfur is underabundant, i.e. the sum of sulfur atoms locked in the sulfur-bearing molecules detected so far only accounts for a fraction of its cosmic abundance.As an example, <cit.> summed up the abundances of SO, CS, SO_2, and H_2S, which are the most abundant S-bearing molecules observed in the gas phase both in low- and high-density molecular clouds. They concluded that these molecules only accountfor a fraction of the sulfur abundance in the cosmos, of the order of10^-3. In the solid phase, only OCS <cit.> and SO_2 <cit.> have been detected so far in icy grain mantles toward high-mass protostars. Their estimated abundances are low, however,and can account for only about 0.5%, and0.8-4.0%, respectively, of the total sulfur abundance <cit.>.Thus, the problem of the missing sulfur is a hot question in astrochemistry. It is obvious to postulate that sulfur is incorporated in some species – either in the solid phase and/or in the gas phase – which for different reasons have not yet been observed. It is important to note that this lack concerns the dense interstellar medium only. In diffuse clouds the amount of gas phase sulfur fully accounts for the total sulfur abundance and rules out the possibility of its depletionon refractory interstellar grains (see e.g. ).In this context some studies have been performed to try to understand which species, although not yet observed in the solid phase, could be present, and then, once expelled to the gas phase by thermal or non-thermal processes, could be searched for in the gas phase. It isbelieved that the desorption of grain mantle species into the gas phase, for example afterwarming by a protostar or sputtering by energetic cosmic ions, gives an important contribution to the gas phase composition <cit.>. As outlined by <cit.> it seemsplausible that an important fraction of the observed S-bearing gas phase species is released from grains because molecules such as H_2S, SO_2, OCS, SO, H_2CS, HCS^+, and NShave abundances that cannot be explained with gas-phase-only chemical models <cit.>.A molecule that has been considered in particular detail is H_2S. This is a very important point because as soon as the medium recondenses (toward the formation of molecular clouds), the high abundance of hydrogen easily produces hydrogenated solid speciessuch as H_2O, CH_4, and NH_3 which are, indeed, well observed. It is then puzzlingthat this is not the case for H_2S <cit.>. At present it is not clear whether this is due to observational difficulties. The next generation of instruments, namely the James Webb Space Telescope (JWST) could clarify the question. In the meantime, a number of laboratory experiments simulating the energetic processing of icy mantles on grains in the ISM have demonstrated that energetic processing of solid H_2S by ions and photons (UV, X-rays) produce sulfur-sulfur bonds (H_2S_2 and HS_2 are easily formed) and also a refractory polymer-like residue<cit.>.It has beensuggested that part of the missing sulfur could be in solid unvolatile refractory grains <cit.> or it could be released in the gas phase as H_2S_2,HS_2,S_2 <cit.>,or CS_2 <cit.>. This would also imply that hydrogen sulfide is not detected since it is easily transformed into different species.These findings stimulated recent efforts to observe H_2S_2, HS_2, and S_2 in the gas phase toward the low-mass warm core IRAS 16293-2422 <cit.>. Estimated upper limit abundances of these molecules are up to two orders of magnitude lower than the H_2S abundance in the source. This possibly indicates that gas-phase chemistry after their desorption from the icy mantles efficiently destroys those species.With the aim ofcontributing to the field, we present here the results of a series of experiments conducted at the laboratories of the Centre de recherche sur les Ions, les MAtériaux et la Photonique (CIMAP)-Grand Accélérateur National d'Ions Lourds (GANIL)in Caen (France) and at theLaboratorio di Astrofisica Sperimentale (LASp) in Catania (Italy). The experiments conducted at CIMAP-GANIL concern the thermal processing of different mixtures H_2O:SO_2 and the implantation of multicharged sulfur ions in water ice; the experiments conducted in Catania are relative to ion bombardment (30 keV He^+) of the same mixed species. The results indicate that ionic species such as HSO_3^-, HSO_4^-, and S_2O_5^2- are produced by the three processes (thermal, S-implantation in pure water ice, and ion bombardment) and we suggest that they have to be searched for in the inter- and circumstellar regions where they could contribute to the inventory of the missing sulfur atoms.Our experiments are also relevant to some objects in the Solar System, namely Jupiter's Galilean satellites <cit.>. In particular, frozen SO_2 is the dominant species at the surface of Io, and it was also observed in cometary comae <cit.>. Our results are therefore discussed also in the light of their relevance for these objects.§ EXPERIMENTAL PROCEDURE The experiments conducted at CIMAP-GANIL concern the thermal processing of different mixtures of H_2O:SO_2. The frozen samples were prepared by condensing opportune mixtures of water and sulfur dioxide gases on a CsI window at 16 K.A fine valve allowedthe deposition rate to be controlled. A nozzle was used to transmit the gas into the high vacuum chamber and onto the cold CsI substrate installed in thecentre of the chamber on a cold finger connected to a closed-cycle helium cryostat. The pressure in the high vacuum chamber was below 10^-7 mbar.The temperature of the substrate was controlled by a carbon resistance and a compound linear thermal sensor (CLTS) situated on the holder, providing a precision of 0.1 K. After deposition the samples wereheated up at a rate of about 1K/min and IR spectra taken at the chosen temperatures in the spectral range 5000 - 600 cm^-1 (2-16.7 μm) with a resolution of 1 cm^-1. To this end, a Nicolet Magna 550 Fourier Transform Infrared Spectrometer (FTIR) was used. The spectra are taken in transmittance, at normal incidence, and were corrected by a background spectrum recorded before deposition <cit.>.In the experiments conducted in Catania,H_2O:SO_2 (1:2) mixtures wereaccreted onto a cold (16 K) silicon substrate in a vacuum chamber (P < 10^-7 mbar).Infrared transmittance spectra (resolution of 1 cm^-1) were obtained, before and after 30 keV He^+ ion bombardment, bya Bruker Equinox 55 FTIR spectrometer. Ion beams were produced by an ion implanter (Danfysik 1080-200) and irradiated the sample on a spot greater than the area probed by the infrared beam <cit.>. As usual in this kind of experiment the molecular ratio of the irradiated mixture is different from that expected in space, which is often dominated by water ice. This is due to the experimental need fora sufficient number of mother molecules to produce the daughter species.It is important to note that the stoichiometry of a deposited mixture can be evaluated only approximately. In all of the studied mixtures (i.e. at CIMAP-GANIL and in Catania) weevaluated the column density of the deposited species (H_2O and SO_2) from infrared spectroscopy. The results significantly differ from the nominal gas mixtures that we prepared before accretion onto the cold finger. This is due to the different thermodynamic properties of the deposited species.The IR bands of a given molecule were usedto measurethecolumn density N in units of molecules cm^-2 through the formulaN=∫τ(ν)dν/A ,where τ(ν) is the optical depth (which is 2.3 times the absorbance plotted in the figures) at wavenumber ν (cm^-1) and A is the band strength (cm molecule^-1).The used band strength values are given in Table <ref>together with band peak positions andassignment. The band strength values are valid for pure species, andusing them to evaluate the column density of each molecule in a mixture introduces a large error that can be as high as 50%. § RESULTS §.§ Energetic processing As an example of the results obtained after ion irradiation of frozen mixtures of water or sulfur dioxide, inFigure <ref> we show the spectra of a deposited (16 K) H_2O:SO_2 (1:2) ice mixture that had a thickness ∼0.53 μm, roughly half of the penetration depth of the incoming 30 keV He^+ ions calculated by the The Stopping and Range of Ions in Matter (SRIM) software <cit.>.Also shown is the spectrum obtainedafter irradiation with 3.5×10^1430 keV He^+/cm^2. By using the stopping power of the incoming ions as calculatedby the SRIM software <cit.> we find that thefluence corresponds to a deposited energy (dose) of 12.3 eV/16 u <cit.>.From Figure <ref> it can be easily seen that the intensity of the two SO_2 bands centred at 1325 cm^-1 and 1150 cm^-1 diminishes after irradiation. More precisely, the column density of sulfur dioxide decreases from about 5×10^17 molecules cm^-2 to about 3×10^17 molecules cm^-2.A fraction of SO_2 molecules (and of H_2O as well) was used to build up new species. In fact, several new bands appeared in the spectrum after ion irradiation (see also Figure <ref>). We can observe the formation of the SO_3 polymeric chains testified by the presence of the broad band centred at 1200 cm^-1 <cit.>. Features of sulfate SO_4^2- and bisulfate HSO_4^- ions areobserved,as are featuresof the counter-ion H_3O^+. The results are consistent with the previous findings of <cit.> who observed the same bands after 800 keV proton irradiation of H_2O:SO_2 = 3:1 and 30:1 mixtures (at T=86 K, T=110 K, and T=132 K). The peak position of the bands of newly formed sulfur-bearing species observed in the present experiments and in those available in the literature <cit.> are listed in Table <ref>.The experiments described so far are relative to ions whose penetration depth is greater than the thickness of the irradiated layers, asusually occurs with ice mantles on dust grains in inter- and circumstellar environments irradiated by cosmic ions. There are, however, many instances in which the thickness of the irradiated icy layers is much greater than the penetration depth of the ions that remain implanted in the target. This is the case of most of the icy objects in the Solar System(e.g. satellites of the giant planets, comets, Pluto). Implanted ions –if they are reactive, likecarbon and sulfur ions – have the chanceto form molecular species that include the projectile <cit.>.Relevant to this paper are the results obtained by <cit.> concerningthe implantation of S^q+ (q = 7, 9, 11) ions at an energy range between 35 and 176 keV in water ice at 80 K and aimed at simulatingthe complexity of the irradiation environment to which the surface of icy satellites of the giant planets, particularlyEuropa, are exposed being embedded in the planetarymagnetospheres. The experiments,performed at the low-energy ion beam facility ARIBE of GANIL in Caen (France), indicate that implantation produces hydrated sulfuric acidwith yields that increase with ion energy. The identification was due to the appearance, in the IR spectra of implanted targets, of a broad featurecharacterized by three maxima around 1135 cm^-1, 1105 cm^-1, and 1070 cm^-1 (see the spectrum in the upper panel of Fig. <ref>).Following <cit.> the three observed peaks were assigned to H_2SO_4, to HSO_4^- in monohydrate, and to SO_4^2- in tetrahydrate.§.§ Thermal processing Three samples of icy mixtures with the H_2O:SO_2 concentration ratios 1:10, 1:1, and 3:1 were deposited at 16 K and then warmed step by step up to T=160 K. A blank experiment of pure SO_2 was alsoperformed. Infrared spectra were taken at low temperature and at various steps during the heating of the samples. Spectra taken at 120 K are plotted in the bottom panel of Fig. <ref>. <cit.>, guided by the works of <cit.> and <cit.>, suggest that the peaks at 1035 and 1011 cm^-1 are probably due to the bisulfite ion HSO_3^- and either one of its reaction products or an isomer.Another absorbance peak present at around 956 cm^-1 (see Fig. <ref>) is attributed to S_2O_5^2-, meta-bisulfite <cit.>. Positions of observed absorption peaks were measured (see Fig. <ref>) and compared with the position of absorption bands identified in the similar heating experiments of H_2O:SO_2 icy mixtures performed by <cit.> and later by <cit.> on different ratios of the same mixture. Our finding for 1:1 and 3:1 mixtures are in excellent agreement with the results of <cit.>. However, in the experiment of <cit.> with the mixture 30:1 (i.e. with the lowest SO_2 concentration), the absorption feature of the bisulfite ion is located at about 1070-1060 cm^-1. We do not observe a peak in this region at low T for a mixture 1:10 (i.e. with the highest SO_2 concentration), but only in the spectra taken at high temperature (120 K and above). This could be explained as being due to the sublimation of SO_2 and to the drop in its initial high concentration (see Fig. <ref>).Because we do not know the band strengths of the newly formed bands, we are not able to measure the column density of the species. However, assuming that the band strength values do not depend on the temperature, we can deduce the fate of sulfur dioxide and sulfite ions in the mixtures by measuringthe band areas with increasing temperature. The band areas of all relevant features and the relative area of the SO_2 band at 1149 cm^-1 were measured and they are plotted in Fig. <ref>. For the newly formed bands, we in fact measured a small initial valuefor the band areas (in agreement with the finding byand ) that we attribute to the thermal reactions induced by the water latent heat of condensation of water ice. When pure SO_2 ice is heated, its sublimation occurs at 120 K as evidenced by a drop in the relative absorption band area (Fig. <ref>). When SO_2 ice is mixed with water ice, the area of the 1149 cm^-1 band decreases well before 120 K. Together with the decrease in the SO_2 band area, the sulfite absorption feature (1035–1065 cm^-1) grows with the temperatureup to 120 K, after which it is lost as the samples are further warmed. Thus, before it sublimates, about 50% of the sulfur dioxide was used by thermal reactionswith H_2O for the formation of sulfur-bearing ionic species. At temperatures ≥120 K, the sublimation of SO_2 takes place and the newly formed species follow the same fate. The rate (efficiency) of the formation of ionic species slightly depends on the relative concentrations of the two ices in the mixture as shown in the two bottom panels of Fig. <ref>. We also notice that because of a higher number of water molecules surrounding the SO_2 molecules, some of these latter are trappedand sublimate at higher temperatures, asis commonly observed for many other icy mixtures <cit.>.§.§ Energetic vs thermal processingThe spectra of H_2O:SO_2 mixtures processed thermally and by ion bombardment, and the spectrum of pure water ice after S^7+ implantation areplotted together inFig. <ref> for comparison. The overview of the bands associated with these molecules for allanalysed experiments is given in Table <ref>. We note that sulfate and bisulfate ions are the result of radiolytic processes, while bisulfite and meta-bisulfite are produced by thermal processing. This finding can be reciprocally confirmed and explained by the comparison with the results ofnon-radiolytic, thermally driven experiments <cit.>. <cit.> performed experiments by heating frozen H_2O:SO_2:H_2O_2mixtures, the last component being the main product of water ice radiolysis. They found that sulfate ions are produced when H_2O_2 is present, in contrast to what happens after thermal processing of the binary mixtureH_2O:SO_2. Thus, for the binary mixture we haveSO_2 + H_2O→HSO_3^-+H^+ . By adding H_2O_2 (product of radiolysis) this is rapidly followed by H_2O_2+HSO_3^-+H^+→HSO_4^2-+H_3O^+It is relevant to mention that<cit.> used these results as a possibleexplanation for some of the observations related to the presence and distribution of hydrogen peroxide across Europa's surface and of its lack on Ganymede and Callisto.Similarly <cit.> studied the thermal processing of the solid mixture H_2O:SO_2:O_3. They demonstrated that thermally driven reactions in solid phase occur below 150 K, and the main sulfur-bearing species is bisulfate. They also suggested thatSO_2 and O_3 on the surface of the icy Jovian satellites will efficiently react making detection of these molecules in the same vicinity unlikely.§ DISCUSSION §.§ Protostellar regions As already said, the only sulfur-bearing molecules observed in the solid phase toward high-mass protostars are OCS<cit.> and SO_2 <cit.>, but they have low abundances andcan account only for a minor amount of the elemental sulfur. Nevertheless, the presence of SO_2 in the icy mantles in protostellar regions has stimulatedthe experiments presented here, aimed at investigating which additional sulfur-bearing species are formed after energetic and/or thermal processing of sulfur dioxide mixed with water ice. Our findingthat ionic species such as HSO_3^-, HSO_4^-, and S_2O_5^2- are produced raises two questions: (1) are they observed/observable in the interstellar medium in the solid phase and/or in the gas phase after they are desorbed from the icy mantles because of thermal and/or non-thermal mechanisms? and (2) can they give a significant contribution to the inventory of sulfur species?In order to find an answer to the question of the presence of the ionic sulfur-bearing species in the solid phase, we have given a look at the literature and made a comparison between our experimental results and the observations by <cit.> relative to the infrared spectra of four embedded protostars in the 750-1230 cm^-1 range. This spectral region is dominated by the very intense silicate bandthat complicates the detection of possible further contributing species. However, <cit.> were able to detect, for NGC 7538 IRS9, a band at 1110 cm^-1 that they attributed to frozen ammonia in a polar water-rich interstellar ice, and several othersnear 785, 820, 900, 1030, and 1075 cm^-1 that were unidentified. Later, the band at 1030 cm^-1 was confirmed by ISO observations and attributed to frozen methanol <cit.>.Unfortunately, a fitting procedure between astronomical and laboratory spectra is not feasible in the present case. In fact, the profile of the observed features, whichoverlap with the dominant silicate band, cannot be well defined. In addition, the relative intensities of the possibly observed features cannot be reproduced by a single laboratory spectrum. Each one should be treated as a single feature, but this would introduce anindetermination that is too strong. Therefore, here wecan only outline that the sulfur species which we identify in laboratory spectra are in a spectral region where there are several unidentified bands whose peak positions are coincident or very close to those measured in the laboratory. Future astronomical observations (e.g. by the JWST) could help to clarify their attribution.In particular, it would be interesting to observe MonR2-IRS3 (or similar sources),a source warm enough to have caused the sublimation of the most volatile speciesand retained the less volatile ones <cit.>. The lack of an appreciable amount of SO_2 in the observed spectra gives some insight into the chemical pathway that drives the formation of sulfur-bearing species on icy mantles. It is in fact thought that sulfur atoms accreting on grains are mostly hydrogenated and should produce H_2S; however, this is not observed. As suggested by <cit.> this could be due to energetic processing by cosmic ions that in presence of oxygen and carbon bearing species (e.g. CO) easily converts hydrogenated sulfur into other species, including SO_2. However,the further ion processing of SO_2 in the presence of oxygen and carbon bearing species reduces the amount of SO_2 in favour of other species such as thoseinvestigated here. In addition, the lack ofSO_2 after sulfur implantation into water ice has already been evidenced by <cit.>. Those authors outlined thatSO_2 is formed by the addition of S to O_2 (and/or HO_2), but it is easily converted to hydrated sulfuric acidvia SO_2 + H_2O_2→H_2SO_4In other words, atomic sulfur that hits a grain surface is efficiently converted by energetic and thermal processes to more complex sulfur-bearing species rather than being accumulated as sulfur dioxide. In this scenario, we can suggest that these species contribute to the inventory of sulfur-bearing species, but at present it is not possible to establish whether their abundance can significantly contribute to solving the question of the missing sulfur because we are not able, due to the lack of suitable band strength values and the paucity of observational evidences, to measure the column density of these species in the solid phase. In addition, these ionic species and the fragments of SO_3 polymers have not yet been observed in the gas phase after their sublimation when the temperature increases, for examplein the regions nearer to the forming star. We hope that the results presented herestimulate the search for these species in the gas phase in opportune environments by usingthe ALMA facility, for example. Adequate laboratory studies on the relevant spectral line parameters of the sulfur-bearing species are also necessary.§.§ Solar System objects The experimental results presented here are relevant to a number of objects in the Solar System where sulfur dioxide has been observed oris presumed to be present. These objects include Jupiter's Galilean satellites Io, Europa, Ganymede, and Callisto. Being embedded in the Jovian magnetosphere, they are exposed to the complex flux of low- (plasma) and high-energy electron and ion bombardment <cit.>.Io's surface is in fact dominated by sulfur dioxide, which is expelled from the very intense volcanic activity triggeredby tidal effects <cit.>. Although it is thought that Io has lost nearly all of its hydrogen <cit.>, the detection of hydrogen pickup ions by Galileo's plasma analyser <cit.> in the space surrounding the satellite raised the question regarding its origin. A first suggestion was hydrogen sulfide <cit.>. However, the abundance of this compound is very low (with an upper limit of 10^-4) with respect to SO_2 <cit.>. The next candidates as hydrogen bearing species are then water ice and/orhydrate materials whose absorption bands around 3150 cm^-1were tentatively observed <cit.>. It is, however, also possible that the detected flux of hydrogen ions comes from the Jovian magnetosphere and not from the satellite.In this scenario, and assuming that hydrogen bearing species are present on Io's surface along with the certain presence of intense fluxes of energetic ions and electrons, radiolytic products are likely to be present. The most abundant should be de-hydrated species such as the fragments of the elemental sulfur residue formed after ion bombardment of pure sulfur dioxide <cit.>. The sulfur residue could be responsible of the observed red slope in the near-infrared/visible spectral region of Io's spectra and of the molecular fragments S_4 and S_8 <cit.>. Much less abundant (and difficult to observe) are the hydrated ions that we have synthesized in the experiments described here; nevertheless,they merit further investigation. The present experiments are of primary relevance to the remaining three water ice dominated Galilean satellites and for the other icy satellites orbiting Jupiter, Saturn (e.g. Enceladus), and Uranus. As already mentioned above,<cit.> have demonstratedexperimentaly that magnetospheric sulfur ions implanted inEuropa's surface produce hydrated sulfuric acid. The production rate is high enough to explain the quantity of hydrated sulfuric acid on the surface of Europa inferred to be present by modelling the near infrared (2 ) water ice band <cit.> as observed by Galileo the Near-Infrared Mapping Spectrometer (NIMS). However, the bands due to hydrated sulfuric acid in the laboratory spectra of sulfur-implanted water ice targetsappear in the 1100 cm^-1 region (see Fig. <ref> and Table <ref>). Such a spectral region has been investigated by instruments on boardVoyager and Cassiniand will be investigated byinstruments on boardthe James Webb Space Telescope (JWST).The data collected by flyby and orbiter missions is incomplete and/or collected under imperfect illumination conditions (e.g. high phase angles). Therefore, the contribution of JWST will be relevant and we suggest that a particular effort should be made to identify the hydrated sulfuric acid features, particularly on Europa which exhibits surface regions exposed to very intense fluxes of energetic sulfur ions <cit.>. sulfur-bearing ions should alsobe searched for in the exospheres of the icy satellites where they could be expelled by thermal andnon-thermal processes.Our experiments are also relevant to comets and to all of the small objects in the outer Solar System (trans-Neptunian objects). Several sulfur-bearing species have already been observed in different families of comets <cit.>.The question of the type and abundance ofsulfur-bearing species was revised by <cit.> based on the results obtained by the Rosetta Orbiter Spectrometer for Ion and Neutral Analysis/Double Focusing Mass Spectrometerin the coma of comet 67P/Churyumov-Gerasimenko. Those authors measured the abundances of the species that were previously known to be present on comets, namely H_2S, OCS, SO, S_2, SO_2, and CS_2. <cit.> detected S_3, S_4, CH_3SH, and C_2H_6S for the first time, and they concluded that the derived total elemental sulfur abundance of 67P does not show any sulfur depletion. Inaddition, those authors presented results indicating that sulfur-bearing species have been processed by radiolysis in the pre-solar cloud and that at least some of the ice from this cloud has survived in comets up to the present. This conclusion is in fact based on experimental results that show how ion irradiation of sulfur-bearingspecies produce asolid unvolatile sulfur-rich residue <cit.> and also molecules originally not present such as CS_2 <cit.> and the ionic species discussed here that we suggest should be searched for. § CONCLUSION In this paper we have described the results of a series of experiments concerning thermal and energetic processing of SO_2 ices mixed with water ice. Theresults indicatethat ionic speciessuch as HSO_3^-, HSO_4^-, and S_2O_5^2- are formed. Ion bombardment also produces SO_3 polymers and a sulfur-rich refractory residue. The results have been discussed in view of their potential relevance to the debate on the missing sulfur in the interstellar and circumstellar regions, and to the chemical evolution of the surfaces of icy objects in the Solar System. The results can be summarized as follows: – We suggest that sulfur-bearing ionic species could be synthesized on interstellar icy grain mantles by energetic or thermal processes. These species and the fragments of SO_3 polymers sublimate when the temperature increases, for example in the regions close to the forming star. This finding should stimulate theoretical, observational, and experimental researches. It is in fact important to theoretically investigate the contribution of the ionic species expelled in the gas phase to the chemistry of those regions. At the same time experimental efforts to measure the rotational spectra of these molecules are necessary in order to allowthe observers to identify them through astronomical observations. – Hydrated sulfuric acid formed after sulfur ion implantation in water ice produces alterations of the shape of its 2 μm band as already observed <cit.>. It also gives origin to a multi-peaked band in the mid-IR spectral region <cit.>, whichshould be searched for in the spectra of water-dominated solid surfaces. sulfur-bearing ionic species should be searched for in the gas phase after being released from the surface by thermal and/or non-thermal processes. - On Io it is possible that non-hydrated sulfur-bearing specieshavealready been observed. Only a small amount of hydrated sulfur-bearing species is predicted to be present on Io's surface, if any. - The inventory of sulfur-bearing molecules recently implementedby the Rosetta finding <cit.> supports evidence for the occurrence of radiolysis. If so, that inventory would be even more implemented including the species discussed here. 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"authors": [
"Z. Kanuchova",
"Ph. Boduch",
"A. Domaracka",
"M. E. Palumbo",
"H. Rothard",
"G. Strazzulla"
],
"categories": [
"astro-ph.IM"
],
"primary_category": "astro-ph.IM",
"published": "20170627125723",
"title": "Thermal and energetic processing of astrophysical ice analogues rich in SO$_2$"
} |
School of Electrical and Computer Engineeringand Birck NanotechnologyCenter, Purdue University, West Lafayette, IN 47907, USAWe present the general analytical theory for Dyakonov surface waves at the interface of abiaxial anisotropic dielectric with an isotropic medium. We demonstrate that these surface waves can be divided into todo distinct classes, with qualitatively different spatial behavior. We obtain explicit expressions for the Dyakonov waves dispersion and the parameter range for theirexistence.42.25.Bs,42.25.Lc,43.35.Pt.Dyakonov Waves in Biaxial Anisotropic Crystals Evgenii E. Narimanov December 30, 2023 ===============================================Electromagnetic surface waves, strongly localized near the interface of two different media, pay an important role in many areas of science and technology – from optical microscopy <cit.> and biosensing <cit.> to nano-optical tweezing <cit.> to photonic integrated circuits. <cit.> Electomagnetic surface waves are responsible for such phenomena as superlensing,<cit.> enhanced Raman scattering <cit.> and extraordinary light transmission through subwavelength holes. <cit.> While there exists a number of different kinds of surface electormagnetic waves, such as e.g. surface plasmons at the interface of a metal and a dielectric, <cit.> orTamm-Shockeley states <cit.> atthe boundary of a photonic crystal, <cit.> a new class of surface electromagnetic modes has recently gained considerable attention. <cit.> These Dyakonov surface waves exist at the interface of an isotropic and anisotropic dielectric media. They can be supported by transparent optical materials, and thus do not suffer from the metallic absorption that plagues surface plasmons. <cit.> Compared to the Tamm-Shockley state, Dyakonov wave does not require any period patterning of the material forming the system, with the resulting light scattering due to the inevitable disorder as a result of an imperfect fabrication of such lattice.The presence of Dyakonov waves at the isotopic-anisotropic interface has been firmly established in the experiment, <cit.> and a number of adequate theoretical methods exists for their quantitative description. <cit.> However, due to the inevitable complexity of the boundary conditions at the interface of a fully-anisotropic dielectric the resulting theoretical description generally leads to a system of nonlinear equations that must be solved numerically. While this may be considered a straightforward task, Dyakonov waves are usually extended over many wavelengths, <cit.>and are therefore closeto the propagation wave threshold – which makes the numerical solution more challenging. What is even more important, with the theoretical “toolbox” limited to numerical methods, the root-finding algorithm may even missan entire class of possible solutions.In this work, we present a complete analytical solution for the Dyakonov surface waves at the interface of an isotropic and a biaxial dielectric medium. We show that, depending on the magnitudes of the dielectric permittivity components in the system, the interface can simultaneously support two different classes of surface waves, with qualitatively different spatial behavior.§ THE MODEL We consider the interface of an isotopic dielectric with the permittivity ϵ_0, with a biaxial anisotropic medium, with the permittivity tensorϵ=( [ ϵ_x 0 0; 0 ϵ_y 0; 0 0 ϵ_z ]).We furthermore assume that one of the symmetry directions of the anisotropic crystal (which will be referred to as the axis z in our coordinate system – see Fig. <ref>) is normal to the interface, as this is generally the case for a high-quality interface. While a non-orthogonal orientation of ẑ with respect to the plane of surfaceispossible, this would lead to a relatively high density of surface defects – thus making the theory for surface waves ata idea planar interface irrelevant for most practical application. For convenience, the coordinate system originz = 0 is chosen at the plane of the interface – see Fig. <ref>.In this work, we focus on guided surface waves with the in-plane momenttum q≡( q_x, q_y),E( r, t )= E_ q(z)·exp(i q_x x + iq_y y - i ω t), B( r, t ) = B_ q(z)·exp(i q_x x + iq_y y - i ω t),whereE_ q(| z | →∞)→ 0,B_ q(| z | →∞) → 0 § ELECTROMAGNETIC WAVES IN A BIAXIAL MEDIUM For an evanescent wave that decays away from the z = 0 interface, we haveE_ q( z) = e·exp( - κ z), B_ q( z) = b·exp( - κ z),Note that for a complex κ, the expressions (<ref>), (<ref>) also describe the propagating waves in the medium.Substituting (<ref>),(<ref>) with (<ref>), (<ref>) into Maxwell's equations, we obtainb_x =c/ω( q_y e_z - i κ e_y), b_y =c/ω( i κ e_x - q_x e_z), b_z =c/ω( q_x e_y -q_y e_x),andM( [ e_x; e_y; e_z ]) =0, whereM ≡ ( [Δ_x(κ) q_x q_y i κ q_x; q_x q_yΔ_y(κ) i κ q_y; i κ q_x i κ q_yΔ_z(κ) ]),andΔ_x(κ) =ϵ_x (ω/c)^2 - q_y^2 + κ^2,Δ_y(κ)=ϵ_y (ω/c)^2 - q_x^2 + κ^2,Δ_z(κ)=ϵ_z (ω/c)^2 - q_x^2 -q_y^2.From(<ref>) we find the electrical field components in terms of the amplitude ae_x = i κ q_x (q_y^2 - Δ_y(κ) ) · a, e_y = i κ q_y (q_x^2 - Δ_x(κ) ) · a, e_z =( Δ_x(κ) ·Δ_y(κ)- q_x^2 q_y^2 ) · a,which together with (<ref>)-(<ref>) define the entire electromagnetic field (e, b)in (<ref>), (<ref>). Also, from Eqn. (<ref>) we obtaindet[M] = 0,which yieldsϵ_z ·κ^4+[( ϵ_x + ϵ_y )·(ω/c)^2- (ϵ_x + ϵ_z)· q_x^2 - (ϵ_y + ϵ_z)· q_y^2]·κ^2 +(ϵ_z (ω/c)^2 - q_x^2 - q_y^2)× (ϵ_x ϵ_y (ω/c)^2- ϵ_x q_x^2 -ϵ_y q_y^2)=0.Eqn. (<ref>) is a quadratic equation for κ^2, with the straightforward solutionκ_±^2 =1/2{ϵ_x + ϵ_z/ϵ_z q_x^2 + ϵ_y+ϵ_z /ϵ_z q_y^2 -( ϵ_x + ϵ_y )·(ω/c)^2 . ± . √(D)},where the DiscriminantD =[ (ϵ_x - ϵ_y) (ω/c)^2 + ϵ_z - ϵ_x/ϵ_z q_x^2 + ϵ_y - ϵ_z /ϵ_z q_y^2 ]^2+4·(ϵ_y - ϵ_z )·(ϵ_y - ϵ_z ) /ϵ_z^2q_x^2q_y^2.When the Discriminant is positive, there are three distinct possibilities for the nature of the waves supported by the anisotropic dielectric. If the right-hand side of Eqn. (<ref>) is positive forκ_+^2 and κ_-^2, both waves with the “in-plane” momentum q≡ (q_x, q_y) are evanescent. In the opposite case, when the right-hand side ofEqn. (<ref>) is negative in both cases, the corresponding two waves are propagating. Finally, when it's positive for one choice of the sign in(<ref>) and negative forthe other, we find that for the given in-plane momentum qthe dielectric interface supports one propagating and one evanescent wave.As follows from Eqn. (<ref>), the Discriminant ispositive-definite (for any q) in each of the following cases: * any uniaxial dielectric (ϵ_x = ϵ_y or ϵ_x = ϵ_z or ϵ_y = ϵ_z),* ϵ_z <min[ϵ_x, ϵ_y],* ϵ_z >max[ϵ_x, ϵ_y].The boundaries that separate different portions of the (q_x, q_y) phase space that respectively support only the propagating waves, or only the evanescent fields, or a mixture of evanescent and propagating waves, are given byq_x^2 + q_y^2 =ϵ_z (ω/c)^2,andq_x^2/ϵ_y+ q_y^2/ϵ_x=(ω/c)^2,This behavior is illustrated in Fig. <ref>.However, ifmin[ϵ_x, ϵ_y] < ϵ_z <max[ϵ_x,ϵ_y],the Discriminant in Eqn. (<ref>) can, and does, for certain ranges of the values of q_x and q_y, become negative. In this case, κ_± is complex, with nonzero values for both its real and imaginary parts. These “ghost waves”, recently described in Ref. <cit.>, combine the oscillatory behavior of the propagating waves with the exponential decay characteristic of the evanescent fields, and represent the third class of the waves that can be supported by a transparent dielectric medium.When the inequality (<ref>) is satisfied, the boundaries of the portion of the (q_x, q_y) phase space of the ghost modes are defined by the four equations√(| ϵ_x - ϵ_z|/ϵ_z) q_x ±√(| ϵ_y - ϵ_z|/ϵ_z) q_y ±√(| ϵ_y - ϵ_x|)ω/c = 0,Fig. <ref> shows the phase space of a biaxial anisotropic dielectric that supports ghosts waves. Note its nontrivial structure near the point corresponding to the intersection of the boundaries described by Eqns. (<ref>) and (<ref>) in the magnified view of its panel (b).When the permittivity ϵ_zin the normal-to-the-interface direction approaches the value of one of thein-plane permittivities ϵ_x or ϵ_y, the ghost regions in the phase space collapse to increasingly narrow strips parallel to eithertheq_x (when ϵ_z →ϵ_x) or q_y (forϵ_z →ϵ_y) axis. This “collapse” is however relatively slow, and substantial ghost regions are still present even when the permittivity is within 1% of the critical value,as seen in Fig. <ref>.Most importantly, ghost regions show substantial presence in actual biaxial anisotropic crystals. This is illustrated inFig. <ref>, where we show the phase space for thesodium nitrite NaNO_2, with the dielectric permittivity tensor components <cit.> ϵ_x = 1.806, ϵ_y = 2.726 and ϵ_z= 1.991.While Eqns. (<ref>) - (<ref>) adequately describe the generalcase of a dielectric crystal with arbitrary degree of anisotropy, the isotropic limit ϵ_x →ϵ_y→ϵ_z →ϵ_0 is singular, as here both κ_+ and κ_- are identical,κ_+(ϵ_x, ϵ_y, ϵ_z →ϵ_0) =κ_-(ϵ_x, ϵ_y, ϵ_z →ϵ_0) = κ_0, withκ_0 = q_x^2 + q_y^2 - ϵ_0 (ω/c)^2,and direct substitution of (<ref>),(<ref>) into (<ref>), (<ref>), (<ref>) and(<ref>), (<ref>), (<ref>) yieldse_x, e_y, e_z, b_x, b_y, b_z →0· a_0,with a_0 →∞. This uncertainty can be removed if we explicitly introduce s- and p- polarizations, correspondingly with e_z^(s) = 0 and b_z^(p) = 0:e_x^(s)= q_y · a_s, e_y^(s)= - q_x · a_s, e_z^(s)= 0, b_x^(s)=i c κ_0/ωq_x · a_s, b_y^(s)=i c κ_0/ωq_y · a_s, b_z^(s)= -c q^2 /ω· a_s,ande_x^(p)= q_x · a_p, e_y^(p)= q_y · a_p, e_z^(p)=i q^2/κ_0· a_p, b_x^(p)=i ωϵ_0/c κ_0q_y · a_p, b_y^(p)= - i ωϵ_0/c κ_0q_x · a_p, b_z^(p)= 0.Hereq ≡√(q_x^2 + q_y^2),while a_s and a_p are the scaled amplitudes of the s- and p-polarized waves respectively. § DYAKONOV WAVE Assuming that the interface at z = 0 separates transparent isotropic medium with the permittivity ϵ_0 at z < 0 from biaxial anisotropic dielectric with the permittivity tensor (<ref>), for the guided surface wave with the in-plane momentum q = (q_x, q_y) we obtainE_q(z) ={[( a_se_s + a_pe_s ) e^κ_0 z, z < 0; a_+e_+ e^ - κ_- z +a_-e_+ e^ - κ_- z, z > 0 ].andB_q(z) ={[( a_sb_s + a_pb_s ) e^κ_0 z, z < 0; b_+e_+ e^ - κ_- z +b_-e_+ e^ - κ_- z, z > 0 ].where (note the sign change κ_0 → -κ_0 from (<ref>) - (<ref>)to (<ref>) - (<ref>)as the evanescent field for z<0 behaves as exp(+κ_0 z)) e_s =(q_y,-q_x,0), b_s = - c/ω( i κ_0 q_x, i κ_0 q_y,q_x^2 + q_y^2 ), e_p =(q_x,q_y, - i/κ_0(q_x^2 + q_y^2 )),b_p =i ωϵ_0/c κ_0 ( - q_y,q_x,0 ),ande_±=( i κ_± q_x (q_y^2 - Δ_y(κ_±) ), i κ_± q_y (q_x^2 - Δ_x(κ_±) ), . .Δ_x(κ_±) ·Δ_y(κ_±)- q_x^2 q_y^2),b_±=ω/c( q_y (ϵ_y Δ_x(κ_±) - ϵ_x q_x^2), . - .q_x (ϵ_x Δ_y(κ_±) - ϵ_y q_y^2), i q_x q_yκ_± (ϵ_y - ϵ_x )).With non-magnetic (μ = 1) dielectric materials at both sized of the interface, at z = 0 we have the continuity of all three components of the magnetic field B_q, and the continuity ofE_x, E_y and D_z ≡ϵ_z E_z. However, as follows from (<ref>), the continuity of both tangential components of the electric field immediately implies the continuity of B_z Furthermore, sinceϵ_z E_z ∝[curl B ]_z ∝ q_x B_y - q_y B_x,the continuity of D_z = ϵ_z E_z is a direct consequence of the continuity of the tangential magnetic field. Therefore, out of six boundary conditions here only four are actually independent, consistent with the four independent amplitudes a_sm a_p, a_+ and a_-. Imposing the continuity of E_x, E_y, ϵ_z E_z and ∂_z B_z ∝( q_x B_y + q_y B_x ), we obtain:N( [ a_s; a_p; a_+; a_- ]) = 0,where the matrix N is defined asN=([ i q_y/q_x iκ_+(q_y^2 - Δ_y^+) κ_- (q_y^2 - Δ_y^-); - i q_x/q_y i κ_+ (q_x^2 - Δ_x^+) κ_- (q_x^2 - Δ_x^-); 0 i q^2 ϵ_0/κ_0 ϵ_z Δ_x^+ Δ_y^+ - q_x^2 q_y^2 Δ_x^- Δ_y^- - q_x^2 q_y^2; i q^2 κ_0/q_x q_y 0 (ϵ_y - ϵ_x) ω^2 κ_+^2/c^2(ϵ_y - ϵ_x) ω^2κ_-^2/c^2 ]),withΔ_x,y^±≡Δ_x,y(κ_±).Introducing the new variable ζ_± coresponding to the z-components of the amplitudes of the electric field in the anisotropic material ( e_+)_z and ( e_-)_z,ζ_±=( Δ_x^±Δ_y^± - q_x^2 q_y^2 ) · a_±,from (<ref>) and (<ref>) we obtainP(ω;q) ( [ ζ_+; ζ_- ]) = 0,where the matrix P is defined byP(ω;q)=( [ α_+ α_-; β_+ β_- ]),andα_±=ϵ_z/ϵ_0 + κ_±/κ_0× (ω/c)^2 (ϵ_x q_y^2 + ϵ_y q_x^2 ) - q^2(q^2 - κ_±^2)/Δ_x^±Δ_y^± - q_x^2 q_y^2,β_±=κ_±·κ_0 + κ_±/Δ_x^±Δ_y^± - q_x^2 q_y^2.The dispersion of the surface wave is then given bydet[P(ω;q) ] = 0,which yieldsκ_0 (κ_+ + κ_-) ·{ϵ_x ϵ_y/ϵ_0( (ω/c)^2 - q_x^2/ϵ_y - q_y^2/ϵ_x) - κ_+ κ_- } +κ_+κ_- {(ϵ_x + ϵ_y ) (ω/c)^2 -ϵ_0 + ϵ_x/ϵ_0 q_x^2 -ϵ_0 + ϵ_y/ϵ_0 q_y^2 } +{ϵ_x ϵ_y/ϵ_0κ_0^2 ( (ω/c)^2 - q_x^2/ϵ_y - q_y^2/ϵ_x) - κ_+^2 κ_-^2 } = 0 .Eqn. (<ref>) uniquely defines the dispersion relation of the Dyakonov surface wave ω( q), and is the primary result of this section.For a guided surface wave, all its components, in both the isotropic and anisotropic sides of the interface, must decay away from the boundary. For z < 0, this implies thatq> √(ϵ_0) ω/c.At the same time, in the anisotopic medium the waves with the in-plane momentum, q can belong to either the evanescent or ghost sub-classes – see Section II. From Eqns. (<ref>) and (<ref>) we therefore obtain q> √(ϵ_z) ω/c,andq_x^2/ϵ_y + q_y^2/ϵ_x > (ω/c)^2.Eqns. (<ref>), (<ref>) and (<ref>) substantially reduce the range of the momentum and frequency that needs to be explored in the numerical solution of Eqn. (<ref>). Furthermore, as shown in Ref. <cit.> (see also Appendix A), the Dyakonov surface wave only exists whenmin(ϵ_x, ϵ_y) ≤ϵ_z < ϵ_0 <max(ϵ_x, ϵ_y). While the numerical solution of Eqn. (<ref>) is generally straightforward, for small - to - moderate anisotropy, the surface waves are known <cit.>to be relatively weakly guided,κ_0≪ω/c, which turns numerical root-finding into a challenging numerical problem <cit.>. In the next section we will therefore develop the method for the analytical solution of Eqn. (<ref>). § ANALYTICAL SOLUTION FOR THE SURFACE WAVE DISPERSION Despite its relative complexity,Eqn. (<ref>) is not transcendental, but only contains algebraic functions. As a result, it can be reduced to a polynomial equation. Furthermore, as we show in the present section, the resulting polynomial equation is of the 4th order, and therefore allows a complete analytical solution.Choosing the y-direction at the one corresponding to the largest permittivity in the plane of the interface, ϵ_y > ϵ_x, we introduce the new variable u =ϵ_x ϵ_y/ϵ_0(q_x^2/ϵ_y + q_y^2/ϵ_x - ( ω/c)^2 ).Note that, as follows from (<ref>), u> 0. Thenκ_+^2 κ_-^2 =ϵ_0/ϵ_z( q^2 - ϵ_z (ω/c)^2) · u,andκ_+ + κ_- =[q^2 +ϵ_0/ϵ_zu -( ϵ_x + ϵ_y - ϵ_x ϵ_y/ϵ_z) (ω/c)^2.+. 2√(ϵ_0/ϵ_z(ϵ_z (ω/c)^2 - q^2 ) · u) ]^1/2We can then express Eqn. (<ref>) as κ_0 (κ_+ + κ_-) ·( u + κ_+κ_-) = κ_+ κ_- + B̂,whereÂ=( ϵ_x + ϵ_y - ϵ_x ϵ_y/ϵ_z) (ω/c)^2 - q^2 - ϵ_0/ϵ_z· u,andB̂= - κ_0^2 u - κ_+^2 κ_-^2.We then square both sides of Eqn. (<ref>), which yieldsĜ κ_+ κ_- =F̂ ,whereĜ=ϵ_0{(ϵ_xϵ_y/ϵ_0 - ϵ_x - ϵ_y + ϵ_0 ) (ω/c)^2 .+[ (ϵ_x/ϵ_0 + ϵ_y - ϵ_0/ϵ_z -ϵ_x ϵ_y/ϵ_0^2) q_x^2 . +. .(ϵ_y/ϵ_0 + ϵ_x - ϵ_0/ϵ_z -ϵ_x ϵ_y/ϵ_0^2) q_y^2] } = 2 ϵ_0 {(1 - ϵ_0/ϵ_z) · u + κ_0^2 ( ϵ_x + ϵ_y - ϵ_0/ϵ_z - ϵ_x ϵ_y/ϵ_0^2) . +.( 1/ϵ_0 - 1/ϵ_z)( ϵ_0^2 - ϵ_0 (ϵ_x + ϵ_y ) + ϵ_x ϵ_y )(ω/c)^2}andF̂= - ϵ_0 (1 - ϵ_0/ϵ_z) {(ϵ_2 (ω/c)^2 - u)^2 . - . ϵ_0 (1 - ϵ_0/ϵ_z) (ω/c)^2u } +κ_0^2 {ϵ_0 ( ϵ_2^2/ϵ_z + (1 - ϵ_0/ϵ_z) (ϵ_1 + 2 ϵ_2 )) (ω/c)^2 . - .(ϵ_1 + ϵ_0/ϵ_z(ϵ_0 + 2 ϵ_2) - ϵ_0^3/ϵ_z^2) u } -κ_0^4 ·{ϵ_0/ϵ_z(ϵ_1 + 2 ϵ_2 ) },withϵ_1 =ϵ_ 0 - ϵ_x - ϵ_y + ϵ_x ϵ_y/ϵ_z,andϵ_2 =ϵ_x + ϵ_y - ϵ_0 - ϵ_x ϵ_y/ϵ_0= (ϵ_0 - ϵ_x ) (ϵ_y - ϵ_0 )/ϵ_0Note that, in addition to the solutions of the original equation (<ref>), the new Eqn. (<ref>) contains spurious roots corresponding to Âκ_+ κ_- + B̂ < 0. We therefore need to constrain the solutions of(<ref>) with the inequality κ_+ κ_- + B̂ > 0.Together, Eqns. (<ref>) and (<ref>) are equivalent to the original equation (<ref>). Since u > 0 and q > √(ϵ_z)ω/c (see Eqns. (<ref>),(<ref>) and (<ref>)), from Eqn. (<ref>) we findκ_+κ_- =κ_z ·√(ϵ_0/ϵ_z· u),whereκ_z≡ √(q^2 - ϵ_z ). Substituting (<ref>) into (<ref>), we obtainκ_z·Ĝ·√(ϵ_0/ϵ_z· u )=F̂.Introducing the new dimensionless variableχ ≡ c/ωκ_z,we can express Eqn. (<ref>) in the forma_4·χ^4 +a_3·χ^3 +a_2·χ^2 +a_1·χ+a_0 = 0,wherea_4 =ϵ_0/ϵ_z(ϵ_1 + 2ϵ_2), a_3 =√(ϵ_x ϵ_y/ϵ_0û)· 2 ϵ_0 (ϵ_x + ϵ_y - ϵ_0/ϵ_z - ϵ_x ϵ_y/ϵ_0^2), a_2 = - ϵ_0 ϵ_2^2/ϵ_z + ϵ_0 (ϵ_1 + 2 ϵ_2) ·(1 - ϵ_0/ϵ_z)+û·ϵ_x ϵ_y/ϵ_0( ϵ_1 + 2 ·ϵ_0 ϵ_2/ϵ_z + ϵ_0^2/ϵ_z(1 - ϵ_0/ϵ_z) ), a_1 = 2 (1 - ϵ_0/ϵ_z) √(ϵ_x^3 ϵ_y^3/ϵ_zû)·(1 - ϵ_z/ϵ_0 + û), a_0 =(1 - ϵ_0/ϵ_z) ·ϵ_x ϵ_y/ϵ_0·û·(ϵ_1 ϵ_z +ϵ_x ϵ_y û),andû ≡ ϵ_0/ϵ_x ϵ_y(c/ω)^2· u = (c/ω)^2( q_x^2/ϵ_y + q_y^2/ϵ_x) - 1.The expression (<ref>) is a quartic equation for χ, and allows an immediate analytical solution via the Ferrari formula, <cit.> so thatχ= F(û; ϵ_0, ϵ_x, ϵ_y, ϵ_z).Then, introducing the polar angle θ that defines the direction of the in-plane momentum q,q_x = q ·cosθ, q_y = q ·sinθ,from (<ref>) and (<ref>) we obtainω/c=q/√(ϵ_z +F^2(û)),sinθ=±√(ϵ_x ϵ_y/| ϵ_y - ϵ_x |·( û + 1/ϵ_z +F^2(û) - 1/ϵ_y)),which parametrically defines the function ω( q, θ).In general, a quartic equation like (<ref>) has four distinct roots. However, in our case χ should satisfy a number of additional constraints. Aside from being a positive real quantity, it must also exceed the value of √(ϵ_0 - ϵ_z),χ>√(ϵ_0 - ϵ_z),since decay of the surface wave away from the interface impliesκ_0 = ω/c√(χ^2 + ϵ_z - ϵ_0) > 0.As we prove in Appendix B, Eqn. (<ref>) only has no more than a single real positive solution that satisfies (<ref>), so there is no ambiguity of choosing the correct root. We therefore obtainF= - a_3/4 a_4 + s_1 S + s_2/2√(- 4· S^2 - 2 p̂ -s_1·q̂/S),wherep̂=a_2/a_4 - 3/8·a_3^2/a_4^2,q̂=a_3^3 - 4 a_2 a_3 a_4 + 8 a_1 a_4^2/8 a_4^3, S =1/2·√(- 2/3p̂ + 1/3 a_4( Q + Δ_0/Q)), Q =√(Δ_1 + √(Δ_1^2 - 4·Δ_0^3)/2),Δ_0 = a_2^2 - 3 a_1 a_2 + 12 a_0 a_4,Δ_1 = 2 a_2^3 - 9 a_1 a_2 a_3 + 27 a_0 a_3^2 + 27 a_1^2 a_4- 72 a_0 a_2 a_4, s_1,2=± 1. While the choice of s_1 and s_2 in Eqn. (<ref>) that leads to a positive real root that satisfies Eqn. (<ref>), is unique, such a solution only exist in a limited range of angles θ. Furthremore, the resulting solution must be tested against the inequality (<ref>) to remove the spurious roots. As a result, for the angular range of θ that supports the Dyakonov surface wave, we obtain (see Appendix C):θ_1 < |θ|< θ_2,orπ - θ_2 < |θ|< π - θ_1,where, assuming ϵ_y > ϵ_x, θ_1 = arcsin[ (ϵ_x ϵ_y/ϵ_y - ϵ_x. . × . .( ϵ_1 + 2 ϵ_2/(ϵ_1 + 2 ϵ_2) + ϵ_2^2 - 1/ϵ_y) )^1/2] and θ_2 = arcsin[( ( 1 - √(1 +4 ϵ_z ( ϵ_0 - ϵ_x )( ϵ_y - ϵ_0 )/ϵ_0^2 (ϵ_0 - ϵ_z)) ) . . × . . ϵ_0/2 ϵ_zϵ_0 - ϵ_z/ϵ_y - ϵ_x+ϵ_y - ϵ_0/ϵ_y - ϵ_x)^1/2 ].Here, θ_1 and θ_2 correspond to κ_- = 0 and κ_0 = 0 respectively. At the same time, θ_1 corresponds to the boundary of the inequality (<ref>), while θ_2 represents the “edge” of the inequality (<ref>) – see Appendix C. Within the angle range (<ref>) for any direction θ and the frequency ω, there is one and only one surface wave, described by the parametric equations (<ref>), (<ref>)with the functionF(û, ϵ_0, ϵ_x, ϵ_y, ϵ_z) fromEqn. (<ref>), while for any angle outside this range, there is no surface wave.In Fig. <ref> we plot the surface wave dispersion for the interface of potassium titanyl phosphate (KTP) and aluminium oxynitride (AlON) (panel (a)), and arsenic trisulfide with aluminum arsenide (panel (b)). The results of the present work can also be applied to uniaxial materials, as illustrated in Fig. <ref> for calcite and CdF_2 (panel (a)), and lithium niobate (LiNbO_3) and KTaO_3 (panel (b)).Following Ref. <cit.>, it is also instructive to project the surface wave dispersion onto the wavevector space (q_x, q_y) that we studied in Section II. In Fig. <ref> we show this projection for the surface wave at the interface of isotropic aluminum arsenide and biaxial arsenic trisulfide. As expected, the magenta curve that represents the Dyakonov surface wave, terminates at the boundaries corresponding toκ_0 = 0(blue line) and κ_- = 0 (green line). Note that, depending on the wavevector of the surface wave, it could be observed both in the “evanescent” and “ghost” portions of the phase space (see panel (c)).§ TWO CLASSES OF DYAKONOV SURFACE WAVES Near the boundary of an isotropic medium with a uniaxial dielectric, the Dyakonov surface wave is formed by evanescent waves on both sides of the interface. However, for a biaxial dielectric that supports both the evanescent and the ghost waves (see Section II), the localized surface wave can be formed from either the evanescent or from ghost waves, depending on its in-plane momentum. As a result, for the interface of a isotropic medium with a biaxial medium, we can have two different types of the Dyakonov surface wave. A “conventional” Dyakonov surface, as originally described by M. Dyakonov in 1988 <cit.>wave monotonically decays on both sides of the interface, while the ghost surface wave, together with the exponential decay also shows oscillatory behavior in the anisotropic medium – see Fig. <ref>.Note that, depending on the magnitude of the permittivity of the isotropic medium ϵ_0 (ϵ_z < ϵ_0< ϵ_y), at a single frequency the isotropic - biaxial interface can either support both the “conventional” and the “ghosts” mode patterns, or only the "conventional" modes. The corresponding critical value ϵ_cof the permittivity ϵ_0 is given by the equation (see Appendix D)(ϵ_c - ϵ_z)^2 (5 ϵ_z - 3(ϵ_x + ϵ_y) + ϵ_x ϵ_y/ϵ_z)+(ϵ_c -ϵ_z) (ϵ_x^2 + 4 ϵ_x ϵ_y + ϵ_y^2 - 8 ϵ_z (ϵ_x + ϵ_y ) + 10 ϵ_z^2 ) + 3 ϵ_z ( ϵ_y - ϵ_z) (ϵ_z - ϵ_x)+2/ϵ_z( (ϵ_c - ϵ_z)^2 + ϵ_z(2 (ϵ_x + ϵ_y) - 5 ϵ_z ) ) × √(ϵ_z ( ϵ_y - ϵ_z ) ( ϵ_z - ϵ_x ) ( ϵ_c - ϵ_z ) ) = 0,which for ϵ_x < ϵ_z < ϵ_y always has a single solution in the interval ϵ_z < ϵ_c < ϵ_y. In scaled variables ϵ_c / ϵ_z, ϵ_z/ϵ_x, ϵ_y/ϵ_z the solution of Eqn. (<ref>) can be expressed asϵ_c/ϵ_z= G(ϵ_z/ϵ_x, ϵ_y/ϵ_z).We plot this function in Fig. <ref>.For ϵ_c < ϵ_0 < ϵ_y, the Dyakonov surface waves that are supported by the interface of isotropic and biaxial dielectric media, belong to the “conventional” class for all allowed propagation angles. However, if ϵ_z < ϵ_0 <ϵ_c, for the propagation angle θ in the range θ_1 < | θ| < θ_3 and π - θ_3< | θ| < π - θ_1 we find “conventional” Dyakonov waves, while for θ_3 < | θ|< θ_2 and π - θ_2 < | θ| < π - θ_3the surface modes belong to the “ghost” class – see Fig. <ref>(c).Here, the angle θ_3 only depends on the dielectric permittivies of the media forming the interface, and is defined as the solution of the system of equations (<ref>) and (<ref>), where the latter taken with the positive signs.§ DISCUSSION The key feature of the Dyakonov surface waves that makes them an ideal platform for experiments on nonlinear optics and strong coupling, is their inherent “lossless“ nature. While the residual linear absorption in the dielectric as well as light scattering due to surface roughness can never be completely avoided, the corresponding contributions to the effective mode loss can be dramatically reduced, as demonstrated in Mie resonance experiments with the measured Q-factors on the order of 10^10. <cit.> As a result, with an evanescent coupling (from e.g. a high-index prism) to the isotropic-biaxial interface, one can observe an enormous increase of the field intensity at this boundary, only limited by the effective loss due to system imperfections (surface and builk disorder, etc.) and ultimately by the non-locality of the dielectric response <cit.>(corresponding to the variations of the dielectric permittivityon the order of (a_0 / λ)^2 ∼ 10^-6, where a_0 is on the order of the atomic size and λ is the wavelength).For the applications to nonlinear optics however, the effective “selection rules” such as the phase-matching conditions <cit.> are defined by the spatial variation of the corresponding optical modes. The qualitative difference between the“ghost” and the `conventional” surface waves, respectively with- and without oscillations away from the interface, that can be simultanenouls supported by the same isotropic-biaxial interface at the same frequency, will therefore have dramatic effect on the nonlinear-optical phenomena in this system. <cit.>§ CONCLUSIONS In summary, we have developed a complete analytical theory of Dyakonov surface waves at the interface of an isotropic medium with a biaxial anisotropic dielectric. As opposed to earlier work on this subject, our approach does not require any numerical root-finding, and offers substantial advantage in the description of the surface waves near the propagation threshold. We have also presented a detailed description of the ghost waves that combine the properties of propagating and evanescent solutions, and of the corresponding surface modes supported by these ghost waves. § ACKNOWLEDGEMENTS This work was partially supported by the Army Research Office (grant W911NF-14-1-0639), National Science Foundation (grants DMR-1120923 and DMR-1629276), andGordon and Betty Moore Foundation.§Some of the necessary conditionsfor the existence of the Dyakonov wave in (<ref>) can be immediately obtained from the general structure of Eqn. (<ref>) and its constituents. Eqn. (<ref>) immediately implies that both the first and the last terms in the curly brackets in Eqn. (<ref>) are negative-definite, thereforeϵ_0 + ϵ_x/ϵ_0 q_x^2 +ϵ_0 + ϵ_y/ϵ_0 q_y^2 <(ϵ_x + ϵ_y ) (ω/c)^2.Sinceϵ_x/ϵ_0 q_x^2 + ϵ_y/ϵ_0 q_y^2 =ϵ_x ϵ_y/ϵ_0(q_x^2/ϵ_y + q_y^2/ϵ_x - (ω/c)^2) + ϵ_x ϵ_y/ϵ_0(ω/c)^2>ϵ_x ϵ_y/ϵ_0(ω/c)^2,from (<ref>) and (<ref>) we obtainϵ_x + ϵ_y > ϵ_0 + ϵ_x ϵ_y/ϵ_0,which implies thatmin(ϵ_x, ϵ_y ) < ϵ_0 <max(ϵ_x, ϵ_y).Similarly, from (<ref>), (<ref>) and (<ref>) ϵ_x + ϵ_y > ϵ_z + ϵ_x ϵ_y/ϵ_0,orϵ_z < max(ϵ_x, ϵ_y) -min(ϵ_x, ϵ_y) ·(1 -max(ϵ_x,ϵ_y)/ϵ_0)<max(ϵ_x, ϵ_y). §First, we consider the number of real positive solutions of Eqn. (<ref>). Sinceϵ_1 + 2 ϵ_2 =(ϵ_0 - ϵ_x)(ϵ_y - ϵ_0)/ϵ_0 +ϵ_x ϵ_y(1/ϵ_z - 1/ϵ_0) > 0,since with our choice of ϵ_x < ϵ_y (see (<ref>)) the requirement (<ref>) reduces to ϵ_x ≤ϵ_z < ϵ_0 < ϵ_y, and thereforea_4 > 0.Similarly, since û > 0,a_1 < 0,anda_3 =√(ϵ_x ϵ_y/ϵ_0û)· 2 ϵ_0 (ϵ_x + ϵ_y - ϵ_0/ϵ_z - ϵ_x ϵ_y/ϵ_0^2)=2/ϵ_z √(ϵ_x ϵ_y/ϵ_0û)·((ϵ_0 - ϵ_x)(ϵ_y - ϵ_0) + ϵ_x ϵ_y/ϵ_0(ϵ_0 - ϵ_z) ) > 0.Therefore, regardless of the sign of a_2,the number of sign changes of the polynomial a_4 χ^4 + a_3 χ^3 + a_2 χ^2 + a_1 χ + a0 is equal to one if a_0<0 and to two if a_0 > 0. According to the Descartes' rule of signs, <cit.> Eqn. (<ref>) has no more than one positive realroot in the former case and no more than two positive real roots in the latter. So, in general Eqn. (<ref>) has no more than two positive real roots.However, the solution of Eqn. (<ref>) must also satisfy the inequality (<ref>). Introducing the new variableξ≡χ - √(ϵ_0 - ϵ_z),to satisfy (<ref>) we need ξ > 0. From (<ref>) we obtaina_4ξ^4 + b_3ξ^3 + b_2 ξ^2 + b_1ξ + b_0 = 0,whereb_3 = a_3 + 4 a_4 √(ϵ_0 -ϵ_1), b_2 = a_2+ 3 a_3 √(ϵ_0 - ϵ_z) + 6 a_4 (ϵ_0 - ϵ_z) , b_1 = a_1 + 2 a_2 √(ϵ_0 - ϵ_z)+ 3 a_3 (ϵ_0 - ϵ_z)+ 4 a_4 (ϵ_0 - ϵ_z)^3/2, b_0 = a_0 + a_1 √(ϵ_0 - ϵ_z) + a_2 (ϵ_0 - ϵ_z)+ a_3 (ϵ_0 - ϵ_z)^3/2 + a_4 (ϵ_0 - ϵ_z)^2.From (<ref>) and (<ref>)b_3 > 0.For b_0 we obtainb_0 = - ϵ_ 0 - ϵ_z/ϵ_0 ϵ_z( ϵ_x ϵ_y û + ϵ_0 √(ϵ_x ϵ_y/ϵ_z(ϵ_0 - ϵ_z) û) - (ϵ_0 - ϵ_x) ·(ϵ_y - ϵ_0) )^2 < 0.Ifeither b_1 >0, b_2 > 0, or b_1 < 0, b_2 < 0, or b1< 0, b_2 > 0, then the number of sign changes of the polynomial a_4 χ^4 + a_3 χ^3 + a_2 χ^2 + a_1 χ + a0 is equal to one, and therefore Eqn. (<ref>) has no more than one real positive root. It is if and only if b1> 0, b_2 <0 that Eqn. (<ref>) can in principle have two positive real roots. For b1> 0, b_2 <0, from Eqns. (<ref>), (<ref>) we obtaina_2 + 3 a_3 √(ϵ_0 - ϵ_z)+ 6 a_4 (ϵ_0 - ϵ_z)< 0,a_1 + 2 a_2 √(ϵ_0 - ϵ_z)+ 3 a_3 (ϵ_0 - ϵ_z) + 4 a_4 (ϵ_0 - ϵ_z)^3/2> 0,Then a_2<- 3 a_3 √(ϵ_0 - ϵ_z)-6 a_4 (ϵ_0 - ϵ_z), and3a_3(ϵ_0 - ϵ_z) + 4 a_4 (ϵ_0 - ϵ_z)^3/2 >- a_1-2 a_2 √(ϵ_0 - ϵ_z)>- a_1 + 2(ϵ_0 - ϵ_z) ·(3 a_3 + 6 a_4√(ϵ_0 - ϵ_z)),which yields- 3 a_3 - 8 a_4 √(ϵ_0 - ϵ_z) > - a_1/ϵ_0 - ϵ_z.With a_1 < 0, a_3 >0 and a_4 > 0, and ϵ_0 > ϵ_z (see (<ref>)), the left-hand side of (<ref>) is negative, while the right-hand size is positive. The system of the inequalities (<ref>),(<ref>) is thereforeinconsistent, and the case b_1 > 0, b_2 < 0cannot be realized. Therefore, Eqn. (<ref>) cannot have more than one positive real root, and Eqn. (<ref>) cannot have more than one real solution with χ > √(ϵ_0 - ϵ_z).§We define θ_1 as the propagation angle that corresponds to the limiting case of the inequality (<ref>). In terms of our parameter û defined by Eqns. (<ref>) and (<ref>), the bound (<ref>) corresponds toû(θ_1)= u(θ_1) =0,which together with (<ref>) implies thatκ_-(θ_1) = 0Sincea_0(û = 0) = a_1(û = 0) = a_3(û = 0) = 0,we obtainF(û=0)=√(- a_2(û = 0)/a_4 (û = 0)) =√(ϵ_0 - ϵ_z + ϵ_2^2/ϵ_1 + 2 ϵ_2).Substituting (<ref>) into (<ref>), we obtainsin^2θ_1 =ϵ_x ϵ_y/ϵ_y - ϵ_x( ϵ_a + 2 ϵ_2/ϵ_0 (ϵ_1 + 2 ϵ_2 ) + ϵ_2^2 - 1/ϵ_y),leading to our definition of θ_1 in Eqn. (<ref>). Since we defined the x- and y- directions with ϵ_y > ϵ_s, the inequality (<ref>) then impliesθ_1 < | θ| < π/2,orπ/2 < | θ| < π - θ_1. The angle θ_2 is defined as the propagation direction of the surface wave corresponding to the limiting case of (<ref>) when the latter turns into the exact equalityÂ(θ_2) κ_+(θ_2) κ_-(θ_2) + B̂(θ_2)= 0.Substituting (<ref>) into (<ref>), we find that eitherκ_0(θ_2) = 0,or- ϵ_x ϵ_y/ϵ_0û(θ_2) (ω/c)^2=κ_+(θ_2)κ_-(θ_2)Since for θ in the range defined by Eqns. (<ref>),(<ref>) we find û > 0, and Eqn. (<ref>) therefore cannot be satisfied – so that(<ref>) is the only option. Then, substituting (<ref>) into Eqn. (<ref>), we obtain(ϵ_2 -ϵ_xϵ_y/ϵ_0·û(θ_2) )·(ω/c)^2=κ_+(θ_2) κ_-(θ_2). From (<ref>) we obtainκ_+(θ_2)κ_-(θ_2) =(ω/c)^2√(ϵ_x ϵ_y/ϵ_z(ϵ_0 - ϵ_z) û(θ_2)).Substituting (<ref>) into (<ref>), we find û(θ_2) =ϵ_0 ϵ_2 /ϵ_x ϵ_y+ ϵ_0^2 (ϵ_0 - ϵ_z)/2 ϵ_x ϵ_y ϵ_z× (1 - √(1 + 4·ϵ_z(ϵ_0 - ϵ_x ) (ϵ_y - ϵ_0 )/ϵ_0^2(ϵ_0 - ϵ_z)) )From (<ref>) and (<ref>)û(θ_2) =ϵ_0/ϵ_ycos^2θ_2 +ϵ_0/ϵ_xsin^2θ_2 -1.Substituting (<ref>) into (<ref>) and using (<ref>), we findsin^2θ_2 =ϵ_y - ϵ_0/ϵ_y - ϵ_x+ ϵ_0/2 ϵ_zϵ_0 - ϵ_z/ϵ_y - ϵ_x× (1 - √(1 + 4 ·ϵ_z(ϵ_0 - ϵ_x ) (ϵ_y - ϵ_0 )/ϵ_0^2 (ϵ_0 - ϵ_z)) ).To satisfy Eqn. (<ref>), we therefore need0 <| θ| < θ_2,orπ - θ_2<| θ| < π,Together, (<ref>), (<ref>) and (<ref>), (<ref>) are equivalent to (<ref>).§The critical angle θ_1 corresponds to the point where the iso-frequency curvecorresponding to the Dyakonov surface wave in the (q_x, q_y) space terminates at the line(<ref>). We can show that the ghost boundary in the first quadrant, √(ϵ_z - ϵ_x/ϵ_z)q_x + √(ϵ_y - ϵ_z/ϵ_z) q_y =√(ϵ_y - ϵ_x) ω/c,can never cross this point. An assumption that such intersection point (q_x^(1),q_y^(y)), that satisfies both (<ref>) and (<ref>),may exist, leads to the equation(q_x^(1) - ϵ_y ω/c√(( ϵ_y - ϵ_z )(ϵ_z - ϵ_x)/ϵ_z (ϵ_y - ϵ_x)))^2+(ω/c)^2 ·ϵ_y^2/ϵ_z·(ϵ_z - ϵ_x)^2/(ϵ_y - ϵ_x)^2 = 0,which cannot be satisfied for any ϵ_x < ϵ_z < ϵ_z. As a result, in the first quadrant (q_x >0, q_y > 0) the ghost boundary is either always above or always below the curve of Eqn. (<ref>). The ellipse of Eqn. (<ref>) intersects the positive half of the q_y-axis at the point of √(ϵ_x), while for the ghost boundary (<ref>) the corresponding crossing point is at √(ϵ_z (ϵ_y - ϵ_x)/ (ϵ_z - ϵ_x)) > √(ϵ_x). In the first quadrant of the q space the ghost boundary is therefore always above the elliptical curve of Eqn. (<ref>). As a result, this boundary, and thus the θ_1 “edge” of the iso-frequency curve of the Dyakonov surface wave, is always in the “conventional” regime, with the field characterized by the exponential decay on both sides of the interface. As a result, for the system to support the “ghost” surface waves, the ghost boundary (<ref>) must cross the iso-frequency curve of the Dyakonov waves, Eqn. (<ref>). The onset of the ghost regime then corresponds to the case when the ghost boundary intersects the iso-frequency curve precisely at its end at the angle θ_2. As follows from Eqn. (<ref>), the critical angle θ_2 corresponds to the point where the iso-frequency lineofthe Dyakonov surface wave in the (q_x, q_y) space terminates at the circleq_x^2 + q_y^2 =ϵ_0ω^2/c^2.For the intersection point (q_x^(2), q_y^(2))of (<ref>) with the ghost boundary(<ref>) in the first quadrant we obtainq_x^(2)=ω/c√(ϵ_z (ϵ_z - ϵ_x ))+√((ϵ_y - ϵ_z )(ϵ_0 - ϵ_z))/ϵ_y - ϵ_x , q_y^(2)=ω/c√(ϵ_z (ϵ_y - ϵ_z ))+√((ϵ_z - ϵ_x )(ϵ_0 - ϵ_z))/ϵ_y - ϵ_x .Substituting (<ref>),(<ref>) into (<ref>) and using (<ref>), we obtain(ϵ_0 - ϵ_z)^2 (5 ϵ_z - 3(ϵ_x + ϵ_y) + ϵ_x ϵ_y/ϵ_z)+(ϵ_0 -ϵ_z) (ϵ_x^2 + 4 ϵ_x ϵ_y + ϵ_y^2 - 8 ϵ_z (ϵ_x + ϵ_y ) + 10 ϵ_z^2 ) + 3 ϵ_z ( ϵ_y - ϵ_z) (ϵ_z - ϵ_x)+2/ϵ_z( (ϵ_c - ϵ_z)^2 + ϵ_z(2 (ϵ_x + ϵ_y) - 5 ϵ_z ) ) × √(ϵ_z ( ϵ_y - ϵ_z ) ( ϵ_z - ϵ_x ) ( ϵ_0 - ϵ_z ) ) = 0,which defines the values of the dielectric permittivity of the dielectric media corresponding to the onset of ghost surface waves in the system phase space. 10SW_microscopy M. Specht, J. D. Pedarnig, W. M. Heckl, and T. 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"authors": [
"Evgenii Narimanov"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170627081305",
"title": "Dyakonov Waves in Biaxial Anisotropic Crystals"
} |
University of Modena and Reggio Emilia Via P. Vivarelli, 10 Modena41125 Italy University of Modena and Reggio Emilia Via P. Vivarelli, 10 Modena41125 Italy University of UdineUdine Italy University of Modena and Reggio Emilia Via P. Vivarelli, 10 Modena41125 Italy Image captioning has been recently gaining a lot of attention thanks to the impressive achievements shown by deep captioning architectures, which combine Convolutional Neural Networks to extract image representations, and Recurrent Neural Networks to generate the corresponding captions. At the same time, a significant research effort has been dedicated to the development of saliency prediction models, which can predict human eye fixations. Even though saliency information could be useful to condition an image captioning architecture, by providing an indication of what is salient and what is not, research is still struggling to incorporate these two techniques. In this work, we propose an image captioning approach in which a generative recurrent neural network can focus on different parts of the input image during the generation of the caption, by exploiting the conditioning given by a saliency prediction model on which parts of the image are salient and which are contextual. We show, through extensive quantitative and qualitative experiments on large scale datasets, that our model achieves superior performances with respect to captioning baselines with and without saliency, and to different state of the art approaches combining saliency and captioning. <ccs2012> <concept> <concept_id>10010147.10010178.10010179.10010182</concept_id> <concept_desc>Computing methodologies Natural language generation</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010178.10010224.10010225.10010227</concept_id> <concept_desc>Computing methodologies Scene understanding</concept_desc> <concept_significance>500</concept_significance> </concept> </ccs2012> [500]Computing methodologies Scene understanding [500]Computing methodologies Natural language generation This work is partially supported by “Città educante” (CTN01_00034_393801) of the National Technological Cluster on Smart Communities (cofunded by the Italian Ministry of Education, University and Research - MIUR) and by the project “JUMP - Una piattaforma sensoristica avanzata per rinnovare la pratica e la fruizione dello sport, del benessere, della riabilitazione e del gioco educativo”, funded by the Emilia-Romagna region within the POR-FESR 2014-2020 program.We acknowledge the CINECA award under the ISCRA initiative, for the availability of high performance computing resources and support. We also gratefully acknowledge the support of NVIDIA Corporation with the donation of the GPUs used for this research.Author's addresses: M. Cornia and L. Baraldi and R. Cucchiara, Department of Engineering “Enzo Ferrari”, University of Modena and Reggio Emilia, Modena, Italy; G. Serra is with the Department of Computer Science, Mathematics and Physics, University of Udine, Udine, Italy. Paying More Attention to Saliency: Image Captioning with Saliency and Context Attention Rita Cucchiara======================================================================================= § INTRODUCTIONA core problem in computer vision and artificial intelligence is that of building a system that can replicate the human ability of understanding a visual stimuli and describing it in natural language. Indeed, this kind of system would have a great impact on society, opening up to a new progress in human-machine interaction and collaboration. Recent advancements in computer vision and machine translation, together with the availability of large datasets, have made it possible to generate natural sentences describing images. In particular, deep image captioning architectures have shown impressive results in discovering the mapping between visual descriptors and words <cit.>. They combine Convolutional Neural Networks (CNNs), to extract an image representation, and Recurrent Neural Networks (RNNs), to build the corresponding sentence. While the progress of these techniques is encouraging, the human ability in the construction and formulation of a sentence is still far from being adequately emulated in today's image captioning systems. When humans describe a scene, they look at an object before naming it in a sentence <cit.>, and they do not focus on each region with the same intensity, as selective mechanisms attract their gaze on saliency and relevant parts of the scene <cit.>. Also, they care about the context using peripheral vision, so that the description of an image alludes not only to the main objects in the scene, and to how they relate to each other, but also to the context in which they are placed in the image. An intensive research effort has been carried out in the computer vision community to predict where humans look in an image. This task, called saliency prediction, has been tacked in early works by defining hand-crafted features that capture low-level cues such as color and texture or higher-level concepts such as faces, people and text <cit.>. Recently, with the advent of deep neural networks and large annotated datasets, saliency prediction techniques have obtained impressive results generating maps that are very close to the ones computed with eye-tracking devices <cit.>.Despite the encouraging progress in image captioning and visual saliency, and their close connections, these two fields of research have remained almost separate. In fact, only few attempts have been recently presented in this direction <cit.>. In particular, Sugano <cit.> presented a gaze-assisted attention mechanism for image caption based on human eye fixations (i.e. the static states of gaze upon a specific location). Although this strategy confirms the importance of using eye fixations, it requires gaze information from a human operator. Therefore, it can not be applied on general visual data archives, in which this information is missing.To overcome this limit, Tavakoli <cit.> presented an image captioning method based on saliency maps, which can be automatically predicted from the input image.In this paper we present an approach which incorporates saliency prediction to effectively enhance the quality of image description. We propose a generative Recurrent Neural Network architecture which can focus on different regions of the input image by means of an attentive mechanism. This attentive behaviour, differently from previous works <cit.>, is conditioned by two different attention paths: the former focused on salient spatial regions, predicted by a saliency model, and the latter focused on contextual regions, which are computed as well from saliency maps. Experimental results on five public image captioning datasets (SALICON, COCO, Flickr8k, Flickr30k and PASCAL-50S), demonstrate that our solution is able to properly exploit saliency cues. Also, we show that this is done without losing the key properties of the generated captions, such as their diversity and the vocabulary size. By visualizing the states of both attentive paths, we finally show that the trained model has learned to attend to both salient and contextual regions during the generation of the caption, and that attention focuses produced by the network effectively correspond, step by step, to generated words.To sum up, our contributions are as follows. First, we show that saliency can enhance image description, as it provides an indication of what is salient and what is context. Second, we propose a model in which the classic machine attention approach is extended to incorporate two attentive paths, one for salient regions and one for context. These two paths cooperate together during the generation of the caption, and show to generate better captions according to automatic metrics, without loss of diversity and size of the dictionary. Third, we qualitatively show that the trained model has learned to attend to both salient and contextual regions in an appropriate way.§ RELATED WORKIn this section, we review the literature related to saliency prediction and image captioning. We also report some recent works which investigate the contribution of saliency for generating natural language descriptions. §.§ Visual saliency predictionSaliency prediction has been extensively studied by the computer vision community and, in the last few years, has achieved a considerable improvement thanks to the large spread of deep neural networks <cit.>. However, a very large variety of models have been proposed before the advent of deep learning and almost each of them has been inspired by the seminal work of Itti and Koch <cit.>, in which multi-scale low-level features extracted from the input image were linearly combined and then processed by a dynamic neural network with a winner-takes-all strategy. The same idea of properly combining different low-level features was also explored by Harel <cit.> who defined Markov chains over various image maps, and treated the equilibrium distribution over map locations as an activation. In addition to the exploitation of low-level features, several saliency models have also incorporated high-level concepts such as faces, people, and text <cit.>. In fact, Judd <cit.> highlighted that, when humans look at images, their gazes are attracted not only by low-level cues typical of the bottom-up attention, but also by top-down image semantics. To this end, they proposed a model in which low and medium level features were effectively combined, and exploited face and people detectors to capture important high-level concepts. Nonetheless, all these techniques have failed to effectively capture the wide variety of causes that contribute to define the visual saliency on images and, with the advent of deep learning, researchers have developed data-driven architectures capable of overcoming many of the limitations of hand-crafted models. First attempts of computing saliency maps through a neural network lacked from the absence of sufficiently large training datasets <cit.>. Vig <cit.> proposed the first deep architecture for saliency, which was composed by only three convolutional layers. Afterwards, Kümmerer <cit.> based their models on two popular convolutional networks (AlexNet <cit.> and VGG-19 <cit.>) obtaining adequate results, despite the network parameters were not fine-tuned on a saliency dataset. Liu <cit.> tried to overcome the absence of large scale datasets by training their model on image patches centered on fixation and non-fixation locations, thus increasing the amount of training data.With the arrival of the SALICON dataset <cit.>, which is still the large publicly available dataset for saliency prediction, several deep architectures have moved beyond previous approaches bringing consistent performance advances. The starting point of all these architectures is a pre-trained Convolutional Neural Network (CNN), such as VGG-16 <cit.>, GoogleNet <cit.> and ResNet <cit.>, to which different saliency-oriented components are added <cit.>, together with different training strategies <cit.>. In particular, Huang <cit.> compared three standard CNNs by applying them at two different image scales. In addition, they were the first to train the network using a saliency evaluation metric as loss function. Jetley <cit.> introduced a model which formulates a saliency map as generalized Bernoulli distribution. Moreover, they trained their network by using different loss functions which pair the softmax activation function with measures designed to compute distances between probability distributions. Tavakoli <cit.> investigated inter-image similarities to estimate the saliency of a given image using an ensemble of extreme learners, each trained on an image similar to the input image. Kruthiventi <cit.>, instead, presented an unified framework to predict both eye fixations and salient objects. Another saliency prediction model was recently presented by Pan <cit.> who, following the large dissemination of Generative Adversarial Networks, trained their model by using adversarial examples. In particular, their architecture is composed by two agents: a generator which is responsible for generating the saliency map of a given image, and a discriminator which performs a binary classification task between generated and real saliency maps. Liu <cit.>, instead, proposed a model to learn long-term spatial interactions and scene contextual modulation to infer image saliency showing promising results, also thanks to the use of the powerful ResNet-50 architecture <cit.>.In contrast to all these works, we presented two different deep saliency architectures. The first one, called ML-Net <cit.>, effectively combines features coming from different levels of a CNN and applies a matrix of learned weights to the predicted saliency map thus taking into account the center bias present in human eye fixations. The second one, called SAM <cit.>, incorporates neural attentive mechanisms which focus on the most salient regions of the input image. The core component of the proposed model is an Attentive Convolutional LSTM that iteratively refines the predicted saliency map. Moreover, to tackle the human center bias, the network is able to learn multiple Gaussian prior maps without predefined information. Since this model achieved state of the art performances, being at the top of different saliency prediction benchmarks, we use it in this work. §.§ Image captioningRecently, the automatic description of images and video has been addressed by computer vision researchers with recurrent neural networks which, given a vectored description of the visual content, can naturally deal with sequences of words <cit.>. Before deep learning models, the generation of sentences was mainly tackled by identifying visual concepts, objects and attributes which were then combined into sentences using pre-defined templates <cit.>. Another strategy was that of posing the image captioning as a retrieval problem, where the closest annotated sentence in the training set was transferred to a test image, or where training captions were split into parts and then reassembled to form new sentences <cit.>. Obviously, all these approaches limited the variety of possible outputs and could not satisfy the richness of natural language. Recent captioning models, in fact, address the generation of sentences as a machine translation problem in which a visual representation of the image coming from a convolutional network is translated in a language counterpart through a recurrent neural network.One of the first models based on this idea is that proposed by Karpathy <cit.> in which sentence snippets are aligned to the visual regions that they describe through a multimodal embedding. After that, these correspondences are treated as training data for a multimodal recurrent neural network which learns to generate the corresponding sentences. Vinyals <cit.>, instead, developed an end-to-end model trained to maximize the likelihood of the target sentence given the input image. Xu <cit.> introduced an approach to image captioning which incorporates a form of machine attention, by which a generative LSTM can focus on different regions of the image while generating the corresponding caption. They proposed two different versions of their model: the first one, called “Soft Attention”, is trained in a deterministic manner using standard backpropagation techniques, while the second one, called “Hard Attention”, is trained by maximizing a variational lower bound through the reinforcement learning paradigm.Johnson <cit.> addressed the task of dense captioning, which jointly localizes and describes in natural language salient image regions. This task consists of generalizing the object detection problem when the descriptions consist of a single word, and the image captioning task when one predicted region covers the full image. You <cit.> proposed a semantic attention model in which, given an image, a convolutional neural network extracts top-down visual features and at the same time detects visual concepts such as regions, objects and attributes. The image features and the extracted visual concepts are combined through a recurrent neural network that finally generates the image caption.Differently from previous works which aim at predicting a single caption, Krause <cit.> introduced the generation of entire paragraphs for describing images. Finally, Shetty <cit.> employed adversarial training to change the training objective of the caption generator from reproducing ground-truth captions to generating a set of captions that is indistinguishable from human generated captions.In this paper, we are interested in demonstrating the importance of using saliency along with contextual information during the generation of image descriptions. Our solution falls in the class of neural attentive captioning architectures and, in the experimental section, we compare it against a standard attentive model built upon the Soft Attention approach presented in <cit.>.§.§ Visual saliency and captioningOnly a few other previous works have investigated the contribution of human eye fixations to generate image descriptions. The first work that has explored this idea was that proposed in <cit.> which presented an extension of a neural attentive captioning architecture. In particular, the proposed model incorporates human fixation points (obtained with eye-tracking devices) instead of computed saliency maps to generate image captions. This kind of strategy mainly suffers of the need of having both eye fixation and caption annotations. Currently, only the SALICON dataset <cit.>, being a subset of the Microsoft COCO dataset <cit.>, is available with both human descriptions and saliency maps. Ramanishka <cit.>, instead, introduced an encoder-decoder captioning model in which spatiotemporal heatmaps are produced for predicted captions and arbitrary query sentences without explicit attention layers. They refer to these heatmaps as saliency maps, even though they are internal representations of the network, not related with human attention. Experiments showed that the gain in performance with respect to a standard captioning attentive model is not consistent, even though the computational overhead is lower.A different approach, presented in <cit.>, explores if image descriptions, by humans or models, agree with saliency and if saliency can benefit image captioning. To this end, they proposed a captioning model in which image features are boosted with the corresponding saliency map by exploiting a moving sliding window and mean pooling as aggregation strategies. Comparisons with respect to a no-saliency baseline did not show significant improvements (especially on the Microsoft COCO dataset).In this paper, we instead aim at enhancing image captions by directly incorporating saliency maps in a neural attentive captioning architecture. Differently from previous models that exploit human fixation points, we obtain a more general architecture which can be potentially trained using any image captioning dataset, and can predict captions for any input image. In our model, the machine attentive process is split in two different and unrelated paths, one for salient regions and one for context. We demonstrate through extensive experiments that the incorporation of saliency and context can enhance image captioning on different state of art datasets. § WHAT IS HIT BY SALIENCY?Human gazes are attracted by both low-level cues such as color, contrast and texture, and high-level concepts such as faces and text <cit.>. Current state of the art saliency prediction methods, thanks to the use of deep networks and large-scale datasets, are able to effectively incorporate all these factors and predict saliency maps which are very close to those obtained from human eye fixations <cit.>. In this section we qualitatively investigate which parts of an image are actually hit or ignored by saliency models, by jointly analyzing saliency and semantic segmentation maps. This will motivate the need of using saliency predictions as an additional conditioning for captioning models.To compute saliency maps, we employ the approach in <cit.>, which has shown good results on popular saliency benchmarks, such as the MIT Saliency <cit.> and the SALICON dataset <cit.>, and which also won the LSUN Challenge in 2017. It is worthwhile to mention, anyway, that the qualitative conclusions of this section can be applied to any state of the art saliency model.Since semantic segmentations algorithms are not always completely accurate, we perform the analysis on three semantic segmentation datasets, in which regions have been segmented by human annotators: Pascal-Context <cit.>, Cityscapes <cit.> and the Look into Person (LIP) <cit.> dataset. While the first one contains natural images without a specific target, the other two are focused, respectively, on urban streets and human body parts. In particular, the Pascal-Context provides additional annotations for the Pascal VOC 2010 dataset <cit.> which contains 10,103 training and validation images and 9,637 testing images. It goes beyond the original Pascal semantic segmentation task by providing annotations for the whole scene, and images are annotated by using more than 400 different labels. The Cityscapes dataset, instead, is composed by a set of video sequences recorded in street scenes from 50 different cities. It provides high quality pixel-level annotations for 5,000 frames and coarse annotations for 20,000 frames. The dataset is annotated with 30 street-specific classes, such as car, road, traffic sign, etc. Finally, the LIP dataset is focused on the semantic segmentation of people and provides more than 50,000 images annotated with 19 semantic human part labels. Images contain person instances cropped from the Microsoft COCO dataset <cit.> and split in training, validation and testing sets with 30,462, 10,000 and 10,000 images respectively. For our analyses we only consider train and validation images for the Pascal-Context and LIP datasets, and the 5,000 pixel-level annotated frames for the Cityscapes dataset. Figure <ref> shows, for some sample images, the predicted saliency map and the corresponding semantic segmentation on the three datasets.We firstly investigate which are the most and the least salient classes for each dataset. Since there are semantic classes with a low number of occurrences with respect to the total number of images, we only consider relevant semantic classes (i.e. classes with at least N occurrences). Due to the different dataset sizes, we set N to 500 for the Pascal-Context and LIP datasets, and to 200 for the Cityscapes dataset. To collect the number of times that the predicted saliency hits a semantic class, we binarize each map by thresholding the values of its pixels. A low threshold value leads to a binarized map with dilated salient regions, while an high threshold creates small salient regions around the fixation points. For this reason, we use two different threshold values to analyze the most and the least salient classes. We choose a threshold near 0 to find the least salient classes for each dataset, and a value near 255 to find instead the most salient ones. Figures <ref> and <ref> show the most and the least salient classes in terms of the percentage of times that saliency hits a region belonging to a class. As it can be seen, there are different distributions depending on the considered dataset. For example, for the Pascal-Context, the most salient classes are animals (such as cats, dogs and birds), people and vehicles (such as airplanes and cars), while the least salient ones result to be ceiling, floor and light. As for the Cityscapes dataset, cars are absolutely the most salient class with a 70% of times in which is hit by saliency. All other classes, instead, do not reach the 40%. On the LIP dataset, the most salient classes are all human body parts in the upper body, while the least salient ones are all in the lower body. As expected, people faces are those most hit by saliency with an absolute number of occurrences near to 90%. It can be observed as a general pattern that the most important or visible objects in the scene are hit by saliency, while objects in the background, and the context itself of the image are usually ignored. This leads to the hypothesis that both salient and non salient regions are important to generate the description of an image, given that we generally want the context to be included in the caption, and that the distinction between salient regions and context, given by a saliency prediction model, can improve captioning results.We also investigate the existence of a relation between the size of an object and its saliency values. In Figure <ref>, we plot the joint distribution of object sizes and saliency values on the three datasets, where the size of an object is simply computed as the number of its pixels normalized by the size of the image. As it can be seen, most of the low saliency instances are small; however, high saliency values concentrate on small objects as well as on large ones. In summary, there is not always a proportionality between the size of an object and its saliency, so the importance of an object can not be assessed by simply looking at its size. In the image captioning scenario that we want to tackle, larger objects correspond to larger activations in the last layers of a convolutional architecture, while smaller objects correspond to smaller activations. Since salient and non salient regions can have comparable activations, the supervision given by a saliency prediction model on whether a pixel belongs or not to a salient region can be beneficial during the generation of the caption. § SALIENCY AND CONTEXT AWARE ATTENTIONFollowing the qualitative findings of the previous section, we develop a model in which saliency is exploited to enhance image captioning. Here, a generative recurrent neural network is conditioned, step by step, on salient spatial regions, predicted by a saliency model, and on contextual features which account for the role on non-salient regions in the generation of the caption. In the following, we describe the overall model. An overview is presented in Figure <ref>.Each input image I is firstly encoded through a Fully Convolutional Network, which provides a stack of high-level features on a spatial grid {𝐚_1, 𝐚_2, ..., 𝐚_L}, each corresponding to a spatial location of the image. At the same time, we extract a saliency map for the input image using the model in <cit.>, and downscale it to fit the spatial size of the convolutional features, so to obtain a spatial grid {s_1, s_2, ..., s_L} of salient regions, where s_i ∈[ 0, 1 ]. Correspondingly, we also define a spatial grid of contextual regions, {z_1, z_2, ..., z_L} where z_i = 1-s_i. Under the model, visual features at different locations will be selected or inhibited according to their saliency value.The generation of the caption is carried out word-by-word by feeding and sampling words from an LSTM layer, which, at every timestep, is conditioned on features extracted from the input image and on the saliency map. Formally, the behaviour of the generative LSTM is driven by the following equations:𝐢_t= σ(W_vi𝐯̂_t + W_wi𝐰_t + W_hi𝐡_t-1 + 𝐛_i)𝐟_t= σ(W_vf𝐯̂_t + W_wf𝐰_t + W_hf𝐡_t-1 + 𝐛_f)𝐨_t= σ(W_vo𝐯̂_t + W_wo𝐰_t + W_ho𝐡_t-1 + 𝐛_o)𝐠_t= ϕ(W_vg𝐯̂_t + W_wg𝐰_t + W_hg𝐡_t-1 + 𝐛_g)𝐜_t= 𝐟_t ⊙𝐜_t-1 + 𝐢_t ⊙𝐠_t𝐡_t= 𝐨_t ⊙ϕ(𝐜_t)where, at each timestep, 𝐯̂_t denotes the visual features extracted from I, by considering the map of salient regions { s_i }_i, and those of contextual regions { z_i }_i. 𝐰_t is the input word, and 𝐡 and 𝐜 are respectively the internal state and the memory cell of the LSTM. ⊙ denotes the element-wise Hadamard product, σ is the sigmoid function, ϕ is the hyperbolic tangent , W_* are learned weight matrices and 𝐛_* are learned biases vectors.To provide the generative network with visual features, we draw inspiration from the machine attention literature <cit.> and compute the fixed-length feature vector 𝐯̂_t as a linear combination of spatial features {𝐚_1, 𝐚_2, ..., 𝐚_L} with time-varying weights α_ti, normalized over the spatial extent via a softmax operator:𝐯̂_t = ∑_i=1^Lα_ti𝐚_i, α_ti = exp(e_ti)/∑_k=1^Lexp(e_tk).At each timestep the attention mechanism selects a region of the image, based on the previous LSTM state, and feeds it to the LSTM, so that the generation of a word is conditioned on that specific region, instead of being driven by the entire image.Ideally, we want weights α_ti to be aware of the saliency and contextual value of location 𝐚_i, and to be conditioned on the current status of the LSTM, which can be well encoded by its internal state 𝐡_t. In this way, the generative network can focus on different locations of the input image according to their belonging to a salient or contextual region, and to the current generation state. Of course, simply multiplying attention weights with saliency values would result in a loss of context, which is fundamental for caption generation. We instead split attention weights e_ti into two contributions, one for saliency and one for context regions, and employ two different fully connected networks to learn the two contributions (Figure <ref>). Conceptually, this is equivalent to building two separate attention paths, one for salient regions and for contextual regions, which are merged to produce the final attention. Overall, the model obeys to the following equation:e_ti = s_i · e_ti^sal + z_i · e_ti^ctxwhere e_ti^sal and e_ti^ctx are, respectively, the attention weights for salient and context regions. Attention weights for saliency and context are computed as follows:e_ti^sal = v_e, sal^T ·ϕ (W_ae, sal·𝐚_i + W_he, sal·𝐡_t-1) e_ti^ctx = v_e, ctx^T ·ϕ (W_ae, ctx·𝐚_i + W_he, ctx·𝐡_t-1)Notice that our model learns different weights for saliency and contextual regions, and combines them into a final attentive map in which the contributions of salient and non-salient regions are merged together. Similarly to the classical Soft Attention approach <cit.>, the proposed generative LSTM can focus on every region of the image, but the attentive process is aware of the saliency of each location, so that the focus on salient and contextual regions is driven by the output of the saliency predictor. §.§ Sentence generationWords are encoded with one-hot vectors having size equal to that of the vocabulary, and are then projected into an embedding space via a learned linear transformation. Because sentences have different lengths, they are also marked with special begin-of-string and end-of-string tokens, to keep the model aware of the beginning and end of a particular sentence.Given an image and sentence (𝐲_0, 𝐲_1, ..., 𝐲_T), encoded with one-hot vectors, the generative LSTM is conditioned step by step on the first t words of the caption, and is trained to produce the next word of the caption. The objective function which we optimize is the log-likelihood of correct words over the sequence max_𝐰∑_t=1^T logPr (𝐲_t | 𝐯̂_t, 𝐲_t-1, 𝐲_t-2, ..., 𝐲_0) where 𝐰 are all the parameters of the model. The probability of a word is modeled via a softmax layer applied on the output of the LSTM. To reduce the dimensionality, a linear embedding transformation is used to project one-hot word vectors into the input space of the LSTM and, viceversa, to project the output of the LSTM to the dictionary space. Pr (𝐲_t | 𝐯̂_t, 𝐲_t-1, 𝐲_t-2, ..., 𝐲_0) ∝exp (𝐲_t^T W_p 𝐡_t) where W_p is a matrix for transforming the LSTM output space to the word space and 𝐡_t is the output of the LSTM.At test time, the LSTM is given a begin-of-string tag as input for the first timestep, then the most probable word according to the predicted distribution is sampled and given as input for the next timestep, until an end-of-string tag is predicted.§ EXPERIMENTAL EVALUATIONIn this section we perform qualitative and quantitative experiments to validate the effectiveness of the proposed model with respect to different baselines and other saliency-boosted captioning methods. First, we describe datasets and metrics used to evaluate our solution and provide implementation details.§.§ Datasets and metrics To validate the effectiveness of the proposed Saliency and Context aware Attention, we perform experiments on five popular image captioning datasets: SALICON <cit.>, Microsoft COCO <cit.>, Flickr8k <cit.>, Flickr30k <cit.>, and PASCAL-50S <cit.>.Microsoft COCO is composed by more than 120,000 images divided in training and validation sets, where each of them is provided with at least five sentences generated by using Amazon Mechanical Turk. SALICON is a subset of this one, created for the visual saliency prediction task. Since its images are from the Microsoft COCO dataset, at least five captions for each image are available. Overall, it contains 10,000 training images, 5,000 validation images and 5,000 testing images where eye fixations for each image are simulated with mouse movements. In our experiments, we only use train and validation sets for both datasets. The Flickr8k and the Flickr30k datasets are composed by 8,000 and 30,000 images respectively. Both of them come with five annotated sentences for each image. In our experiments, we randomly choose 1,000 validation images and 1,000 test images for each of these two datasets. The PASCAL-50S dataset provides additional annotations for the UIUC PASCAL sentences <cit.>. It is composed of 1,000 images from the PASCAL-VOC dataset, each of them annotated with 50 human-written sentences, instead of 5 as in the original dataset. Due to the limited number of samples and for a fair comparison with other captioning methods, we first pre-train the model on the Microsoft COCO dataset, then we test it on the images of this dataset without a specific fine-tuning.For evaluation, we employ four automatic metrics which are usually employed in image captioning: BLEU <cit.>, ROUGE_L <cit.>, METEOR <cit.> and CIDEr <cit.>.BLEU is a modified form of precision between n-grams to compare a candidate translation against multiple reference translations. We evaluate our predictions with BLEU using mono-grams, bi-grams, three-grams and four-grams.ROUGE_L computes an F-measure considering the longest co-occurring in sequence n-grams. METEOR, instead, is based on the harmonic mean of unigram precision and recall, with recall weighted higher than precision. It also has several features that are not found in other metrics, such as stemming and synonymy matching, along with the standard exact word matching. CIDEr, finally, computes the average cosine similarity between n-grams found in the generated caption and those found in reference sentences, weighting them using TF-IDF. To ensure a fair evaluation, we use the Microsoft COCO evaluation toolkit[<https://github.com/tylin/coco-caption>] to compute all scores. §.§ Implementation detailsEach image is encoded through a convolutional network, which computes a stack of high-level features. We employ the popular ResNet-50 <cit.>, trained over the ImageNet dataset <cit.>, to compute the feature maps over the input image. In particular, the ResNet-50 is composed by 49 convolutional layers, divided in 5 convolutional blocks, and 1 fully connected layer. Since we want to maintain the spatial dimensions, we extract the feature maps from the last convolutional layer and ignore the fully connected layer. The output of the ResNet model is a tensor with 2048 channels. To limit the number of feature maps and the number of learned parameters, we fed this tensor into another convolutional layer with 512 filters and a kernel size of 1, followed by a ReLU activation function. Differently from the weights of the ResNet-50 which are kept fixed, the weights of this last convolutional layer are initialized according to <cit.> and fine-tuned over the considered datasets. In the LSTM, following the initialization proposed in <cit.>, the weight matrices applied to the inputs are initialized by sampling each element from the Gaussian distribution of 0 mean and 0.01^2 variance, while the weight matrices applied to the internal states are initialized by using the orthogonal initialization. The vectors v_e, sal and v_e, ctx as well as all bias vectors 𝐛_* are instead initialized to zero.To predict the saliency map for each input image, we exploit our Saliency Attentive Model (SAM) <cit.> which is able to predict accurate saliency maps according to different saliency benchmarks. We note however, that we do not expect a significant performance variation when using other state of the art saliency methods.As mentioned, we perform experiments over five different datasets. For the SALICON dataset, since its images have all the same size of 480×640, we keep the original size of these images, thus obtaining L=15×20=300. For all other datasets, which are composed of images with different sizes, we set the input size to 480×480 obtaining L=15×15=225. Since saliency maps are exploited inside the proposed saliency-context attention model, we resize the SALICON saliency maps to have a size of 15×20 while, for all other datasets, we resize them to 15×15.All experiments are performed by using the Adam optimizer <cit.> with Nestorov momentum <cit.> using an initial learning rate of 0.001 and batch size 64. The hidden state dimension is set to 1024 while the embedding size to 512. For all datasets, we choose a vocabulary size equal to the number of words which appear at least 5 times in training and validation captions. §.§ Quantitative results and comparisons with baselinesTo assess the performance of our method, and to investigate the hypotheses behind it, we first compare with the classic Soft Attention approach, and we then build three baselines in which saliency is used to condition the generative process.Soft Attention <cit.>: The visual input to the LSTM is computed via the Soft Attention mechanism to attend at different locations of the image, without considering salient and non-salient regions. A single feed forward network is in charge of producing attention values, which can be obtained by replacing Eq. <ref> withe_ti = v^T_e ·ϕ(W_ae·𝐚_i + W_he·𝐡_t-1).This approach is equivalent to the one proposed in <cit.>, although some implementation details are different. In order to achieve a fair evaluation, we use activations from the ResNet-50 model instead of the VGG-19, and we do not include the doubly stochastic regularization trick. For this reason, the numerical results that we report are not directly comparable with those in the original paper (ours are in general higher than the original ones).Saliency pooling: Visual features from the CNN are multiplied at each location by the corresponding saliency value, and then summed, without any attention mechanism. In this case the visual input of the LSTM is not time dependent, and salient regions are given more focus than non-salient ones. Comparing with Eq. <ref>, it can be seen as a variation of the Soft Attention in which the network always focuses on salient regions.𝐯̂_̂t̂ = 𝐯̂ = ∑_i=1^L s_i 𝐚_i Attention on saliency: This is an extension of the Soft Attention approach in which saliency is used to modulate attention values at each location. The attention mechanism, therefore, is conditioned to attend salient regions with higher probability, and to ignore non-salient regions.e_ti = s_i · v^T_e ·ϕ(W_ae·𝐚_i + W_he·𝐡_t-1) Attention on saliency and context (with weight sharing): The attention mechanism is aware of salient and context regions, but weights used to compute the attentive scores of salient and context are shared, excluding the v_e^T vectors. Notice that, if those were shared too, this baseline would be equivalent to the Soft Attention one.e_ti = s_i · e_ti^sal + (1-s_i) · e_ti^ctxe_ti^sal = v_e, sal^T ·ϕ (W_ae·𝐚_i + W_he·𝐡_t-1) e_ti^ctx = v_e, ctx^T ·ϕ (W_ae·𝐚_i + W_he·𝐡_t-1) It is straightforward also to notice that our proposed approach is equivalent to the last baseline, without weight sharing. In Table <ref> we first compare the performance of our method with respect to the Soft Attention approach, to assess the superior performance of the proposal with respect to the published state of the art. We report results on all the datasets, both on validation and test sets, with respect to all the automatic metrics described in Section <ref>. As it can be seen, the proposed approach always overcomes by a significant margin the Soft Attention approach, thus experimentally confirming the benefit of having two separate attention paths, one for salient and one for non-salient regions, and the role of saliency as a conditioning for captioning. In particular, on the METEOR metric, the relative improvement ranges from 32.9 - 32.8/32.8 = 0.30% on the PASCAL-50S to 20.3 - 19.8/19.8 = 2.53% of the Flickr8k validation set.In Table <ref>, instead, we compare our approach with the three baselines that incorporate saliency. Firstly, it can be observed that the Saliency pooling baseline usually performs worse than the Soft Attention, thus demonstrating that always attending to salient locations is not sufficient to achieve good captioning results. When plugging in attention, like in the Saliency Attention baseline, numerical results are a bit higher, thanks to a time-dependent attention, but still far from the performance achieved by the complete model. It can also be noticed that, even though this baseline does not take into account the context, it sometimes achieves better results than the Soft Attention model (such as in the case of SALICON, with respect to the METEOR metric). Finally, we notice that the baseline with attention on saliency and context, and with weight sharing, is better than Saliency Attention, further confirming the benefit of including the context. Having two completely separated attention paths, such as in our model, is anyway important, as demonstrated by the numerical results of this last baseline with respect to those of our method. §.§ Comparisons with other saliency-boosted captioning modelsWe also compare to existing captioning models that incorporate saliency during the generation of image descriptions. In particular, we compare to the model proposed in <cit.>, which exploited human fixation points, to the work by Tavakoli <cit.> which reports experiments on Microsoft COCO and on PASCAL-50S, and to the proposal by Ramanishka <cit.> which used convolutional activations as a proxy for saliency.Table <ref> shows the results on the three considered datasets in term of BLEU@4, METEOR, ROUGE_L and CIDEr. We compare our solutions to both versions of the model presented in <cit.>. The GBVS version exploits saliency maps calculated by using a traditional bottom-up model <cit.>, while the other one includes saliency maps extracted from a deep convolutional network <cit.>.Overall, results show that the proposed Saliency and Context Attention model can overcome the other methods on different metrics, thus confirming the strategy of including two attention paths. In particular, on the METEOR metric, we obtain a relative improvement of 4.57% on the SALICON dataset, 5.53% on the Microsoft COCO and 8.94% on the PASCAL-50S.§.§ Analysis of generated captions We further collect statistics on captions generated by our method and by the Soft Attention model, to quantitatively assess the quality of generated captions. Firstly, we define three metrics which evaluate the vocabulary size and the difference between the corpus of captions generated by the two models and the ground-truth: * Vocabulary size: number of unique words generated in all captions;* Percentage of novel sentences: percentage of generated sentences which are not seen in the training set;* Percentage of different sentences: percentage of images which are described differently by the two models;Then, we measure the diversity of the set of captions generated by each of the two models, via the following two metrics <cit.>: * Div-1: ratio of number of unique unigrams in a set of captions to the number of words in the same set. Higher is more diverse.* Div-2: ratio of number of unique bigrams in a set of captions to the number of words in the same set. Higher is more diverse. In Table <ref> we compare the set of captions generated by our model with that generated by the Soft Attention baseline. Although our model features a slight reduction of the vocabulary size on SALICON, COCO and PASCAL-50S, captions generated by the two models are very often different, thus confirming that the two approaches have learned different captioning models. Moreover, the diversity and the number of novel sentences of the Soft Attention approach are entirely preserved.§.§ Analysis of attentive states The selection of a location in our model is based on the competition between the saliency attentive path and the context attentive path (see Eq. <ref>). To investigate how the two paths interact and contribute to the generation of a word, in Figure <ref> we report, for several images from the Microsoft COCO dataset, the changes in attention weights between the two paths. Specifically, for each image we report the average of e_ti^sal and e_ti^ctx values at each timestep, along with a visualization of its saliency map. It is interesting to see how the model was able to correctly exploit the two attention paths for generating different parts of the caption, and how generated words correspond in most cases to the attended regions. For example, in the case of the first image (“a group of zebras graze in a grassy field”), the saliency attentive path is more active than the context path during the generation of words corresponding to the “group of zebras”, which are captured by saliency. Instead, when the model has to describe the context (“in a grassy field”), the saliency attentive path has lower weights with respect to the context attentive path. The same can be observed for all the reported images; it can also be noticed that generated captions tend to describe both salient objects and the context, and that usually the salient part, which is also the most important, is described before the context.§.§ Qualitative resultsFinally, in Figure <ref> we report some sample results on images taken from the Microsoft COCO dataset. For each image we report the corresponding saliency map, and captions generated by our model and by the Soft Attention baseline compared to the ground-truth. It can be seen that, on average, captions generated by our model are more consistent with the corresponding image and the human-generated caption, and that, as also observed in the previous section, salient parts are described as well as the context. The incorporation of saliency and context also help the model to avoid failures due to hallucination, such as in the case of the fourth image, in which the Soft Attention model predicts a remote control which is not depicted in the image. Other failure cases, which are avoided by our model, include the repetition of words (as in the fifth image) and the failure to describe the context (first image). We speculate that the presence of two separate attention paths, which the model has learned to attend during the generation of the caption, helps to avoid such failures more effectively than the classic machine attention approach. For completeness, some failure cases of the proposed model are reported in Figure <ref>. The majority of failures occurs when the salient regions of the image are not described in the corresponding ground-truth caption (as for example in the first row), thus causing a performance loss. Some problems arise also in presence of complex scenes (such as in the fourth image). However, we observe that the Soft Attention baseline fails as well to predict correct and complete captions in these cases. § CONCLUSIONWe proposed a novel image captioning architecture which extends the machine attention paradigm by creating two attentive paths conditioned on the output of a saliency prediction model. The first one is focused on salient regions, and the second on contextual regions: the overall model exploits the two paths during the generation of the caption, by giving more importance to salient or contextual regions as needed. The role of saliency with respect to context has been investigated by collecting statistics on semantic segmentation datasets, while the captioning model has been evaluated on large scale captioning datasets, using standard automatic metrics and by evaluating the diversity and the dictionary size of the generated corpora. Finally, the activations of the two attentive paths have been investigated, and we have shown that they correspond, word by word, to a focus on salient objects or on the context in the generated caption; moreover, we qualitatively assessed the superiority of the captions generated by our method with respect to those generated by the Soft Attention approach. Although our focus has been that of demonstrating the effectiveness of saliency on captioning, rather than that of beating captioning approaches which rely on different cues, we point out that our method can be easily incorporated into those architectures. ACM-Reference-Format | http://arxiv.org/abs/1706.08474v4 | {
"authors": [
"Marcella Cornia",
"Lorenzo Baraldi",
"Giuseppe Serra",
"Rita Cucchiara"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170626164557",
"title": "Paying More Attention to Saliency: Image Captioning with Saliency and Context Attention"
} |
plain =1 | http://arxiv.org/abs/1706.08485v1 | {
"authors": [
"Bing Wang"
],
"categories": [
"math.DG",
"math.AG",
"53C44"
],
"primary_category": "math.DG",
"published": "20170626171521",
"title": "The local entropy along Ricci flow---Part A: the no-local-collapsing theorems"
} |
MPP-2017-126On the Structure of Quantum L_∞ algebras Ralph Blumenhagen, Michael Fuchs, Matthias Traube Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6,80805 München, Germany It is believed that any classical gauge symmetry gives rise to an L_∞ algebra. Based on the recently realized relation betweenclassical W algebras and L_∞ algebras, we analyze how this generalizes to the quantum case. Guided by the existence of quantum W algebras, we provide a physically well motivated definition of quantumL_∞ algebras describing theconsistency of global symmetries in quantum field theories. In this casewe are restricted to only twonon-trivialgraded vector spaces X_0 and X_-1 containing the symmetry variations and the symmetry generators. This quantum L_∞ algebra structure is explicitly exemplified for the quantum W_3 algebra. The naturalquantum product between fields is thenormal ordered one so that, due to contractions between quantum fields, the higher L_∞ relations receive off-diagonal quantum corrections. Curiously, these arenot present in the loop L_∞ algebra of closed string field theory. § INTRODUCTION Derived from closed string field theory <cit.>, the structure of L_∞ algebras were suggested to underly all classical perturbative gauge symmetries and their dynamics. For the first time, they actually appeared in the context of higher spin gauge theories <cit.> andwere also discussed in the mathematics literature (see e.g. <cit.>).Motivated by the study of field theory truncations of string field theory <cit.>, the authors of <cit.> arguedthat the symmetry and the action of any consistent perturbative gauge symmetry is controlled by an L_∞ algebra. For Chern-Simons and Yang-Mills gauge theories as well as for double field theory the symmetries and equations of motion could be expressed in terms of anL_∞ structure.Based on the higher spin AdS_3-CFT_2 duality, a large setof explicit non-trivial L_∞ algebraswere identified recently <cit.> by showing that the well understood class of classical W algebras can also be rewritten in terms ofhigher products satisfying the relations ofL_∞ algebras. Recall that W algebrasappear as extended chiral symmetry algebras of two-dimensional conformal field theories (CFTs)(see <cit.> for a review) and are actually not describing gauge symmetries butinfinitely many global symmetries. These examples are special in the sense that only two graded vector spaces were non-trivial, X_0 contains the symmetry parameters and X_-1 the generators of the W algebra. The special featureof W algebras, namely that the Poisson bracket between the generators closes only non-linearly, implied non-trivial higher products, corresponding e.g. to field dependent symmetryparameters.In <cit.> this correspondence was restricted to the classical case, for which the product of fields is just the point-wise product of holomorphic functions. However, from CFT it is well known that these classical W algebras appear as the classical ħ→ 0 limit of quantum W algebras. Here one is dealing with chiralquantum fields, whose product involves a normal ordering prescription. In addition, the field content of the algebra itself and their structure constants receive ħ corrections.It is an interesting question, how the L_∞structure generalizes to the quantum case. In the context of stringfield theory, this was already analyzed in <cit.> and further elucidated in the mathematical context in <cit.>. In this paper we generalize the analysis of <cit.> to quantum W-algebras. We will see that the higher products now involve the normal ordered product as the fundamental one, and that they also receive ħ corrections. In addition also the quadratic relations among the higher products receivequantum corrections, induced by non-trivial contractions following from the application of Wick's theorem. Since we are dealing with an interacting (non-free) CFT, these contractions are given by the singular part of the operator product expansion (OPE) and, as will be shown, imply off-diagonal terms among the naive classical L_∞ relations. Guided by quantum W algebras we are thus led to a well motivated definition of quantum L_∞ algebras that control the symmetries of a quantum theory. Similar as in the case of classical symmetries the quantum L_∞ algebras we look at are restricted to a graded vector space X = X_0 ⊕ X_-1 and are constructed such that they become the classical L_∞ algebra of the classical symmetry in the ħ→ 0 limit. The paper is organized as follows: In section <ref>we recall the definition of a classicalL_∞ algebra and its connection to the gauge algebra of classical gauge field theories. In section <ref>, after identifying the possible origin of quantum corrections,we first define quantum L_∞ algebras. Then we will compare it to loop L_∞ algebras, the quantum corrected L_∞ algebras arising for closed string field theory (CSFT)<cit.>. In section <ref> we will show in detail that the quantum W_3 algebra is organized in terms of a quantum L_∞ algebra. § THE L_∞ GAUGE ALGEBRA OF A CLASSICAL SYMMETRYIn this section we review how a perturbative classical gauge algebra is actually controlled by an L_∞ algebra. L_∞ algebras are generalized Lie algebras where one has not only a two-product, the commutator, but more general multilinear n-products with n inputsℓ_n: X^⊗n→X x_1, …, x_n↦ℓ_n(x_1, …, x_n) , defined on a graded vector space X = ⊕_n X_n, where n denotes the grading. The products are graded symmetric ℓ_n (…, x_1,x_2, …) = (-1)^1+ deg(x_1) deg( x_2) ℓ_2 (…, x_2,x_1, …) ,with deg( ℓ_n(x_1,…,x_n))=n-2+∑_i=1^ndeg(x_i) . These ℓ_n define an L_∞ algebra, if they satisfy theinfinitely many relations J_n(x_1,…, x_n):=∑_i + j = n + 1 (-1)^i(j-1) ∑_σχ(σ;x) ℓ_j ( ℓ_i (x_σ(1), …, x_σ(i) ), x_σ(i+1) , …, x_σ(n) ) = 0 .The permutations are restricted to the ones withσ(1) < ⋯< σ(i) , σ(i+1) < ⋯< σ(n) ,and the sign χ(σ; x) = ± 1 can be read off from (<ref>).The first relations J_n with n=1,2,3,… can be schematically written as J_1 = ℓ_1 ℓ_1 , J_2 = ℓ_1 ℓ_2 - ℓ_2 ℓ_1,J_3 = ℓ_1 ℓ_3 + ℓ_2 ℓ_2 + ℓ_3 ℓ_1,J_4 = ℓ_1 ℓ_4 - ℓ_2 ℓ_3 + ℓ_3 ℓ_2 - ℓ_4 ℓ_1 , from which one can deduce the scheme for the higher J_n. More concretely, the first L_∞ relations read ℓ_1(ℓ_1 (x) )= 0 ℓ_1 ( ℓ_2(x_1, x_2) )=ℓ_2( ℓ_1 (x_1) , x_2 ) + (-1)^x_1 ℓ_2(x_1, ℓ_1 (x_2) ) ,revealing that ℓ_1 must be a nilpotent derivation with respect to ℓ_2. Denoting (-1)^x_i=(-1)^deg(x_i) the full relation J_3 reads0 = ℓ_1(ℓ_3(x_1,x_2,x_3) ) + ℓ_2(ℓ_2(x_1,x_2),x_3 )+(-1)^(x_2+x_3)x_1ℓ_2(ℓ_2(x_2,x_3),x_1 )+ (-1)^(x_1+x_2)x_3ℓ_2(ℓ_2(x_3,x_1),x_2 )+ ℓ_3(ℓ_1(x_1),x_2,x_3 )+(-1)^x_1ℓ_3(x_1,ℓ_1(x_2),x_3 ) +(-1)^x_1+x_2ℓ_3(x_1,x_2,ℓ_1(x_3) ) and means thatthe Jacobi identity for the ℓ_2 product is mildly violated by ℓ_1 exact expressions. For this reason,L_∞ algebras are also called strong homotopy Lie algebras in the mathematical literature. The framework of L_∞ algebras is quite flexible and it has been suggestedthat every classical perturbative gauge theory (derived from string theory), including its dynamics, is organized by an underlying L_∞ structure <cit.>. For sure, the pure gauge algebra of such theories satisfies the L_∞ identities.To see this, let us assume that the field theory has a standard type gauge structure, meaning that the variations of the fields can be organized unambiguously into a sum of terms each of a definite power in the fields. Defining the space of gauge parameters ε to be X_0 and the field space Φ to be X_-1 and setting all other graded vector spaces to be trivial, the gauge variations can be expressedasδ_ε Φ =∑_n≥0 1n! (-1)^n(n-1)2 ℓ_n+1(ε,Φ, …, Φ_n times ). It was shown in <cit.>, that the closure of the symmetry variations[δ_ε_1,δ_ε_2] Φ=δ_- C(ε_1,ε_2, Φ) Φ,and the Jacobi identity∑_cycl [ δ_ε_1, [δ_ε_2 ,δ_ε_3 ] ] = 0are equivalent to the L_∞ relations with two and three gauge parameters. Here theclosure relation allows for a field dependent gauge parameter which can be written in terms of L_∞ products asC(ε_1,ε_2, Φ) =∑_n≥0 1n! (-1)^n(n-1)2 ℓ_n+2(ε_1,ε_2, Φ, …, Φ_n times) .Since it is precisely these relations that we will extend to the quantum case, let us briefly exemplify the procedure of identifying the constraints arising from the gauge closure with L_∞ relations up to cubic order in the fields. Using (<ref>), the gauge commutatorreads [ δ_ε_1, δ_ε_2 ] Φ= . {ℓ_2( ε_2, ℓ_1(ε_1))+ ℓ_2(ε_2, ℓ_2 (ε_1, Φ) ) - ℓ_3 ( ε_2, ℓ_1(ε_1), Φ)- ℓ_3 (ε_2, ℓ_2(ε_1, Φ), Φ) - 12 ℓ_2( ε_2, ℓ_3(ε_1, Φ,Φ) ) } - {ε_1 ↔ε_2} + O(Φ^3),while the right hand side of the gauge closure condition can beexpanded asδ_-C(ε_1, ε_2, Φ) Φ = δ_- ℓ_2(ε_1, ε_2 )Φ + δ_- ℓ_3(ε_1, ε_2, Φ) Φ + δ_1 2ℓ_4(ε_1, ε_2, Φ, Φ) Φ +O (Φ^3) = -ℓ_1 (ℓ_2( ε_1, ε_2)) - ℓ_2 (ℓ_2(ε_1, ε_2 ) ,Φ ) + 12ℓ_3 (ℓ_2( ε_1, ε_2) , Φ, Φ )= - ℓ_1 (ℓ_3( ε_1, ε_2, Φ)) - ℓ_2 (ℓ_3(ε_1, ε_2, Φ ) , Φ )= +1 2ℓ_1(ℓ_4(ε_1, ε_2, Φ , Φ)) + O(Φ^3) .Comparing (<ref>) with (<ref>) we see that demanding closure yields conditions on the ℓ_n products. For instance, at zerothorder in Φ one obtains the condition ℓ_1 (ℓ_2(ε_1, ε_2)) = ℓ_2(ε_1, ℓ_1 (ε_2) ) - ℓ_2(ε_2, ℓ_1 (ε_1) ) .Upon interchanging the arguments this is exactly the L_∞ relation J_2(ε_1, ε_2) = 0 in (<ref>). At first order in Φ one gets 0 = ℓ_2(ε_2, ℓ_2 (ε_1, Φ) )+ ℓ_2 ( ℓ_2(ε_1, ε_2 ) ,Φ) -ℓ_2(ε_1, ℓ_2 (ε_2, Φ) ) - ℓ_3 ( ε_2, ℓ_1(ε_1), Φ)+ ℓ_3 ( ε_1, ℓ_1(ε_2), Φ) + ℓ_1 ( ℓ_3( ε_1, ε_2, Φ) ) .This is the L_∞ relation J_3(ε_1, ε_2, Φ) = 0 in which the term ℓ_3(ε_1, ϵ_2 , ℓ_1 (Φ)) is missing, as we have set X_-2=0.This result is just a consequence of the generaltwo relations between the classical gauge algebra andthe L_∞ algebra:gauge closure ⇔0 =J_n(ε_1, ε_2, Φ, …, Φ_n-2 times ) ,gauge Jacobi identity ⇔0 = J_n(ε_1, ε_2, ε_3,Φ, …, Φ_n-3 times). As one can check, these are actually the only non-trivial L_∞ relations in case that the graded vector space is given by X=X_0⊕ X_-1.This can be generalized by adding a vector space X_-2 containing the equations of motion, thus allowing the freedom thatgauge closure only holds on-shell <cit.>. § QUANTUM L_∞ GAUGE ALGEBRASIn the last section we recalledhow the L_∞ relations guarantee the consistency of a classical gauge algebra. Recently it was shown that also global classical W algebras arising in two-dimensional conformal field theory yieldnon-trivial examples of L_∞algebras. Driven by the aim to extractphysically well motivated aspects ofa quantum extension ofL_∞ algebras, we analyzewhether a generalized version of this correspondence holds for quantum W algebras.On the way, we encounter a couple of new structuresthat can be traced back to the non-associativity of the normal ordered products appearing in the quantum W algebra. Resolving these issuesguides us to a proposal of a quantum L_∞gauge algebra that we will present in the section.Concretely, in section <ref>,by demanding consistency of the quantized symmetry algebra, weoutline how the usual notion of an L_∞ algebra has to be adjusted for a quantum L_∞ algebra. We find that beyondthe higher products also the L_∞ relations receive quantum corrections, whose origin lies in the necessity to performWick contractions between quantum fields. In <ref> we review the L_∞ algebra of closed string field theory and the quantum corrections appearing there. As it turns out, the quantum corrections due to Wick contractions do not appear there.§.§ The quantum L_∞ algebra of a quantum symmetryGoing from a classical field theory to a quantum field theory, the fields become operator valued. We want to consider quantum symmetries which in the classical limit ħ→ 0 becomea classical symmetry of the kind described in the last section. In particular we are still working only on the graded vector space X=X_0⊕ X_-1, where the symmetry parameters are contained in X_0 and the field operators in X_-1. In the case of W-algebras, the infinitely many symmetry parameters[Note that the holomorphic function ϵ(z)does not parametrize a gauge variation, as the latter would depend on z and z.] are compactly encoded inϵ(z)=∑_n∈ℤ z^n+Δ-1ϵ_n and the infinitely many symmetry generators in W(z)=∑_n∈ℤ z^-n-Δ W_n. Here Δ denotes the conformal dimension of the chiral field W(z).In the classical case it was crucial that the variation of the field could be organized in terms of definite powers in the fields to define the corresponding L_∞ products. In order to adapt the notion of field powers, we have to specify an operator product in the quantum case. Inspired by the analysis of W algebras, to be discussed in detail in section <ref>,we define the operator product to be the symmetrized normal orderedproduct A ⋆B = 1 2 ( N(AB) + N(BA) ) .This is a convenient choice, as by taking the classical limit ħ→ 0, it becomes the usual point-wise multiplication of fields. Let us already point out one subtlety relative to the classical case, that will be one source of quantum corrections. As can be seen from the notion of the normal ordering in 2d CFT, the ⋆ product above while commutative fails to beassociative. There[This can be shown using the general formula 6.227 in <cit.>.],the non-associativity of the normal ordered productis given by(εA) ⋆B - ε( A ⋆B) =ε(ABAB ),where ε is just a c-number symmetry variation and A,B are operator valued fields. Moreover, the last term denotes extra terms arising from the contraction between the two operators defined as lim_y→x( A(x)B(y) - (ABAB )(x,y))=N(A B)(x)which in a CFT is nothing else than the singular part of the operator product expansion.Having defined the product between operators,we assume that variations of the field can be schematically written in the form δ_ε^qu Φ∼∑_nε Φ⋆…⋆Φ_n times ,where for simplicity we consideredbosonicfields andsymmetry parameters. Following the lines of the classical discussion we define graded symmetric multilinear quantum n-productsL_n+1:X^⊗n→Xand rewrite the variation in the form δ_ε^quΦ=∑_n≥0 1n! (-1)^n(n-1)2 L_n+1(ε,Φ, …, Φ_n times ). The quantum L_n products still carry the intrinsic grading deg L_n=n-2. Since the star-product is symmetric, the L-products are automatically symmetric when interchanging two fields. Since in the limit ħ→ 0,the star product becomes the normal field product, the quantum L_n-products will become the classical ℓ_n-products with the right degree and symmetry properties.Following the classical analysis, the question now is which constraints arise from demanding the closure of the quantum symmetry algebra [δ_ε_1^qu ,δ_ε_2^qu ] Φ=δ_- C(ε_1,ε_2, Φ)^qu Φ and theJacobi identity∑_cycl [ δ_ε_1^qu, [δ_ε_2^qu ,δ_ε_3^qu ] ] = 0.Here, the field dependent closure parameter C(ε_1,ε_2, Φ) should still be expressedin terms of the symmetrized normal ordered product C (ε_1,ε_2, Φ)∼∑_n ε_1ε_2 ·Φ⋆…⋆Φ ,allowingto read off the L_n products with two symmetry parametersC(ε_1,ε_2, Φ) =∑_n≥0 1n! (-1)^n(n-1)2L_n+2(ε_1,ε_2, Φ, …, Φ_n times) .To identify potential sources of quantum corrections in the L_∞relations, we write out the first few terms of both sides of the closure condition (<ref>). Up to second order in the fields, the left hand side can be expanded as[ δ_ε_1^ qu , δ_ε_2^ qu] Φ={L_2(ε_2, L_1(ε_1) )+ L_2(ε_2, L_2 (ε_1, Φ)) - L_3 (ε_2, L_1(ε_1), Φ )-L_3 (ε_2, L_2(ε_1, Φ), Φ )- 12 L_2(ε_2, L_3(ε_1, Φ,Φ)) }- {ε_1 ↔ε_2} ,while the right side isδ_-C^ qu (ε_1, ε_2, Φ) Φ = δ_- L_2(ε_1, ε_2 )Φ + δ_- L_3(ε_1, ε_2, Φ) Φ =-L_1 ( L_2( ε_1, ε_2)) - L_2 ( L_2(ε_1, ε_2 ) ,Φ ) + 12 L_3 ( L_2( ε_1, ε_2) , Φ, Φ ) - L_1 ( L_3( ε_1, ε_2, Φ)) - L_2 ( L_3(ε_1, ε_2, Φ ) , Φ ).To read off the quantum L_∞ relations, we now sort (<ref>) and (<ref>) according to the power in Φ .Since now the power of Φ is with respect tothe symmetrized normal ordered product, this is a bit more subtle than in the classical case. One first has to bring all terms into the schematic form (ε_1ε_2) · (Φ⋆…⋆Φ) that also appeared in the definitions of the L-products (<ref>) and (<ref>). While some terms are already of this form, for othersa rebracketing is necessary. Consider for instance the fourth term in (<ref>) that, upon using (<ref>),can be schematically written asL_3 (ε_2, L_2(ε_1, Φ), Φ) ∼ε_2(( ε_1Φ) ⋆Φ) .Using the non-associativity of the ⋆-product (<ref>), this becomes L_3 (ε_2, L_2(ε_1, Φ), Φ) = ε_1 ε_2 ·(Φ⋆Φ) + ε_1 ε_2 ·(ΦΦΦΦ). Let usassume for simplicity a free theory such that ΦΦΦΦ is proportional ħ1. Then the last term in (<ref>) is proportional to ϵ_1 and ϵ_2 and therefore a quantum correction to the L_∞ relation at zeroth order in Φ. Treating the last term in (<ref>) in an analogous way, we find the quantum corrected L_∞ relation at zeroth order in Φ0=L_2( L_1(ε_1) , ε_2) + L_2(ε_1, L_1(ε_2 ) ) +L_1(L_2(ε_1, ε_2) ) - L_3 (ε_2, L_2(ε_1, Φ),Φ Φ), Φ) + L_3 (ε_1, L_2(ε_2, Φ),Φ Φ), Φ) + L_2 ( L_3(ε_1, ε_2, Φ),Φ Φ) , Φ) . Similarly also all other L_∞ relations get correctedby contractions of higher L_∞ relations.Let us summarize: Guided by quantum algebras in 2d CFT, we identified two sources of quantum corrections to L_∞ algebra. First, relativeto the classical products, the higher quantum L_∞ products can receive corrections of higher order in ħ . The second kind of quantum corrections arises from contractions between quantum fields that appearwhen sorting the relations in powers of the field.These contractions change the power of the fieldsso that the classically separated L_∞ relations receive quantum suppressed off-diagonal corrections.We want to stress that the contractions differ severely from theory to theory. While in free theories the contraction is proportional to the identity operator, in interacting theories (like generic CFTs) the contraction of two fields is usually field dependent again. We can therefore not provide a general closed formula for which contraction of which L_∞ relation contributes to which other L_∞ relation. Guided by these observations we suggest to define quantum L_∞ algebras that govern (global) quantum symmetries asfollows: One has a graded vector spaceX= X_0 ⊕ X_-1, where X_n is said to have degree n. In addition there are multi-linear quantum products L_n(x_1,…,x_n) that have degree deg(L_n)=n-2 so thatdeg( L_n(x_1,…,x_n))=n-2+∑_i=1^ndeg(x_i) .Each product can in principle receive quantum corrections at anypower in ħ. The products are graded commutative, i.e.L_n (…, x_1,x_2, …) = (-1)^1+ deg(x_1) deg( x_2) L_n (…, x_2,x_1, …) .Like in the classical case, onedefines J^qu_n(x_1,…, x_n):=∑_i + j = n + 1 (-1)^i(j-1) ∑_σχ(σ;x) L_j ( L_i (x_σ(1), …, x_σ(i) ), x_σ(i+1) , …, x_σ(n) ) . The L_n products define a quantum L_∞ algebra if they satisfy for each m = 2,3 andn∈ℤ^+_0J^qu_m+n(ϵ_1, …, ϵ_m, x_1,…, x_n)+∑_ (y_1,…, y_k) →(x_1, …, x_n) ħ^ξJ^qu_m+k(ϵ_1, …,ϵ_m, y_1,…, y_k_→(x_1,…,x_n))=0Since this is the main formula of the paper we want to explain the formula in more detail. ϵ_i ∈ X_0 is a symmetry parameter and x_i ∈ X_-1 is a field. While the first term is the known one from the classical L_∞ relations, the second term contains the crucial new feature of quantum L_∞ algebras, namely the corrections due to contractions of other L_∞ relations. To cover all such corrections we sum over all L_∞ relations whose field input (y_1, … , y_k) can contract into (x_1, … , x_n). The ξ≥ 1 counts the number of contractions employed to convertthe dependence on (y_1,…, y_m) into a dependence on (x_1,…, x_n). The underbrace signals that only the terms that arise from the particular contraction are to be taken here. To avoid permutation factors we let the sum run only over (y_1, … , y_k) that are not equal under permutation. Furthermore notice that the order of the (y_1, … , y_k) does not play a role since the J^ qu share the permutation property of (<ref>) [Here an obstacle becomes apparent if one tries to generalize the above definition beyond the given case where contractions appear only between elements of X_-1. When contractions appear not only between elements with even parity the order of the y_1, … , y_k does indeed matter. Lacking an example to follow we cannot give a precise ordering prescription to fix this issue here. ]. Let us provide a more general and mathematically precise definition for the quantum L_∞ algebra. Since the quantum corrections mix the different L_∞ relations, we can also define quantum L_∞ algebras very compactly by demanding that for m∈{ 2,3} and ϵ_i ∈ X_0 the sum of allL_∞ relations vanish∑_n=1^∞∑_ (x_1, …x_n) ∈X_-1^n J^qu_m+n (ϵ_1, …, ϵ_m, x_1, …x_n)=0 ,where as before the second sum runs only over distinct (x_1, …, x_n). In case the L products do not change the power of the input, the terms in (<ref>) separate into the classical L_∞ relations (<ref>). On the other hand, using normal ordered products in the L products, (<ref>) reduces to the former definition (<ref>). Nevertheless we want to stress that in general (<ref>) does not need any physical input in form of a contraction. From the mathematical viewpoint the definition (<ref>) might therefore be more appealing. We nevertheless prefer (<ref>) that also makes it manifest that in the ħ→ 0 limit one encounters the classical L_∞ relations and that their off-diagonal quantum corrections arise from the contraction of quantum fields. In section <ref> we show in much detailhow quantum W algebras fit precisely into this definition of quantum L_∞ algebras. Especially in section <ref> we will demonstrate that the quantum relations (<ref>) can be given a precise meaning for the quantum W_3 algebra.§.§ Comparison to the L_∞ algebra of CSFTWe will now compare our definition for a quantum L_∞ algebra with the L_∞ algebra of closed string field theory (CSFT) <cit.>. To distinguish these two different L_∞ definitions, we will follow Markl <cit.> and call the L_∞ algebra of CSFT a loop L_∞ algebra, while the definition from last section will be called quantum L_∞ algebra.In a loop L_∞ algebras one usually expands the quantum products according to their loop level, thus their power of ħ L_n (x_1, …, x_n) = ∑_gL^g_n (x_1, …x_n) , where L_n^g is proportional to ħ ^g.Then, the L_n^g products define a loop L_∞ algebra, if for any level g the following relation holds (we use the notation of <cit.>)0 =∑_g_1 + g_2 = g∑_i + j = n + 1(-1)^i(j-1) ∑_σχ(σ;x)×L_j^g_1 ( L_i^g_2 (x_σ(1), …, x_σ(i) ), x_σ(i+1) , …, x_σ(n) )+ 1 2 ∑_s (-1)^ deg(h_s) + n - gL_n+2^g-1 (h_s, h^s, x_1, …, x_n) . The sum over s in the last term runs over a basis of fields labeled by s. The field with an upper index, h^s, is the conjugate field to h_s with respect to a scalar product ⟨ h^s, h_t ⟩ = δ^s_t. The ∑_s L_n^g-1(h_s, h^s, …) can be interpreted as an identity operator. When contracting h_s, h^s to eliminate this identity operator, we obtain an additional ħ factor such that, together with the ħ^g-1 from the L_n^g-1, the last term is proportional to ħ^g as well. Let us compare the defining relations of (global) quantum and(gauge) loop L_∞ algebras: The first part of the loop L_∞ relation (<ref>) appears in quantum L_∞ algebras as the order ħ^g term, when inserting the expansion (<ref>) into the first term of (<ref>). The second term of (<ref>) does not appear in the quantum L_∞ relations in (<ref>). The reason for this is, that the quantum L_∞ was derived in a setting where the total vector space contained only degree 0 objects, the symmetry parameters, and degree -1 objects, the fields. Therefore X = X_0 ⊕ X_-1 and all objects with a degree other than 0 and -1 were set tozero. Demanding all terms in the defining relation of loop L_∞ algebras (<ref>) to have the same degree, we finddeg(h_s) + deg(h^s) = -3 . Since h_s is a field, its degree is deg(h_s) = -1 and the degree of h^s is bound to be deg(h^s) = -2. Therefore, h^s istrivialand the second term in (<ref>) couldnot appear in the derivation of the quantum L_∞ based entirely onquantum gauge variations. Remarkably, the second term in the quantum L_∞ relation (<ref>) has no counterpart in the loop L_∞ algebras.Therefore the L_∞ relations of the CSFT L_∞ algebra do not receive correctionsfrom contraction terms.The question arises if there exista connection between the two definitions. From the current status, the answer is not completely clear to us and more work or insight is required to fully clarify it. We can only say thatthe structure of(gauge) loop L_∞ arose as a consequence of the quantum master equation of the BV-formalism for the CSFT quantum action. On the contrary,our (global) quantum L_∞ definition is based on the analysis ofbootstrapped and therefore exactly solvable global quantum W algebras in 2d CFT. § THE QUANTUM W_3- L_∞ ALGEBRA In the recent paper <cit.> it was shown that (classical) W algebras are highly non-trivial (classical) L_∞ algebras with field dependent symmetryparameters.In this section we will show that the quantum W_3-algebra fits into the framework of the quantum L_∞ algebra of section <ref> (and was in fact motivating it).We expect that more general quantum W-algebraswill even provide more intricate examples ofquantum L_∞ algebras. §.§W algebrasIn two-dimensional conformal field theories the energy momentum tensor T(z) is a quasi primary field that has conformal dimension two, generates the conformal transformations and obeys the Virasoro algebra. A W algebra is an extension of the Virasoro algebra by chiral primary fields of conformal dimension usually larger than two. The prototype example is Zamolodchikov's W_3 algebra <cit.>, generated by two fields {T(z),W(z)} of conformal dimensions two and three.The (quantum) OPEs among these fields are known to be[Up to some structure constants, theform of the OPE between quasi-primary fields is generally known<cit.> (for a pedestrian derivation see also<cit.>),as has been exploited for the classical W-L_∞ algebra relation in <cit.>.]1ħ T(z)∘T(w) =c/2(z-w)^4 +2( T(w)(z-w)^2+12 ∂T(w)(z-w)) ,1ħ T(z)∘W(w) =3( W(w)(z-w)^2+13 ∂W(w)(z-w)) , 1ħ W(z)∘W(w) =c/3(z-w)^6+α ( T(w)(z-w)^4+12 ∂T(w)(z-w)^3+320 ∂^2 T(w)(z-w)^2+130 ∂^3 T(w)(z-w)) +β ( Λ^qu(w)(z-w)^2+12 ∂Λ^qu(w)(z-w)) .Here the field Λ^ qu denotes the normal ordered productΛ^qu=N(TT)-ħ3 10 ∂^2 Twhere we have indicated the quantum correction linear in T. The corresponding algebra for the modes satisfies the Jacobi-identity for α=2 ,β=325c+22ħ .Following <cit.>, in these formulas we have introduced ħ so that the classical limit and its quantum corrections areclearly visible. In the ħ→ 0 limit, the commutator (singular part of theOPE) becomes the Poisson bracket{·,·}_PB= lim_ħ→0 1iħ[·,·] .There existthree sources of quantum corrections. Two of them are manifest inthe ħ corrections in (<ref>) and (<ref>) [Notice that when expanding the fraction β we get an infinite series with terms at any order in ħ. Separating the different powers of ħ^g in different L_n^g products, as usually done in loop L_∞ algebras, see (<ref>), is therefore not illuminating in this example.]and the third is the appearance of the normal ordered product N(TT) instead of the usual point-wise multiplication (TT) in the classical case.The normal ordered product between two chiral fields is defined asN(ϕ χ)(w)= 12πi∮_γ(w) dz ϕ(z)∘χ(w) (z-w), where γ(w) is a path encircling w counterclockwise once. The normal ordered product is therefore the first regular term in the OPE between the two fields. Note that this product is neither commutative nor associative. Since for the correspondence to an L_∞ algebra one needs graded symmetric products, we use the symmetrized normal ordered product ⋆ from (<ref>) that is still non-associative. To demonstrate this, let us explicitly compute the left hand side of (<ref>) for A=B=T(εT)⋆T - ε(T⋆T) = 14πi∮dz ϵ(z)T(z)∘TT(z)∘T(w)(z-w) = cħ96∂^4 ϵ+ħ2∂^2 ϵT +ħ2∂ϵ ∂T ,where both sides depend on w. Note that these corrections arise from thecontraction of operators below the integral and that they are ħ-suppressed relative to the leading order normal ordered products.The extended symmetry algebra acts with δ_ε_iW_j(w) =12πi ∮_γ(w) dz ε_i(z) 1ħ W_i(z)∘W_j(w), where i, j = { T, W }. Instead of writing ε_T and ε_W from now on we willwrite ε for ε_T and η for ε_W §.§ L_n products with one symmetry parameter Let us now follow the steps outlined in the sections <ref> and <ref> to construct the quantum L_∞ algebra corresponding to the quantum W_3 algebra. The fields {T, W} have degree -1, and the symmetry parameters {ε, η} have degree zero. Therefore the total vector space is X= X_0 ⊕ X_-1 and each X_n = X_n^T ⊕ X_n^W splits into a T and a W part.As in <cit.>, we will use boldface to highlight vectors in this two-dimensional space, for instance W = (T, W) will denote either of the fields. Furthermore we equip all L_n products with an upper index from the set{ T, W, ϵ, η} that denotes in which of the four subspaces of X the image of the higher product L_n is located. Inserting (<ref>) in (<ref>),for thequantum correctedinfinitesimal variations one obtainsδ_ε T =c 12 ∂^3 ε_L^T_1(ε) + (2 ∂εT +ε ∂ T)_L^T_2(ε,T) , δ_ε W =(3 ∂εW +ε ∂ W)_L^W_2(ε,W) , δ_η T =(3 ∂ηW +2 η ∂ W)_L_2^T(η,W) andδ_η W =c 360 ∂^5 η_L^W_1 (η) + α(1 6 ∂^3 ηT + 1 4 ∂^2 η ∂ T+ 3 20 ∂η ∂^2 T+1 30 η ∂^3 T)_L^W_2(η,T)aaaaaaaaa-3ħβ 10(∂η ∂^2 T + 1 2 η ∂^3 T)_L^W_2(η,T)=+β(∂η(T⋆ T) +1 2η ∂ (T⋆ T))_-1 2L^W_3(η,T,T) .Notice that we have already written all terms in the form (<ref>) such that we can directly read off the L_n products.Compared to the classical higher products, the only change is in δ_η W, where L^W_2(η,T) receives an explicit ħ-correction andℓ^W_3(η,T,T) involves thequantum product T⋆ T. §.§ L_n products with two symmetryparameters Recall that the L_n products with two symmetry parameters appear in theclosure condition (<ref>) [δ_ε_i^qu ,δ_ε_j^qu ] W_k =δ_- C(ε_i,ε_j, 𝐖)^qu W_k,upon expanding (<ref>) 𝐂 (ε_i,ε_j, 𝐖) =∑_n≥0 1n! (-1)^n(n-1)2L_n+2(ε_i,ε_j, 𝐖 , …, 𝐖_n times) .To obtain the 𝐂 (ε_i,ε_j, 𝐖) we insert (<ref>) into the symmetryclosure condition and use the generalized Wick theorem for chiral vertex operator algebras <cit.>∮dy 2 π i(y-w)^n A(y)∘(∮dz 2 π i(z-w)^m B(z)∘ C(w)) ======-∮dy 2 π i(y-w)^m B(y)∘(∮dz 2 π i(z-w)^n A(z)∘ C(w)) =∑_j=0^nnj ∮dz 2 π i( ∮dy 2 π i (y-z)^jA(y)∘ B(z))∘ C(w) (z-w)^(m+n-j)in the special case m, n = 0. In this way, for instance we can derive[δ_ε_1,δ_ε_2] T(z) =(1/2 πi)^2 ∮dy1ħ (∮dw ε_1(w)ε_2(y) 1ħT(w)∘T(y))∘ T(z) =1/2 πi∮dy(∂ε_1(y)ε_2(y)-ε_1(y)∂ε_2(y)) 1ħ T(y)∘T(z) , so thatthe C-productcan be read off as𝐂(ε_1,ε_2,𝐖)=ε_1∂ε_2-∂ε_1ε_2:=L_2^ε(ε_1,ε_2) .Similarly we find𝐂(ε, η, 𝐖)=ε∂η-2 ∂εη:=L_2^η(ε,η), 𝐂(η_1,η_2,𝐖) =L_2^ε(η_1,η_2)+L_3^ε(η_1,η_2,T) ,withL^ ε_2(η_1,η_2)=α(130 η_1 ∂^3 η_2-130 ∂^3η_1 η_2 +120 ∂^2η_1 ∂η_2-120 ∂η_1 ∂^2η_2) =-3ħβ10(12 η_1 ∂^3 η_2-12 ∂^3η_1 η_2 -12 ∂^2η_1 ∂η_2+12 ∂η_1 ∂^2η_2),L^ ε_3(η_1,η_2,T)=β(η_1 ∂η_2-∂η_1η_2)T .Please note the explicit first order quantum correction in L^ ε_2(η_1,η_2) and the infinitely many quantum corrections hidden in the ħ dependence of β.§.§ Quantum L_∞ relations with two symmetry parameters Having determined the quantum corrected L_n products for the W_3 algebra, let us now stateand check the quantum L_∞ relations J^qu_m+n(ϵ_1, …, ϵ_m, x_1,…, x_n)+∑_ (y_1,…, y_k) →(x_1, …, x_n) ħ^ξJ^qu_m+k(ϵ_1, …,ϵ_m, y_1,…, y_k_→(x_1,…,x_n))=0when plugging in exactly two symmetryparameters. These are the ones that are equivalent to the quantum closure condition (<ref>). §.§.§ Quantum corrections to the L_∞ relations The distinguished new feature of the definition of quantum L_∞ algebras is the second term in (<ref>) where the contractions appear. Let us therefore first list the L_∞ relations that are non-trivially corrected by such contraction terms. Since we plug in two symmetryparameters and we need at least two fields to be able to contract, we must have at least four inputs in (<ref>). But since the highest L_n product is L_3, all relations J^ qu_6 ,J^ qu_7, … = 0 are automatically satisfied. To further trivialize most cases we can use that the only non-trivial L_3 products are L_3^W(η, T,T) and L_3^W (η_1, η_2, T). Since the first L_3 always maps into the kernel of the second L_3, for J^ qu_5∼ L_3 L_3 one can conclude J^qu_5 (ϵ_i, ϵ_j, 𝐖, 𝐖 , 𝐖 ) = 0 .In a similar vein, evaluating(<ref>) one finds that triviallyJ^qu_4 (ε, ε, W, W)= 0 , J^qu_4 (ε_1, ε_2, 𝐖, 𝐖 )= 0 , J^qu_4 ( ε, η, W, T)= 0. The only non-zero contraction terms cantherefore arise in the termsJ^qu_4 (ϵ, η, T,TT,T ) , J^qu_4 (η_1, η_2 ,W , TW,T ), J^qu_4 (η_1, η_2 ,T , TT,T) .From the form of the OPEs (<ref>), one realizes that the contraction T TTT yields terms proportional to ħ T and the identity ħ1, while the second contraction reads W TWT ∼ħ W.Hence the L_∞ relations that are non-trivially corrected by a contraction of a higher L_∞ relation are0 =J_2^qu(ε,η)+ħJ^qu_4(ε,η,T,T_→1) , 0 =J_3^qu(ε,η,T)+ħJ^qu_4(ε,η,T,T_→T) ,0 =J_2^qu(η_1,η_2)+ħJ_4^qu(η_1,η_2,T,T_→1) , 0 =J_3^qu(η_1,η_2,T)+ħJ_4^qu(η_1,η_2,T,T_→T) , 0 =J_3^qu(η_1,η_2,W)+ħJ^qu_4(η_1,η_2,T,W_→W) . Followingthe logic of section <ref>, wewill now explicitly evaluate the contractions appearingin these quantum L_∞ relations. We start with terms arisingfrom contractions of the L_∞ relationJ^ qu_4 (η_1, η_2 , T,T). In a first step we findJ^qu_4(η_1,η_2,T,T)= -L^T_2(L^ ε_3(η_1,η_2,T),T)+12L^T_2 (η_2,L^W_3(η_1,T,T))- 12 L^T_2(η_1,L^W_3(η_2,T,T) ) .Recall that every L_∞ relation collects the contribution of the form (η_1 η_2) (T ⋆ T). While the terms in the second line are already of this form, the first term is not, so thatthe non-associativity of the ⋆-product (<ref>)is expected to induce contractions. Inserting the explicit expression of the L_n products into the first termyields-L^T_2(L^ ε_3(η_1,η_2,T),T)=-2β(∂(f T)⋆T)-β((fT)⋆∂T), where we abbreviated f:= η_1∂η_2-∂η_1η_2.Using the normal ordering prescription (<ref>) and its function linearity in the second argument we find for the first term-2β(∂(f T)⋆ T)(z) =-β (∮dy2 π if(y) T(y)∘ T(z) (y-z)^2+∂ f(z)N(TT)(z)+f(z) N(T ∂ T)) =-β (c ħ 240 ∂^5 f(z)+ħ 3 ∂^3 f(z)T(z)+ħ 2 ∂^2 f(z)∂ T(z)=====+2∂ f(z) N(TT)(z)+f(z)∂ N(TT)(z)) . Evaluating the second term in (<ref>) similarly gives-β ( (f T)⋆∂ T)(z)= == -β 2(cħ 60∂^5 f(z)+2 ħ 3∂^3 f(z)T(z)+3ħ 2∂^2 f(z) ∂ T(z) +f(z) ∂ N(TT)) .Putting both terms together results in-L^T_2(L^ ε_3(η_1, η_2, T),T)=-βħc 80∂^5 f(z) -βħ3∂^3 f(z) T(z)-5 βħ4 ∂^2 f(z)∂T(z) -βħ2 ∂f(z) ∂^2 T(z)-2β ∂f(z)N(TT)(z)-3β2 f(z)∂N(TT)(z)so that we can directly read offħJ_4^ qu(η_1,η_2,T,T_→1) =-βħ c80∂^5 f(z) ,ħJ_4^ qu(η_1,η_2,T,T_→ T) =-βħ 3∂^3 f(z) T(z)-5 βħ 4∂^2 f(z) ∂ T(z) -βħ 2∂ f(z) ∂^2 T(z). Computing the other contractions is more lengthy, but follows the same steps. Let us therefore only state the resultsħJ^ qu_4(ε,η,T,T_→ T)= -4ħβ 3(∂η ∂^3 ε-12 η ∂^4 ε)T -2ħβ(∂η ∂^2 ε-1 3 η ∂^3ε) ∂ T-ħβ η ∂^2 ε ∂^2 T ,ħJ^ qu_4(ε,η,T,T_→1)= -βħ c40(∂η ∂^5ε+12η ∂^6ε),and finally ħJ_4(η_1,η_2,T,W_→W)= - 3βħ/ 4(∂η_1 ∂^2η_2-∂η_2 ∂^2 η_1)∂W-3βħ2∂^2 f ∂W -9βħ4∂f ∂^2 W .§.§.§ Checking the quantum L_∞ relations We are now in the position tostate and check the quantum L_∞ relation with two symmetryparameters.We will sort them according to their appearance in the quantum closure condition (<ref>) with i,j,k ∈{T, W}. * (TT,T): The closure condition (<ref>) with (ij,k) = (TT,T) is equivalent to0= J^qu_2(ε_1,ε_2)= -L^T_1 (L^ ε_2(ε_1,ε_2))+ L^T_2(L^T_1(ε_1),ε_2) + L^T_2(ε_1,L^T_1(ε_2))and0= J^qu_3(ε_1,ε_2,T)= L^T_2(L^ ε_2(ε_1,ε_2),T)+L^T_2(L^T_2(ε_2,T),ε_1) + L^T_2(L^T_2(T,ε_1),ε_2) .Inserting (<ref>) these relations are readilychecked to be satisfied.* (TT,W): There is only one non-trivial relation0 = J^ qu_3(ε_1,ε_2,W)=L^W_2(L^ ε_2(ε_1,ε_2),W )+L^W_2(L^W_2(ε_2,W),ε_1 ) + L^W_2(L^W_2(W,ε_1),ε_2 ) ,that is also directly satisfied.* (TW,T): One finds the single non-trivial relation0 =J^qu _3(ε,η,W) =L^T_2(L^ η_2(ε,η),W)+L^T_2(L^T_2(η,W),ε) + L^T_2(L^W_2(W,ε),η).As before, a short computation shows that this equation is satisfied without any constraints. * (TW,W): This is the first truly interesting case, asthe closure condition involves a contribution from a contraction0 =J^qu_2(ε,η)+ħJ^qu_4(ε,η,T,T_→1) , 0 =J^qu_3(ε,η,T)+ħJ^qu_4(ε,η,T,T_→T) , 0 =J^qu_4(ε,η,T,T) .When evaluating these relations, the contraction terms computed in (<ref>) are crucial. Like in the classical case, the first equation is satisfied for α=2. Note that terms from the quantum part of L_2(η,T) get exactly canceled by the quantum correction from the contraction. The second equation is indeed satisfied for β=16α 5c+22ħ, the value of the quantum W_3 algebra. The third relationholds without giving any constraintson α, β. * (WW,T):In this case the closure is equivalent to the quantum L_∞ relations0 =J^qu_2(η_1,η_2)+ħJ_4^qu(η_1,η_2,T,T_→1) , 0 =J_3^qu(η_1,η_2,T)+ħJ_4^qu(η_1,η_2,T,T_→T) , 0 =J_4^qu(η_1,η_2,T,T) .Again, the contraction terms (<ref>) are needed. The first equation is satisfied for α=2 and the second for β=16 α 5c+22ħ. Again, the quantum corrected L_∞ relations fix the open constants exactly to the values expected for the quantum W_3 algebra. The third equation holds independently of the numerical values of α, β. * (WW,W): The quantum L_∞ relationsequivalent to closure are0 =J^qu_3(η_1,η_2,W)+ħJ^qu_4(η_1,η_2,T,W_→W) , 0 =J^qu_4(η_1,η_2,T,W) .After inserting the contraction term (<ref>), both equations hold independent of α and β. §.§ L_∞ relations with three symmetryparameters After we have checked the L_∞ relations with two symmetryparameters, it remains to evaluate those with three symmetryparameters. Recallthat these are equivalent to the Jacobi identity ∑_cycl [ δ^qu_ε_i, [δ^qu_ε_j,δ^qu_ε_k]]=0 .For three symmetryparameter insertions,J_n=0 is trivially satisfied for n≥ 5in the case of the W_3 algebra. Therefore,there cannot be any correction terms arising from contractions. Again sorting them according to the triplet (ijk) in (<ref>), the quantum L_∞ relations read as follows: * (TTT):0=L^ ε_2(L^ ε_2(ε_1,ε_2),ε_3)+L^ ε_2(L^ ε_2(ε_3,ε_1),ε_2)+L^ ε_2(L^ ε_2(ε_2,ε_3),ε_1).* (TTW):0=L^ η_2(L_2^ ε(ε_1,ε_2),η)+L^ η_2(L_2^ η(η,ε_1),ε_2)+L^ η_2(L_2^ η(ε_2,η),ε_1).* (WWT):0=L^ ε_2(L_2^ ε(η_1,η_2),ε )+L^ ε_2(L_2^ η(ε,η_1),η_2 )+L^ ε_2(L_2^ η(η_2,ε),η_1 )0=+L^ ε_3(η_1,η_2,L^T_1(ε) ) ,0=-L^ ε_2(L^ ε_3(η_1,η_2,T),ε )+L^ ε_3(L^ η_2(η_1,ε),η_2,T ) ,0= -L^ ε_3(L^ η_2(η_2,ε),η_1,T )+L^ ε_3(L^T_2(T,ε),η_1,η_2 ).The first J_3-type relation requires β=16α 5c+22ħ to holdand, due to the appearance of the non-vanishing last term, featuresthat the two-product L_2 violates itsJacobi identity.* (WWW):0 =L^ η_2(L_2^ ε(η_1,η_2),η_3 )+L^ η_2(L_2^ ε(η_3,η_1),η_2 )+L^ η_2(L_2^ ε(η_2,η_3),η_1 ),0 =L^ η_2(L_3^ ε(η_1,η_2,T),η_3 )+L^ η_2(L_3^ ε(η_3,η_1,T),η_2 )+L^ η_2(L_3^ ε(η_2,η_3,T),η_1 ) ,0 =L^ ε_3(L_2^T(η_1,W),η_2,η_3 )+L^ ε_3(L_2^T(η_2,W),η_3,η_1 )+L^ ε_3(L_2^T(η_3,W),η_1,η_2 ) . § SUMMARY AND CONCLUSIONSThis completes the proof that the quantum W_3 algebra is an example for a quantum L_∞ algebra as defined in section <ref>.Like for the classicalW_3 algebra, the quantum corrected relations with two inputs gave the constraint α=2 and the relations with three inputs J^ qu_3=0 required β=16 α 5c +22ħ. The only other non-trivial higher order relations were satisfied without any further constraint. The L_∞ relations with three symmetry parameters were essentiallythe same as in the classical case. Let us emphasize that the quantum contractionsin (<ref>) are necessary for the L_∞ relations to hold. This means that the quantum W_3 algebra does neither define a classical nor a loop L_∞ algebra (as appeared for CSFT),but this new type of a quantum L_∞ algebra.Of course the higher products in CSFT and for quantum W algebras are different from the onset. In the latter case they involve the non-associative normal ordered product of 2d CFT, whereas in the former casethey are the loop corrected n-vertices of CSFT.Thus, it seems thatfor global and gauge symmetries there does not exist a uniqueversion of a physically reasonable definition of anL_∞ algebra for a quantum theory. We expect that in general the whole class of W algebras yields further examples for quantum L_∞ algebras, since all of them have a closing symmetryalgebra that involves normal ordered products as defined in CFT. As in the classical case, also higher n-products will be non-trivial. Since our analysis of quantum W-algebras is restricted to non-trivial elements in X_0 ⊕ X_-1, it is not obvious whether and how this structure generalizes to more general gradings.3em Acknowledgments: We are grateful to Andreas Deser for discussions.utphys | http://arxiv.org/abs/1706.09034v2 | {
"authors": [
"Ralph Blumenhagen",
"Michael Fuchs",
"Matthias Traube"
],
"categories": [
"hep-th",
"math-ph",
"math.MP"
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"primary_category": "hep-th",
"published": "20170627200614",
"title": "On the Structure of Quantum L$_\\infty$ algebras"
} |
Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371 School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom CNR-SPIN, Viale del Politecnico 1, I-00133, Rome, Italy Bosonic cascade lasers are terahertz (THz) lasers based on stimulated radiative transitions between bosonic condensates of excitons or exciton-polaritons confined in a trap. We study the interaction of an incoming THz pulse resonant in frequency with the transitions between neighboring energy levels of the cascade. We show that at certain optical pump conditions the cascade becomes transparent to the incident pulse: it neither absorbs nor amplifies it, in the mean field approximation. The populations of intermediate levels of the bosonic cascade change as the THz pulse passes, nevertheless. In comparison, a fermionic cascade laser does not reveal any of these properties. Optically induced transparency in bosonic cascade lasers A. V. Kavokin December 30, 2023 ========================================================The concept of bosonic cascade lasers has been introduced a few years ago with the objective of generating THz frequency radiation in a compact semiconductor system <cit.>. The bosonic cascade is defined as a series of bosonic energy levels with equal THz range spacing in energy. A boson excited in the highest level can undergo a series of transitions down the cascade, generating multiple THz frequency photons in analogy to fermionic quantum cascade lasers <cit.>, which were also developed in the THz regime <cit.>. The critical difference of a bosonic cascade is that bosonic final state stimulation enhances the scattering rates such that even in the limit of weak spontaneous scattering rate, particles can reach the ground level of the cascade. A variety of theoretical considerations of bosonic cascades have since been considered, including the quantum statistics of the cascade levels <cit.> and the interplay of double bosonic stimulation coming from a THz cavity and the bosonic particles themselves <cit.>. Physical implementations of bosonic cascades can be based on excitons or exciton-polaritons in parabolic traps <cit.> and are presently under experimental development <cit.>.In this Letter, we discuss the electromagnetically induced transparency (EIT) of bosonic cascades. EIT is currently studied in a large variety of systems including diluted atomic gases <cit.>, solid solutions, electromechanical <cit.>, optomechanical systems <cit.>, etc. EIT is especially promising for the slowing and storing of light <cit.>. EIT in Bose-Einstein condensates has been extensively discussed as well, see e.g. <cit.>. In this context, we find that bosonic cascades offer an interesting peculiarity linked with the interplay of stimulated absorption and emission of light in the cascade that leads to the transparency. This is in contrast with the interference nature of EIT in the most part of the aforementioned works.We define the transparency of a system at a given frequency as the property of allowing electromagnetic radiation to pass at that frequency. This definition can still be applied to a system that is already generating radiation at the chosen frequency. In such case, the definition only means that there should be no change to the generation rate when the system is illuminated, or if some radiation is absorbed, an equal amount should be re-emitted such that there is no net change to the generation-absorption rate.§ OPERATION SCHEMEWe consider a cascade of M bosonic levels, with populations n_k, coupled to a THz mode, with population n_THz, as illustrated in Fig. <ref>.Such a system has been described previously by the system of semiclassical rate equations for the cascade level populations n_k <cit.>:dn_0/dt=-n_0/τ +W[n_1(n_0+1)(n_THz+1)..-n_0(n_1+1)n_THz]dn_k/dt=-n_k/τ +W[n_k+1(n_k+1)(n_THz+1)..-n_k(n_k+1+1)n_THz]+W[n_k-1(n_k+1)n_THz..-n_k(n_k-1+1)(n_THz+1)]∀0<k<Mdn_m/dt=P-n_M/τ +W[n_M-1(n_M+1)n_THz..-n_M(n_M-1+1)(n_THz+1)] Here W is the spontaneous transition rate between neighbouring levels, τ is the lifetime of particles in each level, and P is the pumping rate. For simplicity, we assume that W and τ are independent of the level index. We also neglect higher order scattering processes (e.g., parametric scattering processes <cit.>), which were considered in detail in Ref. <cit.>.To model an incident THz field, we assume the existence of a THz mode, with intensity n_THz, that overlaps with the bosonic cascade modes. This THz mode is driven externally at a rate P_THz, experiences gain Γ from relaxation processes in the cascade, and leaves the system with a lifetime τ_THz: dn_THz/dt=-n_THz/τ_THz+Γ+P_THz In the absence of any confinement of the THz mode (that is, in the absence of any THz cavity), the lifetime can be estimated from the time it takes a THz photon to cross the cascade. For example, considering a 3μm system size, a THz photon would take approximately 0.01ps to cross the system going at the speed of light (assuming a refractive index of the system material equal to 1 at THz frequency). Even though the lifetime of a THz photon in the system is very short, it is important to describe the THz field in the system by a dynamical equation to allow for the possibility of its depletion.We define Γ as the net THz generation-absorption rate, which is given by summing all the stimulated THz emission processes and subtracting the absorption processes: Γ =W∑_k=0^M-1[n_k+1(n_k+1)n_THz-n_k(n_k+1+1)n_THz]=W(n_M-n_0)n_THz Note that only THz photons generated by processes stimulated by n_THz are included in Γ. Spontaneously generated THz photons are accounted for in the equations for the evolution of the cascade levels, however they would be emitted in all directions while it is implicit that n_THz represents a composition of THz photons traveling in a particular direction through the system. Our objective is to study the change in the net THz generation-absorption rate, that is, Γ, when the system is subjected to a THz field.§ STEADY-STATE SOLUTIONEquations <ref>-<ref> are readily solved for the steady-state of the system. Figure <ref> shows the variation of the net THz generation-absorption rate, Γ, as a function of P and P_THz, for typical parameters. It can be seen that for small cascade pumping, the application of a THz field increases the THz generation rate, while for large cascade pumping, the application of a THz field reduces the THz generation rate. This effect can be interpreted by considering that the THz field tends to favour equal populations of all cascade levels, as it can either enhance relaxation of particles in the cascade (through stimulating THz emission) or enhance excitation of particles in the cascade (through THz absorption). Since the cascade pump, P, is applied to the highest level in the cascade it favours population of higher levels at weak intensity. Application of the THz field then favours the relaxation of particles from these high levels resulting in THz emission. On the other hand, a strong cascade pump favours relaxation to the lower levels of the cascade, due to strong stimulated THz emission by cascade particles. In this regime, the THz pump favours excitation of the cascade levels, corresponding to absorption of THz photons.The vertical contour in Fig. <ref> that divides the regions of positive and negative Γ is most important for our purposes. The existence of this contour makes Γ almost independent on P_THz at a selected value of P. Although it does not show on the scale of the plot, the contour is very slightly curved, however, it is still possible to find specific values of P and P_THz for which Γ is exactly zero in the presence and absence of P_THz (these values are marked as black spots in Fig. <ref>). Under these conditions, we would expect a continuous THz field to pass through the system with no change to its intensity.§ DYNAMICSThe dynamics of bosonic cascades subject to pump pulses can be calculated by propagating Eqs. <ref>-<ref> numerically in time. Fig. <ref> considers an initial steady state of the cascade and then the application of a THz pulse. We do not specialize to a pulse of any specific duration, but assume that it is longer than the timescales set by τ and τ_THz. In the time intermediate between switch on and switch off of the square pulse, the system is effectively in a steady state. This allows us to make general statements about square pulses with different durations. Given that polaritons may have lifetimes on the picosecond timescale, it is implied that we consider pulses of duration on the order of tens of picoseconds or longer. The intensity of the square pulse and the cascade pump intensity is taken to correspond to the black spots in Fig. <ref>. Although the relative changes are not very large, they are clearly noticeable. Consequently, the THz pulse is in principle detectable for all times within the pulse duration.At the same time, Fig. <ref> shows the evolution of Γ when the THz pulse is switched on and off. There are transient changes in Γ upon switch on and off of the THz pulse, however, they are extremely small, being on the order of 10^-6 compared to the THz field pump rate. Indeed this can be expected, as in an adiabatic regime one follows the almost vertical contour in Fig. <ref>. In practice, since the contour is not perfectly vertical, small changes in Γ occur when moving between the two steady states, and working in a non-adiabatic regime equally allows for keeping the changes in Γ small. After the transient effects have died out, Γ is exactly zero after switch on or off of the THz pulse, corresponding to a perfect transparency of the system to the THz field.For completeness, Fig. <ref> shows also the evolution of the THz mode population, n_THz. Since Γ is always very small, the THz mode population essentially follows the sigmoid switch on and off of P_THz. While we have considered square shaped pulses (with sigmoid edges), for which the physics of the system can be interpreted as making transitions between the steady states of Fig. <ref>, other pulse shapes could also be considered. More abruptly changing pulse shapes give a larger transient contribution to Γ, and for continuously varying pulse shapes (e.g., Gaussian) there will always be in general a non-zero value of Γ at all times. In principle, with Gaussian shaped pulses it is possible to arrange for the time integrated value of Γ to be zero, through careful choice of P and P_THz. In such case the time-integrated intensity of a THz pulse would be preserved, however, due to non-zero instantaneous values of Γ the THz pulse would become reshaped in time.§ COMPARISON WITH FERMIONIC CASCADESIt is instructive to compare the physics of bosonic cascades with fermionic cascades, which obey a similar set of rate equations to Eqs. <ref>-<ref>, but with the stimulated terms due to cascade occupation removed: dn_0/dt=-n_0/τ +W[n_1(n_THz+1)-n_0n_THz] dn_k/dt=-n_k/τ +W[n_k+1(n_THz+1)-n_kn_THz]+W[n_k-1n_THz-n_k(n_THz+1)]∀0<k<M dn_m/dt=P-n_M/τ +W[n_M-1n_THz-n_M(n_THz+1)] dn_THz/dt =-n_THz/τ_THz+Γ+P_THz where we now define Γ according to: Γ =W∑_k=0^M-1[n_k+1n_THz-n_kn_THz]=W(n_M-n_0)n_THzA direct comparison of fermionic and bosonic cascades using the same parameters is not particularly intuitive. Bosonic cascades are designed to function in the limit Wτ≪1, where they make use of the bosonic stimulation of scattering processes to allow efficient relaxation through all levels of the cascade. Fermionic cascades can not function in the limit Wτ≪1; taking the typical value Wτ=8.3×10^-7 <cit.> for bosonic cascades and substituting into the fermionic case will leave a system with no significant population of all but the highest level.We can nevertheless consider the possibility of a fermionic cascade achieving transparency, in the limit Wτ>1 using a similar analysis to the bosonic case. Figure <ref> shows a contour plot of Γ on P and P_THz.Unlike the bosonic case, the contours Γ(P_THz)=constant increase monotonically in P_THz and it is impossible to find parameters such that Γ is unchanged when turning P_THz on and off.§ CONCLUSIONThe transparency in bosonic cascades has a different physical origin compared to the well-known EIT. It does not rely on the negative interference of pump and probe fields. Rather, it stems from the exact compensation of the absorption and stimulated emission of radiation, and can be described in the mean-field approximation neglecting the phase of light and interference effects. On the quantum optical level, the bosonic cascades are not transparent, strictly speaking. However, at certain conditions, the mean number of photons in a light pulse going through the cascade would remain unchanged. This peculiar property makes bosonic cascades promising for applications as non-destructive photo-detectors. THz light passes going through bosonic cascades would leave traces in the instantaneous redistributions of mean populations of the cascade levels. None of these effects can be found in fermionic cascades.AK acknowledges the support from the EPSRC Programme grant on Hybrid Polaritonics. 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"authors": [
"T. C. H. Liew",
"A. V. Kavokin"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170627010718",
"title": "Optically induced transparency in bosonic cascade lasers"
} |
A vector field is called a Beltrami vector field, if B×(∇×B)=0. In this paper we construct two unique Beltrami vector fields I and Y, such that ∇×I=I, ∇×Y=Y, and such that both have an orientation-preserving icosahedral symmetry. Both of them have an additional symmetry with respect to a non-trivial automorphism of the number field ℚ( √(5) ).fancy[LE]Beltrami vector fields [RO]G. Alkauskas [CE,CO] [2010]Primary 37C10, 37C80, 15B10, 20C05, Secondary 53C65, 58A10, 76W05The research of the author was supported by the Research Council of Lithuania grant No. MIP-072/2015 Re-Evaluating the Netflix Prize - Human Uncertainty and its Impact on Reliability Sergej Sizov 26 December, 2017 ===================================================================================The aim of this paper is to construct a non-zero solution to the linear system of first order PDE's, given by2.5{[ u= w/ y- v/ z,; v= u/ z- w/ x,; w= v/ x- u/ y, ].such that the solution is in a closed-form, is as simple as possible, is defined in the whole ℝ^3, and has an icosahedral symmetry. This has a limited physical applicability, but a huge mathematical interest - see Problems <ref>, <ref>, <ref>. For few comments concerning solutions in bounded domains with a boundary condition ensuring that a particle in the field remains inside the domain, see Section <ref>.§ THE MAIN RESULTSIn the most general case, if (M,g) is a Riemannian 3-manifold, the curl operator on differential 1-forms is defined as *:̣Ω^1(M)↦Ω^1(M), where * is the Hodge star operator, and d is the exterior derivative <cit.>. The interpretation in terms of vector fields is as follows. Namely, let ι_X be tensor contraction with respect to a vector field X, d be an exterior derivative, and ^♯ denote the isomorphism from 1-forms to vector fields derived from g <cit.>. Then we define curl of a vector field X, denoted by ∇× X, by ∇× X=(*ι̣_Xg)^♯ <cit.>. For the functional-analytic and spectral treatment of curl operator, see <cit.>. In this paper we deal with the most basic example M=ℝ^3, and the standard Euclidean metric. But see the end of this paper for questions related to the case M=S^3, the three-dimensional sphere. For a vector field a_x/ x+a_y/ y+a_z/ z we usually write (a_x,a_y,a_z).Thus, let, as usual, ∇×B=curl B. A 3-dimensional vector field B in ℝ^3 is called a Beltrami field, or force-free magnetic field (FFF) in physics, if B×(∇×B)=0; sometimes the condition ∇·B=div B=0 is also included. Thus, ∇×B=fB, where f(x,y,z) is a scalar function. Note that for f constant (such fields are called constant Beltrami fields), ∇·B=0 is automatic (here and henceforth we deal with smooth fields only). An important special case of this is when f≡ 1, or ∇×B=B. This is a curl operator eigenvalue 1 case. Sometimes such vector fields are called Trkalian vector fields <cit.>. The equality ∇×B=B is exactly the topic of the current paper. For general smooth 3-manifolds, Beltrami fields are defined via differential Beltrami 1-forms <cit.>. For motivation from a point of view of mathematical physics, we note that in the setting of a magnetohydrodynamic description of plasmas <cit.>, force-free magnetic fields play a prominent role. Their characteristic property is the collinearity of magnetic field and electric current and therefore the vanishing of the Lorentz-force. Magnetic field B and current density j then satisfy (after normalization){[ j=fB,; curl B=j,;div B=0. ].This shows that B is a Beltrami field. We further quote <cit.>: “The simultaneous appearence of force-free fields in a great diversity of physical domains has increased their importance in the last decades.They are particularly a subject of an intensive research in solar physics; indeed, on the sun's surface, mostoftheobservedstructuresandphenomema (flares, mass ejection, coronal heating, prominences) are mainly due to the strong magnetic field B. When the equilibrium holds - or in a quasi-static evolution - the only significant force, say the Lorentz force j×B, must vanish. Therefore,the electric currents j are parallel to B and, thanks to Ampère's Law, that means that B is force-free". And quoting <cit.>: “Beltrami fields play a prominent role in the theory of exact, closed form solutions to the Euler and Navier-Stokes equations and their relations to the elctromagnetic wave equations. Moreover, Beltrami fields are related to minimum energy plasma fields and have theorefore garnered much attention from the magnetohydrodynamics community". After these motivating remarks, now we will pass the the main topic of this paper. Let ϕ=1+√(5)/2. Define α=[ -100;0 -10;001 ],β=[ 0 0 1; 1 0 0; 0 1 0 ],γ=[1/2 -ϕ/2 1/2ϕ;ϕ/2 1/2ϕ -1/2; 1/2ϕ1/2ϕ/2 ].These three matrices generate the icosahedral group 𝕀⊂ SO(3) of order 60, and α,β generate the tetrahedral group 𝕋 of order 12. {I,α,β^2αβ,βαβ^2} form a Klein four group 𝕂, which we will encounter later; see (<ref>). Polyhedral symmetries are ubiquitous in science, nature, history, and Art - they manifest from octahedral symmetries of carved stone balls from late Neolithic (c. 3000 BC) found in Scotland, to icosahedral symmetry of capsids of adenoviruses <cit.>. The first main result of this paper can be stated immediately as follows.Let us define the vector field V=(V_x,V_y,V_z), V_y=V_x(y,z,x), V_z=V_x(z,x,y), where V_x = 2xsin(x/2)sin(ϕ y/2)sin(z/2ϕ) -2ϕ xsin(x/2ϕ)sin(y/2)sin(ϕ z/2)+2ϕ^-1 xsin(ϕ x/2)sin(y/2ϕ)sin(z/2)+ ysin z+2ycos(x/2)cos(ϕ y/2)sin(z/2ϕ) -2ycos(x/2ϕ)cos(y/2)sin(ϕ z/2)+ zsin y-2zcos(x/2)sin(ϕ y/2)cos(z/2ϕ)+2zcos(ϕ x/2)sin(y/2ϕ)cos(z/2),and the vector field W=(W_x,W_y,W_z), W_y=W_x(y,z,x), W_z=W_x(z,x,y), whereW_x = xcos y-xcos z- √(5)xcos(x/2)cos(ϕ y/2)cos(z/2ϕ)+ϕ xcos(x/2ϕ)cos(y/2)cos(ϕ z/2)+ϕ^-1 xcos(ϕ x/2)cos(y/2ϕ)cos(z/2)- ϕ^-2ysin(x/2)sin(ϕ y/2)cos(z/2ϕ)-ϕ^2ysin(x/2ϕ)sin(y/2)cos(ϕ z/2)+ √(5)ysin(ϕ x/2)sin(y/2ϕ)cos(z/2)- ϕ^2zsin(x/2)cos(ϕ y/2)sin(z/2ϕ)- ϕ^-2zsin(ϕ x/2)cos(y/2ϕ)sin(z/2) +√(5)zsin(x/2ϕ)cos(y/2)sin(ϕ z/2).Then: 1) the vector field I=(I_x,I_y,I_z)=V+W has an icosahedral symmetry: if we treat I as a map ℝ^3↦ℝ^3, for any ζ∈𝕀one has ζ^-1∘I∘ζ=I; moreover, W has the full icosahedral symmetry 𝕀×{I,-I}; 2) as a Taylor series, V contains only terms with even compound degree, W contains only odd-degree terms, ∇×V=W, ∇×W=V; thus I satisfies the identity ∇×I=I; 3) the Taylor series for V starts with a degree 6 vector field 𝐌/768, where 𝐌=(ϖ,ϱ,σ) is given by{[ ϖ=(5-√(5))yz^5+(5+√(5))y^5z-20y^3z^3+(10+10√(5))x^2yz^3+ (10-10√(5))x^2y^3z-10x^4yz,;ϱ=(5-√(5))zx^5+(5+√(5))z^5x-20z^3x^3+(10+10√(5))y^2zx^3+(10-10√(5))y^2z^3x-10y^4zx,;σ=(5-√(5))xy^5+(5+√(5))x^5y-20x^3y^3+(10+10√(5))z^2xy^3+(10-10√(5))z^2x^3y-10z^4xy; ].the Taylor series for W starts from a degree 5 vector field N/768, where 𝐍=(λ,ξ,χ), ξ=λ(y,z,x), χ=λ(z,x,y), andλ=(35-5√(5))xy^4-(35+5√(5))xz^4+60√(5)xy^2z^2 -(70+10√(5))x^3y^2+(70-10√(5))x^3z^2+2√(5)x^5. We can now state one corollary. Suppose, a point x∈ℝ^3, x=(x_0,y_0,z_0)≠0, has a non-trivial stabilizer subgroup G_x<𝕀, which is then a rotation of order 2, 3, or 5 with respect to the axis ℝx. Then, since a vector I(x_0,y_0,z_0) is unchanged under the action of G_x, we immediately get thatI(x_0,y_0,z_0)=C(x_0,y_0,z_0)·(x_0,y_0,z_0). All such points x with non-trivial stabilizer subgroups lie on 60/5+60/3+60/2=62 lines, obtained from ℝ(ϕ,1,0) (12 lines corresponding to centres of faces of a dodecahedron), ℝ(1,1,1) (20 lines corresponding to vertices of a dodecahedron), and ℝ(1,0,0) (30 lines corresponding to centres of edges) under the action of the group 𝕀. In the last section we will see that on every of these lines there exist infinitely many points where the vector field I vanishes. For example, if s_0 is the root of1-ϕcos(s)+ϕ^-1cos(ϕ s)=0,then I(ϕ s_0,s_0,0)=0. The smallest positive non-zero such s_0 is given by s_0=5.1625967944_+. We can call zeros of I lying on these 62 lines as trivial zeros. Are there any non-trivial zeros of the vector field I?Each of these exceptional 62 lines split into disjoint open segments, each being a complete orbit. The endpoints of the segment are two trivial zeros. The dynamics of all the rest orbits is considerably far more complicated.The vector field 𝐌 is the numerator of the vector field for the icosahedral projective superflow <cit.> (there are two related notions - projective superflow and polynomial superflow), whence the motivation and the first step comes from.One has ζ^-1∘M∘ζ=M for any ζ∈𝕀, and this is the unique, up to the scalar multiple, polynomial 6-homogeneous vector field with this property <cit.>. Calculations show that ∇×M=N has a full icosahedral symmetry, and ∇×(∇×M)=0. There exist exactly 5 irreducible projective superflows in dimension 3 <cit.>. However, the orbits of all five 3-dimensional superflows - the tetrahedral (group of order 24, full symmetry), the octahedral (24, orientation-preserving symmetry), the icosahedral (60, also orientation-preserving symmetry), 3-prismal (12), and 4-antiprismal (16) - are algebraic curves, since the corresponding system of differential equations in all cases possesses two independent algebraic first integrals. This shows a deep analogy (and essential differences) between integration of superflows, and integration of dynamical systems whose Lagrangian has infinitesimal symmetries, exactly as E. Noether's 1918 theorem tells (see <cit.>, Chapter IV, 12, and also <cit.>). We recall that 𝒲 is the first integral for a vector field (a_x,a_y,a_z), if𝒲/ x· a_x+𝒲/ y· a_y+𝒲/ z· a_z=0. Two independent first integrals of the flow with the vector field M are given byx^2+y^2+z^2, (ϕ^2x^2-y^2)(ϕ^2 y^2-z^2)(ϕ^2z^2-x^2). Thus, integration of superflows has a strong algebro-geometric and number-theoretic side <cit.>.For example, 12 particular orbits of the flow with the vector field M are shown in Figure <ref>. Note only that differently from the vector field given in <cit.>, there is no denominator (x^2+y^2+z^2)^2. But since x^2+y^2+z^2 is the first integral of the corresponding differential system, that is, xϖ+yϱ+zσ≡0, this does not matter. However, in case of the current paper, for the differential system{[ ẋ(t)=I_x(x(t),y(t),z(t)),; ẏ(t)=I_y(x(t),y(t),z(t)),; ż(t)=I_z(x(t),y(t),z(t)), ].we cannot expect the existence of a single closed-form first integral, not to mention two of them, as Figure <ref> suggests. Thus, differently from Noether's theorem for Lagrangians, symmetry is not the only feature of the superflow (with the vector field M, for example) which guarantees the existence of (polynomial) first integrals - superflows is in essence an algebro-geometric topic; for more on this, see <cit.>. The terms in the Taylor expansion of V_x(x,y,z) are not just of even compound degree, but in fact of even degree in x and odd in y and z; this is a consequence of the fact 𝕂<𝕀. Cyclically so for V_x(y,z,x) and V_x(z,x,y). Equally, W_x is odd in x and even in each of y,z.Note that vector fields V and W have one additional symmetry. Indeed, let us denote by τ the non-trivial automorphism of the number field ℚ(√(5) ). Thus, τϕ=-ϕ^-1. In the above formulas for V_x and W_x (Theorem <ref>), let us expand everything in Taylor series, swap y and z, leave x intact, and apply τ summand-wise. We readily obtainτ V_x(x,z,y)=V_x(x,y,z), τ W_x(x,z,y)=-W_x(x,y,z).The icosahedral group, as a group, is isomorphic to A_5. The latter group has two non-equivalent 3-dimensional representations, and the second one is given exactly via an embedding τα=α, τβ=β, τγ≠γ (<cit.>, Chapter 8, 5, Problem 7). The second result of this paper is completely analogous, only the corresponding Taylor series starts from a degree 9 rather than 5, and due to a careful choice of multiplying parameter, there is the same symmetry with respect to ℚ(√(5) ); see (<ref>) further, where any scalar multiple of the collection given produces the solution to our problem, but only ℚ-scalar multiples have this additional symmetry. The fact that the Taylor coefficients vanish to the order 8 rather than 4 makes this vector field even more exceptional. Formulas are more lengthy, so we write down explicitly only the even part. Let us define the vector field V^0=(V^0_x,V^0_y,V^0_z), V^0_y=V^0_x(y,z,x), V^0_z=V^0_x(z,x,y), whose Taylor series contains only even compound degrees, by V_x^0 = 2xsin(x/2)sin(ϕ y/2)sin(z/2ϕ) -2ϕ xsin(x/2ϕ)sin(y/2)sin(ϕ z/2)+2ϕ^-1xsin(ϕ x/2)sin(y/2ϕ)sin(z/2)+ 2ϕ ysin z+(7-√(5))ycos(x/2)cos(ϕ y/2)sin(z/2ϕ)+ 2ϕ^2ycos(x/2ϕ)cos(y/2)sin(ϕ z/2) +2√(5)ycos(ϕ x/2)cos(y/2ϕ)sin(z/2)- 2ϕ^-1zsin y-(7+√(5))zcos(x/2)sin(ϕ y/2)cos(z/2ϕ)- 2ϕ^-2zcos(ϕ x/2)sin(y/2ϕ)cos(z/2) -2√(5)zcos(x/2ϕ)sin(y/2)cos(ϕ z/2), and its odd counterpart W^0 is defined by W^0=∇×V^0.Then the vector field Y=V^0+W^0 has an icosahedral symmetry, and satisfies ∇×Y=Y. The Taylor series for V^0 starts from the degree 10 vector field P/23224320, where P=(ϖ_0,ϱ_0,σ_0), ϱ_0(x,y,z)=ϖ_0(y,z,x), σ_0(x,y,z)=ϖ_0(z,x,y), andwhere ϖ_0 = -18x^8yz+(84+84√(5))x^6y^3z+(84-84√(5)) x^6yz^3-(126+126√(5))x^4y^5z-(126-126√(5))x^4yz^5+ (36+108√(5))x^2y^7z+(36-108√(5))x^2yz^7-504√(5)x^2y^5z^3 +504√(5) x^2y^3z^5+ (9-5√(5))y^9z+(9+5√(5))yz^9-(120-24√(5))y^7z^3 -(120+24√(5))y^3z^7+252y^5z^5.The Taylor series for the vector field Y starts from a degree 9 vector field Q/23224320=∇×P/23224320≠0. Thus, a one-parameter family of vector fields I_a=I+aY has 𝐍+𝐌/768 as a beginning of its Taylor series expansion, has an icosahedral symmetry, and satisfies ∇×I_a=I_a. If a∈ℚ, the same symmetry with respect to ℚ( √(5) ) applies. As we will soon see in Note in Section <ref>, this family arises from a 1-dimensional setting for the Helmholtz equation. This is what we meant by saying the simplest possible in the very introduction to this paper. Higher dimensional solutions to the Helmholtz equation, which lead to Beltrami vector fields with various polyhedral symmetries, are treated in <cit.>; see Section <ref>. If two n-dimensional vector fields X and Y in ℝ^n are given byX=∑_i=1^nf_i/ x_i,Y=∑_i=1^ng_i/ x_i,then their Lie bracket is defined as <cit.>[X,Y]=∑_i=1^n(X(g_i)-Y(f_i))/ x_i.The Taylor series of [I,Y] starts from a scalar multiple of a vector field [N,Q]. Computer calculations show that this is not identically 0, and so Vector fields I and Y do not commute.§ OVERVIEW One of the motivations to investigate Beltrami fields is the followingclaim which follows from the result by Arnold <cit.>, and which shows that Beltrami flows on a closed 3-manifold might have a complicated topology. Namely, the orbits of a Beltrami flow on a 3-manifold are not always constrained to lie on a 2-torus as is the case for steady Euler flows with a C^ω vector field and which are not everywhere collinear with its curl. For more information on various aspects of Beltrami flows, see <cit.>. For mathematical physics-related aspects of force-free magnetic fields, see <cit.>. It is impossible to give a wide overview on a prolific literature related to Beltrami fields, so we will confine to few papers.For a computational aspect (in bounded simply-connected domains), see <cit.>. Symmetry questions of force-free fields not depending on the variable z (the so called 2-dimensional FFF) are treated in <cit.>. Recall that a planefield ξ on a 3-dimensional manifold is a smooth mapping that assigns to every point p a plane in its tangent space. A planefield ξ is said to be integrable at a point p, if there exists a smooth surface S passing through p such that ξ is tangential to S in some neighbourhood of p. A planefield ξ is called a contact structure if and only if it is everywhere non-integrable. The relation between Beltrami (and Trkalian) fields with contact structures is investigated in <cit.>. A topic, slightly related to our paper, was investigated in <cit.>, where it was shown that non-stationary solution of the Navier-Stokes equations in ℝ^d (d=2,3), if this solution is left invariant under the action of a finite subgroup of the orthogonal group, decays much faster as |x|→∞ or t→∞ than in a generic case. The decay is extremely fast in case d=3 and the full symmetry group of an icosehedron; that is, 𝕀×{I,-I}. In <cit.> the authors prove, via an iteration scheme, the existence of force-free magnetic fields in the exterior domain of some compact simply connected surface S. The huge difference emerges if we consider constant, or non-constant force-free fields. The spherical curl transform for Trkalian fields using differential forms and its Radon transform are investigated in <cit.>. In <cit.> the author investigates the eigenfunctions of the equation ∇×B=λB for finite cylindrical geometry with normal boundary condition n⃗·B=0 for nonaxisymmetric modes. The author investigates also the equation ∇×∇×B=λ^2B, since this reveals the underlying elliptic nature of the initial eigenvector problem. The method of Chandrasekhar and Kendall (which we will also employ) is being used. Double-curl spectral Beltrami equation{[ ∇×(∇×B)+ α∇×B+βB=0(in the domain Ω),; n⃗·B=0,n⃗·(∇×B)=0 (on ∂Ω), ].are investigated in <cit.>. The authors show that if the domain Ω is multiply connected, then this equation has a nonzero solution for arbitrary complexnumbers α and β. In relation to the results of the current paper, we will emphasize one consequence of the result proved in <cit.> (see <cit.> for numerical results in this direction). Consider the simplest example of a Beltrami condition satisfied by a three-dimensional solenoidal vector field, obeying{[ ∇×B=λB (in the domain Ω),; n⃗·B=0 (on ∂Ω). ].Here λ is a real (or complex) constant number, Ω⊂ℝ^3 is a bounded domain with a smooth boundary, and n⃗ is a unit normal vector onto Ω. This system is regarded as an eigenvalue problem with respect to the curl operator. Then one of the results in <cit.> claims that if Ω is simply connected, then the system (<ref>) has a nonzero solution for special λ included in a set of discrete real numbers; these numbers represent the point spectrum of the self-adjoint part of the curl operator. Now, consider a simply-connected domain Ω which has an icosahedral symmetry 𝕀. For example, we can take a domain (see Figure <ref>)Ω={(x,y,z)∈ℝ^3:(ϕ^2x^2-y^2)(ϕ^2 y^2-z^2)(ϕ^2z^2-x^2)+(x^2+y^2+z^2)^3 < 1}.Let (B,λ) solves an eigenvalue problem (<ref>). It is obvious that any orientation-preservingorthogonal change of coordinates γ^-1∘B∘γ(x,y,z), γ∈𝕀, also solves this problem. Thus,B_𝕀=1/60∑_γ∈𝕀γ^-1∘B∘γis a vector field with an icosahedral symmetry which also solves the eigenvalue problem (<ref>). However, without delving deeper into the geometry of the surface ∂Ω and the system (<ref>), we cannot guarantee that B_𝕀 is non-zero. Further, as was shown in the introduction, there exist 62 lines (see also Section <ref>) passing through the origin such that for points on these lines, a vector field B_𝕀 is collinear with a line itself. Suppose now that a domain Ω is a star domain with respect to the origin, which implies that intersection of every line through the origin with Ω is a closed interval. The domain (<ref>) is an example. Then we have the following result. (By icosahedral symmetry we always mean the group 𝕀). If a vector field B_𝕀 has an icosahedral symmetry, and (B_𝕀,λ) solves the eigenvalue problem (<ref>) for the star domain (with respect to the origin) Ω with an icosahedral symmetry, then there exist at least 62 points on Ω where a vector field B_𝕀 vanishes. This gives a new perspective on uniqueness of vector fields I and Y which are constructed in this paper. Next, we formulate one problem which seems to be of a huge interest from the point of view of geometry. In <cit.>, Section 7.2, the author shows that the existence of conformally flat contact metric manifolds corresponds to finding the solutions to the equation (in cartesian coordinates) ∇×B=|B|B, where |B| is vectors length. For the standard Sasakian structure of a constant curvature +1 on S^3, using stereographic projection to ℝ^3, the corresponding vector field is B=8(xz-y)/(1+x^2+y^2+z^2)^2/ x +8(x+yz)/(1+x^2+y^2+z^2)^2/ y +4(1+z^2-x^2-y^2)/(1+x^2+y^2+z^2)^2/ z.Here |B|=4/1+x^2+y^2+z^2. In relation to this, as a more algebro and differential-geometric version of results of the current paper, we pose Does there exist a Beltrami vector field B, meaning ∇×B=f(x,y,z)·B, which is given by rational functions, and which has a 4-Klein group? tetrahedral? octahedral? icosahedral symmetry?Note that, if ∇×B=f(x^2+y^2+z^2)·B, then∇×(η^-1∘B∘η)=f(x^2+y^2+z^2)·η^-1∘B∘η for any η∈ SO(3). Therefore, it as if seems that we can construct Beltrami fields with any cyclic or polyhedral symmetry from any given Beltrami field by averaging, like (<ref>). However, in most cases we will end up with a 0 vector field, and independent methods are truly needed - this is the whole essence of this paper! For example, the Klein four group𝕂={I,diag(-1,-1,1),diag(-1,1,-1),diag(1,-1,-1)}is a subgroup of 𝕀. But yet, for B given by (<ref>),∑_η∈𝕂η^-1∘B∘η=0.Still, if β is given by (<ref>), we get a non-trivial example by calculatingF=1/4∑_j=0^2β^-j∘B∘β^j=(U(x,y,z),U(y,z,x),U(z,x,y)) (factor 4, not 3, is for simplicity), whereU=2xy+2xz-2y+2z+1+x^2-y^2-z^2/(1+x^2+y^2+z^2)^2.Thus, the answer to Problem <ref> is positive at least in a cyclic order 3 subgroup generated by β. Moreover, we have∇×F=4/1+x^2+y^2+z^2·F=4/√(3)·|F|·F.We will address this problem in the next publication.§ CONSTRUCTION One of the main identities of a 3-dimensional vector calculus claims that for a smooth vector field B,∇×(∇×B)=∇(∇·B)-∇^2B; here ∇^2 is a vector Laplace operator, and for a scalar function f, ∇ f=gradf. To prove our two theorems, first we will construct a vector fieldV such that: 1) V satisfies the vector Helmholtz equation ∇^2V=-V; 2) it satisfies ∇·V=0; 3) V has an icosahedral symmetry;4) all elements in the Taylor series are of even compound degree. If the first two are satisfied, then the identity (<ref>) implies∇×(∇×V)=V. Then we put W=∇×V, we will see that I=V+W has the properties described by items 1) and 2) in Theorem <ref>. Such method of deriving solutions to ∇×B=νB from scalar solutions to the Helmholtz equation (∇^2+ν^2)Ψ=0 (then such solution Ψ is called Debye potential) was developed by Chandresekhar and Kendall <cit.>. Our contribution is the fact that we consider vector solutions, and especially the emphasis on part 3), which is new in the theory of Beltrami fields. See the end of Section <ref> where it is shown that orthogonal orientation-preserving symmetries of a vector field carries automatically as symmetries of its curl.To satisfy the requirements 1), 2) and 3) above (we now secure the notation V for a specific vector field given by Theorem <ref>, changing the unspecified vector field to H), we will try to find constants a_i, and homogeneous linear forms L_i, k_i, such that H=(G_x,G_y,G_z), whereG=G_x=∑_i(a_icos(k_i )+L_isin(k_i)), and the two other coordinates G_y and G_z are obtained from G by a cyclic permutation. First, we know that H=(G_x,G_y,G_z) is invariant under conjugation with α=diag(-1,-1,1), βαβ^2=diag(1,-1,-1), and β^2αβ=diag(-1,1,-1). These four matrices, as already mentioned two times, together with the unity I produce the Klein's fourth group 𝕂<𝕋<𝕀. This gives -G(-x,-y,z)=G(x,y,z),G(x,-y,-z)=G(x,y,z). As the first three linear forms k_i, we just take x,y,z. Invariance under 𝕂 gives, respectively, the following sum as a part of G_x in (<ref>); namely, azsin y+bysin z. Next, let us define the following 12 linear forms and vectors as follows.ℓ_x,0(x,y,z)=1/2x+ϕ/2y+1/2ϕz, j_x,0=(1/2,ϕ/2,1/2ϕ), ℓ_x,1(x,y,z)=-1/2x+ϕ/2y+1/2ϕz, j_x,1=(-1/2,ϕ/2,1/2ϕ), ℓ_x,2(x,y,z)=1/2x-ϕ/2y+1/2ϕz, j_x,2=(1/2,-ϕ/2,1/2ϕ), ℓ_x,3(x,y,z)=1/2x+ϕ/2y-1/2ϕz, j_x,3=(1/2,ϕ/2,-1/2ϕ). Similarly we define 8 other linear functions and 8 vectors by cyclically permuting variables. For example,j_y,2=(1/2ϕ,1/2,-ϕ/2), ℓ_y,2=j_y,2·(x,y,z)^T=1/2y-ϕ/2z+1/2ϕx, and so on. Orthogonality of the matrix γ tells, for example, that j_x,2, j_z,1 and j_y,0 are orthonormal vectors. All these relations amount to the same identities ϕ^2+ϕ^-2+1=4 (unit length), or ϕ-ϕ^-1-1=0 (orthogonality). This gives 14 linear forms in total (12 plus two forms y and z), and this is our complete collection in (<ref>). Next, if the coordinate of the form (<ref>) is invariant under conjugating with 𝕂, it is of the form G = azsin y+bysin z +ccosℓ_x,0-ccosℓ_x,3-ccosℓ_x,2+ccosℓ_x,1+ K(x,y,z)sinℓ_x,0+K(-x,-y,z)sinℓ_x,3+K(-x,y,- z)sinℓ_x,2-K(x,-y,-z)sinℓ_x,1+ dcosℓ_y,0-dcosℓ_y,2-dcosℓ_y,1+dcosℓ_y,3+ L(x,y,z)sinℓ_y,0+L(-x,-y,z)sinℓ_y,2+L(-x,y,-z)sinℓ_y,1-L(x,-y,-z)sinℓ_y,3+ ecosℓ_z,0-ecosℓ_z,1-ecosℓ_z,3+ecosℓ_z,2+ M(x,y,z)sinℓ_z,0+M(-x,-y,z)sinℓ_z,1+M(-x,y,-z)sinℓ_z,3-M(x,-y,-z)sinℓ_z,2. Here a,b,c,d,e are arbitrary constants, and K,L,M are arbitrary linear forms.Now, recall that we want ∇^2G=-G to be satisfied. We will achieve this if each of the summands in (<ref>) satisfies the Helmholtz equation. Now, the following Lemma is immediate. Let a,b≠0 be two 3-dimensional vectors-rows, and x=(x,y,z)^T. The function ax·sin(bx) is a solution to the Helmholtz equation if and only if ⟨a,b⟩=0, and |b|=1. The same holds for the cos function.By a direct inspection, two vectors orthogonal to j_x,0 are given by j_z,2 and j_y,1. So,K(x,y,z)=fℓ_z,2+gℓ_y,1, L(x,y,z)=hℓ_x,2+iℓ_z,1, M(x,y,z)=jℓ_y,2+kℓ_x,1.We therefore have 11 free coefficients (a through k) at our disposition. Such a function G(x,y,z) is invariant under conjugation with 𝕂 and satisfies the Helmholtz equation. In vector terms, the vector field (G(x,y,z),G(y,z,x),G(z,x,y)) has a tetrahedral symmetry 𝕋 of order 12 (generated by matrices α and β) and satisfies the vector Helmholtz equation.We will reduce the amount of free coefficients by requiring that a vector field has also a γ-symmetry, and that the divergence vanishes.§ ORDER OF APPROACH The expression (<ref>) is what we mean by the first order approach. The second order functions (x^2-y^2)cos z and 2xycos z, for example, also satisfy the Helmholtz equation. In general, let n∈ℕ_0, and P_n(x,y)=(x+iy)^n, Q_n(x,y)=(x+iy)^n be the standard harmonic polynomials of order n. Then the nth order solutions to the Helmholtz equation are given by P_n(x,y)cos z, Q_n(x,y)cos z. Other solutions are given by sin z instead of cos z, or from any of these by an orthogonal change of variables, and all possible linear combinations. This case is treated in <cit.>, including a construction of Beltramivector fields with tetrahedral and octahedral symmetries. Now, fix N∈ℕ, and consider the expression (<ref>), where this time: i)Linear forms k_i are given by 12 linear forms ℓ_w,a, w∈{x,y,z}, a∈{0,1,2,3}, and 3 linear forms x,y,z;ii)a_i is a polynomial in (x,y,z) of even compound degree and L_i is of odd, respectively, both are of degrees at most N;iii)Each term in (<ref>) satisfies the Helmholtz equation;iv) (G(x,y,z),G(y,z,x),G(z,x,y)) has the icosahedral symmetry 𝕀;v) (G(x,y,z),G(y,z,x),G(z,x,y)) is solenoidal.Find the dimension d_𝕀(N) of linear space of all such functions G.In the next section we will show tat d_𝕀(0)=0, and d_𝕀(1)=2. While dealing with the icosahedral group, we are working with the dimension n=3. The same set of ideas carries to any dimension, with a slight attenuation of requirements - there is no same dimensional analogue of a curl operator in dimensions other than 3 (n=n(n-1)/2 holds for n=3 only, n∈ℕ).Namely, the property ∇×G=G is replaced by the properties 1), 2) and 4) in the beginning of Section <ref>. This leads, for example, to the vector field 𝔇=(T_x,T_y), given byT_x = -cos y+√(3)sin(x/2)sin(√(3)y/2)+cos(√(3)x/2)cos(y/2), T_y = -cos x+√(3)sin(y/2)sin(√(3)x/2)+cos(√(3)y/2)cos(x/2),with these properties. i)The vector field 𝔇 has a 6-fold dihedral symmetry, generated by thematrices[ 0 1; 1 0; ],1/2[-1 -√(3);√(3)-1; ]. ii)the Taylor series for 𝔇 contains only even compound degrees, and it starts from3/8((2xy-x^2+y^2),(2xy+x^2-y^2)); iii)it satisfies the vector Helmholtz equation ∇^2𝔇=-𝔇;iv)div 𝔇=0.So, these properties relate the vector field to ABC flows. The current paper is thus an introduction to a much broader setting dealt with in <cit.>. We remind that the ABC flow, or Arnold-Beltrami-Childress flow, is described by a vector field 𝐛=(Asin z+Ccos y,Bsin x+Acos z,Csin y+Bcos x), A,B,C∈ℝ. Symmetry related question of these are treated in <cit.>. Such a vector fields satisfies 𝐛=∇×𝐛, and from the Helmholtz equation point of view, it represents an order 0 approach. Generally, it does not have any non-trivial orthogonal symmetries, except in some cases, like A=B=C=1, but, of course, there many more profound symmetries apart from those generated by the group 2π·ℤ^3 <cit.>.§ COMPUTER-ASSISTED CALCULATION Now, we will proceed with determining 11 free coefficients in (<ref>). i) First, we will require that the Taylor coefficient of G of degree 6 is exactly equal to ϖ as given by (<ref>), and Taylor coefficients of degrees ≤ 5 are absent. This gives, with computer calculations performed on MAPLE, 6 linear relations among 11 constants. The free coefficients turn out to be a,b,c,d, and f. This is, of course, to some extent a loose choice; one can also arrive at 5 other parameters, other 6 being linear expressions in former ones. This way we obtain a 5-parameter function G given by (<ref>), such that its Taylor series starts with ϖ, it is invariant under conjugation with 𝕂, that is, has a symmetry (<ref>), and satisfies the Helmholtz equation.ii) As the next step, we require that ∇·(G(x,y,z),G(y,z,x),G(z,x,y))=0. This gives two more relations, leaving a,b, and d as free coefficients. In fact, these computations, though they involve only manipulations with polynomials and Taylor coefficients of trigonometric functions, take some time on a modern computer. It is infeasible to do it by hand. (In <cit.>, however, we will show that there is an alternative method to calculate the vector field in Theorem <ref> manually). iii) Finally, a 3-parameter vector field H=(G(x,y,z),G(y,z,x),G(z,x,y)) has the tetrahedral symmetry, satisfies the Helmholtz equation, has a vanishing divergence, and the correct beginning of the Taylor series. If it is invariant under conjugation with γ, it has the icosahedral symmetry, and the problem is solved. We have:H = γ^-1∘(G(x,y,z),G(y,z,x),G(z,x,y))∘γ(x,y,z)= γ^-1∘(G(ℓ_x,2,ℓ_z,1,ℓ_y,0),G(ℓ_z,1,ℓ_y,0,ℓ_x,2),G(ℓ_y,0,ℓ_x,2,ℓ_z,1)):=γ^-1(A,B,C).Since j_x,0 is the first row of γ^-1=γ^T, therefore, we finally require thatA/2+ϕ B/2+C/2ϕ=G(x,y,z). This gives two more relations, and we thus obtain a 1-parameter family of icosahedral vector fields, with a free parameter being a. We will get a particularly symmetric, with respect to a non-trivial automorphism of ℚ(√(5) ), example if a=τ b. Indeed, the last of the 10 linear equations we obtained reads as b=1920-3/2a-√(5)/2a+384√(5). Let us choose a=384· 2. This gives b=384· 2, exactly what we need. This yields c=d=e=0, f=384ϕ^-1, g=-384ϕ, h=384ϕ, i=384· 1, j=-384· 1, k=384ϕ^-1. Thus, if we start from a collection of the coefficients(a,b,c,d,e,f,g,h,i,j,k)=(2,2,0,0,0,ϕ^-1,-ϕ,ϕ,1,-1,ϕ^-1), we arrive at the vector field 1/384(ϖ,ϱ,σ), where (ϖ,ϱ,σ) is given by (<ref>). Now, (<ref>) gives G = 2zsin y+2ysin z+ (-x+y-z)sinℓ_x,0+(x-y-z)sinℓ_x,3+(x+y+z)sinℓ_x,2-(-x-y+z)sinℓ_x,1+ (ϕ x-y)sinℓ_y,0+(-ϕ x+y)sinℓ_y,2 +(-ϕ x-y)sinℓ_y,1-(ϕ x+y)sinℓ_y,3+ (-ϕ^-1x+z)sinℓ_z,0+(ϕ^-1x+z)sinℓ_z,1 +(ϕ^-1x-z)sinℓ_z,3-(-ϕ^-1x-z)sinℓ_z,2. Finally, let us collect all functions in each row as factors of x,y,z, and use trigonometric addition formulas. For example,x(-sinℓ_x,0+sinℓ_x,3+sinℓ_x,2+sinℓ_x,1) = 4xsin(x/2)sin(ϕ y/2)sin(z/2ϕ).Thus we get (in fact, MAPLE does these tedious computations for us) the first displayed formula in Theorem <ref>, where(V_x,V_y,V_z)=1/2(G(x,y,z),G(y,z,x),G(z,x,y)). For the first coordinate of its curl, we have W_x=V_z/ y-V_y/ z, and this gives the second displayed formula in Theorem <ref>. To get immediately formulas in Theorem <ref>, in <cit.> we should set(a,b,c,d,e,f,g,h,i,j,k)=(1,1,0,0,0,ϕ^-1/2,-ϕ/2,ϕ/2,1/2,-1/2,ϕ^-1/2).Some explanation is needed why the icosahedral symmetry holds for W, too. This is clear from a coordinate-free definition of a curl; see, for example, (<cit.>, Chapter XVIII, 4). Indeed, let F=(F_x,F_y,F_z) be a smooth vector field, let M be any point, and let n be any direction from this point. In the plane perpendicular to it and passing through M, let us round the point M with a region Σ, whose smooth boundary is λ and an area is |Σ|. Then the Kelvin-Stokes theorem states that(∇× F)_n=lim_Σ→ M1/|Σ|∫_λF_λλ̣, F_λλ̣=F_xx̣+F_yỵ+F_zẓ;here (∇× F)_n is a projection of a vector ∇× F(M) onto a direction of n, and Σ shrinks to the point M. Thus, we can define this projection coordinate-free, and since n is any direction, this defines the curl. Now it is clear that if a vector field remains unchanged under a certain orthogonal transformation η∈ SO(3), so does its curl. If we instead specialize(a,b,c,d,e,f,g,h,i,j,k)=(-2ϕ^-1,2ϕ,0,0,0,-ϕ^-2,-ϕ^2, -1,ϕ,ϕ^-1,1),we obtain Beltrami vector field Y with icosahedral symmetry whose Taylor series starts at degree 10 rather than 6, and I+aY is our 1-parameter family. Now we can start from the correct choice of parameters a through k, and verify all the needed properties by MAPLE. The codes are provided by <cit.>. Changing parameters to the ones given by (<ref>) verifies formulas and claims in Theorem 2.§ FINAL REMARKS Of course, such questions as i)which of the orbits under the flow with the vector field I are bounded; ii) are closed;need to be investigated. Also,iii)for which points x∈ℝ^3 there exists a bounded increasing sequence {t_i:i∈ℝ}, such that F(x,t_i) tends to infinity;here F(x,t) is the flow with a vector field I. All we can say that these classifications must have an icosahedral symmetry, too.The existence of C^∞ or even C^ω vector fields on a 3-sphere S^3 with no circular orbits where demonstrated by Kuperberg <cit.>, while for Beltrami C^ω vector fields (with non-vanishing curl) this was ruled out in <cit.>. It is interesting to know the answer for the flow F(x,t). At this moment, we can answer partially to these questions as follows. Consider the vector field I. For the vector field Y the results are completely analogous, though the function Υ - see (<ref>) - then is more complicated. Any point x∈ℝ^3∖{0} has 60 different points under the action of the group 𝕀. However, there are three exceptional cases. Let s∈ℝ∖{0}. F- Faces. The point (ϕ s,s,0) has 12 equivalent points under the action of 𝕀;V- Vertices. The point (s,s,s) has 20 equivalent points;E- Edges. the point (s,0,0) has 30 equivalent points.Geometrically, consider 6 planes given by (ϕ^2x^2-y^2)(ϕ^2 y^2-z^2)(ϕ^2z^2-x^2)=0. These 6 planes split the unit sphere into 12 pentagons having arcs of great circles as their sides, their centers being given by F (more precisely, intersection of these lines with the unit sphere), 20 triangles with centers being V, and 30 intersection points given by E. Thus, it is more convenient to describe all these lines using not the geometry of a dodecahedron or an icosahedron, but rather an icosidodecahedron; see Figure <ref>.Each of F,V,E (and its equivalent) defines a line passing through the origin, 62 lines in total. On these 62 lines the vector field M, as given by (<ref>), vanishes. Each of these lines is divided into segments. Each segment is the complete orbit of the flow F(x,t).Every orbit starts and finishes at two fixed points of the vector field I, respectively. Since calculations are analogous in all three cases, we will show this in case F. Indeed, then a point (ϕ s,s,0) has only 12 equivalent points (±ϕ s,± s,0) (signs are independent), and all cyclic permutations. We take (ϕ s,s,0) as a representative. By a direct calculation, I(ϕ s,s,0)=(ϕΥ(s),Υ(s),0),whereΥ(s)=-s√(5)(1-ϕcos(s)+ϕ^-1cos(ϕ s)).This giveslim sup_s→∞Υ(s)/s=2√(5)ϕ^-1,lim inf_s→∞Υ(s)/s=-2√(5)ϕ.Indeed, let s=2πℓ, ℓ∈ℕ. Then ϕ s will hit arbitrarily close to π+2π k, k∈ℕ. For this we should have2πℓϕ=π+2π k+ϵ(ℓ)⟹ϕ=1+2k/2ℓ+ϵ(ℓ)/2πℓ.This will occur if (2ℓ,2k+1) is chosen to be a pair of two consecutive Fibonacci numbers (F_3n,F_3n+1), n∈ℕ. From Diophantine properties of quadratic irrationals we know that then ϵ(ℓ)∼c/ℓ. This gives the lim sup part, and analogously for lim inf.Thus we see that if s_1 and s_2 are two consecutive zeros of Υ(s)=0, the segment joining s_1·(ϕ,1,0) and s_2·(ϕ,1,0) is the full orbit, and I(s_1(ϕ,1,0))=0, F(s_1(ϕ,1,0),t)=s_1(ϕ,1,0). In this case the differential system (<ref>) turns out to be essentiallyẏ(t)=Υ(y(t)).The behaviour of the vector field I on other lines passing through the origin is far more complicated. For example, Figure <ref> plots {I(5s,6s,7s): s∈[0,150]}; the choice of the line is motivated by a wish to further elucidate Figure <ref>. As a final remark of this paper, consider the Riemannian manifold (S^3,g) (a standard 3-sphere), where g is the usual induced Euclidean metric from ℝ^4, and curl operator was defined in the beginning of this paper. Now, consider a finite subgroup of SO(4). For example, let 𝕆_4 the the group of order 4!· 2^3=192, the so called orientation preserving hyperoctahedral group, generated by matrices α↦[ 0 1 0 0; 0 0 1 0; 1 0 0 0; 0 0 0 1; ], β↦[1000;0 -100;0001;0010 ], γ↦[0100;1000;00 -10;0001;]. We may ask for a similar question of constructing Beltrami vector field, equal to its own curl, with a 𝕆_4-symmetry. 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"authors": [
"Giedrius Alkauskas"
],
"categories": [
"math.DG",
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"math.DS",
"math.MP",
"37C10, 37C80, 15B10, 20C05 (Primary) 53C65, 58A10, 76W05 (Secondary)"
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"primary_category": "math.DG",
"published": "20170627135428",
"title": "Beltrami vector fields with an icosahedral symmetry"
} |
Department of Physics, University of Puget Sound, Tacoma, WA 98416-1031In quantum field theory, the photon-fermion vertex can be described in terms of four form factors which encode the static electromagnetic properties of the particle, namely its charge, magnetic dipole moment, electric dipole moment, and anapole moment. For Majorana fermions, only the anapole moment can be nonzero, a consequence of the fact that these particles are their own antiparticles.Using the framework of quantum field theory, we perform a scattering calculation which probes the anapole moment with a spinless charged particle.In the limit of low-momentum transfer, we confirm that the anapole can be classically likened to a point-like toroidal solenoid whose magnetic field is confined to the origin.Such a toroidal current distribution can be used to demonstrate the Aharonov-Bohm effect.We find that, in the non-relativistic limit, our scattering cross section agrees with a quantum mechanical computation of the cross section for a spinless current scattered by an infinitesimally thin toroidal solenoid.Our presentation is geared toward advanced undergraduate or beginning graduate students. This work serves as an introduction to the anapole moment and also provides an example of how one can develop an understanding of a particle's electromagnetic properties in quantum field theory. Scattering from a quantum anapole at low energies David C. Latimer Received *** ; accepted *** =================================================§ INTRODUCTIONIn quantum field theory (QFT), the interaction vertex between a single photon and a spin-1/2 fermion can be characterized in terms of four electromagnetic (EM) form factors.In the limit of vanishing momentum transfer between the photon and fermion, the form factors encode the static EM properties of the particle, namely, its charge, magnetic dipole moment, electric dipole moment, and anapole moment.<cit.> For beginning graduate students in QFT, two of these moments are certainly familiar, and likely, these students have seen how, in the low-energy limit, one can establish a connection between the QFT formalism and simple Coulombic potentials or magentic dipole interactions.In this paper, we will focus our attention upon the anapole moment - probably the least familiar static property.Classically, the anapole moment arises as a contact quadrupole moment in a multipole expansion of the magnetic field.<cit.> The paradigmatic example of a classical EM source with non-zero anapole moment is a toroidal current distribution; this paradigm carries over to the quantum realm.Using the formalism of QFT, we will effect a scattering calculation that, in the low-energy limit, will make clear that the EM analogue of the quantum anapole is a point-like toroidal solenoid. The possibility of the anapole interaction was first noted by Zel'dovich <cit.> after it was suggested that parity symmetry might be violated in the weak interactions.<cit.>He posited that the anapole moment 𝐚, collinear with the particle's spin, interacted only with currents, 𝐉, and not electric or magnetic fields.The low-energy interaction Hamiltonian was thus H_int = - 𝐚·𝐉, which prompted the analogy between the anapole and a classical toroidal solenoid.Experimental confirmation of parity violation <cit.> meant that the anapole moment was not a mere theoretical exercise, and decades later, the nuclear anapole moment ofcesium-133 was definitively extracted from a nearly dominant background of other interactions.<cit.>In contrast with the rich structure of a cesium nucleus, the sole static EM property of a spin-1/2 Majorana fermion is the anapole moment.<cit.> Majorana fermions are self-conjugate fields; that is, there is no distinction between particle and antiparticle.This fact severely constrains their electromagnetic interactions; they must be electrically neutral with vanishing electric and magnetic dipole moments. In terms of the Standard Model (SM) of particle physics, neutrinos are the only possibleMajorana fermions.Experiments are underway to search for neutrinoless double-beta decay which, if observed, would confirm the neutrino's self-conjugate nature.<cit.>Moving beyond the SM, theories are rife with Majorana fermions.For example, in supersymmetry, the fermionic partners of the neutral SM gauge and Higgs bosons are manifested as Majorana fermions known as neutralinos.If these particles are the lightest supersymmetric particle, then they are a natural candidate for the dark matter (DM) of the universe.<cit.> Outside of supersymmetry, there are other models of DM particles whose primary interaction is through the anapole moment,<cit.> so their EM properties are germane to understanding, for instance, the relic density of DM. In what follows, we will discuss, within the framework of QFT, the interaction vertex between a general spin-1/2 fermion and a single photon.Then, narrowing our consideration to Majorana fermions, we will compute the scattering amplitude for a (spinless) charged scalar particle on a spin-1/2 Majorana fermion, which proceeds via the exchange of a single (virtual) photon.In the limit of small momentum transfer, the interaction probes the anapole moment of the fermion. At such low energies, this amplitude should be equivalent to the quantum mechanical scattering amplitude for scalar particles passing near an electromagnetic target, described, in this case, by a particular magnetic vector potential. We will find that the vector potential is consistent with a point-like toroidal solenoid.Outside of this point-like solenoid, the magnetic field vanishes, but the vector potential does not.As such, our scattering calculation can be used to explore the Aharonov-Bohm effect, a quantum mechanical phenomenon in which EM potentials impact the trajectory of the particle in a region with vanishing EM fields.We find that our QFT calculation is consistent with the QM scattering for a spinless charged particle on an infinitesimally thin toroidal solenoid.§ PHOTON-FERMION VERTEX In QFT, the single-photon EM interactions of a spin-1/2 fermion are governed by a Lagrangian term consisting of a contraction between the fermion current J^μ_EM =Ψ̅Γ^μΨ and the photon vector field A_μ.By requiring the current to be Lorentz covariant and the overall interaction to be gauge invariant, one can decompose the operator Γ^μ into four distinct terms; a pedagogical treatment of this procedure can be found in Ref. em_formfactors.In the end, the matrix element for the electromagnetic currentcharacterizing the fermion transition from momentum and spin states |𝐩,s⟩ to |𝐩',s'⟩ is given by ⟨𝐩',s' | J_EM^μ | 𝐩,s⟩ =u̅^s'(p')[f_1(q^2) γ^μ + i/2m f_2(q^2)σ^μν q_ν + f_a(q^2) ( q^2 γ^μ -q^μ) γ^5 + f_e(q^2) σ^μν q_νγ^5] u^s(p),where q=p'-p is the four-momentum transferred to the fermion by the photon.<cit.>The only non-trivial scalar in this process is q^2, and thus the electromagnetic form factors are functions of this scalar.In a Feynman diagram, this vertex will be represented with a shaded circle as in Fig. <ref>.In the limit of vanishing momentum transfer, i.e., q^2 → 0, the four form factors encode the static EM properties of the fermion. The electric charge of the particle is given by f_1(0), and the anomalous magnetic dipole moment is related to f_2(0).For the remaining terms, f_a(0) is the particle's anapole moment, and f_e(0) is its electric dipole moment.If a particle is to be its own antiparticle, then the particle must clearly be neutral, but it is not so obvious as to why a Majorana fermion's magnetic and electric dipole moments must vanish.A rigorous field theoretic proof of this fact can be found in Refs. bk82,nieves. These two papers also contain a more intuitive argument based upon the non-relativistic limit of the Hamiltonian which governs the interaction between the fermion and EM field.<cit.>The interaction Hamiltonian can be constructed by contracting the fermion EM current, Eq. (<ref>), with the EM four-vector potential, A^μ.The factors of the photon's four-momentum, q^μ, that appear in the EM fermion current are mapped to derivatives, ∂^μ, of the four-vector potential in position space.Heuristically, wesee that thef_2 and f_e terms in the Hamiltonian will involve the EM field strength tensor F^μν≡∂^μ A^ν - ∂^ν A^μ, whereas the f_a term will involve a derivative of the field tensor which can be related to a four-current via Maxwell's equations, ∂_μ F^μν = J^ν.In the non-relativistic limit, the Hamiltonian takes the form H_int = - μ·𝐁 - 𝐝·𝐄 - 𝐚·𝐉, at leading order.In a local QFT, the Hamiltonian must be invariant under a 𝒞𝒫𝒯 transformation; that is, it must be unchanged when, jointly,particles are changed to antiparticles, spatial axes are flipped, and time is reversed. Each of the electromagnetic moments (magnetic dipole, μ; electric dipole, 𝐝; and anapole, 𝐚) is given by the product of a scalar and the fermion's spin.As such, for a Majorana fermion, these moments pick up a minus sign under 𝒞𝒫𝒯.The electric and magnetic fields are even under 𝒞𝒫𝒯 so that, overall, these magnetic and electric dipole terms in the Hamiltonian pick up an overall minus sign under 𝒞𝒫𝒯.As a result, the dipole moments must vanish.On the other hand, the current density 𝐉 is odd under 𝒞𝒫𝒯 so that the anapole term in the Hamiltonian is indeed unchanged under the transformation.In what follows, we will consider only Majorana fermions, setting f_1, f_2, f_e ≡ 0.§ LOW-ENERGY SCATTERINGAs we mentioned above, real photons do not couple to the anapole moment. This can easily be ascertained from the operator for the anapole moment, f_a(q^2) ( q^2 γ^μ -q^μ) γ^5. Because a real photon is massless, the square of its momentum vanishes, q^2=0, and because it is transverse, the contraction of its polarization and momentum vectors vanishes, ϵ· q=0.With these facts, it is clear that the anapole vertex operator will vanish when coupled to a real photon. As such, the only EM probe of an anapole is the EM current of some other particle that scatters off the fermion.In QFT, the simplest current with which to probe the anapole is that produced by a scalar (spinless) particle of charge e and mass m_ϕ.Using a scalar particlelets us focus on the physics in the calculations without having spin complications in the current. Given a scalar field ϕ, the flow of charge can be described by the vector J^μ = ie/2m_ϕ [ϕ^* ∂^μϕ -(∂^μϕ^*) ϕ ].Supposing the scalar particle is a plane wave with definite momentum k^μ,then ϕ∼ e^-i k · x yields a current J^μ∼e/m_ϕ k^μ.We compute the scattering amplitude for this scalar particle incident upon a Majorana fermion of mass M_ψ and initial momentum p. The outgoing momenta of the scalar and fermion are k' and p', respectively.Scattering is achieved by the exchange of a virtual photon of momentum q≡ p'-p.The fermion only interacts through its anapole moment.The Feynman diagram for this scattering process is depicted in Fig. <ref>. Following the conventions in Ref. peskin, the amplitude for thisprocess isℳ = e f_a (k'_μ + k_μ) u̅^s'(p') [(q^2 γ^μ -q^μ) γ^5/q^2] u^s(p).A remark is in order about the practical aspects of computingscattering amplitudes for processesinvolving Majorana fermions.Majorana fermions are special solutions of the Dirac equation.Using a particular (Majorana) representation of the Dirac matrices, Majorana fermions are manifestly realwith half the number of independent parameters as a Dirac fermion.(This notion of “realness" for a Majorana fermion can be generalized to an arbitrary representation of the Dirac matrices.<cit.>)The Feynman rules appearing in most QFT texts are appropriate for Dirac fermions, but with some modifications, one can adapt these same rules for Majorana fermions as in, for example,Refs. denner1,denner2.Because Majorana fermions are solutions to the Dirac equation, the spinors satisfy u^s(p) = M_ψ u^s(p) and u̅^s'(p') '= M_ψu̅^s'(p'). Armed with these identities, we can simplify the amplitude, Eq. (<ref>), considerably:ℳ = 2e f_a u̅^s'(p') [- 2 ( k· q) /q^2M_ψ]γ^5u^s(p).This expression is entirely general; however, we wish to focus upon the limit in which the four-momentum transfer q is small, so in what follows we will expand our expression in powers of this momentum transfer, keeping only the leading order terms.Additionally, we may as well work within a specific frame - the rest frame of the fermion target.If we assume the scatterer to be at rest, then its four-momentum isp = (M_ψ, 0). The scattered fermion's 3-momentum is that transferred by the photon, 𝐪, so that p' = (E', 𝐪) with E' = √(M_ψ^2 +|𝐪|^2). Denoting the photon's energy as q^0, the energy of the scattered fermion can be written as E' = M_ψ + q^0 = √(M_ψ^2 + | 𝐪|^2).Working with these two expressions for E', we find that, in the limit of small momentum transfer, q^0 ≈|𝐪|^2/2 M_ψ;this is just the non-relativistic kinetic energy of the scattered fermion.With these approximations, we can estimate the explicit kinematical factor that appears in the amplitude, Eq. (<ref>), ask· q/q^2≈𝐤·𝐪/|𝐪|^2-k^0/2M_ψ+ 𝒪(|𝐪|). Now we evaluate the Dirac bilinears.We note that a Dirac bispinor can be written in terms of Pauli matrices and two-component spinors ξ^s as followsu^s(p) = 1/√(2(p^0+M_ψ))([[(p^0+M_ψ)1 -𝐩·σ ] ξ^s; [ (p^0+M_ψ)1 +𝐩·σ]ξ^s ]).With this expression for the bispinor and our small q^2 assumption, we approximate the bilinears u̅^s'(p') γ^5 u(p)≈- ξ^s'^†[𝐪·σ] ξ^s + 𝒪(| 𝐪|^3), u̅^s'(p') γ^0 γ^5 u(p)≈ξ^s'^†[𝐪·σ] ξ^s + 𝒪(| 𝐪|^3), u̅^s'(p') γ^j γ^5 u(p)≈2M_ψξ^s'^†σ^j ξ^s + 𝒪(| 𝐪|^2). With these approximations, we can then determine the leading order contributions to the scattering amplitude for small momentum transfer to beℳ = 8e M_ψ f_a[-𝐤·𝐒 + (𝐤·𝐪)( 𝐪·𝐒)/|𝐪|^2] +𝒪(|𝐪|^2),where, for shorthand, we define the spin matrix element 𝐒 = 1/2ξ^s'^†σξ^s.Modulo some dimensional normalization factors, the scattering amplitude is proportional to the matrix element for the interaction Hamiltonian mediating the transition from initial to final states, H_f i∼ℳ.At low energies, the field theoretic amplitude should correspond to the quantum mechanical scattering amplitude.Examining the approximate amplitude, Eq. (<ref>), the interaction Hamiltonian has the structure of the scalar's current interacting with the magnetic vector potential associated with the Majorana fermion, H_f i∼ - e𝐤·𝐀_a.Thus, the vector potential associated with the anapole moment of the Majorana fermion is𝐀_a (𝐪) ∼ f_a[ 𝐒 - ( 𝐪·𝐒)/|𝐪|^2𝐪],again, modulo some dimensional factors. We note that both terms in Eq. (<ref>) are both zeroth order in |𝐪| in the Taylor expansion about the momentum transfer, despite the 𝐪-dependence of the second term. The above relation for the vector potential, Eq. (<ref>), is expressed in terms of its momentum representation.By taking its inverse Fourier transform, we can determine the expression for the vector field in position space:𝐀_a (𝐫) = 1/(2π)^3 ∫𝐀_a (𝐪) e^i 𝐪·𝐫 d^3 q ∼f_a{ δ^(3) (𝐫) 𝐒 + 1/4π 1/r^3[ 3 (𝐒·𝐫̂) 𝐫̂- 𝐒] } .The first term in Eq. (<ref>) arises from the q-independent term in Eq. (<ref>).This is the vector potential associated with a classical point-like anapole moment,<cit.> with its moment aligning with the particle's spin vector.The impact of this vector potential is confined to the origin.The second term arises from the 𝐪-dependent term in Eq. (<ref>), and it extends through all of space.We note that the structure of this second term is consistent with the large r behavior of the vector potential (in the Coulomb gauge) outside of aclassical toroidal solenoid, again with the solenoid's symmetry axis lying along the direction of the particle's spin.<cit.>To be concrete, we now suppose that the spin of the Majorana fermion is aligned with the z-axis and that no spin flip occurs during scattering, s'=s. Then, we have 𝐒 = S 𝐳̂ with S=1/2.We cantake the curl of the vector potential to determine the associated magnetic field:𝐁 = ∇×𝐀∼ -f_a S ( ∂/∂ rδ^(3) (𝐫)) sinθϕ̂.The second term in Eq. (<ref>) has no curl (nor divergence) and thus yields no contribution to the magnetic field.This is consistent with the notion that the anapole moment of a Majorana fermion can be thought of as a point-like toroidal magnetic field.The magnetic field in Eq. (<ref>) is only nonzero at the origin, but if smeared out, it would be directed azimuthally, also consistent with a torus whose axis lies on the z-axis.The magnetic field outside the torus would vanish, as does the field for the anapole moment.Up to this point, we have assumed that the momentum transfer to the Majorana fermion was small relative to its mass, M_ψ.Let us make some further simplifying assumptions. Supposethat the charged scalar particles are also non-relativistic, i.e., |𝐤| ≪ m_ϕ. Additionally, we will take the target fermion mass to be much greater than the scalar particle's mass, M_ψ≫ m_ϕ.In this limit, the target recoil is negligible, and the scalar particle's kinetic energy is essentially unchanged |𝐤| ≈ |𝐤'|.Finally, we will align the incoming scalar current with the fermion's spin along the z-axis, 𝐤= |𝐤|𝐳̂.With these assumptions, the scattering amplitude is approximated byℳ≈ -4 e f_a M_ψ S |𝐤| (1+ cosθ),where the scattering angle θ is the usual polar angle.From the amplitude we compute the differential cross section dσ/dΩ = α/π f_a^2 S^2 |𝐤|^2 (1+cosθ)^2,with α the fine structure constant.§ APPLICATION TO THE AHARONOV-BOHM EFFECT At low energies, the EM analogue of a Majorana fermion is a point-like toroidal solenoid with a magnetic field confined to the origin but a nonzero vector potential throughout all space.Given this, we can use the previous calculation to explore the impact of the curl-free vector potential upon the scattering amplitude - a variant of the Aharonov-Bohm (AB) effect as applied to toroidal solenoids.In their seminal paper, Aharonov and Bohm showed, theoretically, that in quantum mechanics the electron state can be measurably impacted by a magnetic vector potential even if the magnetic field vanishes along the electron's trajectory.<cit.> One of their proposed experimental tests was to have an electron beam pass on both sides of a solenoid with a confined magnetic field. The phase of an electron wave function changes passing the solenoid, with a net phase change between that part of the wave that passed on one side compared to the part of the wave that passed on the other side given by ± e∮𝐀·d𝐫 for a line-integration path surrounding the solenoid. This effect was confirmed experimentally,<cit.> but some critics, e.g., Ref. roy, claimed that fringing magnetic fields around the finite solenoidwere responsible for the observed phase shifts.To counter this criticism, it was suggested that the effect be explored with a toroidal solenoid;<cit.> with this geometry, the magnetic flux is completely contained within a region inaccessible to the electron.Years later, the toroidal analogue of the original AB experiment yielded results consistent with the AB effect.<cit.> Returning to the last section, we find that the non-relativistic differential cross section, Eq. (<ref>), strongly depends upon the vector potential for r>0 where the magnetic field is zero.The angular dependence of the differential cross sectionis derived wholly from the second term in the vector potential in Eq. (<ref>).If the only contribution to the scattering were due to the nonzero magnetic field, then the amplitude would be ℳ =8 e f_a M_ψ S |𝐤| with, for example, forward scattering being equally as likely as backward scattering.The physical consequences of the vector potential where the magnetic field vanishes is a pure quantum effect.Because our calculation is entirely non-relativistic, we should be able to compare our QFT result with a non-relativistic quantum mechanical calculation of the scattering amplitude of a spinless charged particle incident upon an infinitesimal toroidal solenoid. In the literature, we find several such calculations.<cit.>Focusing on Ref. AB_torus, the authors compute this amplitude, assuming a thin classical toroidal solenoid of radius R carrying a magnetic flux Φ, in the first Born approximation. The authors also assume that the electron current and the solenoid's axis are both aligned with the z-axis, analogous to our above computation. Converting their results to natural units,they find f(θ) =1/4 e|𝐤| R^2 Φ (1+cosθ),with a differential cross section given by σ/Ω = |f(θ)|^2. This result is not expressed in terms of the anapole moment, so to aid comparison, we first compute the anapole moment for a classical ideal torus centered at the origin, with symmetry axis aligned with the z-axis.We choose the current orientation so that the magnetic field is in the ϕ̂ direction.The simplest solenoid to consider is one with a rectangular cross-sectional area.Let the height of the solenoid be h and its interior and exterior radii be R and R+ϵ, respectively.Supposing a current I runs through N turns of the solenoid, the magnetic field in the interior of the solenoid is 𝐁 =μ_0 N I/(2πρ) ϕ̂, where ρ is the distance from the z-axis. Given this field, we compute the flux through the solenoid to be Φ = μ_0 N I/2π hln[1 + ϵ/R] ≈μ_0 N I A/2π R,where we assume a thin torus with ϵ≪ Rand denote the torus's cross-sectional area as A. Adapting the definition of the anapole moment for a line current,<cit.> we find𝐚 =-∫I r^2 d ℓ, = NI [(R+ϵ)^2-R^2] h 𝐳̂,≈2 N I A R 𝐳̂,where dℓ is a current-directed infinitesimal line element.We note that contributions to the anapole moment from the radial current vanish. Given this, we can rewrite the flux in terms of the magnitude of the anapole moment as Φ =a/(4π R^2), where we move to natural units by setting μ_0 = 1.In terms of the classical anapole moment, the quantum mechanical scattering amplitude in Eq. (<ref>) can be written asf(θ) =e |𝐤|a/16π(1+cosθ).Given our choice of conventions, the quantum mechanical amplitude is related to the QFT amplitude in Eq. (<ref>) via f(θ) = ℳ/(8π M_ψ).Comparing the two, we see that both amplitudes will result in the same differential cross section assumingwe identify the Majorana fermion's anapole moment with the classical one via a = 8 f_a S. § CONCLUSION Because a Majorana fermion is self-conjugate, its sole static electromagnetic property is the anapole moment.In single-photon interactions, we have seen that these particles only couple to virtual (but not real) photons.This corresponds to the notion that anapoles only experience interactions due to the currents of other particles, but not to local external electric or magnetic fields.Through low-energy scattering, we see that the quantum anapole behaves like a point-like toroidal solenoid. Assuming the fermion's spin is aligned with the z-axis, the anapole's magnetic field is azimuthal in direction and confined to the origin. Because the anapole's magnetic vector potential is nonzero throughout space, one can use a Majorana fermion to explore the AB effect with toroidal geometry.In our scattering calculations, we have exclusively explored the low-energy limit, assuming small momentum transfer.But, because we work in a field theoretic framework, one could easily explore the behavior of the anapole moment in the relativistic regime and generalize to currents from particles carrying spin. The only word of caution in extending the discussion to higher energies is to note that the anapole moment represents an effective interaction; that is, the coupling between a Majorana fermion and photon is mediated by charged particles in a more fundamental theory. If one (or both) of these charged particles is more massive than the Majorana fermion, then its stability is assured. If the dominant-mass particle is much heavier than the Majorana fermion, then the details of these charged-particle interactions are irrelevant at low energies. But, as the momentum transfer in a scattering process increases, further underlying physics could become appreciable.As an example, suppose we were to compute the real Compton amplitude for a Majorana fermion, which involves the interaction between two real photons (viz., the incident and subsequently scattered photons)and the fermion. We would conclude that the amplitude vanishes because anapoles do not couple to real photons.However, generically, the coupling between Majorana fermions and two photons is nonzero.<cit.> So, at sufficient energies, Majorana fermions can scatter photons. § ACKNOWLEDGMENTKMW was funded, in part, by a McCormick Scholar award from the University of Puget Sound. 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"authors": [
"Kyle M. Whitcomb",
"David C. Latimer"
],
"categories": [
"hep-ph",
"hep-th",
"quant-ph"
],
"primary_category": "hep-ph",
"published": "20170627040411",
"title": "Scattering from a quantum anapole at low energies"
} |
Fakultät für Physik, Munich Quantum Center, and Center for NanoScience (CeNS), Ludwig-Maximilians-Universität München, Geschwister-Scholl-Platz 1, D-80539 München, Germany We report the observation of ubiquitous contamination of polymethylmethacrylate by organic molecules with optical activity in the visible spectral range. Contamination sites of individual solvent-specific fluorophores in thin films of polymethylmethacrylate constitute fluorescence hot-spots with quantum emission statistics and quantum yields approaching 30% at cryogenic temperatures. Our findings not only resolve prevalent puzzles in the assignment of spectral features to various nanoemitters in polymer matrices, they also identify means for simple and cost-efficient realization of single-photon sources in the visible spectral range.Contamination of polymethylmethacrylate by organic quantum emitters Andre Neumann^1,*, Jessica Lindlau^1,*, and Alexander Högele December 30, 2023 ===================================================================Embedding quantum emitters within chemically and electrostatically inert polymer matrices such as polymethylmethacrylate (PMMA) is a common approach to reduce the fluorescence (FL) intermittency encountered by a broad class of photoactive nanoparticles <cit.> and molecular dyes <cit.> under ambient conditions, thus promoting stable and enhanced FL <cit.>. However, contamination of the polymer matrix by fluorescent constituents can result in controversial assignment of spectral features. In some spectroscopy experiments it has proven difficult to distinguish between the FL stemming from quantum emitters, the polymer matrix, or the supporting substrate <cit.>. This is not surprising given the challenge of unambiguous assignment of the FL to its actual source for photoactive systems with low quantum yields, or individual quantum emitters with high quantum yields but inherently low absolute FL intensities.In the visible spectral range, the realm of photoactive nanoemitters includes single molecules <cit.>, fluorescent nanodiamonds <cit.>, colloidal quantum dots <cit.> and nanoplatelets <cit.>, transition metal dichalcogenide quantum dots <cit.>, or perovskite nanoplatelets <cit.>. The range of related potential applications in light emitting, detecting, and harvesting devices is as diverse as the specific details of the photophysics of the underlying emitters. In absolute terms, however, and depending on the radiative lifetime, some of these systems feature low FL intensities despite high quantum yields, while others suffer from reduced quantum yields due to optically inactive lowest-lying dark states <cit.> with strongly inhibited FL at cryogenic temperatures. Irrespective of the actual reason for low intensity, any contamination of the relevant FL by photoemissive substrates or matrices is clearly detrimental to both fundamental studies of nanoemitters and their related applications.In the following, we present a comprehensive study targeting a quantitative analysis of the FL in the visible spectral range arising from a thin film of PMMA on various dielectric substrates. Surprisingly, we find that PMMA films prepared by standard solution-deposition procedures exhibit optical activity in the visible both at room and cryogenic temperatures. However, the FL is not a characteristic feature of the PMMA itself. It rather stems from fluorescent contaminants in the PMMA matrix that we ascribe to solvent residuals with specific FL intensity and spectra. For individual fluorescent contaminants, pronounced and spectrally stable zero-phonon lines (ZPLs) with red-shifted vibronic satellites and highly non-classical emission statistics emerge as a generic feature at cryogenic temperatures. At room temperature, thermal broadening of both the ZPL and the vibrational sidebands gives rise to a characteristic three-peak spectrum that can be mistaken for phonon replica of silica color centers <cit.> or subject to other interpretations <cit.>. The basics of our experiment are illustrated in Fig. <ref>a. We performed FL spectroscopy in a home-built optical microscope to study sample-specific emission in the spectral range of 560- 770 nm excited with a continuous-wave laser at 532 nm, a wavelength frequently used to excite FL in the visible. By raster-scanning the sample with respect to fixed diffraction-limited confocal excitation and collection spots, we acquired maps of FL intensity as in Fig. <ref>b - e with a single photon counting avalanche photodiode (APD), and hyperspectral maps with spectrally dispersed FL as in Fig. <ref>f recorded at each raster-scan pixel for spectral analysis of individual emission hotspots. The studies were complemented by time-correlated FL, second-order FL coherence and FL excitation spectroscopy experiments performed either at room temperature or at the cryogenic temperature of 3.1 K.In the first stage of the experiments we studied the FL characteristics of bare dielectric substrates. It has been argued recently that silica-based substrates host intrinsic fluorescent centers with sizable FL intensity in the visible <cit.>. Therefore, we first investigated the FL from the surface of bare fused silica substrates exposed to different cleaning procedures (see Methods for details on cleaning protocols). Under ambient conditions and 250 μW irradiation in a full-width at half-maximum (FWHM) spot of 0.5 μm we acquired raster-scan FL maps shown in Fig. <ref>b - d. For fused silica sonicated subsequently in acetone and isopropanol according to a common cleaning procedure we observed FL from the entire sample surface with inhomogeneous intensity and an average APD count rate of ∼ 4 kcts/s (Fig. <ref>b). After an additional sonication step in deionized water the level of FL decreased to an average of ∼ 2 kcts/s away from hotspot emission with ∼ 4 kcts/s (Fig. <ref>c). Most remarkably, additional treatment with oxygen plasma suppressed the FL from the silica surface below the dark count rate of the APD (Fig. <ref>d). This set of data, consistently observed for quartz and sapphire substrates subjected to oxygen plasma treatment (see the Supplementary Information for substrate-specific FL maps), clearly establishes the absence of intrinsic FL defects on silica substrates. Moreover, it provides a first hint at the source of the FL as stemming from organic surface contaminants that do not withstand oxygen plasma treatment.For the second experimental stage we prepared substrates free of FL background and covered them by spin-coating with PMMA dissolved in anisole. On a silica substrate with 200 nm of PMMA, we observed the reappearance of fluorescent hotspots with intensities of up to ∼ 6 kcts/s on a background of ∼ 0.5 kcts/s (Fig. <ref>e) under measurement conditions identical to those of Fig. <ref>b - d. Similar results were found for as-deposited and thermally cross-linked PMMA films fabricated from anisole solutions (see Methods for sample details). For most hotspots, the FL was spatially localized to the diffraction-limited spot and characterized by room temperature spectra as in Fig. <ref>f. The spectrum with maximum FL at 2.02 eV (614 nm) can be reproduced with some success by three overlapping Gaussian peaks with FWHM linewidths of 90 meV, equidistant separations of 155 meV, and intensities that reduce with decreasing emission energy (grey solid lines in Fig. <ref>f). An explanation for the mismatch between this simplistic model fit and the actual spectrum pending, we point out its striking similarity to the spectra ascribed earlier to various sources <cit.>. Moreover, it exhibits a remarkable correspondence with the spectra of individual dyes in PMMA <cit.>, providing a second hint to hydrocarbon molecules as a source for misinterpretation and establishing a link to the visionary association made between the spectra of non-blinking colloidal quantum dots <cit.> and organic dyes <cit.>.To elucidate the correspondence between the FL hotspots found at room temperature in thin films of PMMA and the spectral signatures of organic molecules we carried out spectroscopy studies at the cryogenic temperature of 3.1 K. Fig. <ref>a and b show representative cryogenic FL maps of PMMA films on a fused silica substrate and a perforated silicon nitride membrane, respectively. Both maps were acquired in the hyperspectral mode by recording spectrally dispersed FL with a nitrogen-cooled CCD and color-coding its maximum intensity at each raster-scan pixel. Note the conceptual difference to the raster-scan maps recorded with APDs: hyperspectral mapping emphasizes emitters with sharp FL peaks over spectrally broad FL background. Again, we found spatially localized emission from diffraction-limited hotspots (inset of Fig. <ref>a) analogous to our room temperature experiments. A few hotspots in Fig. <ref>b (with up to 120 cts/s) clearly stem from PMMA regions suspended over holes which can be unambiguously distinguished from the silicon nitride membrane by the respective FL background (grey and blue areas of the map correspond to intensities of 10 and 50 cts/s, respectively). This observation confirms once more that the PMMA film rather than the substrate is the actual host of FL hotspots.A characteristic cryogenic FL spectrum of a hotspot in PMMA is shown in Fig. <ref>c. It features a narrow and intense peak, which we label as ZPL, accompanied by weak red-shifted satellites. More than 60% of localized emission sites exhibited similar spectral characteristics at low temperature. Within this group of emitters with spectrometer-limited ZPLs, 94% of hotspots constitute the class of emitters with a ZPL centered around 2.05 eV emission energy (605 nm emission wavelength). The corresponding normal distribution of the ZPL energy is shown in Fig. <ref>d, where the blue solid line is a Gaussian fit to the histogram with a FWHM of 130 meV. The remaining 6% of the single-site emitters with intense FL were characterized by two sharp ZPLs (see Supplementary Information for the corresponding normal distribution of emission energies) accompanied by red-shifted sidebands.All spectra were remarkably stable over time without significant FL intermittence during the course of observation of 15 h (Fig. <ref>e) and beyond. Throughout the temporal evolution, the ZPL remained spectrometer-limited to one pixel of the CCD corresponding to an upper bound on the FWHM linewidth of 200 μeV. A high-resolution spectrum recorded with a scanning Fabry-Pérot etalon suggests that spectral wandering broadens the ZPL on sub-minutes timescale to an inhomogeneous peak with a FWHM of 3.18±0.13 GHz (Fig. <ref>f). These spectral signatures find their correspondence in the studies of hydrocarbon fluorophores embedded in a polymer host matrix <cit.>. Within this framework, low-temperature FL of single molecules is characterized by a spectrally narrow ZPL associated with the principal electronic transition <cit.> and sidebands stemming from Franck-Condon transitions between vibronically dressed molecular electronic states <cit.>. Stabilized in PMMA, single molecules exhibit FL with low intermittency and ZPLs limited by spectral diffusion to ∼ 1 GHz <cit.>. The red-shifted satellites of the ZPL are equally well pronounced in vibronic spectroscopy <cit.> of molecules with characteristic vibrational degrees of hydrocarbon complexes.The set of data in Fig. <ref> further substantiates the correspondence. With polarization-resolved measurements shown in Fig. <ref>a we confirmed the dipolar character associated with the molecular transition of the ZPL <cit.>. The orientation of the absorption and emission axes measured with linearly polarized excitation and detection, respectively, were determined as collinear within our experimental precision. Furthermore, time-correlated measurements of Fig. <ref>b revealed the characteristic FL decay dynamics of molecules on nanoseconds timescale <cit.>. The single-exponential lifetimes of 3.8 and 3.6 ns for the ZPL within a spectral window of 60 meV and the total FL intensity, respectively, were the same within the temporal resolution of ∼ 0.3 ns in our experiments, identifying red-shifted sidebands as vibronic ZPL replicas. Finally, single photon emission statistics as a hallmark of single-molecule FL <cit.> are presented in Fig. <ref>c and d. With photon correlation spectroscopy we observed pronounced photon antibunching in the normalized second-order coherence function g^(2)(τ) at zero time delay for both the FL within a band-pass interval of 60 meV around the ZPL (with g^(2)(0) =0.24 ± 0.05 in Fig. <ref>c) and the full FL spectrum without spectral filtering (g^(2)(0)=0.32 ± 0.04 in Fig. <ref>d). Thus, within the uncertainty of our measurement, we can rule out simultaneous photon emission into the ZPL and the sideband spectrum associated with the vibronic ZPL satellites.Having identified the fluorescent hotspots in PMMA as single fluorescent molecules, we utilized vibrationally resolved FL spectroscopy <cit.> to shed light on their molecular nature. Fig. <ref>a shows a spectrum of a hotspot that is representative for fluorescent contaminants in PMMA prepared with anisole as solvent. A series of low-frequency vibrational modes contributes to the sidebands below 80 meV (645cm^-1), followed by a group of replicas around 173 meV (1395cm^-1) and a weaker satellite group around 346 meV (2790cm^-1). The latter is in fact a second harmonic of the preceding group as confirmed by correlation analysis between all individual peaks of the two groups upon a spectral shift by 173 meV. All main vibrational features in emission have their broadened counterpart resonances in absorption, as demonstrated by the FL excitation spectra in Fig. <ref>b recorded for two typical emitters with different ZPL energies as a function of laser energy detuning at constant excitation power. For both quantum emitters of Fig. <ref>b, the absorption is enhanced whenever the laser detuning with respect to the ZPL matches the energy of the vibronic sidebands (the dashed lines in Fig. <ref> emphasize the correspondence between the resonances in emission and absorption).The vibrationally resolved spectrum of Fig. <ref>b is typical for fluorescent molecules in PMMA films from anisole-based solutions. It exhibits a striking similarity with the cryogenic FL of anthracene characterized by a ZPL in the ultraviolet (around 3 eV) and a pronounced vibronic satellite group around 1400 cm^-1 redshifts <cit.>. The according vibrational degrees of freedom are related to the intramolecular stretching of adjacent carbon bonds in polycyclic aromatic hydrocarbons <cit.>. The observation of the ZPL emission in the visible (around 2 eV) from anisole-based PMMA suggests that the optical activity of solvent-related contaminants in such films stems from acene chains such as pentacene, or from anthracene-derived dyes such as alizarin.We applied vibrational FL spectroscopy to hotspots in PMMA films derived from other solvents (see Methods for sample preparation details). As highlighted by the raster-scan maps of Fig. <ref>, the areal density and the FL intensity of hotspots in PMMA films formed with chlorobenzene (Fig. <ref>a, b), methyl isobutyl ketone (Fig. <ref>c, d) and toluene (Fig. <ref>g, h) were similar to anisole-based PMMA characteristics (Fig. <ref>e and Fig. <ref>a). The vibronic signatures, however, showed significant differences. Fig. <ref>c, f, and i show normalized average spectra of 25 brightest fluorophore contaminants in PMMA films prepared with different solvents (see the Supplementary Information for the corresponding average spectrum of anisole-based PMMA). The spectrum of a typical hotspot in chlorobenzene-based PMMA features a low-frequency vibronic band around 250 cm^-1 and a pronounced high-frequency band around 1400 cm^-1 discussed earlier (as indicated by the diamond and the hexagon Fig. <ref>c). In contrast, the vibronic FL characteristics of hotspots in PMMA films formed with methyl isobutyl ketone and toluene solutions (Fig. <ref>f and i, respectively) exhibit additional vibrational signatures at 548 and 757 cm^-1. The vibronic modes, labelled with diamonds and upper and lower triangles in Fig. <ref>c, f, and i are characteristic of rylene dyes composed of naphthalenes. While the low-frequency mode (diamonds in Fig. <ref>c, f, and i) is close to that of the long axis stretch of a terrylene molecule, the higher-frequency modes (upper and lower triangles in Fig. <ref>f and i) are consistent with the short axis stretch and ring deformation of outer naphthalenes, respectively <cit.>. Note that naphthalene-related bands of rylene dyes are only very weakly expressed in the averaged vibronic FL spectra observed in anisole- and chlorobenzene-based PMMA films (Fig. <ref>a and Fig. <ref>c).In addition to solvent-specific differences in the spectra of fluorescent hotspots in PMMA, vibrationally resolved FL spectroscopy identifies the normal modes of aromatic hydrocarbons around 170 meV (1400 cm^-1) as a generic feature of FL contaminants at low temperatures. At elevated temperatures, these modes develop into broad vibronic satellites (see the Supplementary Information for FL spectra at different temperatures) that accompany the FL from the thermally broadened principal molecular transition. With this in mind, the interpretation of the three-peak structure of the room-temperature FL spectrum in Fig. <ref>f as arising from an organic fluorophore is straight forward. For an adequate modelling, however, the contributions of all other vibrational modes must be taken into account. The main corrections to the inhomogeneous spectral profile of the ZPL and the vibronic modes of polycyclic hydrocarbons will naturally appear on the low-energy side of the peaks, where the fit with three Gaussians most significantly deviates from the actual spectrum.In concise terms, our comprehensive study of fluorescent spots, ubiquitously present in PMMA films and on contaminated dielectric substrates, leads to the conclusion that organic fluorophores are the actual source of misinterpreted FL signatures. We estimate the quantum yield of such organic quantum emitters to range from ∼ 5% at room temperature up to 30% at 3.1 K (see the Supplementary Information for the estimate of the quantum yield). These values are not remarkably high, however, the corresponding FL intensity can be significant in studies of photoactive systems with reduced quantum yields in cryogenic or ambient environments. In fact, we found the FL intensity of PMMA hotspots to be roughly a third of the emission intensity of individual terrylenediimide (TDI) molecules at cryogenic temperatures, and in many instances even more intense than commercial radiant dyes at ambient conditions. Given the present technological limitations to solvent purity, it seems unlikely that contamination of PMMA and other polymer matrices can be completely avoided in future experiments. On the other hand, the abundance of stable quantum emitters in polymer films could facilitate a range of fundamental studies and technological developments relying on simple and cheap sources of non-classical light.Methods: All samples were prepared in a clean-room environment. Unless stated otherwise, substrates were cleaned by initial sonication in acetone (Technic, acetone Micropur VLSI) for 5 min, followed by isopropanol (Technic, propan-2-ol Micropur VLSI) for 5 min, and finally exposed for 1 min to oxygen plasma. Polymer covered samples were prepared by spin-coating ∼ 10 μl of PMMA onto oxygen plasma-treated fused silica (CrysTec) and other dielectric substrates (quartz and sapphire). An ellipsometer was used to adjust the spin-coating parameters for a film thickness of 200 nm. The films were obtained from commercial PMMA formulated in anisole with a molecular weight of 950K (MicroChem, 950PMMA A4 resist for electron-beam lithography). The spin-coated PMMA film was left to dry at ambient conditions. Optionally, the samples were baked at 180 ^∘C for 90 s on a hot plate. The perforated silicon nitride membrane (PELCO) of Fig. <ref>b was drop-casted and baked to ensure mechanical stability of freely suspended PMMA. Control experiments were carried out with 4% of 450K PMMA resin (DuPont, Elvacite 2041) diluted in 96% of chlorobenzene (Merck, 801791), methyl isobutyl ketone (Technic, MIBK Micropur VLSI), or toluene (Sigma-Aldrich, 179418).FL imaging and spectroscopy measurements were performed with a home-built confocal microscope coupled to single-mode fibers. Room-temperature experiments were conducted with an apochromatic objective with numerical aperture (NA) of 0.82 (attocube systems, LT-APO/VISIR/0.82) and an oil immersion objective (Olympus, UPLFLN 100XOI2) with NA of 1.30 for the data in Fig. <ref>f. Cryogenic experiments were carried out in a helium bath cryostat or a low-vibration closed-cycle cryostat (attocube systems, attoDRY1000) with base temperatures of 4.2 K and 3.1 K, respectively, using a low-temperature apochromatic objective with NA of 0.65 (attocube systems, LT-APO/VIS/0.65).Continuous wave excitation with a solid-state laser at 532 nm (CNI, MLL-III-532-50-1) was used in all experiments except for the measurements of data in Fig. <ref>b and Fig. <ref>b. All FL maps were recorded with circularly polarized excitation except for Fig. <ref>b and Fig. S1, where linearly polarized excitation was used. Time-resolved FL data in Fig. <ref>b were measured with ps-excitation at 532 nm. The FL excitation experiments of Fig. <ref>b were performed with an optical parametric oscillator (Coherent, Mira-OPO with a FWHM spectral bandwidth of 0.5 nm) or a spectrally filtered supercontinuum laser (NKT Photonics, SuperK EXW-12 with a FWHM spectral bandwidth of 5.5 nm). Single photon counting avalanche photodiodes (PicoQuant, τ-SPADs with dark count rates of 35 cts/s and a temporal resolution of 320 ps) or a monochromator equipped with a liquid nitrogen cooled CCD (PI, Acton SP-2558 and Spec-10:100BR/LN with a spectral resolution of 200 μeV and a gain setting of 4 e^-/count) were used for detection. The hyperspectral raster-scan maps in Fig. <ref>a and b were recorded in the spectral range of 1.68 - 2.20 eV. The data in Fig. <ref>f were measured with a home-built monolithic scanning Fabry-Pérot etalon with a spectral resolution of 150 MHz and a scanning rate of 5.5 MHz/s. Conflict of interest: The authors declare no competing financial interest.Acknowledgment. We thank T. Basché, C. Bräuchle, I. Gerhard, S. Götzinger, K. Karrai, J. P. Kotthaus, E. Lifshitz, J. Lupton, G. I. Maikov, M. Pilo-Pais, I. Pugliesi, K. Puschkarsky, E. Riedle, J. Tilchin and S. E. Beavan for helpful discussions and useful input at various stages of the project, P. Maletinsky and S. Thoms for providing samples with NV centers in diamond and TDI molecules in PMMA, respectively, and P. Altpeter and R. Rath for assistance in the clean-room. This work was funded by the Volkswagen Foundation, the German-Israeli Foundation for Scientific Research and Development (GIF), and the Deutsche Forschungsgemeinschaft (DFG) Cluster of Excellence Nanosystems Initiative Munich (NIM) with financial support from LMUinnovativ and the Center for NanoScience (CeNS). A. H. acknowledges funding by the European Research Council under the ERC grant agreement no. 336749.Author Contributions: ^⋆these authors contributed equally to this work.Corresponding author: [email protected] Information provides details on sample fabrication and experiments.41 natexlab#1#1bibnamefont#1#1bibfnamefont#1#1citenamefont#1#1url<#>1urlprefixURL[Nirmal et al.(1996)Nirmal, Dabbousi, Bawendi, Macklin, Trautman, Harris, and Brus]Nirmal1996 authorM. Nirmal, authorB. O. Dabbousi, authorM. G. 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Polymer covered samples were prepared by spin-coating ∼ 10 μl of PMMA onto oxygen plasma-treated fused silica (CrysTec) and other dielectric substrates (quartz and sapphire). An ellipsometer was used to adjust the spin-coating parameters for a film thickness of 200 nm. The films were obtained from commercial anisole based PMMA with a molecular weight of 950K (MicroChem, 950PMMA A4 resist for electron-beam lithography). The spin-coated PMMA film was left to dry at ambient conditions. Optionally, the samples were baked at 180 ^∘C for 90 s on a hot plate. The perforated silicon nitride membrane (PELCO) of Fig. 1b from the main text was drop-casted and baked to ensure mechanical stability of freely suspended PMMA. Control experiments were carried out with 4% of 450K PMMA resin (DuPont, Elvacite 2041) diluted in 96% of chlorobenzene (Merck, 801791), methyl isobutyl ketone (Technic, MIBK Micropur VLSI), or toluene (Sigma-Aldrich, 179418). §.§ Experimental setupA home-built confocal microscope coupled to single-mode fibers was used for FL imaging and spectroscopy. Room-temperature experiments were performed with an apochromatic objective with numerical aperture (NA) of 0.82 (attocube systems, LT-APO/VISIR/0.82) and an oil immersion objective (Olympus, UPLFLN 100XOI2) with NA of 1.30. Cryogenic experiments were carried out in a helium dewar or a low-vibration closed-cycle cryostat (attocube systems, attoDRY1000) with base temperatures of 4.2 K and 3.1 K, respectively, using a low-temperature apochromatic objective with NA of 0.65 (attocube systems, LT-APO/VIS/0.65). The FL was excited with a continuous wave solid-state laser at 532 nm (CNI, MLL-III-532-50-1), an optical parametric oscillator (Coherent, Mira-OPO with a FWHM spectral bandwidth of 0.5 nm), or a spectrally filtered supercontinuum laser (NKT Photonics, SuperK EXW-12 with a FWHM spectral bandwidth of 5.5 nm). Single photon counting avalanche photodiodes (PicoQuant, τ-SPADs with dark count rates of 35 cts/s and a temporal resolution of 320 ps) or a monochromator equipped with a liquid nitrogen cooled CCD (PI, Acton SP-2558 and Spec-10:100BR/LN with a spectral resolution of 200 μeV and a gain setting of 4 e^-/count) were used for detection. A home-built monolithic scanning Fabry-Pérot etalon with a spectral resolution of 150 MHz was used for high-resolution spectroscopy. §.§ Optical characterization of substratesThree dielectric substrates were studied in hyperspectral raster-scan FL:fused silica (Crystec), quartz (Crystec, z-cut, 0001 orientation) and sapphire (MaTecK, z-cut, 0001 orientation). Prior to cryogenic measurements the substrates were exposed to oxygen plasma. Cryogenic raster-maps of FL maxima are shown in Fig. <ref>a, b, and c for fused silica, quartz, and sapphire, respectively. The FL maximum level was identical for fused silica and quartz (Fig. <ref>a and b) with count rates given by the readout noise of the liquid nitrogen cooled CCD. The sapphire substrate exhibited spatially homogeneous FL intensity stemming from a sharp peak at ∼ 694 nm (1.787 eV) of the R-line of Cr^3+ ions in Al_2O_3.§.§ Optical characterization of PMMA prepared with different solventsFig. <ref> compares cryogenic FL characteristics of PMMA films with 200 nm thickness, prepared by spin-coating mixtures of PMMA in different solvents onto oxygen plasma-treated fused silica substrates. Four different solvents were used: anisole (), chlorobenzene (), methyl isobutyl ketone (), and toluene (). §.§ Spectral characteristics of quantum emitters in anisole-based PMMA filmsRepresentative spectra of most common hotspot quantum emitters in PMMA films prepared with anisole solvent are shown in Fig. <ref>. More than 60% of localized emission sites exhibited similar spectral characteristics at low temperature. Within this group of emitters with spectrometer-limited ZPLs, 94% of hotspots constituted the class of emitters with one sharp and intense zero-phonon line (ZPL) and red-shifted vibronic replicas as in Fig. <ref>a. The corresponding normal distribution of the ZPL energy is shown in Fig. <ref>b. In contrast to such emitters that we label here as type 1, 6% of type 2 fluorescent hotspots exhibited two intense and sharp peaks as in Fig. <ref>c with two intense satellites and a much narrower spread in the energy of the blue-most peak (histogram in Fig. <ref>d).The evolution of a typical type 1 hotspot spectrum with temperature in(anisole-based PMMA) is shown in Fig. <ref>. The ZPL and the vibronic satellites broadened upon heating from 4 to 41 K (Fig. <ref>b), and the overall FL decreased gradually for temperatures above ∼ 25 K (Fig. <ref>c).Both the initial intensity and the spectral lineshape were recovered upon successive cooling back to 4 K. The trend of thermal broadening as in Fig. <ref>b eventually results in significant spectral overlap of the ZPL and vibronic satellites at room temperature with predominant contributions from the vibronic group around ∼ 155 meV and its second harmonic around ∼ 310 meV red-shifts (Fig. <ref>a). §.§ Quantum yield estimate The FL quantum yield of an emitter is given by the ratio of emitted photons to absorbed photons per unit time. In our experiments we estimate the quantum yield of single type 1 fluorescent hotspots in PMMA films formed with anisole-based solutions by scaling the FL intensity to a photostable emitter with known optical properties. We used single nitrogen-vacancy (NV) color centers in bulk diamond that exhibit a quantum yield of Φ_NV≃ 70% <cit.> and a dipole averaged absorption cross-section ofσ_NV≃ 3.1 · 10^-17 cm^2 <cit.> for 532 nm excitation. With these quantities, the conversion cross-section of a fluorescent hotspot, given by the product of the corresponding quantum yield Φ_FH and absorption cross-section σ_FH, is determined as: Φ_FH σ_FH = I_FH/I_NV·τ_FH/τ_NV·Ω_NV/Ω_FH·Φ_NV σ_NV. Here, the term I_FH / I_NV is the FL intensity of a hotspot scaled to the FL of a single NV center for the same excitation power in the linear response regime of both emitters. This ratio ranged from 1.9 to 2.4 for room temperature measurements and peaked at ∼ 13.2 for cryogenic temperatures of 3.1 K and 4.2 K. The emitters were excited with continuous wave excitation at 532 nm and circular polarization to ensure averaging over the possible orientations of the transition dipole moments. The factor τ_FH / τ_NV accounts for the different FL lifetimes of the fluorescent hotspots and the NV centers with τ_FH≃ 3.6 ns and τ_NV≃ 12.9 ns determined experimentally. Finally, we also account for the difference in the effective collection solid angles for fluorescent dipoles embedded in different dielectric environments (PMMA and diamond) with respective refractive indices via Ω_NV / Ω_FH which was close to ≃ 0.35 in our experiments.With these values the conversion cross-section of a typical type 1 fluorescent hotspot excited at 532 nm was in the order of ∼ 5 · 10^-18 cm^2 at room temperature and increased to ∼ 3 · 10^-17 cm^2 at cryogenic temperatures. Using σ_FH≃ 1 · 10^-16 cm^2 as a typical absorption cross-section of the second absorption band of common polycyclic hydrocarbon compounds <cit.>, we obtain an estimate for the fluorescence quantum yield Φ_FH of ∼ 5 % at room temperature and up to 30 % at cryogenic temperatures. We obtained similar values, both at room and cryogenic temperatures, from scaling the quantum yield of fluorescent hotspots in anisole-based samples to fluorescence characteristics of individual TDI molecules. 3 natexlab#1#1bibnamefont#1#1bibfnamefont#1#1citenamefont#1#1url<#>1urlprefixURL[Jelezko and Wrachtrup(2006)]Jelezko2006 authorF. Jelezko and authorJ. Wrachtrup, journalPhys. Status Solidi A volume203, pages3207 (year2006).[Wee et al.(2007)Wee, Tzeng, Han, Chang, Fann, Hsu, Chen, and Yu]Wee2007 authorT.-L. Wee, authorY.-K. Tzeng, authorC.-C. Han, authorH.-C. Chang, authorW. Fann, authorJ.-H. Hsu, authorK.-M. Chen, and authorY.-C. Yu, journalJ. Phys. Chem. A volume111, pages9379 (year2007).[Berlman(1971)]Berlman1971 authorI. B. Berlman, titleHandbook of Fluorescence Spectra of Aromatic Molecules (publisherAcademic Press, year1971), edition2nd ed. | http://arxiv.org/abs/1706.08341v1 | {
"authors": [
"Andre Neumann",
"Jessica Lindlau",
"Alexander Högele"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170626123759",
"title": "Contamination of polymethylmethacrylate by organic quantum emitters"
} |
⇀↽1pt1pt Department of Physics, Clark University, Worcester, MA [email protected] of Molecular Biology, Princeton University, Princeton, NJ [email protected] of Physics, Clark University, Worcester, MA 01610We introduce a minimal model for the evolution of functional protein-interaction networks using a sequence-based mutational algorithm, and apply the model to study neutral drift in networks that yield oscillatory dynamics. Starting with a functional core module, random evolutionary drift increases network complexity even in the absence of specific selective pressures. Surprisingly, we uncover a hidden order in sequence space that gives rise to long-term evolutionary memory, implying strong constraints on network evolution due to the topology of accessible sequence space. 82.39.Fk, 82.39.Rt, 87.18.Cf, 87.18.VfHidden long evolutionary memory in a model biochemical networkRanjan Mukhopadhyay December 30, 2023 ================================================================Within even the simplest living cells there is a highly complex web of interacting molecules, with biological function typically emerging from the actions of a large number of differentfactors <cit.>. What is the relationship between the architecture of such interaction networks and the underlying processes of evolution?Much of the theory related to evolution focuses on the evolution of individual phenotypic traits or on population dynamics (see, for example, <cit.>); however, in general, individual genes do not determine individual traits. Rather, many traits arise from the dynamics of interacting components. With this in mind, we formulated and analyzed a minimal physically-based protein-protein interaction model that allows us to map from sequence space to interactions and, consequently, to network dynamics and fitness. Surprisingly, the model reveals a long-term memory of network origins hidden in the space of sequences.Recently, bottom-up approaches to molecular evolution, typically in the context of the folding properties/thermodynamics of individual proteins or RNAs <cit.> have led to new insights into evolutionary outcomes, for example regarding a power-law distribution of protein family sizes.Here we generalize such bottom-up studies to functional networks. We focus on oscillatory networks of interacting enzymes, both due to the relevance of biological oscillators (e.g. cell cycle, circadian rhythms) <cit.> and due to the simplicity of defining function and fitness.As such a network evolves, are the original nodes still both necessary and sufficient or does the network redistribute function over new nodes? If new nodes do become essential, is there stillmemory of the original network?In order to address these questions, we develop a model of protein-protein interaction networks consisting of two classes of enzymes, activators (e.g. kinases) and deactivators (e.g. phosphatases). Each of these can be in either an active state or an inactive state and only function when in the active state. To model cooperativity, we assume that activation or deactivation of a target (either an activator or a deactivator) requires h independent binding/modification events, with partially modified intermediates being short lived. The resulting chemical kinetic processes arehA^*_i+ T_l h A^*_i+ T^*_l ,hD^*_j+ T^*_l h D^*_j+ T _l ,where A/A^*, D/D^*, and T/T^* denote activator, deactivator, and target in inactive/active states respectively. The corresponding chemical kinetic equation can be approximated as (see Supplementary Material (SM) <cit.>, section I for details)d[ T^*_l]/dt = ∑^m_i=1k_il[ A^*_i]^h[ T_l] - ∑^n_j=1k̃_jl[ D^*_j]^h[ T^*_l]+α[ T_l]- α^'[ T^*_l] , where m and n are the number of distinct types of activators and deactivators respectively. In Eq. <ref>, α and α^' are background activation and deactivation rates. We further assume that the total concentration of each species is constant, such that T_l = c_0 -T^*_l.Protein-protein interaction strengths are generally determined by amino-acid-residue interactions at specific molecular interfaces. Moreover, it has been estimated that > 90% of protein interaction interfaces are planar with the dominant contribution coming from hydrophobic interactions <cit.>. For simplicity, we therefore assume each protein possesses a pair of interaction interfaces, an in-face and an out-face, and we associate a binary sequence, σ⃗_ in/out, of hydrophobic residues (1s) and hydrophilic residues (0s) to each interface. The interaction strength between an enzyme (denoted by index i) and its target (denoted by index l) is determined by the interaction energy E_il= ϵσ⃗_ out^(i)·σ⃗_ in^(l) between the out-face of the enzyme and in-face of its target. (All energies are expressed in units of the thermal energy k_ B T.) The effective reaction rate is then given by k_il = k_0 (1+ exp [-(E_il-E_0)])^-h , where E_0 plays the role of a threshold energy, e.g. accounting for the loss of entropy due to binding. The background activation and deactivation rates are set equal and define the unit of time via α =α' = 1. In our simulations we set k_0 = 10^4, ϵ = 0.2, cooperativity h=2, E_0= 5, c_0 = 1, and we take the length of each sequence representing an interface to be N=25. These interaction parameters were chosen to provide a large range for the rate constants k_il as a function of sequence and to keep the background rates small compared to the highest enzymatic rates; cooperativity was introduced to allow oscillations in relatively simple biomolecular networks. For our evolutionary scheme, we assume a population sufficiently small that each new mutation is either fixed or entirely lost <cit.>. We consider only point mutations – namely replacing a randomly chosen hydrophobic residue (1) in the in- or out-face of one enzyme by a hydrophilic residue (0), or vice versa. In this study, mutations are accepted if and only if they satisfy the selection criterion that the network remains oscillatory and moreover that the network exhibits oscillatory dynamics independent of the choice of initial concentrations of the active fractions (global oscillators). For this purpose we identified the fixed points of the chemical dynamics and carried out linear stability analysis (SM <cit.>, section II). In order to address the question of network drift – how function redistributes over the nodes in an evolving network – we start with a 2-component oscillator (one activator and one deactivator) and add a second activator with all 0s for the sequences representingin- and out-interfaces (so that initially Activator 2 has minimal interaction with the other two components). We then let the system evolve, accepting only mutations corresponding to global oscillators.To characterize network drift, we studied the time evolution of the essentiality of each activator for a random sample of starting sequences that corresponded to oscillators, as depicted in Fig. 1A, where we characterize a component as being “essential" if thesystem stops oscillating when the component is removed <cit.>. In Fig. 1B we exhibit the distribution of the number of accepted mutational steps before the second activator become essential fortwo distinct starting sequences. While the two distributions peak at very different values for the number of mutational steps, the interaction strengths for the two initial states do not differ appreciably (Fig. 1B, inset), highlighting the importance of the underlying sequence in governing evolutionary dynamics. Returning to Fig. 1A, we find relatively rapid flips between states where both activators are essential to states where only one of the activators is essential. Surprisingly, we also note the prevalence of much longer time periods where Activator 1 is always essential or where Activator 2 is always essential. This is true independent of initial conditions. These long evolutionary periods presumably reflect the division of sequence space into two regions or “phases": Phase 1 where Activator 1 is always essential and Phase 2 where Activator 2 is always essential. The system starts in Phase 1 (Activator 2 is inessential), then when Activator 1 first become inessential we infer that the system has entered Phase 2, and so on. Can these two phases be distinguished in terms of measurable dynamical quantities or rate constants? Since the two phases presumably relate to an asymmetry in the roles of the two activators, we quantify this asymmetry via the relative peak-to-valley ratio (PVR) of the oscillations of their active fractions, where relative PVR is ((PVR A_1 - PVR A_2)/(PVR A_1 + PVR A_2)). From Fig. 2A (top panel) and Fig. 2B, we see that relative PVR correlates with the phase, and we display the distribution quantifying this correlation. A corollary is that the probability that an activator is essential also correlates with the relative PVR (Fig. 2C), so that if an activator has a relatively larger PVR it is also more likely to be essential. Moreover, we find that the phase-shift between peaks in the active fractions of the two activators also correlates with the phase (Fig. 2D), so that Activator 1 typically leads in Phase 1 and Activator 2 in Phase 2. Finally in order to determine how these observations relate to the underlying rate constants, we constructed the covariance matrix for the covariation of the 9 rate constantsk_ij and carried out a principal component analysis (SM <cit.>, section IV). We find that the projected component of the rates onto the eigenvector with the largest eigenvalue (PC1 = 94.93%) strongly correlates with the phase (Fig. 2A, lowest panel, and Fig. 2E); we find no such correlation for projections onto any of the remaining eigenvectors. On examining the top eigenvector, we find that it primarily consists of a linearsuperposition of the difference in auto-activation rates of the two activators and the difference in their deactivation rates. This suggests that strong auto-activation coupled with strong deactivation produces an activator that peaks first during each oscillation cycle and also has a large PVR (SM <cit.>, section VIII). However, the co-occurrence of these features does not by itself explain the observed long intervals of two distinct phases.What is the origin of the long-term memory? We first quantify the duration of long-term network memory by constructing a histogram of the number of mutational steps that the system spends in each phase before flipping. As shown in Fig. 3A, we find an approximately exponential distribution, P(τ) ∝ e^- τ/τ_0, where τ_0≃ 3200 ± 48 mutational steps. An exponential distribution implies a fixed, history-independent rate of flipping between the two phases, which in turn suggests that flipping corresponds to barrier crossing. Since our model treats all oscillatory states as equally fit, the only barriers are entropic, i.e. there must be relatively speaking very few boundary points connecting phases (SM <cit.>, section V). To check this hypothesis, we studied the neighborhood of states in Phase 1 and Phase 2. In Phase 1, for example, we distinguished between states where only Activator 1 is essential and states where both are essential. For states where only Activator 1 is essential we found no examples of sequences that were Hamming distance 1 away (that is, separated by a single point mutation) for which Activator 1 stops being essential. Of the states in Phase 1 where both activators are essential, for only 3 % of states the Hamming distance 1 neighborhood contained one or more states where Activator 1 was inessential. The relative rarity of such states (which can be considered as boundary states) is consistent with our hypothesis that in sequence space the two phases touch at a relatively small number of boundary points.Interestingly, in contrast to flipping between phases, the distribution of the number of mutational steps that an activator remains essential exhibits a power-law distribution for short times, as depicted in Fig. 3B. For Activator 1, for example, this power-law part of the distribution is dominated by cases where the system is in Phase 2, with Activator 1 switching between being essential and inessential. Thus the power-law distribution is related to the presence of domains within Phase 2 where Activator 1 is also essential (and likewise for Activator 2 in Phase 1). For longer times, the periods of essentiality correspond to the duration of phases, and thus the distribution decays exponentially (Fig. 4B, inset). In contrast to exponential decay, a power-law distribution implies a history-dependent switching rate, with the escape rate from a domain proportional (on average) to the inverse of the time elapsed since the system entered the domain (SM <cit.>, section IX).It is nota priori obvious how the above observations of two phases generalize to more complex networks. We therefore extended our study by starting with a 3-component oscillator and adding a fourth component (Activator 3) with all its sequences initially set to 0s. Once again we find that Activator 3 becomes essential relatively rapidly (typically in ∼100 mutational steps). If we continue to follow the evolution of essentiality for the activators, we find for each activator long periods (∼1000+ mutational steps) where that activator remains essential, separated by similarly long periods where that activator is intermittently essential/inessential (Fig. 4A). This suggests that for each activator, the sequence space of oscillators divides into two regions: one region where that activator is essential at every point and a second region consisting of smaller domains where the activator is essential interspersed with domains where it is inessential. Note that time periods where one activator remains essential sometimes overlap with periods where one of the other activators remains essential, implying that the region where one activator is essential at every point has some overlap with the regions where other activators are essential at every point. This contrasts somewhat with the 3-component system where Phase 1, the region in which Activator 1 is essential at every point, is complementary to Phase 2. By contrast, as shown in Fig. 4B, the distribution of mutational steps over which any one of the activators is essential for the 4-component system is quite similar to that of the 3-component system, being power-law at short times with a similar exponent, and exponential for longer times, albeit with a shorter decay time τ_0≃ 1750± 54 mutational steps. As for 3-component systems, we also find strong correlation between normalized/relative PVR of oscillation, phase-shift, and essentiality for pairs of activators. We find that when the normalized PVR of an activator is higher, the probability that it is essential is also higher (Figs. 4C and 4D); these results generalize to much larger systems of activators and deactivators (SM <cit.>, section X). In this paper, we focused on oscillatory networks and found that for a sequence-based scheme, evolution explores the space of possible oscillators in a manner strikingly different from in parameter-based evolution (see, for example, <cit.>). We studied how function can become distributed over new nodes due to random network drift. For a 3-node network, the typical timescale for the new node to become essential for oscillation is ∼100 point accepted mutations, which, given the total of 150 sites, corresponds to around 66 % accepted mutations <cit.>. Surprisingly, our model also revealed a much longer term memory (around 2000 point accepted mutations per 150 amino acids for a 3-node system) with exponential decay, indicative ofa barrier crossing process in the space of sequences. We expect our model to be broadly useful for exploring principles of protein network evolution. While simple and easy to implement, the model is biologically grounded in sequence-based evolution, and also physically grounded insofar as all proteins interact via binding with all others. Within this approach, network topology emerges from evolutionary dynamics rather than being put in by hand. Moreover, there is no fine tuning and the degree of cooperativity utilized for the studies in this paper is modest and easily achievable in practice by biochemical networks <cit.>. The model provides a natural framework to study the interplay between selection pressure and sequence-based designability/accessibility. It can moreover be readily extended to larger networks, networks with other functions, and also to other mutation-selection regimes (for example, the concurrent mutations regime expected for larger populations <cit.>). We also believe our results for network drift will apply beyond the context of oscillators studied here. It has been suggested that protein networks evolve primarily by two biological mechanisms: (i) gene duplication, and (ii) random mutations in proteins leading to neo-functionalization, that is, thede novo creation of new relationships with other proteins <cit.>. Our studies illustrate the significance of neo-functionalization in the context of functional networks where protein-protein interactions are physically grounded, i.e. described via quantitative interaction strengths rather than Boolean variables. Our discovery of hidden order in sequence space leading to evolutionary long-term memory could also be quite general, highlighting the strong constraints to network evolution that emerge from the topology of accessible sequence space. It will be interesting to see if the presence of “phases" generalizes to other network types.Future studies may profitably include the evolutionary dynamics of nodes, address other network functions (e.g. signal integration), and explore the role of graded selection in thede novo evolution of new functions.We acknowledge helpful discussions with Yigal Meir and Ammar Tareen. The research was supported in part by DARPA Biochronicity program, Grant D12AP00025, National Science Foundation Grant PHY-1305525, and National Institutes of Health Grant R01 GM082938.30Alberts B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell. Taylor and Francis; 2002. Alon U. 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Iwasaki, T. Oyama and T. Kondo, Science308, 414-415 (2005). suppm See Supplementary Material. shakhnovich M. Heo, S. Maslov and E. I. Shakhnovich, Proc. Nat. Acad. of Sci. USA108, 4258-4263 (2011). ProteinInteractions Z. Keskin, A. Gursoy, B. Ma and R. Nussinov, Chem. Rev.108, 1225-1244 (2008). Moran P. A. P. Moran, Math. Proc. of the Cambridge Philosophical Society54, 60-71 (1958).Nowak M. A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life.Belknap Press 2006. footnote1 Since we find that states where both activators are individually inessential are very rare, approximately 0.001% of the total number of oscillatory states, we ignore such states for the purposes of the figure. Siggia P. Francois, N. Despierre and E. D. Siggia, PLOS Comp. Bio., DOI: 10.1371/journal.pcbi.1002585 (2012). PAM J. Pevsner, Bioinformatics and Functional Genomics (2nd ed.).Wiley-Blackwell; (2009). Ferrell J. E. Ferrell, Trends in Biochem. Sci.21, 460-466 (1996). Desai M. M. Desai and D. S. Fisher, Genetics176, 1759-98 (2007).DillG. J. Peterson, S. Presse, K. S. Peterson and K. A. Dill, PLOS One7, e39052 (2012). Supplementary Materials: Hidden long evolutionary memory in model biochemical network § CHEMICAL RATE EQUATIONS FOR SYSTEM OF INTERACTING PROTEIN SPECIES Weconsider a system consisting of m species of activators (e.g. kinases), denoted by letter A, and n species of deactivators (e.g. phosphatases), denoted by letter D, which can be in active or inactive states. Activators (in their active state) act only on inactive targets and deactivators (in their active state) act only on active targets. The chemical kinetic equations governing the system are given by A^*_i+T_lk^f_ilk^r_ilA^*_i T_lA^*_i+T^*_l D^*_j+T^*_lk̃^f_jlk̃^r_jlD^*_j T^*_lD^*_j+T_lwhere A^*, D^* and T/T^* denote activators, deactivators, and targets respectively ( ^* denotes active state). We assume that the sum of the active fraction and inactive fraction is constant and set it to 1. The rate of change of concentrations of the active fraction of the target, an activator, is given byd[T^*_l]/dt = ∑^m_i≠ lr_il[A^*_iT_l] + ∑^n_j=1(k̃^r_jl[D^*_jT^*_l] - k̃^f_jl[D^*_j][T^*_l] ) +((2r_ll+k^r_ll )[T^*_lT_l] - k^f_ll[ T^*_l][ T_l])+ ∑^m_i≠ l ((r_li+k^r_li )[T^*_lA_i] - k^f_li[ T^*_l][ A_i]) +∑^n_j=1 ( (r_lj+k^r_lj )[T^*_lD_j] - k^f_lj[ T^*_l][ D_j]),and for the active fraction of the target, a deactivator, is given byd[T^*_l]/dt= ∑^m_i = 1r_il[A^*_iT_l] + ∑^n_j≠ l (k̃^r_jl[D^*_jT^*_l] - k̃^f_jl[D^*_j][T^*_l]) +( (r̃_ll+2k̃^r_ll )[T^*_lT^*_l] - 2k̃^f_ll[ T^*_l]^2)+ ∑^m_i = 1 ((r̃_li+k̃^r_li )[T^*_lA_i] - k̃^f_li[ T^*_l][ A_i]) +∑^n_j ≠ l ((r̃_lj+k̃^r_lj )[T^*_l D^*_j] - k̃^f_lj[ T^*_l][ D^*_j]) .The last three terms in Eqs. S3 and S4 are the terms when the target itself acts as an enzyme. For inactive fractions of the target, both activator and deactivator, the rate of change of concentrations is given byd[T_l]/dt = ∑^m_i=1 (k^r_il[A^*_iT_l] - k^f_il[A^*_i][T_l]) + ∑^n_j=1r̃_jl[D^*_jT^*_l],and for each intermediate complex, the rate of change of concentration is given byd[A^*_iT_l]/dt = k^f_il[A^*_i] [T_l]-(k^r_il+r_il)[A^*_iT_l] d[D^*_jT^*_l]/dt = k̃^f_jl[D^*_j][T^*_l]-(k̃^r_jl+ r̃_jl)[D^*_jT^*_l] . Under the assumption that the intermediate complex concentrations are at steady state (quasi-static approximation), we obtaink^f_il[A^*_i][T_l]-(k^r_il+r_il)[A^*_iT_l]= 0 [A^*_iT_l] =k^f_il/k^r_il+r_il[A^*_i][T_l] k̃^f_jl[D^*_j][T^*_l]-(k̃^r_jl+r_jl)[D^*_jT^*_l] =0 [D^*_jT^*_l]=k̃^f_jl/(k̃^r_jl+ r̃_jl)[D^*_j][T^*_l] .Substituting Eqs. (S8) and (S9) inEqs. (S3) and (S4) yields same expression for the rate of change of concentration for the active fraction of target whether it is an activator or deactivator and is given byd[T^*_l]/dt =∑^m_i=1k_il[A^*_i][T_l] - ∑^n_j=1k̃_jl[D^*_j][T^*_l].Substituting Eqs. (S8) and(S9) in Eq. (S5), we obtaind[T_l]/dt= -∑^m_i=1k_il[A^*_i][T_l] + ∑^n_j=1k̃_jl[D^*_j][T^*_l],where k_il=k^f_il/1 + k^r_il/r_il,k̃_jl= k̃^f_jl/1 + k̃^r_jl/r̃_jl .As a check for self-consistency, we confirm that the rates of change of concentration of the active and inactive fractions are equal in magnitude but opposite in sign, as expected, since the sum of the concentrations of the active, inactive, and intermediate complexes is constant. This quasi-static approximation is justified in the limit where the concentrations of the intermediate complexes are small compared to the concentrations of the active and inactive fractions, that is, in the limit where the ratios k_il^f/(k^r_il + r_il) and k̃_il^f/(k̃^r_il + r̃_il) are much smaller than 1, corresponding to relatively short-lived intermediate complexes. We also point out that this approximation neglects competition between targets due to sharing of enzymes <cit.>, which could become important for high affinity targets. Under this simplifying assumption, we need only one rate equation for the active fraction of each species, thus reducing the number of rate equations to the number of enzyme species. Without loss of generality, we assume k^f_il (= k̃^f_jl), and r_il (= r̃_jl) are the same constants for all enzyme-target pairs, so that the only rate constants that depend on binding energies are k^r_il and k̃^r_il. For these rates, we assume an Arrhenius-type form, e.g.k^r_il = A e^- E_il/k_B T where A is a constant and E_il is the binding energy between enzyme represented by label i and target represented by label l. If energy E_il is measured in units of k_BT where T is room temperature, we obtain k_il=k^'_0( 1/1 + e^-(E_il - E_0)), where k^'_0 and E_0 are constants.Similarly,k̃_jl=k^'_0/(1 + e^-(Ẽ_jl - E_0)). We have so far ignored background (enzyme-independent) activation and deactivation of target; we can incorporate this by addinga term of the form α [T_l] - α'[T^*_l] to the right-hand side of Eq. (S10). We next incorporate cooperativity within our minimal model. For this purpose, we assume a two-stage (or more generally, h-stage) enzyme-mediated activation of target molecules with relatively short-lived intermediates. This could correspond, for example, to two phosphorylation sites for each target molecule, where both sites have to be phosphorylated for molecules to be active.For simplicity of the discussion, we first consider a two-component system (1 activator, 1 deactivator species), and assume target molecules can be in three states: inactive (T), active (T^*), and partially phosphorylated (T').Thechemical kinetic equations governing the system are then of the form: A^*+T A^* +T'T' T A^*+T' A^* +T^* .Within the assumption of short-lived complexes discussed above, the rates of change of T' and T^* are given byd[T']/dt = k' [A^*][T] -k'_-[T'] - k” [A^*][T']d[T^*]/dt = k” [A^*][T'] -k̃[D^*][T^*] + α [T] - α'[T^*] .Applying the quasi-static approximation for T' (valid for low concentrations of [T']), we obtain [T'] = k' [A^*][T]/(k'_- +k” [A^*]). Ifk” [A^*] ≪ k'_- (high spontaneous decay rate of partially phosphorylated state), we can further approximate [T'] ≈ k' [A^*][T]/k'_-. We thus obtain d[T^*]/dt = k [A^*]^2[T] -k̃[D^*][T^*] + α [T] - α'[T^*],where k = k' k”/k_-. Both k' and k” can be expected to be of the form in Eq. (S15), while k_- as a spontaneous decay rate can be treated as a constant.For simplicity, we assume that the enzyme binding energies for both steps of phosphorylation are the same, and obtaink=k_0( 1/1 + e^-(E - E_0))^2 ,where k_0 is now a new constant. Along similar lines, we can also introduce cooperativity in deactivation via a two-stage enzyme-mediated deactivation process. We then generalize this to multiple activator/deactivator species, with the simplifying assumption that activators/deactivators involved in both stages belong to the same species, giving us Eqs. (2) and (3) in the main text. § LINEAR STABILITY ANALYSISTo check if the system corresponds to a global oscillator, we used linear stability analysis as a tool. We will describe it here for a two component system: this can be generalized for N-component systems. For 1 activator and 1 deactivator, the chemical rate equation is written as d[A^*]/dt= k_AA[A^*]^2[A]-k_DA[D^*]^2[A^*]+α[A]-β [A^*] = f([A^*],[D^*]),d[D^*]/dt= k_AD[A^*]^2[D]-k_DD[D^*]^3 +γ [D]-δ[D^*] = g([A^*],[D^*]),where [ A^*],[ D^*] are concentration of active species of kinase and phosphatase respectively. The rate constants k_ij are caclulated as described in the paper. In our model we assume that the sum of the active and inactive concentration of each species is constant and is set to 1, i.e., [ A^*]+[ A] = 1 and [ D^*]+[ D] = 1. We find steady state fixed points by settingd[ A^*]/dt= 0,d[ D^*]/dt= 0.For a fixed point given by ([A^*_0],[D^*_0]), the Jacobian matrix is J=[ f_[ A^*]([ A^*_0],[ D^*_0]) f_[ D^*]([ A^*_0],[ D^*_0]); g_[ A^*]([ A^*_0],[ D^*_0]) g_[ D^*]([ A^*_0],[ D^*_0]) ]where,f_[A^*],f_[D^*] represent partial derivatives with respect to [A^*],[D^*] respectively. The eigenvalue of the Jacobian matrix is λ = μ± iω. According to linear stability analysis, ([A^*_0],[D^*_0]) is stable or unstable depending on the real part of λ, i.e. whether μ is negative (stable) or positive (unstable). If the system has only one fixed point which is unstable then the system must oscillate. Since we are interested in global oscillators that do not depend on the initial concentration, we look for only those cases where there is only one unstable fixed point. We find fixed points for a set of dynamical equations using routine fsolve in matlab/octave. Hundreds of trials are performed with different initial values of the dynamic variables ([A^*] and [D^*]), and distinct sets of fixed points are sorted out.§ COMPUTING PVR AND PHASE SHIFT BETWEEN THE PEAKS OF ACTIVATORS To solve the ODEs describing the network of activators and deactivators, we used ode45 function in Matlab. The ODEs are solved for a long enough time interval that the system reaches steady-state oscillations. In Fig. <ref>, we have shown an oscillatory behavior for a 2 activator, 1 deactivator network. In order to compute the peak-to-valley ratio (PVR), we find the peak and the valley (minimum) of the oscillations in steady state and take the ratio of the two. To compute the phase-shift between the activators, we find the time difference between the nearest peak positions of the activators and multiply that by 2π/T_ osc, where T_ osc is the period of oscillation.§ PRINCIPAL COMPONENT ANALYSIS (PCA)In order to examine the correlations between the chemical rate constants and the phases discussed in the main text, we carried out a Principal Component Analysis (PCA) of the rate constants. PCA is usually performed to re-express data in a meaningful basis and to reduce dimensionality (in this case, of the space of rate constants). Our goal is to identify the key rate constants that determine the phase of the system (Phase 1 or 2) for a 3-component system (2 activators, 1 deactivator). For the PCA analysis, we start with a N_ T× d matrix K of rate constants, where each row represents one set of rate constants for a particular mutational step, and each column tracks the time evolution of a rate constant. Here N_T is the total number of mutational steps in an evolutionary simulation. For a system of n enzymes, d=n^2 is the total number of rate constants, which serves as the dimension for PCA analysis; for our 3-component system the dimension is d=9. We follow the following standard steps for PCA:*Subtract the mean of each dimension (i.e. each rate constant) from the corresponding column of K to construct a matrix X.*Calculate the covariance matrix C of X : C=1/N_ T-1X^TX.*Find the eigenvalues and corresponding eigenvectors of the covariance matrix C.*To reduce the dimensionality, consider only the largest m eigenvalues and form an orthonormal feature matrix from the corresponding eigenvectors M = [eig_1eig_2 ...eig_m].*Finally multiply the mean-subtracted data X by the feature vector M to obtain a projected data set of reduced dimension N_T× m. i.e. Final Data = XM. The eigenvector with the largest eigenvalue corresponds to the principal-component direction along which the data has the largest variance. For the data obtained from a very long evolutionary trajectory, we find that the eigenvalues of the covariance matrix are, in ascending order, [0.0020.0190.0270.20.2290.5570.8111.6888.214] and the eigenvectors (columns) corresponding to each eigenvalue areeig1 eig2 eig3 eig4 eig5 eig6 eig7 eig8 eig9 k_11 0.015 0.117 0.052 0.399 0.485-0.244-0.358 0.355-0.525 k_12 0.041 0.034 0.036-0.372 0.387 0.802-0.068 0.232-0.081 k_13 -0.149-0.968-0.135 0.039 0.126-0.006-0.032 0.042-0.034 k_21 0.029 0.022 0.031 0.288 0.324-0.004 0.875 0.199 0.067 k_22 0.027 0.009 0.164 -0.3960.4-0.429-0.142 0.311 0.595 k_23 -0.161 0.171-0.954-0.097 0.126-0.071 0.006 0.049 0.043 k_31 -0.05-0.036 0.019-0.457-0.434-0.193 0.21 0.589-0.412 k_32 -0.061 0.006-0.058 0.492-0.356 0.267-0.189 0.575 0.432 k_33 0.971-0.126-0.189 0.009-0.049-0.024-0.019 0.051 0.001Note that the rows here correspond to the rate constants k_11, k_12, k_13, k_21, k_22, k_23, k_31, k_32, k_33, respectively. To produce Fig. 3E of the main text, we employed the eigenvector PC1 corresponding to the largest eigenvalue (the last column of the above matrix).§ EVIDENCE FOR GEOMETRIC BOTTLENECK Here we review briefly our evidence for a geometric bottleneck between the two phases for 3-component oscillators (2 activators and 1 deactivator). The idea was initially introduced in the main text to explain the observed long-term memory related to the order of magnitude difference between flipping of essentiality and flipping of phase. This idea of a bottleneck found support in the exponential distribution of duration in a single phase (between two phase flips). As noted in the manuscript, we checked this hypothesis of the bottleneck by studying the neighborhood of states in Phase 1 and Phase 2. For example, for only 3% of the states in Phase 1 where both activators are essential does the Hamming distance 1 neighborhood contain one or more states where Activator 1 is inessential. The relative rarity of such states (which can be considered as boundary states) is consistent with our hypothesis that in sequence space the two phases touch at a relatively small number of boundary points Due to the difficulty of visualizing a very high-dimensional sequence space, it is also helpful to study the distribution of points belonging to the two phases in the space of the chemical rate constants. As noted in the manuscript, one direction in this space, designated as PC1 (principal component 1), was relatively effective in discriminating between the phases (though due to some overlap, the discrimination was not complete, as can be seen in Fig. 2E in the main text). Thus in order to characterize the geometric bottleneck, we plot the distribution of states in the PC1 direction (Fig. <ref>A). The distribution is bimodal with the two peaks representing the two phases (with the left peak corresponding to Phase 1 and the right to Phase 2) well separated out. In Fig. <ref>B, we plot the probability that a state is a boundary state (as identified in the main text) along the PC1 direction and, as expected, find a single narrow peak at the center. We note that even near this peak the probability of being in a boundary state is very small (∼ 10^-3) which also suggests that there are relatively very few boundary states, supporting the geometrical bottleneck hypothesis. In contrast, the states where both the activators are essential are uniformly distributed along PC1 (Fig. <ref>C) and such states represent ∼66% of all oscillatory states.As an aside, we also checked if there is any significant difference in the probability that a mutation will fail (that is, lead to a non-oscillatory state) for boundary points versus interior points or, alternatively, as a function of PC1, since a higher value of the failure probability for boundary points could potentially further enhance memory by penalizing genotypes near the transition between the two phases. However, perhaps surprisingly, we find no significant difference in the frequency with which mutations fail for boundary points versus points interior to the two phases. § TOY MODEL Due to the high dimensionality of the sequence space and the complex nature of the relationship between the model parameters and observed behavior of flipping of essentiality and phases, we found it helpful to construct a simplified toy model that can reproduce crucial aspects of the observed behavior. Specifically, we introduce a toy model consisting of two activators, Activator 1 (Act1) and Activator 2 (Act2). The essentiality of each activator is determined by a binary string (consisting of 0s and 1s) of even length L. In our toymodel, Act1 (Act2) will be essential if the number of 1s in the first half (second half) of the binary string is greater than or equal a threshold N_c. We evolve the string by randomly flipping bits, and requiring that at least one activator is always essential. We also constrain the total sum of 1s in the whole string to be equal to or less than a cutoff, M_c, which is greater than or equal to 2N_c so that both activators can in principle be essential at the same time. Setting the difference between M_c and 2N_c to be small leads to relatively few states where both activators are essential, and thus creates a “bottleneck”between domains where each activator is essential.The toy model exhibits behavior that is similar in some crucial respects to the evolutionary model presented in the main text, but also displays some important differences. For our toy model, we start the system with binary string such that Act1 is essential and Act2 inessential, and let the system evolve by introducing random point mutations. We depict the time evolution of essentiality of activators in Fig. <ref> for different values of the parameters (N_c, M_c, and L). As with our evolutionary model, we find flips in essentiality of both Act1 and Act2, with a power-law distribution of the number of mutational steps that an activator remains essential (P(T) ∝ T^- α) for short durations, and exponential decay ( P(T) ∝ e^- T/τ_0 ) for longer durations. For the power-law distribution, the exponent α depends on the model parameters, N_c, M_c, and L, as follows. For a fixed value of M_c and L, α increases with the threshold N_c (Fig. <ref> A). The exponent does not vary with M_c for fixed N_c and L except when M_c is equal to 2N_c (Fig. <ref> C). For a fixed value of N_c and M_c the exponent decreases with the length of the sequence (Fig. <ref> E). For the exponential decay at longer durations, P(T) ∝ e^- T/τ_0, the dependence of the constant τ_0 on the model parameters resembles that of the exponent α (Fig. <ref> B,D,F). To understand the behavior generated by our toy model, it is worth noting that the sequence space can be characterized by two variables, N_1 and N_2, where N_1 (N_2) is the number of 1s in the first half (second half) of the string. We note that for any allowed sequence with N_1 > N_2 (N_2 > N_1), Act1 (Act2) has to be essential. Thus we can characterize these points as Phase 1 (Phase 2). In the toy model Δ N=N_1-N_2 plays a role similar to principal component (PC1 in the full model). We choose our parameters such that the distribution of Δ N is bimodal (Fig. <ref> A), similar to the distribution of PC1. Note that the distribution of phases yields similar decay time as the decay of essentiality. In order to understand how the exponent and decay time depends on the model parameters (N_c, M_c, and L), we depict, in Fig. <ref>,the essentiality of each activator in N_1-N_2 space. The green region corresponds to points where both activators are essential, and, for our toy model, it also corresponds to the boundary region between the two phases. For any allowed point in N_1-N_2 space, the designability (the number of distinct allowed sequences) is given by ^L/2 C_N_1× ^L/2 C_N_1. For given values ofN_c, M_c, and L, we can calculate the total designability associated with the boundary region (total designability equals the sum of the designability of all allowed points in the boundary region of N_1-N_2 space).As expected, we find that that dependence of τ_0 on any one of the model parameters (with the other two parameters held fixed) correlates with total designability. A lower designability (which could correspond to a more pronounced bottleneck between the phases) indicates longer phase duration and, correspondingly, larger value of τ_0. For the power-law exponent, we similarly expect the exponent to be related to the boundary of the green region in Fig. <ref>. A measure of the boundary is the constrained sum of the designabilities of only those states in the green region where either N_1 or N_2 equals N_c, and we find that α indeed correlates with this constrained designability. To be more precise, if any of the model parameters is varied holding the other two fixed, higher constrained designability indicates lower value of α and vice-versa. In Fig. <ref>, we have shown the dependence of exponent α and decay time τ_0 on model parameters along with the relevant designabilities.Despite the similarities highlighted in the previous paragraphs, the behavior of the toy model also differs in some crucial respects from the behavior exhibited by the full model in the main text. For example, in the toy model, we typically do not find long durations where one of the activator remains essential while the other flips essentiality. Rather, the mean phase duration and mean duration of essentiality are similar (differing by a factor of 2-3, in contrast to the full model where they differ by an order of magnitude). Moreover, if the distribution of Δ N is chosen to be bimodal, we find that typically the fraction of time where both activators are essential is very small (∼1%), whereas in the full model this fraction is high (∼66%). In the toy model, the fraction of time both activators are essential is determined by the ratio between N_c and L/4. When this ratio is smaller than 1, it is rare for both activators to be essential. As this ratio approaches 1, it becomes increasingly common for both activators to be essential, and if the ratio becomes greater than 1, then both will be essential most of the time. However, when the ratio is greater than 1 the distribution of Δ N is no longer bimodal. Moreover,unlike the results from the full model, in the toy model, the region in N_1-N_2 space where both activators are essential forms a compact domain (shown in green in Fig. <ref>) and lies at the boundary between the two phases. The difference is highlighted by the density of states in the PC1/Δ N( = N_1-N_2)direction. In the full model, the distribution of states where both activators are essential is almost uniform in the PC1 direction (Fig. <ref> A), quite different from the distribution of boundary points which is peaked around the center of the PC1 axis (Fig. <ref> B). This contrasts with the toy model, where the distribution of states where both activators are essential as well as the distribution of boundary states are peaked around the center of Δ N( = N_1-N_2) (Fig. <ref> B, C).In summary, we have shown that the toy model captures some of the features of the full activator-deactivator oscillator model, e.g., the presence of mixed power-law and exponential distributions of flipping of essentiality. However, unlike the full model, the toy model does not exhibit long-term evolutionary memory; the difference in behavior can be related to the distribution of states in sequence space where both activators are essential.§ STARTING SEQUENCE FOR 2 ACTIVATOR, 1 DEACTIVATOR SYSTEMThe starting sequence for the network shown in green in Fig. 2 is 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0where rows 1,3,5 are the out-faces of Activator 1, Activator 2, and the deactivator, and 2,4,6 are the in-faces for the same. The zeros in row 3 and 4 indicates that Activator 2 is minimally interacting with Activator 1 and the deactivator. The interaction energies corresponding to this sequence for ϵ = 0.2 are [E_11E_12E_13E_21E_22E_23E_31E_32E_33] = [ 2.8 0.0 2.4 0.0 0.0 0.0 2.4 0.0 0.8 ]. The subscript 1,2,3 in the energies denotes Activator 1, Activator 2, and the deactivator. The starting sequence for the network shown in blue in Fig. 2 is 11111 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 011111 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000000 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 011111 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0and the corresponding interaction energies are [E_11E_12E_13E_21E_22E_23E_31E_32E_33] = [ 2.8 0.0 2.6 0.0 0.0 0.0 1.2 0.0 0.2 ].We studied the time evolution of the essentiality of each activator for the above two starting sequences. We found by averaging over multiple long runs that on average 61.6 % of the time both activators are essential. We also found that states where both are inessential are very rare, approximately 0.001% of the total number of oscillatory states. For 38.4% of the time only one of the activators is essential.§ CORRELATION BETWEEN RELATIVE PVR, SIGN OF PHASE SHIFT, ESSENTIALITY, AND RATE CONSTANTS Why are relative peak-to-valley ratio (PVR), sign of phase shift, essentiality, and relative rates of activator auto-activation/deactivation correlated? We offer here a qualitative argument for a 3-component system (2 activators and 1 deactivator). If Activator 1 is leading in phase, it likely has a stronger self-activation than the second activator and thus its activated level A_1^* starts increasing once the level of activated deactivator (D^*) is sufficiently low. Once the level of Activator 1 becomes sufficiently high, it starts activating both Activator 2 and the deactivator, whose levels both start rising. Once the D^* level builds up it starts deactivating both the activators, whose active levels then start to drop. However, since A_1^* had started rising first, the phase shift will be positive and, furthermore, A_1^* will also have reached a relatively higher level than A_2^*before both levels start to drop, so that Activator 1exhibits a higher PVR. Moreover in this scenario, Activator 1 might be expected to play a more important role for the oscillations since it is the auto-activation of Activator 1 that drives both its level and the activated level of Activator 2 to rise, and is thus more likely to play an essential role in the oscillations. It is important to note here that this argument is only qualitative and does not necessarily apply to all oscillatory states. § EXPONENTIAL VERSUS POWER-LAW DISTRIBUTION OF DURATION TIMESConsider a system that can be in two phases: Phase 1 and Phase 2. Furthermore, consider the case where the system enters one of the two phases, say Phase 1, at time t=0. We seek to find an expression for the probability distribution of the time of duration in Phase 1 before the system switches to Phase 2. If P_0(t) is the probability that system still persists in Phase 1 at time t without having switched to Phase 2, and P_T(t) is the probability density of duration time t, then dP_0/dt = -P_T(t). If the switching rate from Phase 1 to Phase 2 is a constant k, independent of time elapsed since entry into Phase 1, thend P_0/d t = - k P_0, implying P_0 (t) = e^- k t. In this case, the probability density P_T(t) is also exponential, P_T∝ e^- k t. We expect this to be the case for an entropic barrier separating the two phases. In contrast, if the escape rate k itself depends on time t elapsed from the moment of entry into Phase 1, in particular, if it is of the form k = α /(τ_0 + t), where τ_0 and α are constants, (we might expect such a form for an extended boundary between the two phases, since the longer the system has spent in Phase 1, the deeper, that is further from the boundary, it is likely to be in the phase; also τ_0 is roughly of the order of the timescale for a single mutational step), thend P_0/d t = - α/τ_0 + t P_0,implyingP_0(t) = (τ_0 + t/τ_0)^-α. In this case, for t > τ_0, we obtain a power-law distribution P_T(t) ∼ t^-a, where a = α - 1.§ GENERALIZATION TO 5 ACTIVATOR, 5 DEACTIVATOR SYSTEMIn order to check the applicability of our main results regarding essentiality of activators, relative phase, andPVRfor more complex networks, we studied a network of 5 activators and 5 deactivators. To simply generate a 5 activator, 5 deactivator oscillator, we started with a 2-component (1 activator, 1 deactivator) oscillator and divided both the activator and deactivator into 5 identical copies, each with the same set of sequences as its parent. We set the concentrations of each of the 5 new activators and 5 new deactivators to be 1/5 of its parent, so that initially the dynamics of the system was identical to the starting 2-component oscillator. The system was evolved for∼3000 accepted mutational steps such that the sequences for the activators and deactivators became quite different. The system was further evolved for 25,000 mutational steps from which we obtained results for essentiality, relative phase, and PVR. During network evolution, mutations were accepted if the system continued to oscillate for a given initial concentration (0.5 for each active fraction). Specifically, for each proposed mutation we solved the dynamical equations and accepted only those mutations for which the amplitude of oscillation of the active fraction of at least one of the components remained above a cutoff (0.001 in this case) for 400 units of time. To test the essentiality of each activator we removed that component and checked if the system continued to oscillate. The plot of essentiality of the 5 activators is shown in Fig. S1A. We observed durations where a given activator remains essential and durations where it continues to flip between essential and inessential.This behavior is very similar to what we observed for a 4-component oscillator (Fig. 5). For a given pair of activators we calculated the PVR and also tracked which activator was leading in phase in the oscillations. We found that similar to 4-component oscillators, for any pair of activators, the probability of one activator leading the other is higher if the relative PVR is higher (Fig. S1B), and the probability that an activator is essential is higher when its normalized PVR is higher (Fig. <ref>C). § ROBUSTNESS OF THE MODELTo confirm the robustness of our results with respect to the choice of model parameters, we used different values of the interaction energy between hydrophobic residues ϵ, the rate constant k_0, and background activation and deactivation rates for the 2 activator, 1 deactivator systems. Qualitatively we find that all our results hold as long as the maximum interaction energy E_ max = N ϵ, does not substantially exceed the threshold energy E_0, that is, provided E_ max - E_0 < 1, where energy is expressed in units of k_ BT; otherwise the reaction rates become saturated at their maximum values over a large fraction of sequence space. Keeping all other parameters fixed, if we vary k_0, we find that the system remains in a particular phase longer for smaller k_0. The duration of phases is shorter for k_0 = 10^4 (Fig. <ref> A) compared to k_0=5625 (Fig. <ref> B). For a smaller value of background activation rate (α = 0.25) for both the activators and deactivator, we find that the duration of phases becomes shorter, i.e., the system switches between Phase 1 and Phase 2 more rapidly (Fig. <ref> C). On the other hand for a smaller value of background deactivation rate (β = 0.25) the phase duration becomes longer (Fig. <ref> D).The quantities relative PVR, phase-shift, and PC1 continue to be correlated with the phase of the system (Phase 1 or Phase 2) (Fig. <ref>). Alternately, if we increase the value of ϵ while maintaining E_0 = N ϵ, we find a significant increase in the phase duration, so that for ϵ = 0.5 we hardly seetransitions from one phase to the other (Fig. <ref> E). It is worth noting that if E_ max = N ϵ is allowed to be much greater than E_0, then physically the concentrations of the intermediates corresponding to enzymes bound to targets should become significant, and one would need to explicitly account for their concentrations in the rate equations, thus increasing their complexity. A systematic investigation of this case lies outside the scope of the current paper.§ FIGURE DETAILSAs different size data sets were used for each figure, we provide the details here. The duration of each single simulation is 10^5 accepted mutational steps. Fig. 1A: A single run for 10^5 accepted mutational steps. Fig. 1B: 4000 data points used to produce the histogram for each starting sequence. Fig. 2A-D: Single run for 10^5 accepted mutational steps. Fig. 2E: 100 simulations each running for 10^5 accepted mutational steps. Fig. 3A: Approximately 6000 data points generated by running 150 simulations each running for 10^5 accepted mutational steps. Fig. 3B: Approximately 10^5 data points generated by running 150 simulations each running for 10^5 accepted mutational steps. Fig. 4A: A single run for 10^5 accepted mutational steps. Fig. 4B: Approximately 10^6 data points generated by running 80 simulations each running for 10^5 accepted mutational steps. Fig. 4C-D: Single run for 2×10^5 accepted mutational steps. 10Rowland M. A. Rowland, W. Fontana, and E. J. Deeds, Crosstalk and Competition in Signaling Networks, Biophys J. 2012 Dec 5; 103(11): 2389–2398. | http://arxiv.org/abs/1706.08499v1 | {
"authors": [
"Md. Zulfikar Ali",
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"Ranjan Mukhopadhyay"
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"published": "20170626174458",
"title": "Hidden long evolutionary memory in a model biochemical network"
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Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom [email protected] Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom Inferring a generative model from data is a fundamental problem in machine learning. It is well-known that the Ising model is the maximum entropy model for binary variables which reproduces the sample mean and pairwise correlations. Learning the parameters of the Ising model from data is the challenge. We establish an analogy between the inverse Ising problem and the Ornstein-Zernike formalism in liquid state physics. Rather than analytically deriving the closure relation, we use a deep neural network to learn the closure from simulations of the Ising model. We show, using simulations as well as biochemical datasets, that the deep neural network model outperforms systematic field-theoretic expansions, is more data-efficient than the pseudolikelihoodmethod, and can generalize well beyond the parameter regime of the training data. The neural network is able to learn from synthetic data, which can be generated with relative ease, to give accurate predictions on real world datasets. Inverse Ising inference by combining Ornstein-Zernike theory with deep learning Alpha A. Lee=============================================================================== Drawing meaningful interference from correlations amongst variables is a fundamental problem in science. The central challenge is to decipher the probability distribution that generates the correlations given a set of observations, and then predict properties of unknown samples with the inferred model. Many probabilistic models have been proposed in the literature <cit.>. Focusing on capturing the sample mean and pairwise correlations, the simplest model, in the sense of maximum entropy, is a Boltzmann probability distribution with a Hamiltonian that contains terms linear and bilinear in the variables <cit.>. For continuous variables, this is the multivariate Gaussian distribution; for discrete variable, this model is the inverse problem of the Ising model in statistical physics. The inverse Ising model, also known as Boltzmann learning <cit.>, has found applications in different disciplines <cit.> such as understanding neural spike trains <cit.>, bird flocks <cit.>, and predicting structures of protein <cit.> and RNA <cit.> as well as their fitness landscapes <cit.> using evolutionary data.The inverse Ising model disentangles correlations in the dataset into pairwise interactions, thus could distinguish direct variable-variable interactions from indirect correlations mediated through other variables.However, exact maximum likelihood inference of the inverse Ising model is numerically challenging because it requires computing the partition function at each step of the optimisation <cit.>. To overcome this challenge, many approximate techniques have been developed<cit.>. Those techniques are typically leading terms of asymptotic expansions that relate the sample mean and correlation to the Ising parameters in limits where the coupling is tractable, e.g. asymptotically small correlations <cit.>, small number of coupled clusters <cit.>, a tree-like structure <cit.>. (For comprehensive reviews, see <cit.>.)A relatively recent class of algorithms maximize the pseudolikelihood rather than the likelihood<cit.>. Although the pseudolikelihood method avoids partition function evaluations and converges to the maximum likelihood solution in the limit of infinite data, the algorithmic complexity scales with the number of samples. In this Letter, we eschew analytical asymptotic expansions and instead use a deep learning model to infer the relationship between the Ising parameters and sample mean and correlation. We will motivate an analogy between the inverse Ising model and the Ornstein-Zernike formalism in liquid state physics. We show that a deep neural network, trained using simulations of Ising models, can approximate an Ornstein-Zernike-like closure for the inverse Ising model. The deep neural network is generalizable and achieves an accuracy beyond analytical approximations for biochemical datasets as well as simulations with disparately different parameters compared to the training data. The neural network learns from synthetic data, which can be generated with relative ease, to give accurate predictions on real world datasets with a runtime independent of the dataset size. We begin by stating the inverse Ising model: We are given p sequences of N variables, {𝐬^α}_α=1^p, each variable s_i can take values ± 1. The maximum entropy distribution with the mean and pairwise correlations agreeing with the data, i.e.< σ_i>_𝒫 = < s_i>_data, < σ_i σ_j>_𝒫 = < s_i s_j>_datais the Ising model 𝒫(σ) = 1/Zexp(∑_i h_i σ_i + ∑_i>j J_ijσ_i σ_j ). The inverse Ising problem is the inference of parameters {h_i, J_ij}from data. This is challenging because the correlation between variables i and j, C_ij =<s_i s_j > - <s_i> <s_j>, can be large even if J_ij = 0 if there is an intervening variable k such that J_ik and J_kj are large. In other words, the pairwise correlations C_ij one detect in a dataset is a measure of the interaction between i and j that is mediated by all other variables. To make further progress, we follow the Ornstein-Zernike formalism <cit.> in liquid state physics to deconvolve direct interactions and indirect correlations. We first consider a one-component, homogenous and isotropic liquid. Molecules interact with a pairwise additive potential v(𝐫_12), where 𝐫_12 is the distance between particles 1 and 2. The liquid structure is characterised by the radial distribution function, g(𝐫_12), which is the probability of observing a molecule at distance 𝐫_12 away from a molecule at the origin. Ornstein and Zernike noticed that g(𝐫_12) can be long ranged even when v(𝐫_12) is short ranged because g(𝐫_12) accounts for both the direct interaction between two molecules as well as the indirect interactions with surrounding molecules. Their crucial insight is to introduce a quantity known as the direct correlation function, c(𝐫_12), and write the total correlation function h(𝐫_12) ≡ g(𝐫_12)-1 as h(𝐫_12) = c(𝐫_12) + ∫d𝐫_3 c(𝐫_13) c(𝐫_32) + ⋯where the first term captures the direct influence of molecule 1 on 2, the second term captures the influence of molecule 1 on 2 mediated by molecule 3, and higher order terms capture correlations induced by more intervening molecules. Equation (<ref>) can be rewritten in a compact form h(𝐫_12) =c(𝐫_12) + ∫d𝐫_3 c(𝐫_13) h(𝐫_32) which is known as the Ornstein-Zernike equation (we have scaled the standard correlation functions by the density to make the link with the Ising model more apparent later). To close the problem, we need a relation between c(𝐫_12), h(𝐫_12), and v(𝐫_12). The crucial feature of many closures is that they are local and take the form f(c(𝐫_12),h(𝐫_12),v(𝐫_12);ρ) = 0where f is a real-valued function (not a functional) and ρ is the liquid density <cit.>. The locality of the closure relationship is approximate <cit.>, albeit a very good approximation for a wide variety of inter-particle interactions. Having introduced the Ornstein-Zernike formalism and closure, we return to the inverse Ising problem. The sample correlation, C_ij, is a discrete analogy of h(𝐫_ij) in the Ornstein-Zernike formalism. Therefore, we introduce the direct correlation D_ij – the discrete analogue of c(𝐫_ij) – and replace the integrals in Equation (<ref>) as a sum over lattice sites, i.e. C_ij = δ_ij+ D_ij + ∑_k D_ik D_kj + ∑_k,l D_ik D_kl D_lj + ⋯. In matrix form, C = (𝕀 - D)^-1 or D = 𝕀- C^-1. Taking J = -C^-1 is known as the mean-field approximation or Direct Coupling Analysis <cit.>. Now we introduce the key assumption of this Letter: we posit that the locality heuristic of closure relations for liquids applies to the inverse Ising problem. This crucial assumption will be verified later by comparing with simulations. We posit a function of four real variables F(·, ·,·,·) such thatJ_ij≈ F(C_ij, [C^-1]_ij,<s_i>,<s_j> ).Unlike the case of homogeneous fluids, we need an extra equation to determine the fields h_i. Motivated by the free energy of inhomogeneous fluids <cit.>, we posit a function G(·, ·,·,·) such thath_i ≈ G( tanh^-1<s_i>, [C^-1]_ii, ∑_j≠ i J_ij<s_j>, ∑_j≠ i C_ij<s_j> ),where J_ij is given by F. We note that the first few terms of most approximate analytical inverse Ising theories do take the form of Equations (<ref>)-(<ref>) with different F and G depending on the approximations<cit.>. F and G are complicated functions. Rather than analytically determining what they are, we will use a deep learning approach and approximate them using a richly parameterised interpolating function. Our hypothesis is that the neural network is able to find F and G that are better than mean-field expansions by inferring directly from data.The training data is prepared by running 20 simulations with N=50 Ising sites. In each simulation, the Ising parameters are drawn from a normal distribution, J_ij = J_ji∼𝒩(0,β_J/√(N)) and h_i ∼𝒩(0,β_h), i.e. the Sherrington-Kirkpatrick model <cit.> with a random field. The variances of the distributions are fixed in each simulation but vary across the 20 simulations, with β_J ∼uniform(0.5,2.5) and β_h ∼uniform(0.5,2.5). The one point and two point correlations are computed from N × 10^7 steps of Markov Chain Monte Carlo (MCMC), sampled at every 10 N steps. F and G are approximated using multilayer neural networks. A l layer neural network approximates functions by l successive non-linear compositions, i.e. y = 𝐖_lσ(𝐖_l-1⋯σ(𝐖_1 𝐱))), where 𝐖_i∈ℝ^M_i× M_i-1 is a weight matrix inferred from data, M_i is the number of units in layer i, and σ(·) is a non-linear function. We use a neural network as it is a numerically tractable way of representing complex functions, and the parameters 𝐖_i can be efficiently inferred with backpropagation. The neural network architecture is discussed in the Supplemental Material and released on 𝚐𝚒𝚝𝚑𝚞𝚋. The computational cost of evaluating the neural network to obtain one entry ofJ is independent of the number of variables or data, hence the total complexity is O(N^2). The largest cost is computing C^-1 (O(N^3)). The pseudolikelihood approximation, the state of art method in the literature, has complexity O(N^2 p) thus our algorithm is computationally less intensive than the pseudolikelihood method in the large data limit where the pseudolikelihood method is mathematically exact; we will show that our method is more data-efficient in the intermediate data regime.To test the generality and accuracy of the model, we simulate Ising models with N =70 sites (note that the model is only trained on N =50 sites), J_ij = J_ji∼𝒩(0,β/√(N)) and h_i ∼𝒩(0,0.3β). A large value of β corresponds to stronger coupling thus further away from the perturbative regimes that underlie analytical theories. Figure <ref>A-B shows the root mean square error of recovering J_ij and h_i as a function of β. The neural network model is more accurate than the Thouless-Anderson-Palmer approximation (TAP), a third order high temperature expansion <cit.>. The pseudolikelihood approximation slightly outperforms that neural network at high temperature, where essentially all mean-field methods become accurate, but crucially the neural network outperforms the pseudolikelihood approximation at low temperatures where correlations become non-trivial.Importantly, the neural network is accurate even for β∈ (2.5,3.5), which is outside the coupling strengths in the training data, demonstrating generalizability. The neural network approximation is also robust to sampling noise. Figure <ref>C-D shows that our neural network outperforms the TAP approximation as well as the pseudolikelihood approximation in the low data limit, demonstrating that our method is data efficient. The neural network approximation is generalizable and accurate even when the couplings are non-Gaussian distributed. Figure <ref>A shows that the neural network approximation can accurately recover the coupling parameter for a 1D ferromagnetic Ising model with constant nearest-neighbour coupling J_ij = J_ji = J (δ_i,j+1 + δ_i+1,j). The coupling matrix recovered from the neural network (inset of Figure <ref>A) is strongly localised on the off-diagonal elements despite all the training data has a delocalised coupling matrix. We use the analytically determined correlation matrix as input to the neural network to focus on the error of the locality approximation, and Figure <ref>A reassures that this intrinsic error is low. Inspired by the use of inverse Ising models in protein structure and fitness prediction <cit.>, Figure <ref>B shows that the neural network approximation can accurately recover a model contact map, and is more data efficient compared to the pseudolikelihood method. In Figure <ref>B, we consider the Bovine pancreatic trypsin inhibitor protein (PDB ID: 5PTI), a benchmark example in ref <cit.> with N=58 amino acids. For this example, we assume the “ground truth” couplings are known (to what extent are amino acid interactions pairwise additive is a separate question <cit.>), and take J_ij = e^-d_ij/7Å/√(N), where the d_ij is the C_β distance between residues i and j. The correlation matrix is computed using MCMC; samples are taken every 10 N steps to ensure independence, and in total we acquire p samples. The generalisability of the neural network to non-Gaussian distributed couplings confirms the approximate locality of the true Ornstein-Zernike-like closure. The functions F and G can be accurately approximated as long as the training data spans the four-dimensional input space. To interrogate what has the neural network learnt, we focus on the case h_i =0 and <s_i>=0 for all i so that the closure is a 3D surface.Figure <ref> shows that the neural network learnt non-trivial corrections to pathologies in analytical theories: mean-field theory predicts J_ij = - C^-1_ij, yet this approximation significantly overestimates J_ij <cit.>. The neural network automatically corrects this by learning a sub-linear function to relate J_ij to C^-1_ij. This is conceptually akin to phenomenologically imposing a large regularisation, a technique discussed in the physics literature <cit.>, except the appropriate regularisation is inferred directly from data. Moreover, the neural network learns to use C_ij, another way to estimate the coupling, and only predicts a large value of J_ij if both C_ij and C^-1_ij are large (c.f. the contours on the C_ij–C^-1_ij plane). More generally, analytical approximate methods generally have some bias depending on aggregate quantities, e.g. inverse temperature and sparsity. The neural network learns those quantities by comparing C and its inverse, and then applies an appropriate correction.We will now go beyond synthetic data where the ground truth is known and apply our method to two problems in computational biology and chemistry. Fitness landscape of HIV-1 Gag: Developing a predictive model for the fitness of HIV virus as a function of the amino acid sequence of the Group-specific antigen (Gag) protein is a challenge in vaccine development. Recent works showed that the fitness landscape can be inferred from the statistics of sequences found in patients <cit.>. The hypothesis is that the frequency of observing conserved sites and sets of correlated mutations reflect the contribution of those residues to fitness <cit.>. Therefore, the fitness of an unknown sequence can be predicted by its log probability computed using a generative model for the sequences observed in patients. We replicate the analysis of ref <cit.> for the HIV-1 Gag protein using the Ising representation for sequences, except the Ising parameters are inferred using the neural network. Figure <ref> shows that the log probability predicted by our model is highly correlated with experimental measurements of replication capacity, with a correlation coefficient r = 0.86, slightly higher than the state-of-the-art model <cit.> (r = 0.83).Tox21 Challenge: A key challenge in drug discovery is predicting whether an unknown molecule will bind to a particular receptor.We consider the Tox21 challenge which reported binding affinities of small molecules against a panel of 12 toxicologically relevant receptors <cit.>. Molecules are represented as a vector recording the presence (1)/absence (-1) of chemical groups (c.f. Supplemental Material). Each receptor is treated independently. As there are more variables compared to the number of data points, we undress finite sampling noise from the correlation matrix using an eigenvalue thresholding method inspired by random matrix theory<cit.> (c.f. Supplemental Material). Separate Ising models are inferred for active ({J^b_ij,h^b_i}) and inactive ({J^n-b_ij,h^n-b_i}) molecules. We score a molecule 𝐟 by the log probability ratio E(𝐟) = ∑_i<jf_i f_j (J^b_ij - J^n-b_ij) + ∑_if_i (h^b_i - h^n-b_i) and the molecule is predicted to bind if E<ϵ, where ϵ controls the false/true positive tradeoff.Figure <ref> shows that the inverse Ising model accurately predicts binding (mean out-of-sample AUC across 12 receptors = 0.85). The state-of-the-art model achieves mean AUC of 0.83 but only when data from every receptor is pooled together <cit.>, thus our model modestly outperforms the state-of-the-art, is approximately 12-times more data efficient, and clearly interpretable in terms of pairwise correlations between chemical features.In conclusion, we demonstrate a method that combines Ornstein-Zernike theory with a highly accurate closure parameterised using deep learning to solve the inverse Ising problem. We illustrate how our method can be used in real world datasets by considering examples in computational biology and chemoinformatics. We anticipate the strategy of parametrising Ornstein-Zernike closure with data to be applicable also in liquid state physics. AAL thanks M. P. Brenner and R. Monasson for insightful discussions and comments, and J. P. Barton for making available the data of Ref <cit.>. AAL acknowledges the Winton Programme for the Physics of Sustainability for funding. Data availability: Codes and data to reproduce results in this paper are available in <cit.>. | http://arxiv.org/abs/1706.08466v2 | {
"authors": [
"Soma Turi",
"Alpha A. Lee"
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"published": "20170626163726",
"title": "Inverse Ising inference by combining Ornstein-Zernike theory with deep learning"
} |
Functionalizing Fe adatoms onCu(001)as a nanoelectromechanical system Michael Schüler^1, Levan Chotorlishvili^1, Marius Melz^1, Alexander Saletsky^2, Andrey Klavsyuk^2, Zaza Toklikishvili^3, Jamal Berakdar^1 December 30, 2023 =============================================================================================================================================== Nested weighted automata (NWA) present a robust and convenientautomata-theoretic formalism for quantitative specifications. Previous works have considered NWA that processed input words only in theforward direction.It is natural to allow the automata to process input words backwards as well,for example, to measure the maximal or average time between a response andthe preceding request. We therefore introduce and study bidirectional NWA that can process inputwords in both directions. First, we show that bidirectional NWA can express interesting quantitativeproperties that are not expressible by forward-only NWA.Second, for the fundamental decision problems of emptiness and universality,we establish decidability and complexity results for the new framework whichmatch the best-known results for the special case of forward-only NWA. Thus, for NWA, the increased expressiveness of bidirectionality isachieved at no additional computational complexity. This is in stark contrast to the unweighted case, where bidirectional finiteautomata are no more expressive but exponentially more succinct than theirforward-only counterparts. § INTRODUCTION We study an extension of nested weighted automata (NWA) <cit.>that can process words in both directions.We show that this new and natural framework can express many interestingquantitative properties that the previous formalism could not. We establish decidability and complexity results of the basicdecision problems for the new framework. We start with the motivation for quantitative properties,then describe NWA and our new framework, and finally the contributions.Weighted automata. Automata-theoretic formalisms provide a natural way to express quantitative properties of systems. Weighted automata extend finite automata where every transition is assigned an integer number called weight.Thus a run of an automaton gives rise to a sequence of weights. A value function aggregates the sequence of weights into a single value. For non-deterministic weighted automata, the value of a word w is the infimum value of all runs over w. First, weighted automata were studied over finite words with weightsfrom a semiring, and ring multiplication as value function <cit.>, and later extended to infinite words with limit averaging or supremum asvalue function <cit.>.While weighted automata over semirings can express severalquantitative properties <cit.>, they cannot express long-run average properties that weighted automata with limitaveraging can <cit.>. However, even weighted automata with limit averaging cannot expresssome basic quantitative properties (see <cit.>).Nested weighted automata. A natural extension of weighted automata is to add nesting, which leads to nested weighted automata (NWA) <cit.>.A nested weighted automaton consists of a master automaton and a setof slave automata. The master automaton runs over input infinite words. At every transition the master can invoke a slave automaton that runsover a finite subword of the infinite word, starting at the position wherethe slave automaton is invoked. Each slave automaton terminates after a finite number of steps and returnsa value to the master automaton.Each slave automaton is equipped with a value function for finite words,and the master automaton aggregates the returned values from slave automatausing a value function for infinite words.Advantages of NWA. We discuss the various advantages of NWA. * For Boolean finite automata, nested automata are equivalent to thenon-nested counterpart, whereas NWA are strictly more expressive thannon-nested weighted automata <cit.>. It has been shown in <cit.> that NWA provide a specification frameworkwhere many basic quantitative properties can be expressed, whichcannot be expressed by weighted automata. * NWA provide a natural and convenient way to express quantitativeproperties. Every slave automaton computes a subproperty, which is then combined using the master automaton. Thus NWA allow to decompose properties conveniently, and provide a naturalframework to study quantitative run-time verification. * Finally, subclasses of NWA are equivalent in expressive power withautomata with monitor counters <cit.>, and thus they provide arobust framework to express quantitative properties.Bidirectional NWA. Previous works considered slave automata that can only process inputwords in the forward direction (forward-only NWA).However, to specify quantitative properties, it is natural to allow slave automata to runbackwards, for example, to measure the maximal or average time between a response andthe preceding request. In this work we consider this natural extension of NWA, namelybidirectional NWA, where slave automata can process words in the forwardas well as the backward direction. Natural properties.First, we show that many natural properties can be expressed in thebidirectional NWA framework.We present two examples below (details in Section <ref>). * Average energy level. Consider a quantitative setting where each weightrepresents energy gain or consumption, and thus the sum of weights representsthe energy level.To express the average energy level property, the master automaton haslong-run average as the value function, and at every transition it invokes a slave automaton that walks backward with sum value function for the weights.Thus the average energy level property is naturally expressed by NWA withbackward-walking slave automata, whilethis property is not expressible by NWA with forward-walking slave automata. * Data-consistency property (DCP).Consider the data-consistency property (DCP) where the input letters correspond toreads, writes, null instructions, and commits. For each read, the distance to the previous commit measures how freshis the read with respect to the last commit, and this can be measured witha backward-walking slave automaton. For each write, the distance to the next commitmeasures how fresh isthe write with respect to the following commit, and this can be measured witha forward-walking slave automaton. Thus the average freshness, called DCP,is expressedwith bidirectional NWA. Moreover, the DCP can neither be expressed by NWA with only forward-walkingslave automata nor by NWA with only backward-walking slave automata.Our contributions.We propose bidirectional NWA as a specification framework for quantitativeproperties. First, we show that the classes of forward-only NWA and backward-only NWAhave incomparable expressiveness, and bidirectional NWA strictly generalizeboth classes.Second, we establish complexity of the emptiness and universality problemsfor bidirectional NWA, where we consider the limit-average value functionfor the master automaton and for the slave automata we consider standardvalue functions for finite words (such as min, max, and variants of sum). The obtained complexity results coincide with the results for forward-only NWA, and range from -complete,to -complete to . However the proofs for bidirectional NWA are much more involved than forward-only NWA. Thus bidirectional NWA have all the advantages of NWA but provide a more expressiveframework for natural quantitative properties. Moreover, the added expressiveness ofbidirectionality is achieved with no increase in the computational complexityof the decision problems (Table <ref>). We highlight two significant differences as compared to the unweighted case: (1) In the unweighted case bidirectionality does not change expressiveness, whereas we show for NWA it does; and(2) in the unweighted case for deterministic automata bidirectionality leads to exponential succinctness and increase in complexity of the decision problems, whereas for NWA bidirectionality does not change the computational complexity. Thus the combination of nesting and bidirectionality is very interesting inthe weighted automata setting, which we study in this work.Related works. Quantitative automata and logic have been extensively studied in recent years in many different contexts <cit.>. The book <cit.> presents an excellent collection of results of weighted automata on finite words.Weighted automata on infinite words have been studied in <cit.>. Weighted automata over finite words extended with monitor countershave been considered (under the name ofcost register automata) in <cit.>. A version of nested weighted automata over finite words has beenstudied in <cit.>, and nested weighted automata overinfinite words has been studied in <cit.>. Several quantitative logics have also been studied, such as <cit.>.However, none of these works consider the rich and expressive formalism of quantitative properties expressible by NWA with slaves that walk both forward and backward,retaining decidability of the basic decision problems. In the main paper, we present the key ideas and main intuitions of the proofs of selected results, and detailed proofs are relegated to the appendix. § DEFINITIONS §.§ Words and automata Words. We consider a finite alphabet of letters Σ. A word over Σ is a (finite or infinite) sequence of letters from Σ. We denote the i-th letter of a word w by w[i], and for i < j we define w[i,j] as the word w[i] w[i+1] … w[j]. The length of a finite word w is denoted by |w|; and the length of an infinite wordw is |w| = ∞. For an infinite word w, word w[i,∞] is the suffix of wwith first i-1 letters removed. For a finite word w of length k, we define the reverse of w, denoted by w^R, as the word w[k] w[k-1] … w[1].Labeled automata. For a set X, an X-labeled automatonis a tuple Σ, Q, Q_0, δ, F,, where (1) Σ is the alphabet,(2) Q is a finite set of states,(3) Q_0 ⊆ Q is the set of initial states,(4) δ⊆ Q ×Σ× Q is a transition relation, (5) F is a set of accepting states, and(6) : δ↦ X is a labeling function. A labeled automaton Σ, Q, {q_0}, δ, F, isdeterministic if and only ifδ is a function from Q ×Σ into Qand Q_0 is a singleton. Semantics of (labeled) automata.A runof a (labeled) automatonon a word w is a sequence of states ofof length |w|+1 such that [0] belongs to the initial states ofand for every 0 ≤ i ≤ |w|-1 we have (π[i], w[i+1], π[i+1])is a transition of . A run π on a finite word w is accepting if and only if the last state π[|w|] of the runis an accepting state of . A run π on an infinite word w is accepting if and only if some accepting state ofoccurs infinitely often in π.For an automatonand a word w, we define (w) as the set of accepting runs on w. Note that for deterministic automata, every word w has at most one accepting run (|(w)| ≤ 1).Weighted automata and their semantics. A weighted automaton is a -labeled automaton, whereis the set of integers.The labels are called weights.We define the semantics of weighted automata in two steps. First, we define the value of arun. Second, we define the value of a word based on the values of its runs. To define values of runs, we will considervalue functions f thatassign real numbers to sequences of integers. Given a non-empty word w, every run π ofon w defines a sequence of weightsof successive transitions of , i.e.,(π)=((π[i-1], w[i], π[i]))_1≤ i ≤ |w|;and the value f(π) of the run π is defined as f((π)). We denote by ((π))[i] the weight of the i-th transition, i.e., (π[i-1], w[i], π[i]). The value of a non-empty word w assigned by the automaton , denoted by(w), is the infimum of the set of values of all accepting runs; i.e., inf_π∈(w) f(π), and we have the usual semantics that the infimum of the empty set is infinite, i.e., the value of a word that has no accepting run is infinite. Every run π on the empty word has length 1 and the sequence (π) is empty, hencewe define the value f(π) as an external (not a real number) value .Thus, the value of the empty word is either , if the empty word is accepted by , or ∞otherwise. To indicate a particular value function f that defines the semantics, we call a weighted automatonwith value function f an f-automaton. Value functions. For finite runs we consider the following classical value functions: for runs of length n+1 we have * Max and min:(π) = max_i=1^n ((π))[i] and(π) = min_i=1^n ((π))[i].* Sum and absolute sum: the sum function(π) = ∑_i=1^n ((π))[i],the absolute sum^+(π) = ∑_i=1^n(((π))[i]), where (x)is the absolute value of x.* Variants of bounded sum: we consider a family of functions called the (variant of) bounded sum value function B.Each of these functions returns the sum if all thepartial sums are in the interval [L,U], otherwise there are many possibilities which lead to multiple variants. For example, we can require that for all prefixes π' of π we have (π') ∈ [L,U]. We can impose a similar restriction on all suffixes, all infixes etc. Moreover, if partial sums are not contained in [L,U], a bounded sum can return ∞, the first violated bound,etc.For infinite runs we consider: * Limit average: (π) = lim inf_k →∞1/k·∑_i=1^k ((π))[i]. Silent moves. Consider a (∪{})-labeled automaton. We consider such an automaton as an extension of a weighted automaton in which transitions labeled byare silent, i.e., they do not contribute tothe value of a run. Formally, for every function f ∈ we define f as the value function that applies f on sequences after removingsymbols. The significance of silent moves is as follows: it allows to ignore transitions, and thus provides robustness where properties could be specified based on desired events rather than steps.§.§ Nested weighted automataNested weighted automata (NWA)have been introduced in <cit.> and originally allowed slave automata to move only forward. The variant we define here allow two types of slave automata, forward walking and backward walking.The original definition of NWA from <cit.> is versatile and hence it can be seamlessly extended to the case with bidirectional (forward- and backward-walking) slave automata. We follow the description of <cit.>. Informal description. A nested weighted automaton consists of a labeled automaton over infinite words,called the master automaton, a value function f for infinite words, and a set of weighted automata over finite words, called slave automata.A nested weighted automaton can be viewed as follows:given a word, we consider the run of the master automaton on the word, but the weight of each transition is determined by dynamically runningslave automata; and then the value of a run is obtained using thevalue function f. That is, the master automaton proceeds on an input word as an usual automaton,except that before taking a transition, it starts a slave automatoncorresponding to the label of the current transition.The slave automaton starts at the current position of the master automaton in the input word and works on some finite part of it.There are two types of slave automata: (a) forward walking, which move onward the input word (toward higher positions), and (b) backward walking, which move towards the beginning of the input word. Once a slave automaton finishes, it returns its value to the master automaton, which treats the returned value as the weight of the current transition that is being executed. The slave automaton might immediately accept and return value , which corresponds to a silent transition, i.e., transition with no weight. If one of slave automata rejects, the nested weighted automaton rejects. We present two examples of properties expressible by NWA.Additional examples are presented in Section <ref>. [Average response time and its dual] Consider infinite words over {r,g,#}, where r representsrequests, g represents grants, and # represents idle.A basic and interesting property is the average number of letters between a request and the corresponding grant, which represents thelong-run average response time (ART) of the system. This property cannot be expressed by a non-nested automaton <cit.>. ART can be expressed by a deterministic nested weighted automaton,which basically implements the definition of ART. This automaton invokes at every request a forward-walking slave automaton with ^+ value function, which counts the number of events until the following grant.On the other events the NWA takes silent transitions. Finally, the master automaton appliesvalue function to the values returned by slave automata. Figure <ref> presents a run of the NWA computing ART. We define the average workload property (AW), which measures the average number of pending requests. The average is computed over all positions in a word.Intuitively, if we pick a position in word w at random, the expected number of pending requests is the average workload of w. Formally, we define the workload at i in w, denoted wl(w,i), as the number of letters r among w[j,i], where j is the lastposition in w[1,i] where g occurs or 1 if such a position does not exist. The average workload of w is the limit average over all positions i of wl(w,i).AW can be expressed by a deterministic (;^+)-automaton with backward-walking slave automata.Basically, the NWA invokes at every position a slave automaton, which counts the number of r letter from its current position to the firstposition containing letter g, where it terminates. Sinceslave automata run backwards, each of them computes the workload at the position of its invocation. Figure <ref> presents a run of the NWA computing AW.Now, we present a formal definition of NWA and their semantics.Nested weighted automata.A nested weighted automaton (NWA) with bidirectional slave automata is a tuple ; f; _-m,…, _0, …, _l, with m,l ∈ where (1) , called the master automaton, is a {-m, …, l}-labeled automaton over infinite words(the labels are the indexes of automata _-m,…, _l),(2) f is a value function on infinite words, called the master value function, and (3) _-m, …, _l are weighted automata over finite words called slave automata. Intuitively, an NWA can be regarded as an f-automaton whose weights are dynamically computed at every step by the corresponding slave automaton. The automata _-m, …, _-1 (resp., _1, …, _l) are called backward walking(resp., forward walking) slave automata. We refer to NWA with both forward and backward walking slave automata asbidirectional NWA.The automaton _0 immediately accepts and returns no weight; it is used to implement silent transitions, which have no weight. We define an (f;g)-automaton as an NWA where the master value function is f and all slave automata are g-automata.Semantics: runs and values. A run ofon an infinite word w is an infinite sequence(, _1, _2, …) such that(1) is a run ofon w; (2) for every i>0 the label j = ([i-1], w[i], [i]) pointers at a slave automaton and(a) if j < 0, then _i is a run of the automaton _j on some prefix of the reverse word (w[1,i])^R, and (b) if j ≥ 0, then _i is a run of the automaton _j on some finite prefix of w[i,∞]. The run (, _1, _2, …) is accepting if allruns , _1,_2, … are accepting (i.e.,satisfies its acceptancecondition and each _1,_2, … ends in an accepting state) and infinitely many runs of slave automata have length greater than 1 (the master automaton takes infinitely many non-silent transitions). The value of the run (, _1, _2, …) is defined asf( v(π_1) v(π_2) …), where v(π_i) is the value of the run π_i inthe corresponding slave automaton, and f is the value function that takes its input sequence, removes symbols and applies f to the remaining sequence. The value of a word w assigned by the automaton , denoted by (w), is the infimum of the set of values of all accepting runs. We require accepting runs to contain infinitely many non-silent transitions as f is a value function over infinite sequences, hence the sequencev(π_1) v(π_2) … withremoved must be infinite.Deterministic nested weighted automata. An NWAis deterministic if (1) the master automatonand all slave automata are deterministic, and (2) in all slave automata, accepting states have no outgoing transitions.Intuitively, a slave automaton in an accepting state can choose (non-deteministically) to terminate or continue running; condition (2) removes this source of non-determinism.Width of NWA.An NWA has width k if and only if in everyrun at every position at most k slave automata are active.§ EXAMPLESIn this section we present several examples of quantitative properties that can be expressed with bidirectional NWA. [Average energy level]We consider the average energy level property studied in <cit.>. Consider W ∈ and an alphabet Σ_W consisting of integers frominterval [-W,W].These letters correspond to the energy change, i.e., negative values represent energy consumption whereas positive valuesrepresent energy gain.For w ∈Σ_W we define the energy level at i as the sumw[1] + … + w[i]. The average energy property (AE) is the limit average of the energy levelsat every position.For example, the average energy level of 2 (-1) 3 ((-1) 1)^ω is 4.AE can be expressed by a (;)-automatonwithbackward-walking slave automata, but it is not expressible by(;)-automata with forward-walking slave automata.To express AE, a (;)-automatonwith backward-walkingslave automata invokes at every position a slave automaton, which runs backward to the beginning of the word and sums up all the letters.In contrast, (;)-automata with forward-walking slave automata can use finite memory of the master automaton, but finite prefixes influence only finitely many values returned byslave slave automata and the limit-average value function neglects finite prefixes.Formally, we can show with a simple pumping argument that for every(;)-automaton withforward-walking slave automata, among words w_i = 1^i 0^ω there exists a pair of words withthe same value. In contrast, all these words have different AE (AE of w_i is i).AE property is often considered in conjunction with bounds on energy values.Typically, energy should not drop below some threshold, in particular, it should not be negative.In addition, the energy storage is limited, which motivates the upper bound on the stored energy, where the excess energy is released. These two restrictions lead to the interval constraint on energy levels, i.e., we require theenergy level at every position to belong to a given interval [L,U], which results in avariant of the bounded sum L,U. [Data consistency]Consider a database server, which processes instructions grouped into transactions.There are four instructions: read r, write w, void # and commit c.The commit instruction applies all writes, finishes the current transaction and starts a new one.The read instructions refer to writes applied before the previous commit. In the presence of multiple clients connected to the database, there are two options to achieve consistency.One option is to use locks that can limit concurrency. A second approach is optimistic concurrency which proceeds without locks,and then rolls back in case there was a collision between transactions. In ordered to limit the number of roll backs, it is preferred that the read instructions occur shortly after commit,while write instructions are followed by the commit instruction as quickly as possible. Formally, we define (a) consistency (or freshness) of a read instruction as the number of steps tothe first preceding commit instruction, and (b) consistency of a write instruction as the number of steps to the following commit instruction. The data consistency property (DCP) of w is the limit average of consistency of reads and writes in w.DCP is expressed by the following deterministic (;^+)-automatonwith bidirectional slave automata.On every read r (resp., w), the NWAinvokes a slave automaton which walks backward (resp., forward) and countsthe number of steps to the first encountered c. On the remaining instructions c,#, the NWAinvokes a dummy slave automaton which corresponds to a silent transition.Consider the framework of Example <ref>.For every position with read r or write w we define a regret at position i as the minimal distance tothe preceding or the following commit c.Intuitively, the regret corresponds to the number of instructions by which we have to prepone or postponethe commit to include the instruction at the current position.We consider the minimal regret property (MR) on words over { r,w,c,#} defined the limit average overpositions with r and w of the regret at these positions.MR can be expressed by a non-deterministic (;^+)-automaton with bidirectional slave automata,which basically implements the definition of MR (the non-deterministic guess is whether it is the preceding or the following commit).The NWA invokes at every r or w position oneof the following two slave automata _B, _F.The automaton _B counts the number of steps to the preceding grant, while_F counts the number of steps to the following grant.§ DECISION QUESTIONS For NWA with bidirectional slave automata, we consider the quantitative counterparts of the fundamental problems of emptiness and universality. The (quantitative) emptiness and universality problems are defined in the same way for weighted automata and all variants of NWA; in the following definitiondenotes either a weighted automaton or an NWA. Emptiness and universality. Given an automatonand a threshold λ, the emptiness (resp. universality) problem askswhether there exists a word w with _(w) ≤λ (resp., for every word w we have _(w) ≤λ).The emptiness and universality problems have been studied for forward-only NWA in <cit.>. * For NWA the value functions considered for the master automaton arethe infimum (or limit-infimum), the supremum (or limit-supremum), and the limit-average.For all the decidability results for the infimum (limit-infimum) and the supremum (limit-supremum)value functions the techniques are similar to unweighted automata <cit.>, which can be easilyadapted to the bidirectional framework.Hence in the sequel we only focus on bidirectional NWA with the limit-average value function for themaster automaton. * Moreover, we study only the emptiness problem for the following reasons. First, for the deterministic case the emptiness and the universality problems are similar and hence we focus on the emptiness problem. Second, in the non-deterministic case the universality problem is already undecidable for -automata evenwith no nesting <cit.>. §.§ The minimum, maximum and bounded sum value functions First, we show thatfor g being ,, or a variant of the bounded sum value function B, the emptiness problem for (;g)-automata with bidirectional slave automata is decidable in .To show that, we prove a stronger result, i.e., every (;g)-automaton can be effectively transformed to a -automaton of exponential size. Key ideas. Weighted automata with value functions ,,B are close to (non-weighted) finite-state automata. In particular, these automata have finite range and for each value λ from the range, the set of words of value λ is regular.Thus, instead of invoking a slave automaton, the master automaton can non-deterministically pick value λ and verify that the value returned by this slave automaton is λ. For backward-walking slave automata the guessing can be avoided as the master automaton can simulate (the reverse of) runs of all backward-walking slave automata until the current position. Thus, we can eliminate slave automata from NWA, i.e., we transform such NWA to weighted automata. Formally, we show that for g ∈{,,B},every (;g)-automaton with bidirectional slave automata can be transformed into an equivalent -automaton of exponential size. The emptiness problem for non-deterministic -automata is in (assuming weights given in unary) and hencewe have the containment part in the following Theorem <ref>. The hardness part follows from -hardness of the emptiness problem for (;g)-automata with forward-walking slave automata only <cit.>.theoremRegularForwardAndBackward Let g ∈{,,B}. The emptiness problem for non-deterministic (;g)-automata with bidirectional slave automata is -complete.Note. The complexity in Theorem <ref> does not depend on encoding of weights in slave automata, i.e., the problem is -hard even for a fixed set of weights, and it remains infor weights encoded in binary. The average energy property from Example <ref> with bounds on energy levels can be expressed with (;B)-automata.The emptiness problem for these automata is decidable by Theorem <ref>.If we assume that the size of slave automata in Theorem <ref> is bounded by a constant, the complexityof the emptiness problem drops to -complete. -hardness follows from-hardness of the emptiness problem for -automata, which can be considered as a special case of NWA.The results of this section apply to general bidirectional NWA.In the following section we consider bidirectional NWA with the sum value function, where we consider additional restrictions of finite width (Section <ref>)and bounded width (Section <ref>). We also justify in Remark <ref> that the finite widthrestriction is natural.§ FINITE-WIDTH CASE In this section we study NWA satisfying the finite width condition.First, we briefly discuss the finite-width condition and argue that it is a natural restriction.Next, we show that the emptiness problem for (finite-width) (;^+)-automata with bidirectional slave automata is decidable in . We conclude this section with the expressiveness results; we show that classical NWA with forward-walking slave automata and NWA with backward-walking slave automaton have incomparable expressive power. Hence, (finite-width) (;^+)-automata with bidirectional slave automata are strictly more expressive than NWA with one-direction slave automata.§.§ The finite-width condition Finite width. An NWAhas finite width if and only if in every accepting run ofat every position at mostfinitely many slave automata are active. Classical NWA with forward-walking slave automata only have finite width.Indeed, in any run, at any position i at most i slave automata can be active. Consider an NWA over {a,b} such that the master automaton accepts a single word a b^ω and all slave automata are backward walking and accept words b^* a. All slave automata terminate at the first position of a b^ω and hence this NWA does not have finite width. The automata expressing properties from Examples <ref>, <ref> and <ref> are finite-width (;^+)-automata with bidirectional slave automata.Observe that an NWA does not have finite width if and only if it has an accepting run, in which at some position i infinitely many backward-walking slave automata terminate.Letbe a (;^+)-automaton with bidirectional slave automata. Except for degenerate cases, runs of , which do not have finite width, have infinite value. Indeed,consider a run π and a position i_0 at which infinitely many automata are active. Since only finitely many forward-walking slave automata are active at i_0, infinitely many of them are backward-walking andfor some position i < i_0, infinitely many slave automata S terminate at position i.Then, one of the following holds: either that value of this run is infinite or one of the following two degenerate cases happen: (a) The slave automata from S are invoked with zero density (i.e., if consider the long-run average of thefrequency of invoking slave automata, then it is zero). This situation represents that monitoring with slave automata happens with vanishing frequency which is a degenerate case. (b) The values returned by the slave automata from S are bounded. It follows that these automata take transitions of non-zero weight only in some finite subword w[i,j] of the input word w. This situation represents monitoring of an infinite sequence, in which all events past position j are irrelevant. This is a degenerate case in the infinite-word case.The finite-width property does not depend on weights and hence we can construct an exponential-sizeautomaton , which simulates runs ofa given NWA . Having , we can check whether it has a run corresponding to an accepting run of , in which infinitely many backward-walking slave automata terminateat the same position. This check can be done in logarithmic space and hence checking the finite-width property is in .A simple reduction from the non-emptiness problem for NWA shows -hardness of checking the finite-width property.theoremFiniteWidthDecidable The problem whether a given NWA has finite width is -complete. §.§ The absolute sum value function We present the main result on NWA of finite width.theoremForwardAndBackwardThe emptiness problem for finite-width (;^+)-automata with bidirectional slave automata is -hard andit is decidable in . Key ideas.-hardness follows from -hardness of the emptiness problem for (;^+)-automata with forward-walking slave automata only. Containment inis shown by reduction to the bounded-width case, which is shown decidable in the following section (Theorem <ref>). We briefly describe this reduction.Consider a finite-width (;^+)-automatonwith bidirectional slave automata. First, we observe that without loss of generality, we can assume thatis deterministic. Second, we observe that in every word accepted by , at almost every position i there existsa barrier, which is a word u such that(a) the word w' = w[1,i] u w[i+1,∞], i.e., w with u inserted at position i, is accepted by , and the runs on w and w' coincide except for positions in w' corresponding to u, (b) in the run on w', backward-walking slave automata active at the end of uterminate within u, (c) in the run on w', forward-walking slave automata active at the beginning of u terminate within u, and (d) u has exponential length.Basically, active slave automata cannot cross u in w' and in the effect insertion of u bounds the number of active slave automata.Existence of barriers follows from the finite-width property of . We insert barriers in w to reduce the number of active slave automata. While inserting u at a certain position may increase the partial average,we show that if at position i in w, exponentially many active slave automata accumulates exponential weight past crossing i (some slave automata walk forward while other backwards), all partial averages(of values returned by slave automata)in w' are bounded by the corresponding partial averages in w. We conclude that for every word w, there exists a word w' such that (i) at every position at most exponentially many slave automata accumulate exponential values, and(ii) the value of w' does not exceed the value of w. Thus, to compute the infimum over all runs of , we can focus on runs in which at every position at most exponentially many slave automata accumulate exponential value. Runs of slave automata in which they accumulate bounded (exponential) values can be eliminated as in Theorem <ref>, i.e.,we can construct an exponential size NWA ', which simulates , and such that its slave automata run as long as they can accumulate value exponential (in ||) andotherwise they non-deterministically pick the remaining value and the master automaton verifies that the pick is correct.Therefore, the infimum over all runs ofcoincides with theinfimum over all runs of ' of width exponentially bounded.If we assume that the size of slave automata in Theorem <ref> is bounded by a constant, the complexityof the emptiness problem drops to -complete. -hardness follows from-hardness of the emptiness problem for -automata, which can be viewed as a special case of NWA. §.§ Expressive power DCP defined in Example <ref> can be expressed by a deterministic finite-width (;^+)-automaton with bidirectional slave automata. We show that both forward-walking and backward-walking slave automata are required to express DCP.That is, we formally show that DCP cannot be expressed by any (non-deterministic) (;^+)-automaton with slave automata walking in one direction only.Classes of NWA.We defineas the class of all finite-width (;^+)-automata with bidirectional slave automata. We define(resp., ) as the subclass ofconsisting of NWA with forward-walking (resp., backward-walking) slave automata only. We establish that classesandhave incomparable expressive power and hence they are strictly less expressive than class .Key ideas.Consider a wordw = (c #^Nr^2K c #^2N r^K )^ω for some big K and much bigger N. An NWA fromcomputes DCP of w by invoking (non-dummy) slave automata at every r letter and taking silent transitionson letters #,c. We show that an NWAfromcannot invoke the right number of slave automata, even if it uses non-determinism.More precisely, we show thatcomputing DCP has to invoke at most O(K) non-dummy slave automata on average on subwords c #^Nr^2K c #^2N r^K. Since N is much bigger than K, we conclude thathas a cycle over # letters at which it takes only silent transitions and a cycle over r letters on which it increases the multiplicity of active slave automata. Using these two cycles, we construct a run of value smaller than DCP, which contradicts the assumption thatcomputes DCP.Similarly, we can show that an NWA fromcannot compute correctly DCP of words of the form w = (cw^2K#^N c w^K #^2N) ^ω, while on these words DCP is expressible by an NWA from .lemmaIncomparableForwardAndBackward (1) DCP restricted to alphabet {r,#,c} is expressed by an NWA from , but it is not expressible by NWA from . (2) DCP restricted to alphabet {w,#,c} is expressed by an NWA from , but it is not expressible by NWA from . The above lemma implies that DCP over alphabet {r,w,#,c} is not expressible by any NWA fromnorfrom .In conclusion, we have:(1) andhave incomparable expressive power. (2) are strictly more expressive thanand .§ BOUNDED-WIDTH CASEIn this section, we study(;)-automata with bidirectional slave automata, which have bounded width. The bounded width restrictionhas been introduced in <cit.> to improve the complexity of the emptiness problem and to establish decidability ofthe emptiness problem for (;)-automata.NWA considered in <cit.> have onlyforward-walking slave automata, while we extend these results to NWA with bidirectional slave automata.This extension preserves the complexity bounds from <cit.>, i.e., the emptiness problem is infor constant width and -complete for width given in unary.The bounded width restriction emerges naturally in examples presented so far. If we bound the number of pending requests, we can express ART and AW (Examples <ref> and <ref>) by automata of bounded width.If we bound the number of writes and reads between any two commits, then DCP and MR (Examples <ref> and <ref>) can be expressed by NWA of bounded width. These natural restrictions lead to more efficient decision procedures. The decision procedure in this section differs from the one from <cit.>. The key step in the decidability proof from <cit.> is establishing the following dichotomy: either the infimum over values ofall words is -∞ or the infimum is realized by dense runs. A run is dense if for the values v_1, v_2, … returned by slave automata invoked at positions 1,2, … we have v_i/i converges to 0, i.e., values returned by slave automata are sublinear in the positions of their invocation. Properties of dense runs allow for further reductions, which lead to a decision procedure. However, we show in the following Example <ref> that in case of NWA with bidirectional slave automata, dense runs may not attain the infimum of all runs.Consider a (;)-automatonwith bidirectional slave automata over Σ = {a,b,c}.The NWAaccepts words (a b^* c)^ω and it works as follows.On letters a,invokes a forward-walking slave automaton _a, which returns the number of the following b letters up to c. On letters c,invokes a backward-walking slave automaton _c, which returns the number of the preceding b letters since a. Finally, on b letters,invokes a slave automaton _b, which takes a single transition and returns value 0. The NWAhas width 3. We can show that the value of any dense run, is 2. However, the infimum over values of all words is 1.The partial average of the values returned by slave automata on finite word u = (a b^* c)^* is 2, while the partial averageover u a b^N is 2 |u| + N/|u| + N. Therefore, the value, which is limit infimum over partial averages, of worda b^n_1 c … a b^n_i c … is 1 if sequence n_1, … is grows rapidly (e.g. doubly-exponentially).Main ideas. In Example <ref>, the words attaining the least value contain long blocks of letter b,at which the NWAis (virtually) in the same state, i.e., it loops in this state.On letters b, the sum of all weights collected by all active slave automata is 2, i.e.,automata _a, _c collect weight 1 and _b collect 0. However, in computing limit infimum over partial averages, we pick positions just before letter c as they correspond to the local minima, i.e., we compute the partial average over prefixes u a b^N, and hence the weights collected by _c do not contribute to this partial average. Then, the sum of all weights collected by slave automata _a, _b over a letter b is 1, which isequal to the least value of the limit infimum of the partial averages.In the following, we extend this idea and present the solution of allbounded-width (;)-automata with bidirectional slave automata. We show that the infimum over all words of a given NWA is the least average value over all cycles.In the following, we define appropriate notions of cycles of NWA and their average with exclusion of some slave automata. The graph of k-configurations. Letbe a non-deterministic (; )-automaton of width k. We define a k-configuration ofas a tuple(q; q_1, …, q_k) where q is a state of the master automaton, and each q_1, …, q_k is either a state of a slave automaton of or . Given a run of , we say that (q; q_1, …, q_k)is the k-configuration at position i in the run if q is the state of the master automaton at position i and there are l ≤ k active slave automata at position i, whose states are q_1, …, q_lordered by position of invocation (backward-walking slave automata are invoked past position i). If l < k, then q_l+1, …, q_k =.We say that a k-configuration C_2 is a successor of a k-configuration C_1 if there exists an accepting run ofand i>0 such that C_1 is the k-configuration at i and C_2 is the k-configuration at i+1 The graph of k-configurations ofis the set of k-configurationsof , which occur infinitely often in some accepting run,with the successor relation.Characteristics of cycles. Letbe a cycle in a graph of k-configurations of . Let F (resp., B) be the set of forward-walking (resp., backward-walking) slave automata, which are active throughout , i.e., which arenot invoked nor terminated within . A focus Fc (for ) is a downward closed subset of F, i.e., it contains all automata from F invoked before some position. We define a focused gain (,Fc) as the sum of weights which automata from Fc accumulate over . A restriction R (for ) is an upward closed subset of B, i.e., it contains all automata from B invoked past certain position. We define an average weight ofexcluding R, denoted by (,R), as the sum of weights of all transitions of slave automata within ,except of transitions of slave automata from R, divided by the number of slave automata invoked within . Intuitively, a focused gain refers to the value, which forward-walking slave automata invoked before some position i, accumulate overthe part of run corresponding to(see Figure <ref>). If the focused gain (,Fc) is negative, then by pumpingwe can arbitrarily decrease the partial average of the values of slave automata invoked before i. In consequence, we can construct a run of the value -∞. Formally, we define condition (*), which implies that there exists a run of value -∞, as follows: (*) there exists a cyclein the graph of k-configurations ofanda focus Fc such that (,Fc) < 0. If the focused gain of every cycle is non-negative, we need to examine averages of cycles, while excluding some backward-walking slave automata. The average weight with restriction corresponds to the partial average of values aggregated overby all slave automata invoked before position j (which can be past ).Backward-walking slave automata in the restriction correspond to automata invoked past j, andhence their values do not contribute to the partial average (up to i) (see Figure <ref>). In Example <ref>, we compute the average of slave automata over letters b, but we exclude the backward-walking slave automaton invoked at the following letter c. Observe that for any cycleand any restriction R, having a run containingoccurring infinitely often,we can repeat each occurrence of cyclesufficiently many times so that the partial average of values of slave automata up to position corresponding to j becomes arbitrarily close to the average (, R). The resulting run contains a subsequence of partial averages convergent to (, R) and hence its value does not exceed (, R). We can now state our key technical lemma.This lemma is a direct extension of an intuition behind computing the infimum over values of all words of the NWAfrom Example <ref>.lemmaTechnicalBoundedWidthLetbe a (;)-automaton of bounded width with bidirectional slave automata.(1) If condition (*) holds, then has a run of value -∞.(2) If (*) does not hold, thenthe infimum inf_w (w) equals the infimum inf_∈Λ, R(, R), whereΛ is the set of all cyclesin the graph of k-configurations of . If the width ofis constant, then the graph of k-configurations has polynomial size in || and it can be constructed in polynomial time by employing reachability checks on the set of all k-configurations w.r.t. to relaxation of the successor relation. Therefore, for every focus Fc and every k-configuration c we can check in polynomial time whether there exists a cyclesuch that [1] = c and (, Fc) < 0.Thus, condition (1) can be check in logarithmic space assuming that weights are given in unary. If weights are given in binary, condition (1) can be checked in polynomial time. Checking condition (2) has the same complexity as condition (1). If the width k is given in unary in input,the graph of k-configurations is exponential in || and conditions (1) and (2) can be checked in polynomial space.Weights in this case are logarithmic in the size of the graph and hence changing representation from binary to unary does not affect the (asymptotic) size of the graph.theoremBoundedWidthForwardAndBackwardThe emptiness problem for (;)-automaton of width k with bidirectional slave automata is(a) -complete for constant k and weights given in unary,(b) infor constant k and weights given in binary, and (c) -complete for k given in unary. § EXTENSIONS In this section we briefly discuss some extensions of the model of bidirectional NWA, i.e., we discuss the possibility of invoking multiple slave automata in onetransition and two-way walking slave automata.Invocation of multiple slave automata. The master automaton of an NWA invokes exactly one slave automaton at every transition.We can generalize the definition of NWA and allow the master automaton to invoke up to someconstant k slave automata at every transition.We call such a model k-NWA.First note that k-NWA contain NWA.Conversely,we briefly describe a reduction of the emptiness problem for k-NWA to the emptiness problem forNWA.First, observe that without loss of generality we can assume that k-NWA always invokes exactly k-slave automata as it canalways invoke a dummy slave automaton, which immediately accepts.Invocation of such a slave automaton is equivalent to taking a silent transition.Next, given a k-NWAwith bidirectional slave automata over the alphabet Σ, we can construct an NWA ' with bidirectional slave automata over the alphabet Σ∪{#}, which accepts words of the form w[1] #^k-1 w[2] #^k-1….The runs of ' on the word w[1] #^k-1 w[2] #^k-1 w[3] … correspond to all runs ofon w; the ' invokes at letters w[i] #^k-1 exactly k slave automata whichinvokes at the corresponding transition over letter w[i].Observe that the emptiness problems forand ' coincide.Two-way walking slave automata.For the ease of presentation we focus on bidirectional NWA where each slave automaton is either forward walking or backward walking.However, in general, we can allow slave automata that change direction while running, i.e.,allow two-way slave automata and obtain the same complexity results. Observe that in case of two-way ^+-automata, we can assume that such an automaton does not visit the same position in the same state twice.Indeed, such a cycle can be eliminated without increase of the value of the run.Thus, without loss of generality we assume that every two-way slave automaton visits every position at most || times. Therefore, instead of invoking a two-way slave ^+-automaton the master automaton invokesmultiple forward-walking and backward walking slave automata, two automata, a forward walking _f and a backward walking _b such that _f (resp., _b)simulates the run ofpast its invocation position (resp., before its invocation position). This reduction shows that every (;^+)-automatonwith two-way walking slave automata is equivalentto a 2-NWA with bidirectional slave automata.This reduction however involves exponential blow-up as slave automata _f, _b can have exponential size in .This follows from the fact that due to reversals each visited position bycan be visited || times in different states.To simulate that in one run, _f and _b have to simulate || instances ofin different states.This exponential blow-up can be avoided by dividing _f, _b into multiple slave automata,each of which tracks only one loop in the run ofas shown in Figure <ref> with automata _1, …, _4.Each of these automata have to track at most two instances ofand hence it involves only quadratic blow-up.The resulting automaton is an ||-NWA as up to ||/2 slaveautomata have to be invoked at every position.Still, the emptiness problems for ||-NWA and for NWA have the same complexity and hence we conclude thatallowing two-way slave automata does not increase the emptiness problem for (;^+)-automata. § DISCUSSION AND CONCLUSION Discussion. We established decidability of the emptiness problem for classes of bidirectional NWA, which include allNWA presented in the examples. An NWA from Example <ref> is covered by Theorem <ref>, while NWA from Examples <ref>, <ref> and <ref> are covered byTheorem <ref>. The lower bounds in the presented theorems follow from the lower bounds of the special case of forward-only NWA. The established complexity bounds (Table <ref>) coincide with the boundsfor forward-only NWA.Concluding remarks. In this work we present bidirectional NWA as a specification formalism for quantitativeproperties. In this formalismmany natural quantitative properties can be expressed, andwe present decidability and complexity results for the basic decision problems. There are several interesting directions for future work. The study of bidirectional NWA with other value functions is an interesting direction. The second direction of future work is to consider other formalism (such as a logicalframework) which has the same expressive power as bidirectional NWA. Acknowledgements.This research was supported in part by the Austrian Science Fund (FWF) under grants S11402-N23,S11407-N23 (RiSE/SHiNE) and Z211-N23 (Wittgenstein Award),ERC Start grant (279307: Graph Games), Vienna Science and Technology Fund (WWTF) through project ICT15-003 and by the National Science Centre (NCN), Poland under grant 2014/15/D/ST6/04543. § APPENDIX In the appendix we recall statements of theorems and lemmas from the main body of the paper keeping their original numbering.Lemmas introduced in the appendix have subsequent numbers. For this reason, the numberingof theorem and lemmas inthe appendix is mixed. § PROOFS FROM SECTION <REF>*The -hardness part follows from -hardness of the emptiness problem for (;g)-automata with forward-walking slave automata only <cit.>. Therefore, we focus onthe containment in . We begin with a definition of a unifying framework of regular value functions. Regular weighted automata and regular value functions.Following <cit.>, we say that a weighted automatonover finite words is a regular weighted automatonif and only if there is a finite set of rationals { q_1, …, q_n} andthere are regular languages _1, …, _n such that[(i)]* every word accepted bybelongs to ⋃_1 ≤ i ≤ n_i, and* for every w ∈_i, its value w.r.t.is q_i.A value function f is a regular value function if and only if all f-automata are regular weighted automata. Examples of regular value functions are , and variants of the bounded sum B with regular conditions, i.e., the partial sum of every prefix, suffix, infix of the run belongs to interval L,U, e.t.c. Having the definition of regular value function, we can easily check whether our variant of the bounded sum is admissible.We define the description size of a given regular weighted automaton , as the size of automata _1, …, _n recognizing languages _1, …, _n that witnessbeing a regular weighted automaton.Let g be a regular value function.Every (;g)-automatonwith bidirectional slave automata is equivalent to an exponential-size-automaton . The automatoncan be constructed implicitly in polynomial time.Since the emptiness problem for -automata can be solved in ,Lemma <ref> implies Theorem <ref>. Moreover, Remark <ref> follows directly from the construction in the following proof. Let Q_m be the set of states of the master automaton of . Since g is a regular value function, every slave automatonhas a finite range R_ and for each value v ∈ R_ there exists a finite-state automaton_,v recognizing the set of words of value v.Let(resp., ) be the union of the sets of states of all automata _,v, whereif a forward-walking (resp., backward-walking) slave automaton and v ∈ R_. We assume that sets of states of automata _,v are disjoint. We define a -automatonwith generalizedcondition as follows.The set of states ofis Q_m × 2^× 2^× 2^. States ofare of the form (q,F_1, F_2,B), whose objectives are as follows.Component q is used to simulate the run of the master automaton.Components F_1, F_2 are used to simulate finite-state automata corresponding to forward-walking slave automata. Accepting states correspond to termination of a slave automaton and hence they are removed from F_1, F_2. Every newly invoked forward-walking slave automaton is added to component F_2, i.e., automatonpicks a transition invoking slave automatonand picks weight of this transition v, then it adds to F_2 an initial state of finite-state automaton _, v.The weight of such a transition is v,If every slave automaton has finite run than F_1 becomes empty at some point of time. Then, we move all states from F_2 to F_1, and put F_2 = ∅.Observe that runs of all forward-walking slave automaton are finite if and only if F_1 is empty infinitely often.Component B is used to simulate finite-state automata corresponding tobackward-walking slave automata. Accepting states of backward-walking slave automata correspond to their termination.Sincemoves in the opposite direction w.r.t. , the automaton adds to component B some subset of accepting states from . There is only one position in the run ofat which it adds states to B. Whenever a backward-walking slave automaton is invoked in a state q_i, we require that q_i belongs to B. This state q_i may be removed from the corresponding set but does not have to.Removal corresponds to a situation when only a single slave automaton is in the state q_i, while leaving q_i in B corresponds the situation when more than one slave automaton is in state q_i.Due to the construction we have (i) for every run of , the automatonhas a run of the same weight, and conversely (ii) for every run of , there exists a runof the same value. § PROOFS FROM SECTION <REF> §.§ The proof of Theorem <ref> *Let(resp., ) be the set of states of forward-walking (resp., backward-walking) slave automata ofand let Q_m be the set of states of the master automaton of . We define a (generalized) -automaton (with no weights)as follows.The set of states ofis Q_m × 2^× 2^× 2^× 2^. Initiallystarts with (q,∅,∅,∅,∅), where q is some initial state of the master automaton. We use sets of states to simulate runs of slave automata and to ensure that every forward-walking slave automaton runs for finitely many steps.We treat backward-walking slave automata in a similar way to forward-walking slave automata except that backward-walking slave automata are started at the termination step of the corresponding slave automata, and they can terminate (which correspond to invocation) multiple, but finitely many times.More precisely, states ofare of the form (q,F_1, F_2, B_1, B_2), whose objectives are as follows.Component q is used to simulate the run of the master automaton. Components F_1, F_2 are used to simulate forward-walking slave automata. Accepting states correspond to termination of a slave automaton and hence they are removed from F_1, F_2. A newly invoked forward-walking slave automaton is added to component F_2. Thus, if every slave automaton has finite run than F_1 becomes empty at some point of time. Then, we move all states from F_2 to F_1, and put F_2 = ∅. Observe that runs of all forward-walking slave automaton are finite if and only if F_1 is empty infinitely often.Components B_1, B_2 are used to simulate backward-walking slave automata. Component B_1 contains the states of slave automata that all finish at the same position, while B_2 contains states of other slave automata.We require that B_1 and B_2 are disjoint at every position. Accepting states of backward-walking slave automata correspond to their termination.However, as the simulating automatonmoves in the opposite direction, it guessesat every step whether some (backward-walking) slave automaton have been terminated at the current position and it may add some accepting states to B_1 or B_2. There is only one position in the run ofat which it adds states to B_1. Whenever a backward-walking slave automaton is invoked in a state q_i, we require that q_i belongs to B_1 or B_2. This state q_i may be removed from the corresponding set but does not have to.Removal corresponds to a situation when only a single slave automaton is in the state q_i, while leaving q_i in B_1 or B_2 corresponds the situation when more than one slave automaton is in state q_i. Consider an infinite run π of , in which (a) component q equals infinitely often to some accepting state of the master automaton, (b) component F_1 is empty infinitely often, and (c) component B_1 is never empty and infinitely often contains an initial state of the slave automaton invoked at the current position. The run π corresponds to an accepting run ofin which infinitely many backward-walking slave automata terminate at the same position (which is the position at which B_1 becomes non-empty for the first time). Conversely, having a run ofof infinite width, the corresponding run ofsatisfies (a), (b) and (c). Checking existence of a run ofsatisfying(a), (b) and (c) can be done in non-deterministic logarithmic space.Thus, checking whetherhas infinite width can be done in polynomial space. We show -hardness of checking finite-width, by reduction from the emptiness problem for NWA with forward-walking slave automata only, which is -complete <cit.>. Letbe an NWA with forward-walking slave automata only over the alphabet Σ. We extend the alphabet Σ by $,#. Next, we construct an NWA ' whose every run has infinite width and which accepts precisely words of the form $ w[1] # w[2] # such thataccepts w[1] w[2] ….Basically, forward-walking slave automata ofignore letters # andevery transition (q,a,q') of the master automaton ofis replaced with two transitions(q,a,q'_#) and (q'_#, #,q'). On the first transition, ' invokes the same automaton asand on transitions labeled by #, the automaton ' invokes a backward-walking slave automaton that runs until $. Now, ifdoes not have an accepting run, ' does not have an accepting run and it has trivially finite width. Otherwise, ifaccepts w, the automaton ' accepts $ w[1] # w[2] # and it has a run of infinite width on it. Therefore, has an accepting run if and only if ' does not have finite width. §.§ The proof of Theorem <ref> *The -hardness part follows from -hardness of the emptiness problem for (;^+)-automata with forward-walking slave automata only <cit.>. Therefore, we focus onthe containment in . Overview. The proof is by reduction to the bounded-width case, which is decidable due to Theorem <ref>.First, we show that without loss of generality we can assume that a given NWAisdeterministic (Lemma <ref>). Next, we define words, called barriers, upon which all active (backward- and forward-walking) slave automata terminate. We show that for finite-width NWA such words do exist (Lemma <ref>).Properties of barriers ensure that if as some position i in the input word, exponentially many slave automata accumulate exponential values, then inserting a barrier actually decreases the partials sum of values returned by slave automata (Lemma <ref>), and hence we can insert barriers even at infinitely many positions and the value of the resulting word does not exceed the value of the original word. Thanks to barriers, we can show that for every word w, there exists a word w' such that at every position at most exponentially many slave automata aggregateexponential values andthe value of w' does not exceed the value of w.Slave automata which aggregate bounded (exponential) values can be eliminated, i.e., we construct an NWA ' which simulates ' in such a way thatruns of slave automata that accumulate at most exponential values are compressed into a single transition. Observe that ' on word w' as above has exponential width. Hence, the infimum ofover all words coincides with the infimum of ' over words on which it has a run of exponential width. We can encode the bound on width into ' and decide the emptiness problem of ' in non-deterministic logarithmic space (Theorem <ref>). The size of ' is doubly-exponentially bounded in the size of , and hence the emptiness problem for finite-width (;^+)-automata with bidirectional slave automata is in . Configuration and multiplicities. Invocation of a slave automaton in an NWA is a form of a universal transitionin the sense of alternating automata.We adapt the power-set construction, which is used to convert alternating automata to non-deterministic automata, to the NWA case.Given a (non-deterministic) NWAwith bidirectional slave automata, we define configurations and multiplicities ofas follows. Letbe the disjoint union of the sets of states of all slave automata of . For a run of , we say that (q_m, A) is the configuration at position p if q_m is the state of the master automaton at position p and A ⊆ is the set of states of slave automata at position p. We denote bythe number of configurations of . We define the multiplicityat position p as the function : ↦, such that (q) specifies the number ofslave automata in the state q at position p. The configuration together with the multiplicity give a completedescription of the state ofat position p. We observe thatwithout loss of generality we can assume that NWA are deterministic.Basically, non-deterministic choices of the master automaton and slave automata can be encoded in the input alphabet.More precisely, the proof consists of two steps.First, we define simple runs as follows. A run of an NWA is simple if at every position in the run slave automata that are in the same state take the same transition. We show that (i) for every run (, _1, _2, …) ofthere exists a simple run ofof the value not exceeding the value of (, _1, _2, …).Second, we show that (ii) there exists a deterministic (; ^+)-automaton ' over an extended alphabet such thatthe sets of accepting simple runs ofand accepting runs of ' coincide and each run has the same value in both automata. The proof of the following lemmais virtually the same as in the case of NWA with forward-walking slave automata only <cit.>.lemmaWLOGDeterministicLimAvgSum Given a (; ^+)-automatonover Σ with bidirectional slave automata, (i) for every run of , there exists a simple run of at most the same value, and (ii) one can compute in polynomial space a deterministic (; ^+)-automaton ' over an alphabet Σ×Γ such that: (1) inf_w ∈Σ^+(w) = inf_w' ∈ (Σ×Γ)^+'(w'), and (2) = '. (i): Consider a run (, _1, _2, …) of . Suppose that π_i, π_j are runs of the same slave automatoninvoked at positions i and j, such that its both instances arein the same state at position s in the word, i.e., π_i[i'] = π_j[j'],where i', j' are the positions in π_i, π_j corresponding to the position s in w. We pick from the suffixes π_i[i', |π_i|], π_j[j', |π_j|] the one with the smaller sum, and in case of the equal sum we pick the shorter. Then, we change the suffixes of both runs to the picked one.Such a transformation does not increase the value of the partial sums and does not introduce infinite runs of slave automata.Indeed, a run of each slave automaton can be changed by such an operation only finitely many times. Thus, this transformation can be applied to any pair of slave runs toobtain a simple run of the value not exceeding the value of (, _1, _2, …). (ii):Without loss of generality, we can assume thatfor every slave automaton infinal states have no outgoing transitions.Letbe the disjoint union ofthe sets of states of the master automaton and all slave automata of . We define Γ as the set of all partial functions h : ↦. We define a (; ^+)-automaton ' over the alphabet Σ×Γby modifying only the transition relations and labeling functions of the master automaton and slave automata of ;the sets of states and accepting states are the same as in the original automata. The transition relation and the labeling function of the master automaton ' of ' are defined as follows:(q, ah,q') is a transition of ' if and only if h(q) = q' andhas the transition (q,a,q'). The label of the transition (q, ah,q') is the same as the labelof the transition (q,a,q') in .Similarly,for each slave automaton _i in , the transition relation and the labeling function of the corresponding slave automaton _i' in ' are defined as follows: (q, ah,q') is a transition of_i' if and only if h(q) = q' and _i has the transition (q,a,q'). The label of the transition (q, ah,q') is the same as the labelof the transition (q,a,q') in _i. First, we see that = '. Second, observe that the master automaton ' and all slave automata _i' are deterministic. Moreover, since we assumed that for every slave automaton infinal states have no outgoing transitions, slave automata _i' recognizeprefix free languages.Finally, it follows from the construction that (i) every simple run (, _1, _2, …) ofis arun of ' of the same value. Basically, we encode in the input word all transitions in functions h ∈Γ. The value of each transition is the same by the construction. Conversely, (ii) every run (, _1, _2, …) of ' is a simple run ofof the same value. Indeed, the fact that transitions are directed by functions h ∈Γ implies that the run is simple.In the following definition we introduce barriers, which are words on which all active slave automata terminate, i.e., if u is a barrier, then forward-walking slave automata terminate while reading u, and backward-walking slave automata terminate while reading u^R (word u from right to left). Barriers have additional properties, which allow us tho show that if exponentially many slave automata accumulate exponential values,then inserting a barrier decreases multiplicities of slave automata and does not increase the partial sums of values returned by slave automata. Let w be an infinite word, i be a position and k > i be the first position such that all backward-walking slave automata invoked past k terminate past i. A finite word u is a barrier at i in w if in word w' = w[1,i] u w[i+1, ∞] we have * backward-walking slave automata invoked past position i+|u| terminate past position i (between positions i+1 and i+|u|) * forward-walking slave automata invoked before position iterminate before position i+|u|, * the configurations at i and i + |u| in w' are the same as the one at i in w,* the length of u is bounded by = (|Q_s + 2|) ·· |Q_s|^2 · |Q_s|,* for every state q of some backward-walking (resp., forward-walking) slave automaton,the multiplicity mult(q) at i in w' (resp., (q) at i+|u| in w') is bounded by mult(q) at i in w, and* every backward-walking (resp., forward-walking) slave automaton active at position i, accumulates over w[1,i] (resp., w[i,∞]) a value greater or equal to the value accumulated over w[1,i]u (resp., uw[i+1,∞]).The above conditions simply state that a barrier terminates all slave automata active and reduces their multiplicities, i.e., the multiplicity of backward-walking slave automata, which are invoked in the suffix w[i+1,∞] is reduced by word u (BC1) and the multiplicity of forward-walking slave automata invoked in prefix w[1,i] is reduced by word u (BC2).Property BC3 ensures that insertinga barrier at position i does not change the run essentially (except for u it only reduces the multiplicities of slave automata). These properties and the bound on the length of barriers (BC4) allow us to reduce the multiplicity of slave automata along words.Properties BC5, BC6 are necessary to show that such a reduction of multiplicities does not increase the values of words. Letbe a deterministic (; ^+)-automaton of finite width with bidirectional slave automata. Then, for every word w,at almost every position a barrier exists.Let w be a word. We consider the unique run ofon w and refer to the positionsin w and the corresponding positions in the run, i.e., we refer to “the configuration at i inw” as the unique configurationofat i while processing word w.We define the profile at position j in w as a pair ofthe configuration at j and multiplicities at j bounded by , defined for everyq ∈ Q_s as min((q), ). Let c_0 < c_1 positions in wsuch that every profile that occurs infinitely often in w occurs between c_0 and c_1.Next, we define c_2 as the minimal position past c_1 such that every backward-walking slave automaton invoked past c_2 terminates at some position past c_1, i.e., any backward-walking slave automaton invokedpastc_2 terminates before it reaches c_1. The NWAhas finite width and hence such c_2 exists.We show that there exists a barrier in w for every position i > c_2.Construction of a barrier u at position i. Let i > c_2. We pick positions a < i < b such that all slave automata active at i terminate within w[a,b] and the profiles at positions a,i,b and letters in w are the same.Since i > c_2 > c_1 > c_0 such a,b exist.Observe that w[a,b] satisfies first three conditions of the barrier definition, but it does not have to satisfy the remaining conditions.Consider word w[b,i]w[a+1,i]. It satisfies all barrier conditions except for BC4. We take w[b,i]w[a+1,i] and transform it into a barrier u by removing certain subwords corresponding to cycles in , i.e., subwords such that at the beginning and at the end of this subwordis in the same configuration. Removal of such subwords does not change runs of the master automaton or slave automata in the suffix of the word. However, to show condition RC5 we need to ensure that the removal operation does not change the profile, i.e.,the profiles at positions i and i+|u| in w[1,i] u w[i+1, ∞] are the same as the profile at i in w.We define an extended configuration in a finite word x at position p as the pair of the configuration at p and the equivalence relation R_pon states of slave automata active at position p such that q_1 R_p q_2 if and only if either * q_1, q_2 are states of forward-walking slave automataandslave automata in states q_1 and q_2 at position p reach the samestate at the end of word x, or* q_1, q_2 are states of backward-walking slave automata andslave automata in states q_1 and q_2 at position p reach the samestate at the beginning of word x.The slave automata that do not reach the end o word x (resp., the beginning of x) are in the same equivalence class ∅. Given two positions p < p' in x with the same extended configuration, we define transformation from p to p' as a functionfromthe set of equivalence classes of R_p (which is equal R_p') into itself such that * for a state q of a forward-walking slave automaton, theequivalence class [q] is transformed into a class [q'] if some slave automaton in state q at position p reaches state from [q'] at position p', and* for a state q of a backward-walking slave automaton, theequivalence class [q] is transformed into a class [q'] if some slave automaton in state q' at position p' reaches state from [q] at position p. Due to determinicity of , the transformation is, indeed, a function and a permutation.Now we describe the subword removal process. First, we markpositions j_1, …, j_nin w at which each of slave automata active at position i in w terminates.Observe that these belong to the interval [a,b].Now, starting with word w[b,i]w[a+1,i] we iteratively remove subwords y such that (a) the extended configurations at the first and the last position of y are the same, (b) the transformation between these positionsis the identity, and (c) y does not contain anyposition corresponding to positions {j_1, …, j_n}. The last word, from which no such a subword can be removed is u. We show that u is a barrier.Word u is a barrier. The positions at which slave automata are terminated are not removed from w[a,b]and hence u satisfies conditions BC1, BC2. Removal of a word satisfying (a) and (b) preserves profile at every step and hence condition BC3 holds for u.Condition BC4 follows from the fact that there are at most · |Q_s|^|Q_s|different extended configurations. Transformations are permutations of equivalence classes, i.e., they are permutations of sets of size at most|Q_s|. Therefore, if k > |Q_s|^|Q_s|, thenamong positions p_1 < … < p_k with the same extended configuration there exists a pair such thatthe transformation between these positionsis the identity.Thus, words of length at least · |Q_s|^2 · |Q_s| contain a subword y satisfying(a) and (b). Furthermore, words of length at least (|Q_s + 2|) ·· |Q_s|^2 · |Q_s| =, contain a subword y satisfying (a), (b) and (c). Therefore, the length of v is bounded by .We show that u satisfies BC5. Let q be a state of some slave automaton and let mult_1(q) be the multiplicity of q in w at position i and mult_2(q), mult_3(q) be multiplicities of q in w[1,i] u w[i+1, ∞] at positions i and i+|u| respectively.Observe that (a) and (b) imply thethe profiles at positions i and i+|u| in w[1,i] u w[i+1, ∞] are the same as the profile at i in w. It follows that min(mult_1(q), N) = min(mult_2(q), N)= min(mult_3(q), N).Now, if q is a state of a backward-walking slave automaton, then all such automata active at position i inw[1,i] u w[i+1, ∞] have been invoked in v and hence mult_2(q) < |u| ≤. It follows thatmult_2(q) ≤ mult_1(q).Otherwise, similarly if q is a state of a forward-walking slave automaton we have mult_3(q) < N and mult_3(q) ≤ mult_1(q). Therefore, condition BC5 holds.Finally, word u satisfies condition BC6. Consider a backward-walking slave automatonactive at position i in w. This automaton terminates before position a and hence it accumulates equal values onsubwords w[a,i] and w[1,i]w[b,i]w[a+1,i]. Now, word u results fromw[b,i]w[a+1,i] by deletion of certain subwords, while it preserves positions corresponding to termination of slave automata. Moreover, the deletion process only shortens runs; the transitions taken by slave automata correspond to the transitions in the original run. Thus,accumulates over w[1,i]u a value smaller or equal to the value accumulated over w[1,i]w[b,i]w[a+1,i]. The case of forward-walking slave automata is symmetric.Now, we show the key property of barriers, i.e., they decrease the partial sum of values if inserted at a position whereexponentially many slave automata accumulate exponential values. This enables us to reduce the emptiness problem for finite-width NWA to the bounded-width case. Letbe a deterministic (;^+)-automaton with bidirectional slave automata. Let C be the maximal weight of slave automata of . Let w be a word and let u be a barrier for w at position i. If more than 2 · || · slave automata accumulate the value exceeding 4 · C · past i in w (i.e, for backward-walking slave automata it is at w[1,i]),then for almost all K, the sum of values returned by slave automata invoked up to k in w is greater than the sum of values ofreturned by slave automata invoked up to k+|u| in w[1,i]uw[i+1,∞]. Let k be the first position past i such that all backward-walking slave automata invoked past k terminate before position i. We show that the partial sum of values returned by slave automata invoked up to k in w is greater than the partial sum of values ofreturned by slave automata invoked up to k+|u| in w[1,i]uw[i+1,∞]. This argument works for every k_0 ≥ k.The partial sum of values returned by slave automata invoked up to k in w consists of the sum of weights: (1) accumulated by slave automata before they reach position i, and (2) accumulated by slave automata after they reach position i. In (1) we include values of backward-walking (resp., forward-walking) slave automata that do not reach position i. Let A be the set of states of slave automata active at position i in w.For q ∈ A, we define (q) (resp., val(q)) as the multiplicity at i (resp., the value accumulated past i) in w by slave automata that are in the state q at i. Observe that (2) = ∑_q ∈ A mult(q) · val(q). The partial sum of values returned by slave automata invoked up to k+|u| in w consists of four components, which are the sum of weights(1') aggregated by slave automata before they reach any of positions i, … i+|u|,(2'a) aggregated over positionsi, … i+|u| by backward-walking slave automata invoked past i+|u| and forward-walking slave automata invoked before i,(2'b) aggregated past positionsi, … i+|u| by slave automata invoked on positions i, … i+|u|, i.e., the values aggregated by backward-walking slave automata over w'[1,i] = w[1,i] andforward-walking slave automata over w'[i+|u|+1, ∞] = w[i+1,∞], and (2'c) aggregated over positionsi, … i+|u| by slave automata invoked on positions i, … i+|u|.We observe that (1)=(1') and we show that (2) > (2'a)+(2'b)+(2'c). Observe that (2'c) is bounded by C · |u|^2. In (2'a), the multiplicities of slave automata are the same as the multiplicities ofthe corresponding slave automata active at i in w while the values they accumulate are bounded by the minimum ofthe values accumulated within w and C · |u|. Indeed, consider a backward-walking slave automaton, which is in state q at position i+|u| in w'. The multiplicity of such automata is equal tothe multiplicity of slave automata that are in state q at position i in w. Moreover, due to condition BC6 satisfied by u, the value which such an automaton accumulates along w'[1,i+|u|] is bounded by the value it accumulates at w[1,i]. Also, such an automaton terminates within |u| and hence its value is bounded by C · |u| as well. The similar reasoning holds for forward-walking slave automata active at position i in w' and hence (2'a) is bounded by ∑_a ∈ A(q) min(C· |u|, val(q)). In (2'b), slave automata accumulate the same values as the corresponding slave automata in w, but multiplicities are bounded by |u| and the values of the corresponding slave automata at position i in w. Indeed, consider a backward-walking slave automaton, which is in state q at position i in w'. Since the profile at i in w and w' is the same, the multiplicity '(q) of such automata is bounded by the multiplicity of slave automata that are in state q at position i in w. Moreover, all backward-walking slave automata active at position i in w' have been invoked within positions i, …, i + |u| and hence the sum of their multiplicities is bounded by |u|. Since w[1,i] = w'[1,i], a backward-walking slave automaton in state q at position i in w' accumulates value val(q) at past position i, i.e,the value which accumulates the same automaton in w. Similar estimates hold for forward-walking slave automata past position i+|u|. Therefore, (2'b) is bounded by ∑_q ∈ A mult'(q)val(q), where ∑_q ∈ A mult'(q) ≤ |u|. Due to condition BC5 of barriers, for every q ∈ A we have '(q) ≤ mult(q). Now, we estimate (2) - (2'a) - (2'b) - (2' c), which equals ∑_q ∈ A ((q) · (val(q) -min(C· |u|, val(q))) - ∑_q ∈ A mult'(q)val(q) ) - C · |u|^2. We partition A into A_1, the states of slave automata at position i, which accumulate the value at least 4 · C · past position i in w, and A_2 the reaming states from A. Then,(2) - (2'a)+(2'b)+(2'c) ≥∑_q ∈ A_1 ((q) - '(q)) · (val(q) - C· |u|) -∑_q∈ A_1'(q) C · |u|- ∑_q ∈ A_2'(q)4 · C · - C · |u|^2. Since ∑_q ∈ A mult'(q) ≤ |u| and|u| ≤, we get(2) - (2'a)+(2'b)+(2'c) ≥∑_q ∈ A_1 ((q) - '(q)) · (val(q) - C· |u|) - 5 · C ·^2. Recall that ∑_q ∈ A_1(q) ≥ 2 · and for every q ∈ A_1 we have val(q) ≥ 4 · C ·.Therefore,∑_q ∈ A_1 ((q) - '(q)) · (val(q) - C· |u|) > 8 · C ·^2 and hence (2) > (2'a)+(2'b)+(2'c). This concludes the proof that insertion of a barrier decreases the partial sum.Using the above lemma we can reduce the emptiness problem for finite-width (;^+)-automata with bidirectional slave automata to the bounded-width case. Letbe a deterministic (; ^+)-automaton of finite width withbidirectional slave automata. There exists a deterministic (; ^+)-automaton ' over an extended alphabet Σ_1with bidirectional slave automata of width bounded exponentially in ||, such thatthe emptiness problems forand ' coincide, i.e., inf_w ∈Σ^ω(w) = inf_w ∈Σ_1^ω'(w). The size of ' is O(· ||). We define an exponential size (; ^+)-automaton ' with bidirectional slave automata of width bounded by exponentially in || such that ' simulates a subset of runes of which satisfy the following condition (*): at almost every position s, among slave automata active at s, at most 2 · || · will accumulate value greater than 4 · C ·. Next, we show that for every run ofthere exists a run satisfying (*) of at most the same value. It follows that the emptiness problems forand ' coincide.Definition of .The alphabet Σ_1 consists of Σ and additional marking letters described below. First, the automaton ' invokes only dummy slave automata before the initiation marking.Past initial marking the master automaton keeps track of the number of invoked slave automata and rejects if the number of slave automata exceeds 2 · || ·. Second, we modify each slave automaton ofso that they work only as long as they can accumulate the value exceeding 4· C ·, where C is the maximal weight among all slave automata of . In particular, slave automata, which accumulate the value below4· C ·, are invoked for a single transition only. We encode in the alphabet whether a slave automaton is invoked for a single transition and the weight of this transition. Moreover, we encode in the alphabet that slave automata in state qought to terminate as in the following run they accumulate the value below4· C ·. The master automaton checks whether the external markings are correct.These constructions involve a single exponential blow up of the master automaton, slave automata and the alphabet. Observe that accepting runs of ' correspond to accepting runs of , which satisfy condition (*). The values of the corresponding runs are the same. Now, we need to show that while computing the infimum over all accepting runs of , we can restrict ourselves to runs satisfying (*). The emptiness problems forand ' coincide. Since ' simulates a subset of runs ofwe haveinf_w ∈Σ^ω(w) ≤inf_w ∈Σ_1^ω'(w). We show how to transform any word w ofinto a word whose unique run satisfies (*), which means that is can be simulated by ', and whose value does not exceed the value of w. It follows that inf_w ∈Σ^ω(w) ≥inf_w ∈Σ_1^ω'(w).Let w be a word accepted by . Let s be a position such that on every position p≥ s in w a barrier exists (Lemma <ref>). We start with i=0, word w_0 = w and p_0 = s. We iteratively at step i, pick first position p > p_i at which more than 2 · || · slave automata accumulate the values exceeding 4 · C · and we insert a barrier u_i at position p. Barrier u_iexists as a barrier for w at the corresponding position.Then, we start the next iteration i+1 with word w_i+1 = w_i[1,p] u_i w_i[p+1, ∞] and p_i+1 = p +|u_i|. Observe thatiterations past i change only positions past p_i in words w_i, w_i+1, …, i.e., all words w_i, w_i+1, … share the prefix w[1,p_i]. Thus, there exists the limit word w_B, which is the result of the iterative process.We argue that the resulting word w_B satisfies (*) and its value does not exceed the value of w. First, we show that w_B satisfies (*).Let u_i be a barrier at position p in w_i and let w' = w_i[1,p] u_i w_i[p+1, ∞]. The forward-walking slave automata active at p in w_i are terminated within u_i in w' and hence accumulate the value below |u_i| < C · N in w', where C is the maximal weight of slave automata of .The number of forward-walking slave automata active between positions p and p+|u_i| which are not terminated within u is boundedby |u_i| <. Therefore, at every position between i and p+|u| in w',at mostforward-walking slave automata accumulate the value exceeding 4· C ·. The similar argument applies to backward-walking slave automata, which shows that at positions p to p+|u| condition (*) is satisfied. It remains to comment on the value of w_B. Barriers inserted in the iterative process satisfy conditions of Lemma <ref>, and hence the partial sums decrease from w_0 to w_1, w_2 and so on. It follows partial sums in w_B are bounded by partial sums in every word w_0, w_1, … and hencethe value of w_B does not exceed the value of w.Algorithm. Letbe a non-deterministic finite-width (;^+)-automaton with bidirectional slave automata. Lemma <ref> reduces the emptiness problem ofto the emptiness problem of _d, which has the same properties as , but it is deterministic. The reduction takes exponential time and produces an exponential-size automaton, while it does not change the number of configurations.Therefore,the value offorand _d is the same. Next, Lemma <ref> reduces the emptiness problem for _d to the emptiness problem of _B which is a deterministic(;^+)-automaton with bidirectional slave automata and the width of _B is linear in , i.e., it is exponential in ||.The size of _B is O(· |_d|), i.e., it is exponential in ||. Therefore, the emptiness problem for _B can be solved in polynomial space in · |_B| (Theorem <ref>) andin the exponential space in the size of .§.§ The proof of Theorem <ref> In the finite-automata framework, the pumping lemma is a standard tool to show inexpressibility results.It is difficult to state a pumping lemma for NWA as their state space is infinite. While there is finitely many configurations of NWA, the multiplicity of running slave automata is unbounded. We consider the graph of configurations of an NWA to reduce the infinite-space case to the finite-state case. Recall that we assume that slave automata do not have outgoing transitions from accepting states. We also assume (without loss of generality) that initial states of slave automatado not have ingoing transitions.Graph of configurations. Letbe an NWA with bidirectional slave automata over an alphabet Σ.We define the graph of configurations G of an NWAas a Σ-labeled graph whose nodes are configurations of . For a ∈Σ, there exists an a-labeled edge from configuration (q_m, A_1) to (q_m', A_2) if and only if* the master automaton ofhas a transition (q_m,a,q_m') invoking a slave automatonin an initial state q_I,* for i=1,2, states A_iare partitioned into states of backward-walking slave automata A_i^B and forward-walking slave automata A_i^F,* there exists a function h_F, which transforms all non-final states from A_1^F onto A_2^F ∖{q_I} such that for every q ∈ dom(h_F) tuple (q,a,h(q)) isa transition of some slave automaton,* there exists a function h_B, which transforms all non-final states from A_2^B onto A_1^B ∖{q_I} such that for every q ∈ dom(h_B) tuple (q,a,h(q)) isa transition of some slave automaton, and* ifis forward-walking q_I ∈ A_2^F; otherwise q_I ∈ A_1^B. Paths in G are related to simple runs of . Consider a simple run of . Its sequence of configurations is an infinite path in G.Conversely, given a path π in G starting in the initial configuration (q_I, ∅), we can specify regular conditions ensuring (a) existence of a simple run corresponding to π and(b) existence of a simple accepting run corresponding to π. Regularity of these conditions follows from the fact that unweighted parts of runs ofcan be simulated by an alternatingautomaton. This automaton checks whether runs of all slave automata are finite; this suffices to ensure that a path corresponds to a valid simple run.It can also check acceptance condition, i.e., whether runs of all slave automata terminate in accepting states and the master automaton visits some accepting state infinitely often. The graph of configurations ofenables us to construct accepting runs ofwith desired properties.Having an accepting run ofwith a sequence of configurations αβγsuch that β is a cycle in the graph of configurations of , we know that for every n>0 there exists an accepting run ofwhose sequence of configurations isαβ^n γ. This is a key observation used in the following lemma. *We show (i) in detail and next we comment on (ii).Suppose thatis a (;^+)-automaton with forward-walking slave automata which computes DCP (Example <ref>) restricted to the alphabet {r,#,c}. First, we show that in the graph of configurations ofthere exist two cycles τ_r, τ_# such that τ_r is an r-labeled cycle in which at least one (non-dummy) slave automaton is invoked, and τ_# is an #-labeled cycle in whichtakes only silent transitions, i.e., it invokes only slave automata that immediately accept returning no value.Next, we use τ_r, τ_# to construct of a runπ_0 on some word u such that the value of π_0 is smaller than DCP of u, which contradictsthe assumption thatcomputes DCP. Existence of τ_r and τ_#. Let K be greater than the number of configurations ofand let N > 15 K^2. To simplify the calculations, we denote by (2K) some natural number from interval [0,2K].For example, we write N + 3K-1/2 = N + (2K). Consider a word w = (c #^Nr^2K c #^2N r^K )^ω. DCP of w is4/3· N+ (2K).Let π be a run ofon w of the value 4/3· N + (2K).We can assume that π is simple (Lemma <ref>). In every block #^2K in w, there exist positions i_1 < i_2 such that the configurations in π at i_1 and i_2 are the same and i_2 - i_1 > K. We remove these parts of π. The resulting sequence π' is a run ofon some wordw' = c #^Nr^L_1 c #^2N r^K c #^Nr^L_2 c #^2N r^K … such that L_1, L_2, … are at most K-1. Observe that DCP of w' is at least 3/2· N and hence the value of π' is at least 3/2· N. However, the partial sums of the values returned by slave automata in π' are bounded by the corresponding partial sums in π. Therefore, the value of π' increases due to the fact that the removed parts of π contain invocations of slave automata returning small values and removal of these parts of π increase partial averages. It follows that infinitely often at least one (non-dummy) slave automaton is invoked over the block r^2K. Consider the sequence of configurations π of run π.There exists infinitely many subsequences τ of π, which correspond totransitions over letters r and satisfy: (A1) the first and the last configuration of τ is the same, (A2) along τ at least one slave automaton is invoked and (A3) the length of τ is bounded by K. Such a sequence corresponds to an r-labeled cycle in the graph of configurations ofof length at most K. There are finitely many such cycles and hence there exists a cycle τ_r satisfying condition (A1), (A2) and (A3) which occurs infinitely often inπ. In a similar we show thatπ contains infinitely often a subsequence τ_#, which corresponds to transitions over letters #,such that(B1) the first and the last configuration of τ_# is the same,(B2) slave automata invoked along τ_# return no value (correspond to silent transitions), and(B3) the length of τ_# is bounded by K. To see that, we divide runs π and π' into blocks separated by letter c. In transformation from π to π', the average number of invocations of (non-dummy) slave automata per block decreases by at most 2K/3. Yet, the value of π' increases by at least 1/6N - (2K) w.r.t. the value of π.Therefore, the average number of invoked slave automata per block cannot exceed 9 K in π.It follows thatat most 14 · K non-dummy slave automata are invoked on average in a block of N letters #. Thus, there exists infinitely many occurrences of subsequences of πsatisfying (B1), (B2) and (B3), and hence there exists a #-labeled cycle τ_# satisfying conditions (B1), (B2) and (B3), which occurs infinitely often in π.Observe that there exist infinitely many subwords c #^Nr^2K c #^2N r^K of w such that in the corresponding positions in π occur both τ_# and τ_r. Thus, there exists a path α_A in the graph of configurations ofsuch that α_A leads from the last configuration of τ_# to the first configuration of τ_r over letters #, r. Moreover, all slave automata in π terminate after finite number of steps, while τ_# and τ_r occur infinitely often. Therefore, there exists a path α_Bfrom the first configuration of τ_r to the last configuration of τ_# over letters #,r,c such that (C1) at least one transition is over letter c, and (C2) all slave automata active at the first configuration of α_B are terminated before the end of α_B, (C3) the master automaton ofvisits an accepting state within α_B. Let u_A (resp., u_B) be a subword of w at which configurations of π form the sequence α_A (resp., α_B). Next, we show the construction of π_0 using τ_r, τ_#, α_A and α_B. The construction of π_0. Let M,L be natural numbers, which we fix later.We define π_0 as some simple accepting run that corresponds to the sequence of configurations α_0 ((τ_#)^L α_A (τ_r)^M α_B)^ω,where α_0 is a sequence of configurations from an initial configuration to the first configuration of τ_#.Such a run exists as we can ensure that at positions corresponding to α_B all slave automata terminate in accepting states and the master automaton visits an accepting state. Let u_0 be a word at which there exists a run with the sequence of configurations α_0. We define u = u_0 (#^L· |τ_#| u_A r^M · |τ_r| u_B )^ω.The run π_0 is an accepting run on u. Observe that DCP of u exceeds |τ_#| · L. However, we show that the value of π_0 is smaller.Run π_0 is a lasso and its limit average is the average of the cycle, which corresponds tothe average of α_B (τ_#)^L α_A (τ_r)^M α_B excluding values of slave automata invoked in the second occurrence of α_B. The non-dummy slave automata are invoked only in α_B and inα_A (τ_r)^M.All slave automata invoked within this cycle terminate by the end of it, and hence(a) the values of slave automata invoked in α_B are bounded by the length of the cycle multiplied by C, the maximal weight of , i.e., S_1 = C · (|α_B| + |α_A| + L· |τ_#| + M · |τ_r|), and (b) the values of slave automata invoked in α_A (τ_r)^M are bounded byS_2 = C · (|α_A| + |α_B| + M · |τ_r|). We have S_1 > S_2, however there are at most |α_B|slave automata invoked in α_B, which accumulate value at most S_1. The remaining slave automata are invoked in α_A (τ_r)^M and there are at least M of them. Thus, the average value of the cycle is at most S_1 · |α_B| + S_2 · M/|α_B| + M. Now, for M = 2 · C · |α_B| · |τ_#|, we have S_1 · |α_B| /|α_B| + M < |α_B| + |α_A| + L/2 + C· |τ_r|· |α_B| andS_2 · M/|α_B| + M < S_2. Let L > 2 · (S_2 + |α_B| + |α_A| + C· |τ_r|· |α_B|), then the average of the cycle, which is bounded byS_1 · |α_B|+ S_2 · M/M+ |α_B|, is smaller than L.However, DCP of u exceeds L, which contradicts the fact thatcomputes DCP.Backward-walking slave automata. The proof for backward-walking slave automata is similar.We consider numbers K,N and a word w = (c w^2K#^Nc w^K #^2N)^ω; we show that there exist cyclesτ_w, τ_#' in , with similar properties to τ_r, τ_# from the forward case. Moreover, there exist sequences of configurations α_A' from the last configuration of τ_w to the first configuration of τ_#', and α_B' from the last configuration of τ_#' to the first configuration of τ_w, with the properties similar to (C1), (C2) and (C3). To show (C2) we use the fact that τ_w and τ_#' occur infinitely often andhas finite-width, and hence for every positioni there exists position j>i such that every (backward-walking) slave automaton active at position j terminates before i (i.e., at some position within [i,j]). Next, we constructfrom τ_w,τ_#',α_A', α_B' a run π_0' of the value lower than DCP of the corresponding word. The construction is virtually the same as in the forward case. § PROOFS FROM SECTION <REF> *(1): Assume that (*) is satisfied.Consider an accepting runπ with configuration [1] occurring infinitely often. Let i be a position at which configuration [1] occurs.Let i' < i be the last position at which any automaton from Fc is invoked.Consider the run resulting from inserting cyclerepeated N times at position i in π. The only slave automata active past position i + ||, which has been invoked before i' are the automata from Fc. Therefore, the partial sum of values returned by slave automata up to position i' decreases by at least(N-1) ·(, Fc) + C < - (N-1) +C, where C is the value of forward slave automata invoked before i', which terminate in . The number of slave automata invoked before i' does not change and henceby picking N large enough we can decease the partial average up to i' arbitrarily. We can apply such a pumping step at every position with configuration [1] obtaining a run whose limit infimum of partial averages diverges to -∞. (2): (⇒):Assume that there exists a cyclein the graph of configurations ofanda restriction R such that (, R) andthere exists an accepting run π with configuration [1] occurring infinitely often.Let i be a position at which configuration [1] occurs. We insertat position i and obtain run π'.Let i' be the last position in π' such that i' ≥ i + || and all automata active at position i, which are not in R, are invoked before i'.Due to presence of backward-walking slave automata i' can be strictly grater than i+ ||. Consider the run resulting from inserting cyclerepeated N times at position i in π'. Then, the partial average up to position i' + N || is given by the expression a + N· p - Δ/b + N · q, where* a is the partial sum of values returned by slave automata invoked up to i' in run π,* b is the number of slave automata invoked up to i' in run π,* p/q = (, R) and q is the number of slave automata invoked in , and* Δ is the value accumulated by backward-walking slave automata invoked past i'+N ||, which terminate within interval [i + N ||, i + (N+1) ||].Now, by taking N large enough we can bring the partial average arbitrarily close to a/b. Using that and simple iteration, we can construct an accepting run of the value (, R).(⇐): Assume thatdoes not satisfy (*) from (1).It follows that in every accepting run offor almost every position i,slave automata invoked before i, accumulate past i the value exceeding value D defined as -C · k^2 ·, whereC is the maximal weight occurring inand is the number of configurations of .Indeed, if there exists such a position i, there exists a cycle past i which we can pump to lower the sum of value returned by slave automata invoked before i. Hence, existence of infinitely many such positions i, implies that condition (*) holds.Let π be an accepting run ofof value λ. Consider ϵ > 0. There exists a prefix of π up to position i such thatthe partial average of values returned by slave automata up to i is at most λ + ϵ,the sum of values accumulated by slave automata invoked before i exceeds D andand the number of slave automata before i exceeds ϵ/-D. Then, the partial average of the values accumulated by slave automata invoked before i within positions 1, …, i is at most λ + 2·ϵ.Now, we can decompose the prefix up to i into simple cycles one by one, i.e., having a prefix τ of π up to position i,we pick a simple cycle, remove it from τ and repeat the process. We terminate when we end up with run τ_E which has no simple cycle to remove; the remaining run τ_E has length bounded by the number of configurations and therefore its sum of values is greater than D. Thus, the partial average of the values accumulated by slave automata invoked before i within positions 1, …, i equals (a) the weighted average of average weights of simple cycles excluding backward-walking slave automata invoked past i,plus (b) the average of τ_E bounded by D ·ϵ/D. It follows that there exist a simple cycleand a restriction R such that (, R) ≤λ + 3 ·ϵ;otherwise the weighted average in (a) exceedsλ + 3 ·ϵ, which contradicts the choice of prefix of π.However, there are infinitely many ϵ >0, while there are finitely many simple cycles. Therefore,there exist a simple cycleand a restriction R such that (, R) ≤λ.*It suffices to show that conditions from Lemma <ref> can be checked (a) in logarithmic space for constant k and unary weights,(b) polynomial time for constant k and binary weights, and (c) polynomial space for k given in unary. Observe that these conditions reduce to weighted reachability, which can be computed in logarithmic space in the size of the graph of k-configurations of , provided that weights can fit in logarithmic space. Otherwise, if weights are represented in binary and are of length greater than logarithmic in the size of the graph, weighted reachability can be implemented using Dijkstra algorithm in polynomial time. The size of the graph of k-configurations is polynomial in the size ofand exponential in k. Thus,the graph of k-configurations ofis polynomial if k is constant, and exponential if k is given in unary. Finally, we comment how to compute the successor relation. We define a presuccessor relation R on k-configurations as follows.We have (q; q_1, …, q_k) R (q'; q_1', …, q_k') if and only if for some a ∈Σ the master automaton ofhas a transition (q,a,q') invoking a slave automatonin an initial state q_I, andfor every component j ∈{1, …, k} one of the following holds * q_j is a non-final state of a forward-walking slave automaton and (q_j,a,q_j') is a transition of this automaton,* q_j is a final state of a forward-walking slave automaton, and q_j' = or q_j' = q_I,* q_j' is a non-final state of a backward-walking slave automaton and (q_j',a,q_j) is a transition of this automaton,* q_j' is a final state of a backward-walking slave automaton, and q_j = or q_j = q_I,ifis forward-walking (resp., backward-walking) slave automaton, then for exactly one component j we have q_I= q_j' (resp., q_I = q_j). Observe that R encodes a local consistency of transitions of the master and slave automata. A sequence of k-configurations consistent with R satisfying the following conditions (a) and (b) corresponds to an accepting simple run. These conditions are:(a) the master automaton visits one of its accepting states infinitely often, and (b) every slave automaton terminates after finitely many steps. Now observe that the successor relation defined in Section <ref> is the presuccessor relation restricted to k-configurations C, which are(1) reachable through R from the initial configuration and (2) a cycle w.r.t. R satisfying conditions (a) and (b) is reachable through R from C.Indeed, for such configurations C_1, C_2 satisfying C_1 R C_2 there exists a sequence of k-configurations consistent with R, which corresponds to an accepting run. It follows that the successor relation in the graph of k-configurations can be computed based on reachability w.r.t. presuccessor relation, which is computable in logarithmic space. | http://arxiv.org/abs/1706.08316v1 | {
"authors": [
"Krishnendu Chatterjee",
"Thomas A. Henzinger",
"Jan Otop"
],
"categories": [
"cs.FL",
"F.1.1"
],
"primary_category": "cs.FL",
"published": "20170626105408",
"title": "Bidirectional Nested Weighted Automata"
} |
Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, University of Stuttgart, 70550 Stuttgart, GermanyInstitute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, University of Stuttgart, 70550 Stuttgart, Germany We present a method for computing the full probability distribution functionof quadratic observablesfor the Fermi-Hubbard model within the framework of determinantal quantumMonte Carlo. Especially, in cold atoms experiments with single site resolution,such a full counting statistics can be obtained from repeated projective measurements. We demonstrate, that the full counting statistics can provide important informationon the size of preformed pairs. Furthermore, we compute the full countingstatistics of the staggered magnetization in the repulsive Hubbard model at half filling andfind excellent agreement with recent experimental results. We show that current experimentsare capable of probing the difference between the Hubbard model and the limiting Heisenberg model.Full Counting Statistics for Interacting Fermions with Determinantal Quantum Monte Carlo Simulations Hans Peter Büchler December 30, 2023 ============================================================================================================ Full counting statistics has emerged as a very powerful tool to characterize and obtain information about a quantum mechanical system bygaining knowledge on the full probability distribution function of anobservable rather than just its expectation value.These concepts have been pioneered by Lesovik and Levitov<cit.>for transport measurements in nano-structures <cit.>; a remarkable application being the demonstration of the fractional charge of quasiparticles in a fractional quantum Hall fluid<cit.>. The concept of full counting statistics turns out to be very powerful inthe context of cold atomic gases, wherethe observable of interest is the number of particles on a set of lattice sites,which is accessible with site-resolved quantum gas microscopes <cit.>. Especially, it has been applied to characterize quantum states of cold atomic gasesin equilibrium <cit.>as well as non-equilibrium states<cit.>.Several cold atoms setups for fermionic atoms are currently equipped with a quantumgas microscope <cit.> and have achieved single-site and single-atom detection,which is required for the measurement of probability distributions in Fock spaceby accumulating histograms of particle configurations over independentmeasurement realizations <cit.>. Notably, both the repulsive <cit.> and the attractive <cit.> Hubbard model have been realized in cold atoms experiments at temperature scales that may be already relevant in the context of normal-state properties of high-T_c cuprates. While for bosonic systems, the full counting statistics is accessible withinpath integral quantum Monte Carlo simulations, in the fermionic situation the notorious sign problem renders such pathintegral approaches inefficient. In turn,for auxiliary field QMC methods such as determinantal quantum Monte Carlo (DQMC) <cit.>,suitable to study such fermionic systems in certain parameter regimes,there is no direct correspondence between the computational configurationspace of auxiliary fields and states in Fock space.In this letter, we demonstrate a method to compute the full counting statistics (FCS)within the framework of determinantal quantum Monte Carlo simulations and provide first comparisons with experiments for fermionic cold atomic gases. The main observation forthis ability is the fact that DQMC simulations decomposes the interacting fermionic system into an incoherent sum over density matrices for free fermions inan external potential <cit.>. For such free fermions, the spectrum ofthe reduced density matrix required for the determination of the full counting statistics is related to the eigenvaluesof a one-particle correlation function <cit.>.Furthermore, the generating function for the FCS of the particle number,magnetization, and thestaggered magnetization is still quadratic in the fermionic field operators.This allows us to calculate the relevant trace over theexponentially large Hilbert space of fermionic Fock states in all particle number sectors as the determinant ofa single-particle operator.We demonstrate that full countingstatistics is a powerful tool for the detection ofpairing and the size of the pairing wave function by an odd-even effect in the probability distributionfor the particle number. Furthermore, we present the first comparison of the FCS in therepulsive Hubbard model at half filling and demonstrate that the experiments clearly observe signatureswhich are not captured by the Heisenberg model.We start with the detailed description of the numerical methods for the determination of the FCS. The quantity of interest is the probability P(N^↑_A, N^↓_A) that there are N^↑_A fermions with spin up and N^↓_Afermions with spin down on a subsystem A with N_s sites,see Fig. <ref>(a). The observable of interest is therefore given by the spin density operator N̂^σ_A = ∑_i∈ Aĉ_i,σ^†ĉ_i,σ withσ∈{↑, ↓} and eigenvalues N^σ_A = 0,…,N_s. Its distribution function is most conveniently derived by the generating function χ(ϕ^↑,ϕ^↓) = ⟨ e^i ϕ^↑N̂^↑_A + i ϕ^↓N̂^↓_A⟩ = Tr(ρ_A e^i∑_σϕ^σN̂^σ_A). Here, ρ_A = Tr_B = A̅( ρ) denotes the reduced density matrix with B=A̅ thecomplement of subsystem A. Then, the joint probabilityP(N^↑_A,N^↓_A) is determined as the coefficient of the Fourier series ofthe generating functionP(N^↑_A,N^↓_A) = ∑_n,m=0^N_se^-iϕ^↑_n N^↑_A -iϕ^↓_m N^↓_A/(N_s+1)^2χ(ϕ^↑_n,ϕ^↓_m) withϕ^σ_n = 2 π n/N_s+1 . Note that the maximal number of fermions with a given spin in the subsystem A is limited by N_s, and therefore, there are only N_s+1 coefficients in the Fourier series.In general, one is mostly interested in the probability distributionsP_N(N_A)for the total particle number N_A = N_A^↑ + N_A^↓ and P_M(M_A)for the magnetization M_A= N_A^↑ - N_A^↓ on subsystem A. These quantities derive directly from the joint probability distribution via P_N(N_A) = ∑_N_A^↓=0^N_s P(N_A - N_A^↓, N_A^↓) with N_A = 0,…, 2 N_s.An analogous expression follows forP_M(M_A) =∑_N_A^↓=0^N_s P(M_A+N_A^↓,N_A^↓) with M_A= -N_s, …, N_s.In the following, we demonstrate that we can efficiently determine the generating function χ within DQMC. The standard procedure of DQMC <cit.> discretizes the inverse temperature βand then decouples the interactions with a Hubbard-Stratonovich (HS) transformation at the expense of introducing an auxiliary field at every site and time slice. Then, thepartition function can be written as a sum over free fermion systems coupled to an (imaginary) time-dependent Ising field.The expectation value of an observable Ô is⟨Ô⟩ = Tr(Ô e^-βĤ) / Tr( e^-βĤ)= 1/Z∑_{s} w_{s} O_{s},where Z = ∑_{s} w_{s} is the partition sum andw_{s} is the weight of one auxiliary field configuration { s }. The latter takes the form <cit.>w_{s} = ( 𝒢^↑_{s})^-1( 𝒢^↓_{s})^-1,where 𝒢^σ_{s} = ⟨ĉ_i, σĉ_j, σ^†⟩_{s}denotes the single-particle Green's function for spin species σ. Such a decomposition naturally carries over to more complex quantities such as the generating function <cit.> χ(ϕ^↑,ϕ^↓) =∑_{s} w_{s}χ^↑_{s}(ϕ^↑)χ^↓_{s}(ϕ^↓) with χ^σ_{s} (ϕ^σ) =Tr( ρ^σ_{s}, Ae^iϕ^σN̂^σ_A) andρ^σ_{s}, A the reduced density matrix for the free fermions with auxiliary field configuration { s } and spin σ. It is important to stress that within a Hubbard-Stratovonich configuration the reduced density matrix factorices into a spin up and spin down part. Each can be writtenin the form of a “Boltzmann weight” as <cit.> ρ_{s}, A^σ = 𝒦^σ_{s},Ae^-H_A^σwith the entanglement Hamiltonian for spin species σH_A^σ = - ∑_i,j ∈ Aĉ_i,σ^†log([G^σ_{s},A]^-1 -1)^i j ĉ_j,σand normalization 𝒦^σ_{s},A= ( 1 - G^σ_{s}, A). Here, [G^σ_{s},A]^ij = ⟨ĉ_i,σ^†ĉ_j,σ⟩_{s}; i,j ∈ A is the one-body densitymatrix (OBDM) for a given HS fieldconfiguration {s}and with sites i and j restricted to subsystem A <cit.>.Therefore, the generating function for a fixed auxiliary field configuration{ s } reduces to χ^σ_{s}(ϕ) = ∏_α=1^N_s (1- λ^σ_α) ∏_α=1^N_s( 1 + e^-ϵ^σ_α + i ϕ), where {ϵ^σ_α} are the eigenvalues of the entanglement Hamiltonian H^σ_A and {λ^σ_α} those of the OBDM. In deriving the above expression, we have used thatthe particle number operator N̂^σ_A = ∑_i∈ Aĉ_i,σ^†ĉ_i,σ is also a quadratic operator,and commutes with the entanglement Hamiltonian H^σ_A, i.e.,they have a common eigenbasis. Within this basis the grand canonical trace can be performed analytically. Finally, the eigenvalues {ϵ^σ_α} and {λ^σ_α} are related through ϵ^σ_α = log(1/λ^σ_α-1) <cit.>, and we obtain the important result χ^σ_{s}(ϕ) = ∏_α=1^N_s(1 + (e^iϕ - 1)λ^σ_α),which allows for the efficient determination of the generating function χ by quantum Monte Carlo sampling of the auxiliary field configurations. If the transformations (<ref>) and (<ref>) are performed in every Monte-Carlo measurement step, error bars for P_N(N_A) and P_M(M_A)can be obtained in the standard way.Note that the equal-time OBDM G^σ_{s},A(τ) depends explicitly on imaginary time τ, which is suppressed in our notation.Due to the cyclic property of the trace in Eq. (<ref>)all imaginary time slices are equivalent and it is possible to average over them to acquire additional statistics.In the following, we demonstrate our approach for the determination of thefull counting statistics in the two-dimensional Fermi-Hubbard model <cit.> on a square lattice.The Hamiltonian is given byℋ = - t ∑_⟨r, r^'⟩, σ(ĉ_r,σ^†ĉ_r^',σ + h.c.) +U ∑_rn̂_r,↑n̂_r,↓. The fermionic operators ĉ_r,σ obey canonical commutation relations {ĉ^†_r,σ, ĉ_r^',σ^'} = δ_r, r^'δ_σ, σ^', and n̂_r,σ = ĉ^†_r,σĉ_r,σ are the density operators while t denotes the nearestneighbour hoppingand U the on-site interation. The model can be efficiently simulated with DQMC for the attractive case U<0at arbitrary fillings, and for repulsive interactions at half-filling. Especially, at half filling and on bipartite lattices, a spin-down particle-hole transformationĉ_r,↓→ (-1)^r_x + r_yĉ^†_r,↓ maps the attractive Hubbard model into the repulsive one.We start with the presentation of the results for the attractive Hubbard model. The full counting statistics P_N(N_A) for the particle number N_A on the sublattice A is derived withinthe setup illustrated in Fig. <ref>(a). The finite-temperaturephase diagram <cit.> of (<ref>) features, away from half filling, a Berezinskii-Kosterlitz-Thouless (BKT)transition at temperature T_BKTto a quasi long-range s-wavesuperconducting state. At half-filling, there is a degeneracy of s-wave superconductivity and charge density wave order, and the SU(2) symmetry of the order parameter suppresses the transition temperature to zero according to the Mermin-Wagner theorm.In both cases, there is a second clearly separated temperature scale T^⋆∼ |U| which marks the onset of pair formation without long-range phase coherence. In Fig. <ref>(b-d), we present the behavior of the full counting statistics in the cross-over from theBCS-type superfluid of large overlappingCooper pairs for weak interactions to a BECof hardcore bosonic on-site pairs <cit.> for strong interactions. The temperature is chosen well below the characteristic temperature T^⋆ for pair formation. We observe a strong even-odd effect for increasing interactions. We will argue in the following that this phenomenon can be well understood through the size of the pairing wave function: in the extreme limit of very strong interactions, all fermions are paired up with a pair wave function localizedon a single lattice site.Then, we expect avanishing probability to find an odd number of fermions on subsystem A. An odd number of fermions can only appear due to unpaired fermions or a pairing functionspreading over several lattice sites. The latter provides only a contribution for pairs along the boundary of subsystem A. In order to quantify this effect, we introduceP_odd = ∑_N_A odd P(N_A) as a measure of the even-odd splitting. We expect a scaling behavior P_odd(L_A) = 1/2 - 1/2(1 - 2 p_deloc)^4 L_A ·⟨ n ⟩ with increasing subsystem size L_A and atomic density ⟨ n ⟩; 4 L_A ·⟨ n ⟩ estimates the number of pairs along the boundary. This ansatz is motivated by the idea of independent pairs randomly distributed in the sample. Then, for a pair centerednextto the boundary, p_deloc denotes the weight of the pair wave functionto be on the other side of the boundary. Therefore, we can interpretp_deloc as a measure forthe size of the wave function.Indeed, we find a perfect fitting of P_odd to (<ref>) for a large parameter regime,see dashed lines in Fig. <ref>(a), which allows us to extract the valuep_deloc for different interactions and densities.The resulting p_deloc for the Hubbard model are shown in Fig. <ref>(b)at density ⟨ n ⟩ = 1 and as a function of |U|; note that sufficiently low temperatures are required to suppress thermal pair breaking.In 2D the binding energy of a bound state is exponentially small in the attractive interaction <cit.>. Therefore for U/t=-3, L=12, a temperature as low as β t=24 was necessary to achieve a fit to Eq. (<ref>),whereas for large |U|, e.g. U/t = -9, “low enough” temperature is β t=4. In Fig. <ref>(c), the behavior ofp_deloc is shown for decreasing densities.The minimum of the pair size at half filling, ⟨ n ⟩ = 1, is a lattice effect (see <cit.> for the 2d continuum model), due to the logarithmically diverging van Hove singularity in the tight-binding density of states on the square lattice. As Fig. <ref>(c) demonstrates, BCS theory at T=0 gives a reasonably accurate estimate of the pair size even for intermediate values of U and densities ⟨ n ⟩.From this quantitative analysis of the even-odd splitting we conclude that the FCS can serve as a powerful tool to observe the formation of pairing in a superconducting state and provides useful information about the size of the pairing wave function. We expect that this analysis carries over to systems where DQMC simulations are hard, such as the repulsive Hubbard model away from half-filling,and can serve as a powerful experimental detection tool. Finally, we study the FCS for the staggered magnetization in the repulsive Hubbard model at half-filling.Recently, this model was extensively studied experimentally using cold atoms in optical lattices <cit.> at interactions U/t=7.2 with a circular central region of homogeneous densityinvolving approximately 80 sites surrounded by a diluteparticle bath. Given the single-siteresolution in the experiments <cit.>, histograms of the staggered magnetization M^st = ∑_i (-1)^i(n_i,↑ - n_i,↓)inside the circular region were accumulated over more than250 experimental realizations <cit.>. Here, we show the original data points of Ref. <cit.> asthe red histograms in Fig. <ref>.As the generating term for the staggered magnetization no longer commutes with the entanglement Hamiltonian in Eq. (<ref>) our method to determine the FCS has to be modified: it is required to explicitly determine the entanglement Hamiltonian and subsequentlydiagonalize the Hamiltonian with the staggered term of the generating function added; the details of this procedure are discussed inthe supplement material. For optimal comparison with the experiment, we determinethe FCS of the staggered magnetization inside the same circular geometry as in the experiment, which is embedded in a large system of size L × Lwith L=20 and periodic boundary conditions. As shown in Fig. <ref>, we find excellent agreement without any adjustable fit parameter. Furthermore, we find strong deviation from the predictions of the staggered magnetization from theS=1/2 antiferromagnetic Heisenberg model. The latter takes the formℋ_SO = J ∑_⟨ i,j ⟩Ŝ_i ·Ŝ_j with super-exchange coupling J = 4 t^2/U, and follows as limiting theory of the Hubbard model for large interactions U ≫ t; the corresponding temperature is given by T/J=(T/t) U/(4t). The Heisenberg model is simulated at the corresponding temperatures with the same circular geometryof N_s = 80sites which is embedded in a larger system of linear dimension L=32. For all temperatures, the halfwidth of the distribution of the staggered magnetizationfor the Hubbard modelis systematically smaller than that of the Heisenberg model.We expect charge fluctuations, which already play an important role at the given interaction strength U/t=7.2, to be the cause of this difference: it is well established, that the next order correction to the Heisenberg model in t/U leads to second and third neighbour exchange as well as four-spin ring exchange interactions of order t^4/U^3<cit.>. Furthermore, the operator measuring thestaggered magnetiztion is reduced bya renormalization factor (1 - 8t^2/U^2) due to doublon-hole pairs<cit.>.We therefore conclude that at the intermediate interaction strength U/t=7.2 the differences between the Hubbard model and the Heisenberg model already play a significant role. The perfect agreement of the experimental data with our DQMC analysis demonstrates that the experiments are indeed capable of probing these corrections to the Heisenberg model.However, there is an important comment of caution in order for the comparison with our theoretical predictionsand the experimental results: As a result of the parity projection, the measurement technique of <cit.>counts doublons and holessitting on the same sublattice (i.e. on diagonally opposite corners of a plaquette) incorrectly to the staggered magnetization.On the other hand, doublon-hole pairs on neighbouring sites, which are thedominant charge fluctuations, are counted correctly to the staggered magnetization as a consequence of the staggering factor. Therefore, the experiments are currently exactly performing the measurement to observe the leading deviations between Heisenberg and Hubbard model. However, the observation of further corrections for lower interactions will require the precise experimental detection of the staggered magnetization.In conclusion, we present a method to compute full quantum mechanical probability distributions of quadratic operatorsin interacting fermion systems which can be simulated with DQMC.We find that for the attractive Hubbard model the dependenceof an even-odd splitting in the particle number distribution function allows one to infer the size of a preformed pair or Cooper pair from in situ images.Furthermore, we apply our method to the repulsive Hubbard model, which has recently been studied experimentally<cit.>. The excellent agreement with our analysis demonstrates that the experimentsare capable of observing corrections to the Heisenberg model for the intermediate interaction strengthsU/t=7.2. Our method is also suitable for the study of the high-temperature phase of the repulsive Hubbard model away from half filling or the investigation of the universality of distribution functions <cit.> at a quantum critical point <cit.>.§.§ AcknowledgmentsWe acknowledge M. Greiner for providing the raw experimental data for Figure <ref>. S.H. thanks T. Roscilde and Feiming Hu for fruitful discussions as well as S. Wessel for providing benchmark results in one dimension and D. Huerga for comments on the manuscript.Computations were performed on JURECA, Jülich Supercomputing Center. Support by the German Science Foundation (DFG) through SFB TRR21 is acknowledged.39 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Levitov et al.(1996)Levitov, Lee, and Lesovik]Levitov1996 author author L. S. 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Otsuka,and author S. Yunoki,@noopjournal journal Scientific Reports volume 2, pages 992 (year 2012)NoStop @#1@@@(A.#1@italiccorr) § APPENDIX§.§ FCS of the staggered particle number To compute the full counting statistics (FCS) of the staggered particle numberN_A^st = ∑_i ∈ A (-1)^i (n̂_i, ↑ + n̂_i,↓) (which corresponds to the staggered magnetizationplus a constant,N_A^(st, U<0)→ M_A^(st, U>0) = ∑_i ∈ A (-1)^i (n̂_i, ↑ - n̂_i,↓) + ∑_i ∈ A (-1)^i, in the half-filled repulsive Hubbard model) a modification of the method described in the main text is necessary. The generating function of P(N_A^st) in one Hubbard-Stratonovich sample isχ_{s}^st(ϕ)= Tr_↑( e^-H_A^↑e^i ϕ∑_ı∈ A (-1)^in̂_i, ↑)Tr_↓( e^-H_A^↓ e^i ϕ∑_i ∈ A(-1)^in̂_i, ↓).Unlike the particle numberN̂_A^σ = ∑_i ∈ Aĉ_i, σ^†ĉ_i, σ, the operator i ϕ∑_i ∈ A (-1)^iĉ_i, σ^†ĉ_i, σ does not commute with the single-particle entanglement Hamiltonian H_A^σ due to the staggering factor (-1)^i. Therefore, there is no common eigenbasis in which one can simply add the eigenvalues of the two operators as was done in the last step leading to Eq. (8) of the main text. Instead, it is necessary to compute H_A^σ explicitlyin the site basis, add the non-commuting operators[ H̃_A^σ]_i, j = [ H_A^σ]_i, j + i ϕ (-1)^iδ_i, jand diagonalize the resulting modified entanglement Hamiltonian H̃_A^σ, which gives the eigenvalues ϵ̃^σ(ϕ).The peculiarity of DQMC that the equal-time Green's function G^σ_{s},A is non-Hermitianand does not necessarily have a spectral decomposition leads to the complication that the entanglement Hamiltonian H_A^σ in Eq. (7) of the main text,H_A^σ = - ∑_i,j ∈ Aĉ_i,σ^†log([G^σ_{s},A]^-1 -1)^ij ĉ_j,σ,has to be computed explicitly throughthe power series of the matrix logarithm <cit.>. This needs to be doneonly once per Hubbard-Stratonovich sample as H_A^σ can be reused in (<ref>) for different values of ϕ.The grand-canonical trace (<ref>) over fermionic degrees of freedom results in a single-particle determinant whichis expressed in terms of the eigenvalues ϵ̃^σ(ϕ) asχ_{s}^st(ϕ) = ∏_σ= ↑, ↓[ ∏_α=1^N_s (1-λ_α^σ)(1 + e^-ϵ̃^σ_α(ϕ))].The computation of the determinant (<ref>) may be severely affected by numerical inaccuracies.For low temperatures the occupation numbers λ^σ_α in the free fermion system tend to zero or one and, consequently, the OBDM G^σ_{s},A is a nearly singular matrix. The evaluation of (<ref>) in finite-precision arithmetic is beset with numerical instabilities and, even if a high-quality implementation of the matrix logarithm <cit.> is used, not all eigenvalues ϵ^σ_α of the entanglement Hamiltonian H_A^σ can be obtained with sufficient accuracy across the entire spectrum. Consider small (negative) “entanglement energy” ϵ^σ_α for which according to the Fermi-Dirac statisticsλ^σ_α = 1/1 + e^ϵ^σ_α≈ 1. The evaluation of ϵ^σ_α = log(1/λ^σ_α -1) ≈log(1 - 1), or equivalentlythe evaluation of (<ref>),is very inaccurate for the small (negative) real part of the entanglement spectrum since log(x) varies greatly for x → 0^+.In view of round-off and cancellation errors it is thereforecrucial to compute (<ref>) with the mathematically equivalent formula [H_A^σ]_i,j = - [log( G^σ_{s},A (1 - G^σ_{s},A)^-1)]_i,j.This expression shifts the accuracy to small (negative) ϵ^σ_α (large λ^σ_α≈ 1) since log(x) is not much affected by errors in its large argument x = λ^σ_α/1 - λ^σ_α.For the computation of (<ref>) errors in the large (positive) part of the entanglement spectrum ϵ^σ_α are not problematic because large ϵ̃^σ_α(ϕ) are irrelevant in the factor (1 - λ^σ_α)( 1 + e^-ϵ̃^σ_α(ϕ)).Here it is assumed that the spectrum ϵ̃^σ_α(ϕ)of the modified entanglement Hamiltonian (<ref>) is qualitatively similar to that of the entanglement Hamiltonian (<ref>) in the sense that the accuracy of the relevant small ϵ̃^σ_α(ϕ) is not affected by the inaccuracy of large ϵ^σ_α.§.§ FCS for a BCS mean-field state The Hamiltonian of the single-band Fermi-Hubbard model in momentum space readsℋ_U<0 = ∑_k,σ (ε_k - μ) ĉ_k,σ^†ĉ_k,σ-|U|/N_sites∑_k, k^', qĉ_k + q, ↑^†ĉ_-k + q, ↓^†ĉ_k^' + q, ↑ĉ_-k^' + q, ↓, with the single-particle band structure ε_k = -2t (cos(k_x a) + cos(k_y a)) for the square lattice. The standard BCS mean-field analysis starts with the BCS reduced Hamiltonian which neglectsthe sum over q in (<ref>) and considers only scatteringbetween pairs of zero center-of-mass momentum (q = 0).Under the assumption of a non-zero s-wave pairing order parameterΔ = ∑_k^'⟨ĉ_k,↑ĉ_-k,↓⟩ a mean-field decoupling is performed. The resulting quadratic Hamiltonian is thensolved with a Bogoliubov transformation <cit.> with coefficientsv_k^2 = 1 - u_k^2 = 1/2( 1 - ε_k^(HF) - μ(n,U)/ E_k),which are parametrized by the chemical potential μ and Δ, the gap parameter. E_k^2 = (ε_k^(HF) - μ(n,U))^2 + Δ^2(n,U)are the excitation energies of the fermionic Bogoliubov quasiparticles. In order to take into account density-density interactions at the mean-field level, it is necessary to includethe Hartree-Fock potential in thesingle-particle energies:ε_k^(HF) = ε_k - |U| n/2 where n = (N_↑ + N_↓) / N_sites is the filling. The mean-field ground state takes the standard form in terms of the BCS coefficients (<ref>)|BCS⟩ = ∏_k( u_k(μ,Δ) + v_k(μ,Δ) ĉ_k, ↑^†ĉ_-k, ↓^†) | vac⟩,where μ(n, U) and Δ(n,U) are self-consistent solutions <cit.> of the gap equation 1/|U| = 1/N_sites∑_k∈1st BZ1/2 √( (ε_k⃗ - μ - |U| n/2)^2 + Δ^2),and the number equation n = 1 - 1/N_sites∑_k∈1st BZε_k⃗ - μ - |U| n/2/√( (ε_k⃗ - μ - |U| n/2)^2 + Δ^2).For general filling n and Hubbard interaction U these self-consistent equations need to be solved numerically to obtain μ and Δ. At half filling (n=1) the number equation has the solution μ = -|U|/2 for any choice of Δ provided that the single-particle dispersion relation ε_k is particle-hole symmetric around ε_|k|=k_F = 0 such that the integral in (<ref>) vanishes.Thus, the inclusion of the Hartree shift in the single-particle energies has preserved,at the mean-field level, the particle-hole symmetry which the Hubbard model posseses for halffilling. In the atomic limit t/U=0 the solutions areμ = -|U|/2 and Δ = |U| √(n(2-n))/2.The BCS state is the ground state of a quadratic (mean-field) Hamiltonian and therefore Wick's theorem can be used to factorize any correlation function into sums of products of single-particle Green's functions. Following closely the method of Ref. <cit.>, the generating function for the total particle number iswritten as χ^(N)(ϕ) = ⟨ e^i ϕN̂_A⟩ = ⟨∏_i ∈ A∏_σ=↑, ↓ e^i ϕ (1 - n̂_i, σ)⟩ = ⟨∏_i ∈ A∏_σ=↑, ↓ E_i σ F_i σ(ϕ) ⟩,where E_i σ = ĉ_i σ + ĉ_i σ^† and F_i σ = ĉ_i σ + e^i ϕĉ_i σ^†. In the second equation of (<ref>) the particle-hole symmetry of the Hubbard Hamiltonian was used to replace the total particle number by the total hole number, which simplifies the analysis <cit.> as the identity e^i λ (1 - ĉ_i^†ĉ_i) = E_i F_i(ϕ) can then be used. The multipoint correlation function (<ref>) is contracted according to Wick's theorem, and, taking into account the fermionic anticommutation relations, the non-zero full contractions <cit.> constitute a determinant:χ^(N)(ϕ)= _i,j=1,N_s σ, σ^' = ↑, ↓( ⟨ E_i σ F_j σ^'(ϕ) ⟩) = ∏_k=1^2 N_s (μ_k + (1 - μ_k) e^i ϕ).In the last step we have introduced the eigenvalues μ_k of the matrix of normal and anomalous Green's functions:M = [⟨ĉ_i↑^†ĉ_j ↑⟩_i,j ∈ A ⟨ĉ_i ↑ĉ_j ↓⟩_i,j ∈ A; ⟨ĉ_i ↓ĉ_j ↑⟩_i,j ∈ A ⟨ĉ_i ↓^†ĉ_j ↓⟩_i,j ∈ A;].The Green's functions, which are restricted to subsystem A, are computed for the BCS state (<ref>). As described in detail in <cit.>, complex conjugate pairs of eigenvalues of M can interfere in Eq. (<ref>) leading to a suppression of odd versus even values in the particle number distribution P(N_A). Complex eigenvalues can appear as soon as non-zero values of the anomalous Green's functions ⟨ĉ_i ↑ĉ_j ↓⟩_i,j ∈ A = - ⟨ĉ_j ↓ĉ_i ↑⟩_i,j ∈ A≠ 0 destroy the hermiticity of M, which makes evident that BCS pairing is at the origin of the even-odd effect inthe distibution P(N_A). With increasing |U| thelocal number fluctuations of the BCS state (<ref>) acquire a large unphysical extensive (Poissonian) contribution (see Figs. 1(c) and (e) of the main text), which is partly due to the fact that the mean-field Hamiltonian does not conserve the total number of particles.Furthermore, the wave function (<ref>),when applied to a lattice model, fails to account for the nearest-neighbour repulsion <cit.> of tightly bound pairs in the limit of strong correlations |U| ≫ t, where interactions between Bogoliubov quasiparticles need to be included <cit.>. This is different from the equivalent model in the continuum where (<ref>) becomes the exact ground state wavefunction both in the extreme BCS and BEC limit <cit.> and reproduces the correct particle number variance and higher cumulants <cit.>.§.§ Trinomial distributionA simple model that is capable of describing the distribution of the staggered magnetizationM_A^st in the limit of high temperatures (see Fig. 3(d) of the main text) regards the staggered magnetization at each lattice siteas an independent random variable m_i^st which can take on the values 0,+1,-1 withthe probabilitiesp_1 ≡ p(m_i^st=+1) = p(↑, +) + p(↓,-)p_2 ≡ p(m_i^st=-1) = p(↑, -) + p(↓,+)p_3 ≡ p(m_i^st=0) = p(d) + p(h) = 1 - p_1 - p_2,with p(σ, f) denoting the probability of the elementary event that a spin σ∈ (↑, ↓) isplaced on a lattice site with staggering factor (-1)^i≡ f ∈ (+,-) and d and h signifying the placementof a doublon and hole, respectively. In the presence of particle-hole symmetry and equal chemical potentials for spin up and down, p(d) = p(h) and p(↑, f) = p(↓, -f), such that the probabilities of all three elementary events are described by the single parameter p_d, which we set to the average double occupancy p_d ←⟨ d ⟩ = 1/N∑_i ⟨n̂_i ↑n̂_i ↓⟩ as computed with Monte Carlo. The total staggered magnetization on a subsystem with N_s sites is given by a sum over a trinomial distributionP(M^st_A =k-l) = ∑_k=0^N_s∑_l=0^N_s-kδ_M^st_A, k-l P_3(X=k, Y=l)where P_3(X=k, Y=l) = N_s !/k!l! (N_s - k - l )! p_1^kp_2^l (1 - p_1 - p_2)^N_s - k - l.For the distribution function in Fig. 3(d) of the main text, the value ⟨ d ⟩ = 0.066(1) as extracted from the Monte Carlo simulations at temperature T/t=1.14 was used. §.§Additional datasets: FCS of magnetization, small subsystems The FCS of the magnetization M̂_A = ∑_i ∈ A(n̂_i,↑ - n̂_i,↓) (see Fig. <ref>) agrees much better between Hubbard and Heisenberg model than that of the staggered magnetization. This can be traced back to the fact that the total magnetization M̂^tot = ∑_i (n̂_i ↑ - n̂_i ↓) commutes with the Hubbard Hamiltonian ℋ and the magnetization on a subsystem A commutes up to an operator Ô_∂ A that has support only on the boundary of A, [ℋ, M̂_A] = Ô_∂ A, which comes from the change of magnetization due to particles hopping into and out of the subsystem. A possible renormalization factorof the magnetization that derives from the canonical transformation of the Hubbard model into a spin-only Hamiltonian <cit.> should then be at most a boundary effect.For temperatures T/t=0.25 and T/t=0.35 a tiny even-odd modulation is visible in the FCS of the magnetization. This is the precursor of the full suppression of odd magnetizations in the FCS of the Heisenberg model. Unlike in Fig. 3 of the main text, in Fig. <ref>odd magnetizations have not been binned together with the nearest even magnetization to give a smooth histogram. Instead, the probability distribution P(M_A) for the Heisenberg modelhas been divided by the bin width, i.e. by two, to aid the visual comparison with the probability distribution for the Hubbard model which has twice as manypossible values for M_A on the abscissa. In Fig. <ref> the distribution P(M_A) of the magnetizationin the repulsive Hubbard modelat half filling is shown for small subsystems of size 3 × 3 and 4 × 4. The interaction strength ranges from U=4 to U=14 and the inverse temperature from β t = 0.5 to β t = 2. Already at these high temperatures an even-odd splitting is visiblefor large enough Hubbard repulsion U.Fig. <ref> shows the distribution P(M^st_A) of the staggered magnetization M̂^st_A = ∑_i ∈ A(-1)^i(n̂_i,↑ - n̂_i,↓)in the half-filled repulsive Hubbard model at U/t=7.2 for small subsystems A. In the left column of Fig. <ref>, the FCS is shownfor the highest and lowest temperatures realized in the experiment of Ref. <cit.>. For the highest temperature T/t=1.14, we find good agreementwith a simple model based on the trinomial distribution (magenta line) which treatseach lattice site as independent (see previous section) and is parametrized solely by the average double occupancy p_d = 1/N∑_r⟨n̂_r, ↑n̂_r, ↓⟩ = 0.066(1) as computed with Monte Carlo for the given temperature. Interestingly, the even-odd splitting in P(M^st_A) is smeared out much more quicklywith increasing subsystem size than in the case of P(M_A). The originof the even-odd asymmetry is in both cases the formation of localmoments; however, contrary to M_A (N_A for ℋ_U<0 and half filling), which can only changeby Δ M_A = ± 1 when a particle crosses the boundary of subsystem A,M^st_A (N^st_A for ℋ_U<0 and half filling) is additionally affected by hopping processes in the bulk of A, which change the staggered magnetization by Δ M^st_A = ± 2. At this level of analysis and at half filling, the even-odd splitting in P(M^st_A) provides no additional informationbeyond the parameter p_deloc extracted earlier from the even-odd splitting in P(M_A) (or rather P(N_A) for H_U<0). §.§ Benchmarking In order to verify the correctness of our numerical implementationwe compare our DQMC methodwith Stochastic Series Expansion<cit.> (SSE) QMC for a Fermi-Hubbard chain <cit.>, where the FCS can be obtained simply by accumulating histograms over Monte Carlo measurement steps.Open boundary conditions are necessary to ensure that theSSE QMC has no sign problem. The agreement between the two methods is excellent (see Fig. <ref>).13 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Al-Mohy and Higham(2012)]Al-Mohy2012 author author A. H. 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"authors": [
"Stephan Humeniuk",
"Hans Peter Büchler"
],
"categories": [
"cond-mat.str-el",
"cond-mat.quant-gas"
],
"primary_category": "cond-mat.str-el",
"published": "20170627173034",
"title": "Full Counting Statistics for Interacting Fermions with Determinantal Quantum Monte Carlo Simulations"
} |
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